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index 0000000..8869612 --- /dev/null +++ b/contents.html @@ -0,0 +1,189 @@ + + + + + + + Contents - Physics 299 + + +

+

+
+
+

CONTENTS

+
+
+
+
+
"All science is either physics or stamp collecting"
+ Ernest Rutherford
+
+ +
+   + + +


+

+

+
+ "Physics is, + hopefully, simple.  Physicists are not"
+ Edward Teller 
+

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+ +


+

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+
+

Capacitors
+

+
+

+
+ " + + I used to wonder how it comes about that the electron is + negative. Negative-positive—these are perfectly symmetric in + physics. There is no reason whatever to prefer one to the + other. Then why is the electron negative? I thought about this + for a long time and at last all I could think was 'It won the + fight!' "
+ Albert Einstein
+
+
+
+

Calculating Capacitance

+
+
    +
+
    +
  • A capacitor is a system of two insulated conductors. 
    +
  • +
+
    +
+
    +
  • elec cap fig1The parallel plate capacitor is the + simplest example.  When the two conductors have equal but + opposite charge, the E field between the plates can be + found by simple application of Gauss's Law.
  • +
+
Assuming the plates are large enough so that the E + field between them is uniform and directed perpendicular, then + applying Gauss's Law over surface S1 we find,
+
elec cap eqn1
+
where A is the area of S1 + perpendicular to the E field and σ is the surface + charge density on the plate (assumed uniform).  + Therefore,
+
elec cap eqn2
+
+
everywhere between the plates.
+
+
+
+
+
+
+
+
+
+
    +
  • The potential difference between the plates can be + found from
  • +
+
elec cap eqn3
+
+
where A and B are points, one on each + plate, and we integrate along an E field line, + d is the plate separation, A the plate area and q the + total charge on either plate.
+
+
+
+
    +
  • The capacitance (capacity) of this capacitor is + defined as,
  • +
+
elec cap eqn4
+
+
    +
  • The expression for C for all capacitors is the + ratio of the magnitude of the total charge (on + either plate) to the magnitude of the potential + difference between the plates.
  • +
+
    +
  • Units of + C:       + Coulomb/Volt = Farad,    1 C/V = + 1 F
  • +
+
exclamation Note that since the Coulomb is a + very large unit of charge the Farad is also a very + large unit of capacitance.  Typical + capacitors in circuits are measured in μF (10-6) + or pF (10-12).
+
+
    +
  • exclamation Note that the expression for the + capacitance of the parallel plate capacitor + depends on the geometric properties (A and + d).  Even though it appears that there is + also a dependence on the charge and potential + difference (q/ΔV), what happens is that whatever + charge you place on the capacitor the pd adjusts + itself so that the ratio  q/ΔV remains + constant.   This is a general rule for + all capacitors.  The capacitance is set by + the construction of the capacitor - not the + charge or voltage applied.
  • +
+
    +
  • exclamation The above expression for the + parallel plate capacitor is strictly only true + for an infinite parallel plate capacitor - in + which "fringing" (see above) does not + occur.  However, so long as d is small + compared to the "size" of the plates, the simple + expression above is a good approximation.
  • +
+
    +
  • exclamation The parallel plate capacitor + provides an easy way to "measure" ε0 +
    +
  • +
+
+
elec cap eqn5
+
+
+
+
+
    +
  • As indicated above the parallel plate + capacitor is the most basic capacitor.  + You should also be able to determine the + expressions for the capacitance of spherical + and cylindrical capacitors,
  • +
+
elec cap fig3         +             +             +             +            elec cap fig2
+
+ divider
+
+
+

Energy and Capacitors

+
+
+
+
    +
  • One of the most important uses of + capacitors is to store electrical + energy.
  • +
+
If a capacitor is placed in a + circuit with a battery, the potential + difference (voltage) of the battery will + force electric charge to appear on the + plates of the capacitor.  The work + done by the battery in charging the + capacitor is stored as electrical + (potential) energy in the capacitor.  + This energy can be released at a later + time to perform work.
+
+
elec cap
+                                  fig4
+
+
+
+
    +
  • The work necessary to move a + charge dq onto one of the plates is + given by, dW = Vdq, where V is the + pd (voltage) of the battery (= + q/C).  The total work to place + Q on the plate is given by,
  • +
+
elec cap
+                                    eqn6
+
+
which is equal to + the stored electrical potential + energy, U.
+
+
+
+
    +
  • The electrical energy actually + resides in the electric field + between the plates of the + capacitor.  For a parallel + plate capacitor using  C = + Aε0/d and  E = + Q/Aε0 we may write + the electrical potential energy, +
    +
  • +
+
elec
+                                        cap eqn7
+
+
(Ad) is the + volume between the plates, + therefore we define the energy + density,
+
+
elec cap eqn8
+
+
+
+
+
+
+
    +
  • Although we have + evaluated this + expression for the + energy density for a + parallel plate capacitor + it is actually a general + expression.  + Wherever there is an + electric field the + energy density is given + by the above.
  • +
+
divider
+
+
+

Combinations of + Capacitors

+
+
+
+
It is common + to find multiple + combinations of + capacitors in + electrical + circuits.  In the + simplest situations + capacitors can be + considered to be + connected in series + or in parallel.  + + + + +
+
+
    +
      +
    • Capacitors + + + + + in Series
    • +
    +
+
+
When + different capacitors + are connected in + series the charge on + each capacitor is + the same but the + voltage (pd) across + each capacitor is + different
+
elec cap
+                                                        fig5
+
+
+
+

+
+
+
In + + + + + this + situation, + using the fact + that V = V1 + + V2 + +V3  + + + + + we can show + that, as far + as the voltage + source is + concerned, the + capacitors can + be replaced by + a single + "equivalent" + capacitor Ceq  + + + + + given by,
+
+
+
+
elec cap
+                                                        fig9
+
+
+
    +
      +
    • Capacitors + + + + + in Parallel
    • +
    +
+
+
For + capacitors + connected in + parallel it is + the voltage + which is same + for each + capacitor, the + charge being + different.
+
+
elec cap
+                                                          fig6
+

+ Using the fact + that QTotal= + Q1 + + Q2 + + Q3 + we can show + that the + equivalent + capacitor, Ceq  + + + + + is given by,
+
+
elec cap
+                                                          eqn10
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
+
+
+
+
+
+
+
+
+
+
    +
+
+
+
+
+
+
+
+
+
+
+
+
+

+
+

+ + At the electric company: "We would be delighted if you + send in your bill. However, if you don't, you will be."
+
+

+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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+

+
+
+

Charged Particle Motion in an Electric Field
+

+
+

+
+ " + + The hardest thing in the world to understand is the income tax"
+ Albert Einstein
+
+
+
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  • Having determined the electric field we now want to determine + the behaviour of a point charge, q0, placed in this + field.
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  • The force on the charge is given by  F = q0E.  + + + + But Newton's second law tells us that F = ma, so + that the acceleration of the particle can be written, a + = (q0/m)E.
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+
    +
  • Once we have an expression for the acceleration it is usually + possible to determine the trajectory of the particle, although + in the general case this will involve solving differential + equations.  However, when E is constant the + acceleration is constant which allows us to use the kinematic + equations describing motion under constant acceleration from the + beginning of the first semester of this course (Physics + 298).  Important equations in Physics should never be + forgotten sadface
    +
  • +
+
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  • In two dimensions, with E constant in one direction + and zero in the other, charged particle motion can be treated in + the same way as projectile motion of a particle under the + influence of a (constant) gravitational field.
  • +
+
charged particle motion
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+
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+
+
+
+ +
+ What do you get if you have Avogadro's + number of donkeys? + Answer: molasses (a mole of asses)
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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+

+
+
+

RC Circuits
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+
+
+
+

CHARGING

+
+
    +
  • fig1An example of a series RC circuit is + shown at right.  With the emf included in the circuit, + applying the loop theorem we find
  • +
+
eqn1
+
+
where q/C is the voltage drop across the + capacitor and i is the current in the circuit.
+
+
+
+
    +
  • Using the fact that i = dq/dt, we obtain
  • +
+
eqn2
+
+
    +
  • This is a "simple" differential equation the solution + of which can be written
  • +
+
eqn3
+
+
or
+
eqn4
+
+
+
+
+
    +
  • The voltage across the capacitor, VC = + q/C and the voltage across the resistor, VR + = iR.  Using the equations above we find that + the dependence of these voltages on time is shown + below
  • +
+
fig2
+
+
+
    +
  • exclamation Note that the time on the + horizontal axis is measured in units of τ = RC, + the capacitative time constant.
  • +
+
    +
  • exclamation After one time constant VC + has reached 63% (1 - e-1) of its + maximum value and VR has 37% (1/e) of + its final value.
  • +
+
divider
+

DISCHARGING

+
+
    +
  • fig1Now + switch the emf out of the circuit and + reapply the loop theorem
  • +
+
eqn5
+
+
which gives
+
eqn6
+
+
which has the solution
+
eqn7
+
and
+
eqn8
+
+
+
+
+
where Cε is the + initial charge on the capacitor + and ε/R is the initial voltage + across the capacitor.
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • fig3The time + dependence of VC and + VR (where VR + is proportional to the current + in the capacitor) are shown at + right.
  • +
+
    +
  • exclamation Once + again the time axis is measured + in units of RC.
  • +
+
    +
  • exclamation After + one time constant VC + has decreased to 37% (1/e) of + its initial value and |VR| + has decreased to 37% (1/e) of + its initial value.
  • +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
+
+
+
+
+

+
+
+
+
+
+
+
+
    +
+
+ +
+

+ +

+ Q: What did one quantum physicist say + when he wanted to fight another quantum physicist?
+ A: Let me atom.

+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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+

+
+
+

Kirchhoff's Laws
+

+
+

+
+ + + + + +
"An expert is a man who has made all + the mistakes which can be made, in a narrow field. + "
+
+ Niels Bohr
+
+
+
    +
  • +
    The most common general method to analyze + electrical circuits is by use of Kirchhoff's Laws.kirchoff
    +
  • +
+
+

staricon Junction Theorem

+
+
+
At any + junction in a circuit the current entering the junction + must equal the current leaving the junction.
+
+
+
(This is nothing more than a statement of + conservation of charge)
+
+
+
+

staricon Loop Theorem

+

The sum of the + changes in potential when traversing any complete + loop is zero.
+

+

(This is equivalent to conservation of energy)
+

+
+
    +
+
divider
+
    +
  • +

    Conventions

    +
  • +
+
As usual, in order to ensure consistent results from + application of these laws, we must adhere to several conventions + concerning the currents and potentials in circuits.
+
+ Potentials
:
+
    +
  1. When a resistive device is traversed in the direction of + current flow the change in potential is -iR.  Conversely, + if the resistance is traversed opposite to the direction of + the current the potential change is +iR.
  2. +
  3. When an emf is traversed in the direction of the emf the + change in potential is +ε.  Conversely, if the emf is + traversed opposite to the emf direction the change in + potential is -ε.
  4. +
+


+ Currents:
+

+
+

In setting up a problem, the current direction in any + particular circuit element is assigned arbitrarily.  + Kitchoff's laws are then applied to the circuit using these + current directions.  After solving the resulting + equations if a current is negative that means the "actual" + current direction is opposite the arbitrarily chosen + direction.
+

+
+
    +
+
divider
+
+
+
    +
  • +

    Application of Kirchhoff's Laws

    +
  • +
+
+

Kirchhoff's laws can be applied to any circuit to + obtain a set of equations relating the currents, resistances and + emfs in the circuit.  These equations can then be solved + for the unknown quantities in the circuit.  For any circuit + follow the steps below.
+

+
+
+
    +
  1. Label the current flowing in each part of the circuit, + bearing in mind that current will "split" on reaching a + junction.  The direction of the defined direction of the + current does not matter - see current convention above.
  2. +
  3. At each junction in the circuit use the junction theorem to + write down the equations relating the currents entering and + leaving. 
  4. +
  5. Define all possible loops in the circuit and label.
  6. +
  7. For each loop choose a starting location then use the loop + theorem to write down the equation relating changes in + potential which must be zero after traversing the complete + loop.
  8. +
  9. Solve the set of equations from 2. and 4. to obtain the + unknown parameters of the circuit.
  10. +
+


+ As an example, consider the circuit below.  With the 3 emfs + we cannot use the series/parallel analysis.
+

+
fig1
+
+
Junctions:
+
a:   I1 = I2 + I3 +
+ b:   I3 + I2 = I3
+
+
+ Loops:
+
1 (including ε1 starting at a traversing + clockwise):  - I3R4 - ε3 - + I1R2 + ε1 - I1R1 + = 0
+ 2 (including ε2 starting at a traversing clockwise): +   - I2R3 - ε2 + ε3 + + I3R4 = 0
+ 3 (including ε1 and ε2 starting at a traversing + clockwise):  - I2R3 - ε2 - + I1R2 + ε1 - I1R1 + = 0
+
+ Looking at these equations it is clear that the two junction + equations are equivalent, and that loop equation 3 is simply the + sum of loop equations 1 and 2.  Therefore there are only 3 + independent equations (a, 1 and 2), which we can solve for, say, + the currents I1, I2 and I3.
+
+ exlamation Note that in more complicated circuits there will be + many more junctions and a large number of possible loops.  + You only need apply the loop theorem to as many loops to obtain + the number of independent equations necessary to determine the + unknown parameters.  That is if you have 3 unknown + quantities, you'll need a total of 3 independent equations.
+
+
    +
+
+ +
+

+ +

+ What do you get if you have Avogadro's + number of donkeys?
+  Answer: molasses (a mole of asses)

+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/elec_circuits_sp.html b/elec_circuits_sp.html new file mode 100644 index 0000000..5e17902 --- /dev/null +++ b/elec_circuits_sp.html @@ -0,0 +1,140 @@ + + + + + + + Electricity - Series and Parallel Circuits - Physics 299 + + + +
+

+
+
+

Series and Parallel Circuits
+

+
+

+
+ " + + It would be better for the true physics if there were no + mathematicians on earth."
+
Daniel Bernoulli

+
+
+
    +
  • You may already be familiar with the idea of circuits with + resistors in series or parallel or some combination of the two.
  • +
+
    +
      +
    • +

      Series

      +
    • +
    +
+
+
fig1The same current flows + in each resistor, the voltages across them are typically + different, where V = V1 + V2 + V3 + which leads to the equivalent resistance formula
+
+
Req = R1 + R2 + + R3
+
+
+
+
+
    +
      +
    • +

      Parallel

      +
    • +
    +
+
+
The potential difference across each resistor is the + same, but the currents through them are typically different, + where I = I1 + I2 + I3.  + This lead to the equivalent resistance formula,
+
eqn1
+
+
Note that both of the diagrams below + represent resistors in parallel.
+
+
+ fig2             +                  + fig3
+
+
+
+
    +
      +
    • +

      Combinations

      +
    • +
    +
+
+
Some circuits can be analysed as combinations of + series and parallel circuits. In the circuit below R2 + and R3 are in parallel, their equivalent resistance + is then in series with R1.
+
fig4
+
+
+ exclamation Note that it is not possible to represent all + circuits as combinations of series and parallel elements, this + is most obvious in many cases where there is more than one + battery in the circuit, see example below.  To analyse this + type of circuit we must use Kirchhoff's Laws.
+
+
fig5
+
+

+
+
+
+
+
+
+ +
+

+ +

+ Got mole problems? Call Avogadro at + 602-1023.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/elec_condins.html b/elec_condins.html new file mode 100644 index 0000000..320942d --- /dev/null +++ b/elec_condins.html @@ -0,0 +1,147 @@ + + + + + + + Electricity - Conductors and Insulators - Physics 299 + + + +
+

+
+
+

Conductors and Insulators

+
+

+
+"What is the use of a new-born child ?"
+Benjamin Franklin
+(when asked what was the use of  a new invention)
+
+
+
+
+
    +
  • Moving electric charges constitute what is know as an electric +current.  It is the electric currents in semi-conductor devices +which are responsible for the electronic technology in today's society.
    +
    +
  • +
      +
    • Conductors are materials which allow the free movement +of electric charge.  Examples include,
    • +
        +
      • Metals
      • +
      • Some liquids
      • +
      • Gas plasmas
        +
        +
      • +
      +
    • Insulators (or non conductors) are materials which +provide +significant resistance to the flow of electric charge.  Examples +include,
    • +
        +
      • Non metals - plastic, wood, glass, rubber etc.
      • +
      • Gases
        +
        +
      • +
      +
    • Semi-conductors are materials whose resistance to +current flow +falls between conductors and insulators.  There are very few such +materials, but their importance in electronic technology cannot be +emphasized enough.  Examples,
    • +
        +
      • Silicon
      • +
      • Germanium
        +
        +
      • +
      +
    +
  • Mechanisms of conduction:
  • +
      +
    • Metals (solid)
    • +
        +
      • Each atom in the solid is "fixed", forming a lattice.
      • +
      • Outer electrons in a metal are weakly bound to the atomic +nucleus.
      • +
      • When an external electric field is applied these outer +electrons move +through the material creating an electric current.
      • +
      +
    • Liquid conductors and gas plasmas
    • +
        +
      • Conducting liquids and gases are comprised of positive and +negative +ions (charged particles).
      • +
      • Both positive and negative ions move when an external +electric field +is applied, thus creating the current.
      • +
      • A positive charge moving to the right creates the same +current as +an equal negative charge moving to the left.
      • +
      +
    • Insulators
    • +
        +
      • All electrons in these materials are tightly bound to the +atomic nuclei.  External electric fields are typically not large +enough to cause any flow of charge.
      • +
      +
    • Semi-conductors
    • +
        +
      • These materials have a small number of weakly bound +electrons, the +number of which is very  dependent on the temperature and +potential +difference applied across the material.
      • +
        +  +
      +
    • It is important to realise that because sustained electric +currents +only occur when a potential difference is maintained in a closed +circuit, +as many charge carriers enter as leave any part of the circuit.  +In +other words electric current is not "used up"; it has the same value +everywhere in the circuit.
      +
      +
    • +
    +
+


+

+
+

Marilyn Monroe +suggests to Einstein: What do you say, +professor, shouldn't we marry and have a little baby together: what a +baby it +would be - my looks and your intelligence!
+Einstein: I'm +afraid, dear lady, it might be the other way around...
+Albert Einstein
+

+
+  +

Dr. C. L. Davis
+Physics Department
+University of Louisville
+email: c.l.davis@louisville.edu +

+

+

+
+


+

+ + diff --git a/elec_contdist.html b/elec_contdist.html new file mode 100644 index 0000000..e8e15ac --- /dev/null +++ b/elec_contdist.html @@ -0,0 +1,96 @@ + + + + + + + Electricity - Electric Field Due to Continuous Charge + Distributions - Physics 299 + + +
+

+
+
+

Electric Field Due Continuous Charge Distribtuions
+

+
+

+
+ " + + Common sense is the collection of prejudices acquired by age + eighteen"
+ Albert Einstein
+
+
+
    +
+
    +
  • Electric charge is a property of individual particles - + protons, electrons etc.  But since these particles are + extremely small it is often convenient to consider charge to be + continuously distributed. 
    +
  • +
+
    +
  • These distributions can be over a line (one dimension), an + area (two dimensions) or a volume (three dimensions).
  • +
+
    +
  • In order to determine the electric field due to a continuous + charge distribution we "sum" the fields due to the individual + "elements" that comprise the distribution, by integrating over + the line, area or volume in question.  For example in the + example below charge is distributed uniformly over the rod on + the x axis.  To determine the electric field at point P, we + write down the expression for the field at P due to the "point" + charge dq located at "x" as shown, then integrate over x from x + = 0 to x = x.
    +
  • +
+
contin charge dist
+
+
    +
+
+
    +
+
+
+
+ +
+ Overheard after a student failed a + physics test miserably: + Nuclear, Hydrogen, Atomic, My test- They can all be bombs.
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/elec_continchgdist.jpg b/elec_continchgdist.jpg new file mode 100644 index 0000000000000000000000000000000000000000..c5d04b3c9efd0c6f742d9cb53e05252aa0bc0776 GIT binary patch literal 4371 zcmZux2UJr_w>}{uAZSd0fHWlu2qGW}J#>`NL z6|g|)O`0^3D(!;sgWmt$_ttvftT|`Tx98h?uQRjH(a_N(09Pld5daVs09YRn;Aj~@ zT=Tr`;0=HP2mk;~$J7ddQnbEnWdm5hJ7z#&01l=CLjeH70szn<06-v)MuEow0}KX( z!;o+|@;noQ={$xFiDbi|Sy|awS<$@bF(?$~JTI@1u!yh_ubiBs;>r9!05LH!F|#tW zv#_wUF&_^U8z(0l3X4K#iRECf)@A*f`dV5fC>Qu zBS1$jzz_fe0pR2LFCbK4Y7i9k2mjK?UjYoEh65nTu>pmG|B?J=0R;q{OBlt#h!lfj z711~*QEe9y9t1#j>;*$<=;)y|CxMPv2nc{;z$!8dD-bQwI5<~8G^X4`d(AbyNlC{| zgsHU0DljINB5GYhJyq6w+r2p>D3e?7@e76%E%?-*Fuy|q2o#nIuHY)Hy{1TvPG?Z! zvJ7bI!MRMK-C{DpMR>M8@IO=~X+U4*sv{$ahKlvYuihggLiP|!;=Z(F>e@W0s z6^%K+q$tO;=z7Q}d1cFgJfN;6#)Q7WpHP3WVf{sQCI5c}aPu?o0ARC63@@S?K!|l? z>por{L)p>59{^5WKKG61_WYs$E9}=nE3m?e2V&4_cl?6B74M9r4j(E!;Q#{nCu5i^ApFEI~_YK#S}SRar(5A0>%+`fO{P`H7N7`Q~pNop5Aq_A4p_rxy4D@ z5XF<{F%V%E+5m9&_|RyNTMa}Fj6JfgL6%&P>@<6yuze~z^lD;-fInrF*{uC$)HSaQ z>b2YRG}7+XyU%}1z$cP-4}4}BRN57*!0GraA?EXg+^1=lrmFQK*8o`1{O33Rk!#%G z+^yA})Qw-;sWBOn%+-s%1oK?eqR|CkXOSEqy<)Bb3Fi`s>;M~qo=+0ICuQw8`q z#w=x17(m~KTntCkVVk}Ec9-ClI%kesX=#k-r zVUx$pk~z+ar{&=@;?Wl#Xx*_#F_jo?KNtCUzq=`Fsf91x@}tb(u7dlti&oE5b;vxS ztJSC!<=*;ASsLVdI!_h_w$#9wxGcO$Xrye%L}n&VVz!)9>ryI?N99>!Y{;bOcP)vu z>Yu4Erce!}feD8vuYI4kbqe}-5Oo=Z0>Ae9IIA6yWK3y;`=QPY#tX5ivCdGdl&+ZGnKj<3o~W#6{4$ty0Z zh7TR&)H?A?#3_}gT7^~dH3_v_e<(pQ%)PuGX2C*sycMHAEyo-uHE#H*E@MjG*ic+A zn9P6mQx)ZNNw7_aZouU+jFpKROb*>3ghI@%V-un_N#%*TW4-CEiDThDv<3AhT0&(I zdY*SkuTy9GK92QkYKz6*U>km7W+L<2v%CsPDKY%CO3PTS$9^W3%67x|?WLXAAKCBZ zhTT0LxVA`tD{^)imvl%SGmx4Al}vsV6*rx)S9+0S**dJ}qeaw_E?dtw_GTdaNt>EK zta$TRq4=hqO_p(fb)NE_mwHpRxvca0QuCft=S@lITQ3I$nb)l3Qw%3-0$og%q~_rp zU&rl=P1)_L444gy+Fd^SWaPOQU#vX>d@cwx7rbP)lagLtqNJm<<2vo0!3?La#modb zkdRTY84ZEin&q})AAvic>917h6BlKb9wXOu;H7Qtcj=Fpc*i8~& zdn78QwsvMI!rrDYjh}~*95si$NJjSW2|l3cY9W>a4uf)J6uW*jJ+PP5QIER{C)oXI zxfA_0OIHa-w}yYiXIGF|ghHaFWSZ*BC zcH=2GOJ;f(K=oDY1EGkTIY+eu)+(Gn6k=u1bNy0w*=?Ty8rs=RN5{C})jxd(I!?3p z=mzEp;7@(v9n!S|vgP9Ijx7NNmUPCz`IT^`*-_K)V%2Z)#^<6B-b`wWX6m^TR=%tE z6V)&mdSfJwAprx^8u5GuQ$cbsXw~@tUU0Zx=j!CZWIu;a5bC}${f-BUs*fPrD)jx$ z>$5WCHoxl7X2)=yYz@{$FhQU1bsa@K-Vk@sud(Eju zH|d|69s%1K>pBJs*5O&=6LrC{R|kZ7)FV3U#5Fm>?K+yt7ks4#%;-`AA6Pw1z~1kj zV3l;m8F+|JEosJV!81b0x(l1_829%NDeW_es*I$mLb|0&MH6pNy}a0MZjOES=RIj7 zB2B>zy^>A>`bK&ElHuK6)RyD&()T10tryPwo)tkkvyriM<@LneFSzb$G1)oQIZ-w{ z+a%l9_{#gM2wP2iLt{>%N_M>!%-Opxw6)b49VF#wEie77`Ru6vdscHKBbjPk5@!li;9Z^zt^a-_y~L3frZ_Tu zwYxS?Tyeo(r~c{bfHC*?A2~g0hTLp0=mH728^(2B!nWe4<}QAZIf3vvUV&1xk}pi( z;#brGIdtvJemP-?zlz$tBon3riVHYR)`NQmVmb5GPvPUKt-vo#-$yKsu=tJGdbchd z0cfqTQ?nJ&sm^RuT!1de({T%V+6jJf4pa)2raC@IzYoWwr8pu%aXK58=dB}<6kG;X z<=2#;Fz>MRsxXyvMa?&x4&DOXv~0Zg*2-GGb2}y1MRlqWIofuHzil~vc-==m8j$nB zxpqC%-X8G9suwRDEKTu5EIm^`9U+lV7c_O4_9Yv!qQi%>PwTm>rJirh7W1)MFlnWg zQkT2mtvzC#+=F7d(j2W?=wQGakV0dGFv-np3HSe{n4LJWFY~6bR@Kr&uj+-}GsUQz z!5tmLBb*bZnlmkxH8vRGB| z^*oN;5n`km>}8uT`}XrQ3FWOzgGuJ+pL#c%ST$s*(bKM2;xpX%MOHC>8mFH}{QyJ% zs@fH3Oub^z(78W6x8?NMvLr{&wX?7fsBlLD1qY(s$~WF|-*4?M`i?q9h0WvE-iAik z@`<9tJO4D}Nw|fp_4rRBfE@;rj_XsF6?1 z7qun$3ht-Zc_F#W3Ka6TB59B&wH#u!Y*vQaUa<>nw}i;iH(JnQ_5+W1phi5-*3s1V ze92oLu#Vwisug1Icxv0H4_*5%Z+933Uf}gAwV%bt^OtuKZ(&7JznV9NP?2sb;@heR z&7O!g8+cdzxWgcfrFf};dppiOCrr&=Qg?c|=zp!;-ZakGkiu)`=+e_J(*szKn(?HXlX%U(a{E!?|Il8emEt#1#nMMZP+GV)Xlz-*Xr=fw4uRY`fcfrI?BwMdS`j|B`O6`5t@AA+=!hT(BWd z4{P9g_F`%JKAwfRTWytgT}8FJ=2xn8^@W2$`LcJ>U}|Y_CSlkk|B2;X%YY21C9J1% zY1wx`N!rBTyfN-l{CM2@V?%5~X;98(TYh1im`)rB;LobQbNjWv%1y~S2M50CmnvwI zT|alYMQ}hJ%wjLJwU)Wz+Elv2$8)k@JnvOKybqNOr{E$MJViX0hJ;z8iIMbWZ|_I; zWx*<+6ImOa_3Ndf4jYp(nEpgV161Iit}~3-Z;$c44!ccaa{(EtW;-&=mJcqhlLed$ ze9;MxZ;Ee2^AYS!-z_>X)vT@AqfvezE35K)Q;Jn0jjdq(YK9gZ6v3+@0m{LJX8Z|m z**QKk+M(}t0z9$3_^r_{@IK z3m%$&&=cX9f+fx~cJKwqYX4^WM+SVHqHOUxOS9ebw1meJ__}vDf(BS1dtcj5JEB zeIiI*@?Pls583+G;dp~rPW-Lr7xQ+bgp$L^Q?X{NmdS6|HX@nw&WkhINYl2%cxe6E zK!Z=OJs3g`(8k5Rj=+^D&~jDD`vz_0?ym38$`_coKcAT15SIM4BrkWai7QG-Q*+1! z`Q)5wn{m&QdyLGOS|8NgY1njQdJJBtFpWIjOib$l-KtrAyOebVKykQNfu}z2-|vd> zqkr&xc*gor!y)!?fXi;u+cH)=N{ypxnQk#%mme6tVN{(=9nBITNzTy~;buSUS>?=I z%&0=EEtn?3*wL)bxx!%qWvz+wX?vy-8gCfffSWN$(JKUX zc?pKB?VK + + + + + + Electricity - Coulomb's Law - Physics 299 + + +
+

+
+
+

Coulomb's Law

+
+

+
+"When man wanted to make a machine that would +walk +he created the wheel, which does not resemble a leg"
+Guillaume Apollinaire
+
+
+  +
    +
  • The magnitude of the force of attraction (or repulsion), F12 +between two point charges q1 and q is +given by Coulomb's Law.
  • +
    +


    +
    +

    +
    +

    where R12  is the distance between the +charges.  k is a constant of proportionality known as the Coulomb +constant, having the value 9 x +109  N.m2 / C2  in a +vacuum. 

    +

    exclamation Note that the Coulomb constant, k, is +often replaced with (1/4π ε0), where +ε0is the permittivity of the vacuum (more later).
    +

    +
  • The direction of this force is along the line joining the two +charges +with the sense determined by the relative signs of the charges
  • +


    +

    +
    +

    +
    +
  • Note that the force on each charge has the same magnitude (as +required by Newton's third law of motion).
  • +
    +  +
  • For two 1 Coulomb charges separated by 1 metre the +magnitude +of the force is given by, +

    +F = (9 x 109  x 1 x 1 )/ 1  =  9 x 109 +  Newtons
    +

    This is an extremely large force (sufficient to +move Mt. Everest with an acceleration of 1cm/s2).  The +Coulomb +is a very large unit.  Typical macroscopic charges +are measured in micro-coulombs (10-6 C).

    +
  • +
  • To handle situations with more than one charge, the charges must +be treated in pairs, so that the overall force on one charge will be +the vector sum of the force +due to each of the other charges.  For example the force on q1 +due to all other charges q2, q3 , q4... +would +be +given +by,
  • +
+
F1 = F21 + +F31 + F41 + ...
+
+
    +
  • hotNotice +the +similarity +of +Coulomb's Law to Newton's Law of Gravitation
  • +
+
eqn1
+

+both are "inverse square" laws.  Substitute charge for mass and +"k" for "G" and you have Coulomb's law.
+exclamation The relative magnitudes of the Coulomb +constant, k = 9 x 109 and the gravitational constant, G = +6.67 x 10-11, is an indication of the relative strengths of +the two forces.  The electrical force of attraction is much, much +stronger than the gravitational force of attraction.
+
+
+
+
+
    +
+


+

+
"The +wireless +telegraph +is +not +difficult +to +understand. The ordinary telegraph is like a very long cat. You pull +the tail +in New York, +and +it +meows +in +Los Angeles. The wireless is +the same, only without the cat."
+Albert Einstein
+

+
+  +

Dr. C. L. Davis
+Physics Department
+University of Louisville
+email: c.l.davis@louisville.edu +

+

+

+
+


+

+ + diff --git a/elec_coulomb_eqn1.gif b/elec_coulomb_eqn1.gif new file mode 100644 index 0000000000000000000000000000000000000000..87001de9bf5bd35120c930800417a18dc8f22a9c GIT binary patch literal 272 zcmV+r0q_1tNk%w1VNd`s0CxZ}A^sXuZ)S9NVRB_UAWdmwa&L2QW^^D=W@c$)WdI@h z1OWg5001ll00jU_04D%|000000008XgpaAq7}27fv^tBkUU2&_b>K)E(DNLrn5-pv zwkE62FBo4YSKlA1+sS&qO-=|%B|9RetvG2Wr%N4^#e{}b*CQ23Lvm40tpa=IrHSUC zm^xg)+dy``7M8Z7<@22wwpT;b^(Y{HBK*-EyZy W16|FKz1^(p9pxGxJo{@<0028dKy$bN literal 0 HcmV?d00001 diff --git a/elec_current.html b/elec_current.html new file mode 100644 index 0000000..c8b4eba --- /dev/null +++ b/elec_current.html @@ -0,0 +1,391 @@ + + + + + + + Electricity - Electric Current - Physics 299 + + + +
+

+
+
+

Electric Current, Resistance and Power
+

+
+

+
+ "When I find myself in the company of + scientists, I feel like a shabby curate who has strayed by + mistake into a drawing room full of dukes"
+ W. H. Auden
+
+
+
+
+

Electric Current
+

+
+
    +
  • Electric current is equal to the rate at which charge passes + a fixed point in space.
  • +
    +
    eqn9
    +

    +
    + Current is measured in Amperes: +
    +
    +
    1 + Ampere = 1 Coulomb/second
    +
    + Although from the above definition it looks as though the Ampere + is defined in terms of the Coulomb in fact it is the Ampere which + is the basic unit, the Coulomb is the dervived unit. The Ampere is + defined in terms of the force between two parallel wires carrying + current as we will see later.
    +
    +
  • It is important to realize that the value of the current is + constant, whatever the cross section of the conductor.  If + this were not so then charge would "pile up" at points along a + conductor.
  • +
    +
  • When you flip a switch a light bulb turn on instantly.  + In fact the current moves at speeds close to the speed of + light.  However, the charge carriers, electrons in a + metallic wire, travel at a much slower velocity - the drift velocity.
    + Consider a wire of length l, cross section A, with n conduction + electrons per unit volume.  The current in the wire can be + written,
  • +
+
eqn1
+
where e is the + charge on the electron and vd is the drift velocity.
+
+
+
    +
  • Current Density, J + (A/m2) is defined by,
  • +
+
eqn2
+
+
physically, + J represents charge movement at a particular place within a + conductor, e.g. when A is large J is small, when A is small + J is large.
+ The general relationship between I and J is
+
eqn3
+
The current is the flux of + J through a surface.
+
+ exclamation Important:  +The +current, +I, +is + + + + + + + + + + + + a scalar quantity, whereas J is a vector.  I has a + "sense" in that we draw arrows to represent its + "direction", but does not obey the rules of vector + algebra.
+
+
+
+
+
+
+
    +
    +
  • confused Historical quirk.  + The direction of current flow is defined as the direction in + which a positive charge will move.  But in solid metallic + conductors the charge carriers are electrons (negative charges) + which actually move in the opposite direction.  Negative + charges moving right to left are exactly equivalent to positive + charges moving left to right.
  • +
+
divider
+
+
+

Resistance

+
+
+
+
    +
+
    +
  • In metallic conductors the electric field and current density + are in the same direction and are found to be proportional to + each other,
  • +
+
eqn4
+
+
where ρ is the + resistivity of the conductor - characteristic of the + conductor.  The conductivity of a conducting material is + defined by, σ = 1/ρ.
+ For a uniform conductor, length l, cross section A, we have E = + V/l and J = i/A, so that
+
+
eqn5
+
+
The resistance of the conductor + R, is defined by,
+
eqn6
+

+ Resistance is measured in ohms (Ω), then resistivity has + units ohm.metre and conductivity (ohm.metre)-1 +
+
+
+
+
+
+
+
+
+
+
+
    +
  • exclamation Important: The relationship V = + IR is NOT + Ohm's Law !
  • +
+
OhmOhm's + Law:
+
"If the ratio of + voltage across a conductor to the current through + it is constant for all voltages then that + conductor obeys Ohm's Law"
+

+ Ohm's law holds for metallic conductors, but not + for devices such as transistors, diodes etc.  + The relationship V = IR can always be used to + determine the resistance at some particular I and + V for any device.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
    +
  • Even in conductors current will only flow between two points + A and B when
  • +
    +
      +
    1. There is a potential difference between A and B (producing + the electric field which forces the charges to move) and,
    2. +
    3. A and B form part of a complete circuit.
      +
    4. +
    +

    + divider
    +
    +
    +

    Power

    +
    +
+
    +
  • Suppose a charge dq moves from point A to point B, where the + potential difference between A and B is VAB, then the + energy released in time dt is given by
  • +
+
elec current eqn7
+
+
+
so that the rate at which energy is + transferred (power), P, is given by,
+
elec current eqn8
+
+
In terms of units we can state that  + Amps x Volts = Watts.
+
+
+
+
+
+
+
+
    +
  • The form of the energy "released" depends on the + electrical component placed between A and B, for + example,
  • +
+
    +
      +
    • Motor - mechanical energy (work) released 
    • +
    • Battery - chemical energy stored in the battery
    • +
    • Resistance - thermal energy (heat) released
      +
    • +
    +
+
+
+
+
+
    +
+
divider
+
+
+

Electro-motive Force - "emf"

+
+
+
+
    +
  • fig2In discussing electric circuits + you may come across the term "emf" - electro-motive + force.  It is important to realize that an "emf" is + NOT a force !
  • +
+
    +
  • If a device has an "emf" it has the ability to maintain a + potential difference (voltage).  Thus, for example, a + battery maintains an emf between its positive and negative + terminals.
  • +
+
    +
  • The emf of a device can be defined by ε = dW/dq, where dW + is the work done on a positive charge dq in taking it + acrosss the potential difference of the device.  In the + case of a simple circuit with a battery (see above) as a + charge traverses the external (to the battery) circuit it + loses energy.  In the circuit above the energy appeara + as heat and light in the light bulb.  When the + charge  returns to the battery the emf of the battery + replenishes its energy.
  • +
+
    +
  • At this introductory level we can consider the emf of a + "source" (battery, generator etc) to be exactly equivalent + to the voltage provided by the source.
  • +
+
    +
  • The direction of the emf always represents the direction a + positive charge would move in the external circuit.  + See circuit at right.  The emf direction is an + important factor when we use Kirchoff's laws to analyze + circuits.
  • +
    +
    +
+
    +
+
+
+
+
divider
+
+
+

Internal Resistance

+
+
+
+
    +
  • All emfs - batteries, generators etc - and electrical + measuring devices - ammeters, voltmeters etc - have an + "internal resistance".
  • +
+
    +
  • fig4As far as circuit analysis is + concerned these internal resistances can simply be + considered as resistors in series with the "ideal" + emf/meter.
  • +
+
    +
  • For ammeters (current measuring devices) the goal is to + have as low an internal resistance as possible so that the + current is not affected.
  • +
+

fig3

+
    +
  • For a voltmeter the internal resistance should be as large + as possible.
    +
  • +
+
+
+

+
+
+ +
+

Q: Does light have mass?
+ A: Of course not. It's not even Catholic!!!

+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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Dielectric Materials
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"Basic research + is like shooting an arrow into the air and, where it lands, + painting a target."
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+ Homer Burton Adkins
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  • In all our discussions to date we have implicitly assumed that + our charges have been located in a vacuum or on the surface of + conductors.  We now need to consider how to take into + account the presence of non-conducting material in the real + world.  Dielectric material is simply another way of saying + non-conducting material.
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  • elec dielec fig2Imagine a parallel + plate capacitor in which a dielectric material is placed between + the plates (at right below).  The dielectric is composed of + atoms/molecules which contain positive and negative + charges.  The applied electric field between the plates, E0, + will cause the positive and negative charges of the constituent + atoms/molecules to move slightly in opposite directions + (right).  Electric dipole moments will be "induced" in the + material, as shown.  The net effect will be for charge to + appear on the surface of the dielectric material as shown.  + The dielectric is said to have been polarized, leading to a + polarization electric field, EP.
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exclamation  In + conductors (metals) there are (almost) free electrons which will + move through the material when an electric field is applied, + generating an electric current.
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  • elec dielec fig1The net E field + between the plates has been reduced,
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elec dielec eqn1
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where κ is called the dielectric constant or + relative permittivity of the medium.
+
+ Note that for a vacuum, since EP = 0,  κ = 1 + and since  EP < E0 for all other + materials κ > 1.
+
+
+
+
    +
  • It is easy to show that for the parallel plate capacitor + the voltage (p.d) between the plates and the energy stored + are reduced by a factor κ, whereas the capacitance is + increased by a factor of κ.
  • +
+
    +
  • By application of Gauss's Law to a parallel plate + capacitor with a dielectric between the plates it can be + shown that to account for the presence of the dielectric + Gauss's Law becomes,
  • +
+
elec dielec eqn2
+
+
As a general rule when dielectric media is + present wherever ε0 appears it must be replaced + by ε0κ.
+
+
+
+
+
+
+
+

+ +

+

A chemist, a biologist and an + electrical engineer were on death row waiting to go in the + electric chair.

+ +

The chemist was brought forward first. + "Do you have anything you want to say?" asked the + executioner, strapping him in. "No," replied the chemist. + The executioner flicked the switch and nothing happened. + Under this particular State's law, if an execution attempt + fails, the prisoner is to be released, so the chemist was + released.

+ +

Then the biologist was brought + forward. "Do you have anything you want to say?" "No, just + get on with it." The executioner flicked the switch, and + again nothing happened, so the biologist was released.

+ +

Then the electrical engineer was + brought forward. "Do you have anything you want to say?" + asked the executioner. "Yes," replied the engineer. "If you + swap the red and the blue wires over, you might make this + thing work."

+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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+

+
+
+

Electric Dipole in an External Electric Field
+

+
+

+
+ " + + When you look at yourself from a universal standpoint, + something inside always reminds or informs you that there are + bigger and better things to worry about."
+ Albert Einstein
+
+
+
    +
+
    +
  • We have already considered the electric field created + by an electric dipole.  Now we consider the behavior of an + electric dipole placed in a uniform (constant) electric field.
  • +
+
elec dip in external field
+
+
+
    +
  • Note that since the force on each of the charges are equal + in magnitude but opposite in direction there is no net + force on the dipole.
  • +
+
    +
  • However, since the two forces are not concurrent, there is + a non-zero torque about the center of the dipole given by,
  • +
+
elec dip ext e eqn1
+
+
+
    +
  • Using the definition of the work done by a torque + (rotational force), it can be shown that the  + electrical potential energy stored by a dipole in an + external field is given by,
  • +
+
elec dip in ext e eqn2
+
+
    +
  • hotA dipole placed in a + uniform electric field will rotate until it is + aligned "-" to "+" along the field - this is the + lowest energy configuration.
  • +
+
    +
  • exclamation  If the external + field is not uniform, the net force will not be + zero.
    +
  • +
+
+
+
+
+
+
+
    +
+
    +
+
+
+
+ +
+ What is a quantum particle? + The dreams that stuff is made of!
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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+

+
+
+

Electric Field due to a Dipole
+

+
+

+
+ " + + To punish me for my contempt for authority, fate made me an + authority myself"
+ Albert Einstein
+
+
+
    +
+
    +
  • An electric dipole consists of two point charges of equal + magnitude, but opposite sign, separated by a short distance.
  • +
+
    +
  • The dipole  is electrically neutral, but due to the + separation of its charges gives rise to an electric field in its + vicinity.
  • +
+
electric dipole
+
+
    +
  • The electric field at the "field point" is given by  + E = E+q + E-q.  + + + + + Note that in adding the two electric fields the y-component + cancels leaving only an x-component given by,
  • +
+
elec dipole eqn1
+
+
where R is the distance from the centre of + the dipole to the field point and the approximation is + valid when r and R are almost equal.  In this case + the dimension of the dipole (a) is small compared to the + field point distance.
+
+
+
+
    +
  • p (=2aq) is called the electric dipole moment.  + It's actually a vector pointing from the negative to the + positive charge in the dipole so that,
  • +
+
elec dipole eqn 2
+
+
    +
  • hotMany molecules have + charge distributions which can be approximated as an + electric dipole, water being one of the most common.
  • +
+
water molecule
+
+
    +
+
+
+
+
+
    +
+
+
+
+

+ "How many Astronomers does it take to change a light + bulb ?
+ None, astronomers prfeer the dark"

+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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+

+
+
+

Electric Field

+
+

+
+"Discovery consists of seeing what everybody +has +seen and thinking what nobody has thought"
+Albert von Szent-Gyorgyi
+
+
+  +
    +
  • When two charges exert a force on each other, through what means +is the force transmitted ?
  • +
+
The simplest assumption is "action-at-a-distance".  +They just "know" +about each other's presence.  If one of the charges moves the +other charge is aware of this immediately.  This sounds like +"magic" with today's scientific understanding. 
+The mechanism to remove the magic was proposed by
+Michael Faraday  faraday +- the electric field.  +Every charge creates its own electric field in the space around it +(actually the space around it means all space); other charges then +interact with this field.  When a charge moves it creates a +disturbance in its electric field which is propagated away from the +charge at the speed of light.
+
+exclamation In developing his theory of gravitation +Newton was aware of the same "action-at-a-distance" +problem.  To solve the problem, in a similar manner, we introduce +the concept of the gravitational field.
+
+exclamation Note that the concept of the electric +field is a convenient construct to describe electromagnetic +phenomena, +but its true existence is neither proven nor essential.
+
    +
    +  +
  • The electric field vector, E,  +is defined in the following way.  If a charge, q, feels a force F, +then +the +electric field, E, at the location of the charge +is +given by
  • +
    +


    +
    +

    +
    +
    +
  • Units of electric field are  Newtons/Coulomb (N/C) or (as +we shall see later) Volts/metre (V/m).
  • +
    +  +
  • If the electric field felt by the charge q is due to +a point +charge Q, located a distance R from q, then its magnitude is given by
  • +
    +
    +


    +

    +
    +
    +
  • The electric field due to many point charges is given by the +vector +sum of the fields due to the individual charges
  • +
    +
    +

    E  =  E1  + + +E 2E 3 +  +.....

    +
    +
    +

    In performing this sum the direction of each E  +is +the +same as the force felt by a positive charge.

    +
  • The electric field in a region of space can be represented by +electric field lines otherwise known as "lines of force".
  • +
    +
    +

    +
    +
    +

    The direction of the electric field line is the same as that of +the force +felt by a positive charge.  The density of the field lines +provides a +measure of the magnitude of the field.  FIeld lines alway begin on +positive +charges and end on negative charges; they cannot be left 'hanging' in +empty +space.
    +

    +
+
efield_lines3.jpg
+
+
    +


    +

    +
    +

    The diagrams above display the electric field lines in the +vicinity of +two equal point charges.

    +
    +
+
+
+

+"Two things are +infinite: the universe and human +stupidity; and I'm not sure about the universe."
+Albert Einstein

+
+
+  +

Dr. C. L. Davis
+Physics Department
+University of Louisville
+email: c.l.davis@louisville.edu +

+

+

+
+


+

+ + diff --git a/elec_field_eqn1.gif b/elec_field_eqn1.gif new file mode 100644 index 0000000000000000000000000000000000000000..c3e8cc3ee66be546ad1ac8efb0631494b200d94b GIT binary patch literal 165 zcmZ?wbhEHb)Mn6OC}%KK{3jKhpH@vk zCI%J;H3n6N1`v;dfhn$~f2H)>y^DGnuXa8^8}r>?R{Dmm^2}WIR~w(aO4{&phg$N{ zP5%zsv||Egu#2Px;n Rk0MLeT&K literal 0 HcmV?d00001 diff --git a/elec_field_eqn2.gif b/elec_field_eqn2.gif new file mode 100644 index 0000000000000000000000000000000000000000..77abb94012cf7a95ddd2f2b426711b02b7a31ba0 GIT binary patch literal 204 zcmV;-05ktbNk%w1VK)FN0CxZ}A^sXuZ)S9NVRB_UAWdmwa&L2QW^^D=W@c$)WdI@h z1OWg5001ll00aOq03rZ@000000007ogpa9f5e%Gzw4%3gvz+Oxd>tol;fOvGRz~Qk zg4s*Da?;Y_boaVvJCt%ip^>HG;Z{nR@Ux;oi!;lVx=f0a*i)lvG<9Fh9h0czj>%!# z`3)D#Bky%^Zuivl7c}4N_va<}Br!-Rc7@o+71(tZwHW017byvN351uamf5z6XbF~3 G0029SQ&wpJ literal 0 HcmV?d00001 diff --git a/elec_gauss.html b/elec_gauss.html new file mode 100644 index 0000000..a2c4337 --- /dev/null +++ b/elec_gauss.html @@ -0,0 +1,227 @@ + + + + + + + Electricity - Gauss's Law - Physics 299 + + +
+

+
+
+

Gauss's Law
+

+
+

+
+ " + + Equations are just the boring part of mathematics. I attempt + to see things in terms of geometry."
+ Stephen Hawking
+
+
+
    +
  • gaussGauss's Law is the first of Maxwell's equations we + will consider.  At first the whole concept of Gauss's Law + will seem to be very abstract and confusing, confused   hopefully at least some of the confusion + will pass as you become more familiar with the idea.  
  • +
+


+
+

+
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+
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  • At the outset it is important to realize that Gauss's Law + and Coulomb's Law are different statements of the same + physical concept.  Which of the two is used in any + particular situation depends on the particular application and + what you are asked to determine.
  • +
+
    +
  • Before stating Gauss's Law we must first define the concept of + FLUX - in particular the flux of the electric field.
  • +
+
+
    +
  • Electric fluxAt every point on a + surface we can calculate an "element" of the electric flux + given by
  • +
+
defn of E flux
+
+
so that the total electric flux passing + through a surface, S, is given by,
+
+
elec gauss eqn2
+
+
+
+
+
+
+
    +
  • Gauss's Law then states that,
  • +
+
red tick    elec gauss 3    red tick
+
+
where the circle on the integral + means that the surface is closed and + qinside is the net charge inside this + closed surface.
+
+
+
+
    +
  • exclamation A closed surface has a definite + inside and outside differentiated by the surface, + e.g. the surface of a sphere.
  • +
  • exclamation The dA vector of a closed + surface is always directed from the inside to the + outside of the surface.
  • +
  • exclamation The exact location of the charges + inside the closed surface is not important, all + that matters is the net charge.
  • +
  • exclamation ε0 is the "Permittivity + of the Vacuum" a constant whose value is 8.85 x 10-12 + C2/(N.m2) where the Coulomb + constant, k = 1/(4πε0).  Note that + if the charges are not located in vacuum ε0 + must be replaced by the permittivity of the medium + in question. 
  • +
  • exclamation The proof of Gauss's Law is beyond + the scope of this course.  Suffice to say the + inverse square dependence on distance of Coulomb's + Law is critical.
  • +
+
    +
  • Before using Gauss's Law to evaluate electric + fields a brief qualitative discussion is + worthwhile.  Consider the situation of two + point charges below.  Application of Gauss's + Law over each of the closed surfaces:
  • +
+
+
    +
  • elec gauss figure 2 S1:  + At every point on this surface both E and + + + + + + + + + + dA are directed "outwards", such that the + scalar product E·dA = EdAcosθ is always + positive.  Thus the integral over the + surface S1 will be positive, as it + must be if Gauss's Law is to be satisfied, since + the net charge enclosed is positive.
  • +
  • S2E is directed + "inwards", dA "outwards", leading to a + negative value for the flux through S2, + consistent with the fact that the net charge + enclosed is negative.
  • +
  • S3:  Some of this surface has + E directed "inwards" the remainder has E + directed "outwards".  dA is + "outwards" everywhere on the surface.  + Therefore the flux integral has both positive + and negative contributions.  Since there is + no net charge enclosed by S3 by + Gauss's Law the net flux will be zero.
  • +
  • S4: Once again there are negative + and positive contributions to the flux integral, + so that we can write Gauss's Law,
  • +
+
elec_gauss_eqn4
+

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  •  Electric flux through a closed box animation.
  • +
+
    +
  • Coulomb's Law, Electric Field, Electric Flux and Gauss's Law video.
    +
  • +
+
+
+ +
+ Did you hear about the French post-doc + who went to work at the Fermi Lab, but never went in because + the sign over the door always said it was closed.
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/elec_gauss_apps.html b/elec_gauss_apps.html new file mode 100644 index 0000000..f34b597 --- /dev/null +++ b/elec_gauss_apps.html @@ -0,0 +1,517 @@ + + + + + + + Electricity - Quantitative use of Gauss's Law - Physics 299 + + +
+

+
+
+

Quantitative Use of Gauss's Law
+

+
+

+
+ " + + It has been said that democracy is the worst form of + government except all the others that have been tried."
+ Winston Churchill
+
+
+
 
+    elec gauss 3    
+
    +
  • Gauss's Law is valid for any closed surface (a Gaussian + surface) and any distribution of charges.  If the electric + field is known at every point on the surface S the integral can + in principle be evaluated and will be seen to be equal to the + sum of the enclosed charges divided by ε0. +    However, only in certain very symmetric situations, + where we can infer a great deal of information about the + electric field, can it be used to actually calculate E.  + + + + + + + + + + + + + In such cases Gauss's Law provides a short cut to determining E.  + + + + + + + + + + + + + The key is to be able to "extract" the E from the flux + integral.
  • +
+
    +
  • We will consider three possible geometric situations in which + we can obtain E from Gauss's Law:
  • +
      +
    • Spherical symmetry - three dimensions
    • +
    • Rectangular symmetry - two dimensions
    • +
    • Cylindrical symmetry - one dimension
    • +
    +
+
divider bar
+
+
SPHERICAL + SYMMETRY
+
    +
+
    +
  • Single Point Charge
  • +
+
elec gauss figure 5Consider a single point charge +Q and a spherical + surface, S,  of radius r and center at the location of + +Q.  From the symmetry of this situation we can conclude + that, everywhere on the surface S, E has the same value + and is directed radially outwards (normal to the surface).  + This is the same as the direction of dA.  Therefore,
+
elec gauss eqn5
+
so that,
+
elec gauss eqn6
+
which is exactly Coulomb's Law !!
+
+ As has already been stated - Gauss's + + + + + + + + + + + + + Law and Coulomb's Law are different statements of + the same physical principle.
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • Spherical Charge Distribution with Uniform + Charge Density
  • +
+
Charge is distributed uniformly + throughout the volume of the sphere (this means that the + sphere must be a non-conductor since as we have seen the + charge on a conductor must reside on the surface) such + that the total charge Q is given by,
+
+
elec gauss eqn7
+
+
where ρ is the (volume) charge + density, in units of Coulombs/m3.
+
+ What is the electric field at any point either outside + or inside the sphere ?
+ Due to the symmetry of this configuration we can + conclude that E is directed radially outwards + everywhere and can (at most) depend only on the + (radial) distance from the center of the sphere.  + There are two distinct regions to consider:
+
+ elec gauss figure 5Outside the + sphere,  r > R
+
+ Applying Gauss's Law over a Gaussian surface (sphere) + of radius r, then,
+
elec gauss eqn 5
+
+
so that,
+
elec gauss eqn 6
+
+
In other words, for points + outside the sphere, the sphere behaves as a + point charge located the sphere's center.
+ hot  We saw + exactly the same type of behavior when + considering the gravitational effect of a + spherical mass.
+
+
+
+
+
+ elec gauss figure 7Inside the sphere, r + < R
+
+ Applying Gauss's Law over a Gaussian surface (sphere) + of radius r, then,
+
+
elec gauss eqn8
+
Or in terms of Q and R,
+
+
elec gauss eqn9
+
+
Note that for r < R only + the charge inside a sphere of radius r + contributes to E.  The charge + between r and R has no effect.
+
+ exclamation It is important to realize that + without using Gauss's Law, these results could + be obtained via Coulomb's Law, but would + involve considerably more work - setting + up  a non-trivial multiple integral to + consider every point charge in the sphere....
+
+
divider bar
+ CYLINDRICAL + + + + + + + + + + SYMMETRY
+
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+
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+
+
+
    +
  • Infinite Line Charge + with Linear Charge Density λ
  • +
+
elec gauss figure8Determine + + + + + + + + + the E field a distance r from the + line charge.  (Note that the units of + λ are Coulombs/meter)
+
+ Symmetry tells us that E can only + have a component perpendicular to the line + charge, that is perpendicular to the + cylindrical surface shown.
+
+ Applying Gauss's Law over the cylindrical + Gaussian surface, radius r and length l, + as shown, there will in principle be three + contributions - one from the curved + surface and one from each of the two + ends.  However, on the ends E + and dA are perpendicular, so that + E·dA = 0, therefore there is no + contribution to the flux through S.  + On the curved surface E and dA + are parallel, thus,
+
+
elec gauss
+                                  eqn10
+
so that,
+
elec
+                                      gauss eqn11
+

+ We can extend this analysis to the + case of a uniformly charged + infinite cylinder in a similar + manner to the extension of the + point charge to the spherical + charge distribution above.
+
+
divider bar
+ RECTANGULAR + + + + + + + + SYMMETRY
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • Infinite plane of + charge
  • +
+
elec gauss
+                                            figure 9Determine the E + field at any distance above or + below an infinite plane with + charge density σ (Coulombs/m2).
+
+ Symmetry dictates the E + must be perpendicular to the + surface everywhere.
+
+ Applying Gauss's Law over the + cylindrical surface shown, + then the curved surface of the + cylinder  contributes + nothing to the flux since E + and dA are + perpendicular.  But on + the ends E and dA + are parallel.  Therefore,
+
+
elec gauss eqn12
+
+
so that,
+
elec gauss eqn13
+
+
+
+
+
+
That is the electric + field is constant - it does + not depend on how far the + field point is from the plane + !! 
+
+ exclamation Note + that this is only true for an + infinite plane of + charge.  If the distance + of the field point from the + plane is small compared to the + "size" of the plane, the above + expression is a good + approximation.
+
+
divider
+
+
+
+
+
    +
  • In all the above + situations the key to + using Gauss's Law is SYMMETRY.  + There must be enough + symmetry in the problem + to know the direction of + E everywhere in + the vicinity of the + charge + distribution.  + Knowing the direction of + E the trick is + then to choose a + Gaussian surface over + which to apply Gauss's + Law such that E + can be "taken out" of + the flux integral.  + So when using Gauss's + Law to determine E + there are three key + steps:
  • +
+
    +
      +
    1. +

      State what you are + assuming about E + based on the + symmetry of the + problem.

      +
    2. +
    3. +

      State clearly the + Gaussian surface(s) + you will use - often + most easily done by + sketching the + surface(s) on a + diagram.

      +
    4. +
    5. +

      Evaluate the + surface (flux) + integral to + determine E.  + The symmetry of E + and choice of + Gaussian surface + should allow "E" to + be "taken out" of + the integral and + thus be determined.
      +

      +
    6. +
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    + +
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+
An engineer + friend of mine told me of a group of scientists that were + nominated for a Nobel prize. Using dental tools, they were + able to sort out the smallest particles that mankind has yet + discovered. The group became known as " the Graders of the + Flossed Quark." + +
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/elec_gauss_cond.html b/elec_gauss_cond.html new file mode 100644 index 0000000..75d38c2 --- /dev/null +++ b/elec_gauss_cond.html @@ -0,0 +1,157 @@ + + + + + + + Electricity - Gauss's Law and Conductors - Physics 299 + + +
+

+
+
+

Gauss's Law and Conductors
+

+
+

+
+ " + + We shouldn't be surprised that conditions in the universe are + suitable for life, but this is not evidence that the universe + was designed to allow for life."
+ Stephen Hawking
+
+
+
    +
  • In electrostatic conditions - no electric current flow - + Gauss's Law applied to conductors (typically metallic objects) + leads to some important conclusions.
  • +
+
exclamation Note that since a conductor contains + "free" charges if an electric field exists anywhere in the + conductor a current will flow.  Thus, in electrostatic + conditions ("static" means all charges are at rest) there can be + no electric field anywhere in the conductor.
+
+
    +
  • elec gauss figure 4Insulated solid + conductor having a net charge
  • +
+
Under electrostatic conditions, E = 0 throughout + the object.  Applying Gauss's Law to the closed surface A, we + conclude that there can be no charge inside A.  But the + conductor has a net charge.  The only possibility is that the + charge resides outside the surface A.  If we gradually + increase the size of A, so that eventually it lies just below the + surface of the conductor, the charge must still reside outside + A.  Therefore, as a consequence of Gauss's Law, any charge + + + + + + placed in a conductor must reside on its surface.
+
+
+
    +
  • Insulated hollow charged conductor (conducting shell)
  • +
+
+

We now hollow out the conductor, changing nothing else.  + Thus, there is no charge inside the hollowed out conductor, so + that E = 0 inside.  This fact leads to the + necessity for an antenna to pick up radio signals inside a + car.  Radio waves are comprised of electric and magnetic + fields (electromagnetic waves - much more later), which must be + received by the radio.  But the car is approximately a + hollow metallic conductor, which means E = 0 + inside.  Without an antenna the radio waves cannot be + received by the radio.  The antenna provides a "shielded + channel" to direct the radio signal into the car.
+
+

+
elec gauss figure 4
+
+
+
+
+
    +
  • Faraday Cage
  • +
+
+

A Faraday + + + + cage is a metal container, which is used to shield + sensitive electronics from stray electric fields.  + Fields outside the container cannot penetrate due to the + above explanation.
+
+

+
+
    +
  • Proof of inverse square nature of Coulomb's Law
  • +
+
+

It can be shown mathematically that if Coulomb's Law is not + exactly of the inverse square form - 1/r2 then + the electric field inside a closed conductor would not be + exactly zero.  All experiments to date have failed to + measure such an electric field, with an accuracy such that + we know that the inverse component of r in Coulomb's Law is + 2 with an accuracy of 16 decimal places.
+

+
+
+
+
+
+
+ +
+ A Simpleton's Guide to Science (stolen + from UK magazine) +
+ Relativity : Family get-togethers at Christmas +
+ Gravity : Strength of a glass of beer +
+ Time travel : Throwing the alarm clock at the wall +
+ Black holes : What you get in black socks +
+ Critical mass: A gaggle of film reviewers +
+ Hyperspace : Where you park at the superstore

+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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z=r;t(*Gw6wfe&&!5>}Jne<#*s6(F9=Q~@&LMaBSarcG$QK3_vq$Z6u4@7LIFe>mQt zF>o)y2&K}3p3C82Z->=mY0Nf$EdS%mV+pib_Z|40{`}md$vaa~wU? z5XN3G_=BinidJ-Z-^so1c4qqZfln7%{4Mv4_RLgfwj{aF27h5EBDuiUF=;TZ#O7)R zIBlbS*!hJ0HxreX4>#{zyD1RspPFyVCg+bU`jkI@|k&?8>x z*htpEFsBW^;4SCGVn$`r*4L2%gQB||pK7DHB)sNBl^9|&i&5h}sB5=NwKJ~ysi?Pz-P|Xm z#8J!&z^ + + + + + + Electricty - Electric Potential Energy - Physics 299 + + +
+
+

Electric Potential Energy
+

+
+

+
+ "Atomic power will make electricity too + cheap to meter"
+ Glenn Seaborg
+
+   +
+
    +
  • It is important to realize that electric potential (potential + difference - measured in Volts) is not the same thing as + electric potential energy (measured in Joules).  We already + know how to calculate the pd dues to a system of charges,
  • +
+

eqn2

+
+

or for a point charge a distance rA + from the charge
+

+

  eqn6
+

+
+
    +
  • What about the Potential Energy, PE ?
  • +
+
+

elec pot energy fig1Imagine two charges Q + and q , with Q at rest.  Now gradually bring q from + + + infinity towards Q to point P.  Then the work + done by an external agent in performing this task must be equal + to the negative of the work done by the electric field due to Q,
+

+

elec pot energy eqn1
+

+
+
+

This work is stored in the two charge system as + Potential Energy, UE ,
+

+
elec pot energy eqn2
+
+
For systems of more than 2 charges we must + consider all possible pairings to determine the total + potential energy.
+
+ exclamation  For U > 0 energy must be provided to + create the system
+           For U < 0 energy must be + provided to break up the system (initially this system is + "bound")
+
+ exclamation  Don't forget  - Change in PE = work + needed to accomplish the change.
+
+
+
+

+
+

+ + Do you realize if it weren't for Edison we'd be watching TV by + candlelight?
+ Al Boliska
+

+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email:
c.l.davis@louisville.edu +
+  

+

+
+


+

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zt3U~nBt0)%gPZq>IW1;$u|Q%6(uSIq%|YhW>C2Pq>-iTKW1rG9BN#Q;WX+Gi?|?n_ zj>-%-r!m3HFugPykY1SxHm8_RZa+DIfANgCg9|qYgAt>uG~Q^{QaRwWfI>B_lLIcP zicHSzsmr@X0m1Ti_tao($cwt;3#i4+v4b`z9u387(T8PJ=yy246*p4ZZZ+ z0sI%MWkbOyGE$QRH19VI6@pZr`if3Hnmzuz7g2$B{ZCh6i%tuB{`v0tHK~H~WqNHX zicosX2^k7vU<8V*C$pg3#qwd#g|X<`7pkwGx(vTm7{VRg1ahk3f4o$6TL7PzzMBK( z4B_tS-pCLX$_?X4Cz| + + + + + + Electricty - Electric Potential and Potential Difference - + Physics 299 + + +
+
+

Electric Potential and Potential Difference

+
+

+
+ "The outcome of any serious research can + only be to make two questions grow where one question grew + before"
+ Thorstein Veblen
+
+ +   +
    +
  • The field near a system of charges can + also be described by a scalar quantity known as the "Electric + Potential".
    +
    + " A potential difference of one volt exists between two + points when one Joule of work is required to move one + Coulomb of charge from one point to the other"
    +
    + Between two points A and B we may write +

    + WAB  =  -VAB q
    +

    where VAB = VB + - VA  is the potential difference between A + and B.

    +
  • +
+
exclamation Note that WAB + is the work done by the electric field in moving the charge.  + The work done by the "external agent"  is -WAB.
+
    +
  • Units of potential difference are volts
    +
    +

    1 Volt  =  1 Joule/Coulomb (J/C)

    +
    +
  • +
  • In a region of space where there is an electric field the work + done by the electric field, dW, when a positive point charge, q, + is displaced by ds is + given by,
  • +
+
eqn1fig1 +
Therefore,
+
eqn2
+
+
For a uniform electric field we + obtain,
+
eqn3
+
+
where an arbitrary path can always be + split into sections along E and sections + perpendicular to E.
+
+
+
+
+
+
+
+
+
+
+
exclamation Note that this means that + the electric field can be expressed in the units + V/m.  [ 1 N/C = 1V/m ]
+
fig2divider
+
+
+
hotThe + electric field is a conservative field + hot
+
    +
+
    +
  • This means that the potential difference between + two points is independent of the path taken.  + Every point in space has a single value of V and E.
  • +
+

+ hamburger Food + for thought....
+
+ The gravitational field behaves in exactly the same + way.  Changes in gravitational energy are + independent of the path taken.  Climbing stairs + from one floor to another involve the same amount of + work against gravity as riding an elevator.
+ Note that in simple gravitational applications we + don't usually define a gravitational potential only + gravitational potential energy (mgh).  In this + case the gravitational potential is defined as gh.
+
+
+
    +
  • ideaExactly equivalent to gravity, it + is CHANGES + in potential difference, delta V, + which are defined.  To obtain absolute values + of V physicists usually define V = 0 at + infinity.  But this is an arbitary definition; + in engineering applications it is often convenient + to define the earth as V = 0.
  • +
+
divider
+
+
    +
  • Potential due to a point + charge
    +
    + For a single point charge Q the potential difference + between A and B is given by,fig3
  • +
+
eqn4
+
+
where + + + + + + + + + E is the field due to a point charge and ds = dr , + so that,
+
+
eqn5
+

+ If we assume rB= ∞ then VB + = 0 and,
+
+
eqn6
+
+
Note that the + potential is inversely proportional to r, + rather than r2 as in the case of + the electric field.
+
+
+
+
+
+
+
+
+
+
+
    +
  • Since V is a scalar the potential due + to multiple point charges is found by + adding the potential due to the + individual charges (taking into account + the sign of the charge). 
  • +
+
    +
  • Continuous Charge + Distributions
  • +
+
    +
+
For continuous distributions of + charge we may write,
+
+
+
eqn7
+
+

The electric potential due to a + continuous charge distribution can + be calculated in a similar manner to + the electric field due to such a + distribution.  For example the + potential at point P due to a + uniform ring of charge (below).
+

+
elec pot
+                                      fig4
+
divider
+
+
+
+
+
+
+
    +
  • Equipotential + Surfaces
    +
    + An Equipotential Surface is a + surface in space on which all points + have the same potential.  Since + all points have the same potential + it requires no work to move a charge + on such a surface.  This means + that there is no component of E in + the plane of the surface, in other + words E must be at right angles to + the surface.
  • +
+
Electric + Field Lines and Equipotential + Surfaces are at right angles
+
(Red dotted lines below)
+
equip2                                                      + + + + + + + + + equip1
+
+
+
+
+
+
+
+
+

+
+
+
+
+
+
+
+
+
    +
+

+
+

+
+

In the period that Einstein was active as + a professor, one of his students came to him and said: "The + questions of this year's exam are the same as last years!" + "True," Einstein said, "but this year all answers are + different."
+ Albert Einstein
+

+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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z;tpVD9{kUCEkoebePcZ2t};L_T(G9Tu?M({068caSOvu+aWhLI6ak`vCJ`Td0pJ*D h;UCII + + + + + + Electricty - Electric Potential due to an Electric Dipole - + Physics 299 + + +
+
+

Electric Potential due to an Electric Dipole
+

+
+

+
+ " + + We will make electricity so cheap that only the rich will burn + candles"
+ Thomas Edison
+
+
+
+
    +
  • Just as we calculated the electric field due to a dipole it is + possible to calculate the electric potential.  However, in + this case it is convenient to use polar co-ordinates. 
  • +
+
elec dip pot fig1Assuming no azimuthal variation we may write,
+
+
elec pot dip eqn1
+
+
When r >> a, then r1, r2 + and r are approximately equal and r2- r1 + = 2acosθ.  Therefore,
+
+
elec pot dip eqn2
+
+
+
+
+
+
    +
  • exclamation Note that the dipole potential decreases as 1/r2 + whereas the dipole E field decreases as 1/r3 + .
  • +
+
    +
  • exclamation Notice that when θ = 900 VP + = 0, that is it would require no work to bring a charge from + infinity to the center of the dipole if it moves along the + dipole bisector.
    +
  • +
+
+
+
+
+
+ +
+

+ + "There are three kinds of men. The one that learns by reading. + The few who learn by observation. The rest of them have to pee + on the electric fence for themselves."
+ Will Rogers
+

+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/elec_potential_efromV.html b/elec_potential_efromV.html new file mode 100644 index 0000000..c2cdd50 --- /dev/null +++ b/elec_potential_efromV.html @@ -0,0 +1,114 @@ + + + + + + + Electricty - Electric Field from Electric Potential - Physics + 299 + + +
+
+

Determining the Electric Field from the Electric Potential
+

+
+

+
+ " + + + + I've found out so much about electricity that I've reached the + point where I understand nothing and can explain nothing"
+ [Describing his experiments with the Leyden jar.]

+ Pieter van Musschenbroek
+
+
+
+
    +
  • Suppose we are given the electric potential, V, as a function + of position.  Is it possible to determine the E + field from this V ?
  • +
+
Starting from the definition of the potential difference + in terms of the E field,
+
elec pot eqn2
+
+
then if the path A to B is along the x axis we + obtain,
+
elec pot efromV eqn1
+
+
+
+
+
+
+
+
+
    +
  • This expression can be written in the alternative + form,
  • +
+
eqn2
+
+
+
If the path A to B is in an arbitrary + direction we obtain the most general expression of "E + from V",
+
+
elec pot efromV eqn3
+
+
+
+
+
+
+
+
+
+
+
+
+
+ +
+

+ + " + + I am an expert of electricity. My father occupied the chair of + applied electricity at the state prison."
+ W. C. Fields
+

+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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+

+
+
+

Electric Charge and Matter
+

+
+

+
+ "A new scientific truth does not triumph + by + convincing + its opponents and making them see the light, but rather + because its + opponents + eventually die, and a new generation grows up that is familiar + with it"
+ Max Planck
+
+ +
+
+
exclamation Electric + charge is "quantized" + in + units of the charge on the electron, e = 1.6 x 10-19 + C.  That is charge always appears in integer multiples of + "e".  Unless we are dealing with individual atoms or + sub-atomic + particles, since the value of "e" is so small, this quantization + is not + apparent and need to be considered. 
+
    +
  • Conservation of charge
  • +
    +
    +
    +
    +

    "The net electric charge of an isolated system + is + constant"
    +
    +

    +
    +

    However, Note that in calculating + the net charge + the sign of + the charges must be taken into account, e.g. the net charge of a + system + containing objects with +2 C  and -1 C is +1 C.

    +

    The conservation of charge is a basic physical "law" in the + same + manner as mass, energy and momentum conservation. 

    +
  • The force between two charges can be attractive or repulsive, + depending on the relative signs of the charges.  This fact + can be + expressed simply,
  • +
    +
    +

    "Like charges repel - unlike charges + attract"

    +
    +


    +

    +
  • The magnitude of the force between charged objects is given + by + Coulomb + 's Law (see later section for detailed description).
  • +
+
    +
  • Most useful/interesting electrical phenomena occur as electric + charge is transferred from one place to another.  Where + possible + electric charge will "flow"  to minimize the total energy + of a + system.  However, initially, we will discuss static + situations, + where the charges are held in place by some external mechanism.
    +
  • +
+


+

+
+

"If + I + were + as + rich + as + Rockerfeller, I'd be richer than + Rockerfeller" 
+ "How's that?" 
+ "I'd do a bit of window cleaning on the side"

+ Ronnie Barker
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+

+
+


+

+ + diff --git a/elec_strmatt.html b/elec_strmatt.html new file mode 100644 index 0000000..7284ede --- /dev/null +++ b/elec_strmatt.html @@ -0,0 +1,253 @@ + + + + + + + Electricity - Structure of Matter - Physics 299 + + +
+

+
+
+

Structure of Matter

+
+

+
+ "If I have seen further it is by standing + on the shoulders of giants"
+ Isaac Newton
+
+
+
    +
  • cartoon It is well known today that all + matter is comprised of atoms.  The concept of + atoms was postulated by the ancient + Greeks , but their existence was not confirmed until a + little more than 100 years ago.  All atoms consist of a + nucleus made of protons and neutrons around + which "orbits"atom
+          spinning one or more electrons. +  
  • +
    +
    +
  • The basic properties of protons, neutrons, electrons and the + atom as a whole are described in the following table.
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    Particle
    +
    +
    +
    Electric charge
    +
    +
    +
    Mass
    +
    +
    +
    Approximate Size
    +
    +
    +
    electron (e-)
    +
    +
    +
    - 1.6 x 10-19 C
    +
    +
    +
    9.1 x 10-31 kg
    +
    +
    +
     < 10-18 m
    +
    +
    +
    proton  (p)
    +
    +
    +
    + 1.6 x 10-19 C
    +
    +
    +
    1.6 x 10-27 kg
    +
    +
    +
    10-15 m
    +
    +
    +
    neutron  (n)
    +
    +
    +
    0
    +
    +
    +
    1.6 x 10-27 kg
    +
    +
    +
    10-15 m
    +
    +
    +
    Atom
    +
    +
    +
    0 (neutral)
    +
    +
    +
    (Np + Nn) x + 1.6 x 10 -27 kg
    +
    +
    +
    10-10 m
    +
    +
    +
    + where Np, Nn are the number of protons and + neutrons, respectively, in the nucleus of the atom.
  • +
+
    +
    + atomic structure +
  • Other important properties:
    +
      +
      +
    • Mass of proton (mp)  is approximately 2000 + times the mass of the electron (me)
    • +
      +
    • More than 99% of the volume of an atom is empty space. +  If the nucleus of the Hydrogen atom was one inch in + diameter the orbit of the electron would have a radius of +  ½ mile.
    • +
      +
    • Current experiments indicate that the electron has no + structure, it is a true point particle.
    • +
      +
    • Protons and neutrons are each comprised of quarks. +  With the same experimental limit as the electrons, + quarks also have no sub-structure.
    • +
      +
    • The wavelength of visible light is of order 5 x 10-7 + m.  Atoms are typically more than 1000 times smaller + than this wavelength.
      +
    • +
    +
  • +
    +
  • The properties of the different elements in nature, e.g. + Hydrogen, Sodium, Uranium etc. are defined by their "atomic + structure".  These elements can be arranged in a form + known as the + "periodic table" which illustrates the + similarities and differences between them.   Dmitri + Mendeleev mendeleev is generally + considered the creator of the modern periodic table.
    + Nottingham University has created the Periodic Table of + Videos, which features a short video clip describing the + properties, history, production etc of each element.
  • +
+
    +
  • Starting with the + Hydrogen atom, which consists of a single proton nucleus + and a single electron, different elements can be created by + adding additional protons to the nucleus while at the same time + adding the same number of electrons to keep the atom neutral. +  For light elements (excluding Hydrogen) the number of + neutrons in the nucleus is equal to the number of protons; for + heavier elements a neutron excess develops.  For example + the most common + Lead atom comprises 82 protons and 126 neutrons.  The + heaviest naturally occuring element is Uranium , having + 92 protons and 146 neutrons.  Certain elements with more + protons can be created but undergo radioactive decay with a + lifetime shorter than that of the earth.
    +
    atomic
+            examples
    +
  • +
+
+
    +
+
    +
  • For this course, the it is important to + realize that the negatively charged electrons are held in + "orbit" around the positively charged nucleus by the + electric force of attraction.  In fact all atomic and + molecular forces of any significance are electrical (or + electromagnetic) in nature.  It could be said that + all of Chemistry is due to the electromagnetic + force.  In fact virtually all everyday physical + phenomena are electromagnetic in nature, friction, + thermodynamics, elasticity etc etc.  Only those + phenomena involving very large masses (where the + gravitational force becomes significant) and certain very + small scale processes (e.g. radioactive decay, fission, + fusion) are not primarily electromagnetic in nature.  + Understanding electricity and magnetism - electromagnetism + -  is critical to our understanding of nature. 
    +
  • +
+ +
A + neutron walks into a bar. "I'd like a beer" he says. The + bartender promptly serves up a beer. "How much will that be?" + asks the neutron. "For you?" replies the bartender, "no charge"
+
+

+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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z2>Zqj%Is#Lm?0Io!01U(BB;LtgL0)xl@c21c)4(NRH(Hq_{DO%BuQC + + + + + + Light and Optics - Apparent Depth - Physics 299 + + +
+

+
+
+

Apparent Depth and Distortion

+
+
+
+
+"Give me but one firm spot on which to stand, +and +I will move the earth"
+Archimedes
+
+ +
+
+
    +
  • The depth of a swimming pool always appears to be less +than it actually is.  This effect is due to the refraction of +light rays as they traverse the boundary between water and air. 
  • +
    +
  • In the following graphic, the chest appears to be closer to the +surface than it actually is.
    +
    Apparent depth
    +
  • +
    +
  • By using Snell's law and simple geometry it can be shown that
    +
    +

    +
    +
  • +
    +
  • Spearfishing is particularly difficult due to the apparent depth +phenomenon !!
    +
    Spear fishhing
    +

    +
    +
    +
    +
  • +
  • The discontinuous bending of a stick held partly in and partly +out of water is also a result of light refraction at the air-water +interface.
  • +
+
Bent stick +
+
+ +
"Physics +is +like sex: sure, it may give some practical results, but that's not why we do it."
+Richard Feynman.

+
+
+  +

Dr. C. L. Davis
+Physics Department
+University of Louisville
+email: c.l.davis@louisville.edu +

+

+

+
+


+

+ + diff --git a/lo_appdepth_eqn1.gif b/lo_appdepth_eqn1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_brewster.html b/lo_brewster.html new file mode 100644 index 0000000..ac78f3d --- /dev/null +++ b/lo_brewster.html @@ -0,0 +1,115 @@ + + + + + + + Light and Optics - Brewster's Law - Physics 299 + + +
+

+

+
+
+

Brewster's Law
+

+
+

+
+ "If aliens + visit us, the outcome would be much as when Columbus landed in + America, which didn't turn out well for the Native Americans."
+ + + + + Stephen Hawking
+
+
+
+
+
    +
  • By applying the boundary conditions for electromagnetic waves + striking the boundary between two media it can be shown that the + intensity of the reflected and refracted waves depend on the + polarisation of the incident wave. 
    +
  • +
+
    +
  • In particular, for unpolarised light incident on the boundary + at a specific angle, θp , the reflected light + contains only one polarisation.  When this occurs the + reflected and refracted light rays are at 900 to each + other,
    +
  • +
+
brewster eqn1
+
+
    +
  • Using Snell's law we can show that,
  • +
+
brewster eqn2
+
+
This relationship is known as brewsterBrewster's + Law.
+
+
brewsters fig1
+
+
+
+
+
    +
  • For light incident on water (from air) we find that θp= + 530.  Thus, at this angle the light + reflected from the surface is (linearly) + polarised.  If this reflected wave is viewed with + polarised sunglasses (which block this particular + polarisation), no reflective glare is observed - + allowing the observer to see clearly underwater.
    +
  • +
+
+
+
+
+
    +
+
+
+
"Outside + of a dog, a book is a man's best friend. Inside of a dog + it's too dark to read."
+ + + Groucho Marx
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/lo_brewster.jpg b/lo_brewster.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_brewster_eqn1.jpg b/lo_brewster_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_brewster_eqn2.jpg b/lo_brewster_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_brewster_fig1.jpg b/lo_brewster_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_colour.gif b/lo_colour.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_coneofdarknessforfish.jpg b/lo_coneofdarknessforfish.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_dispersion.html b/lo_dispersion.html new file mode 100644 index 0000000..2e330aa --- /dev/null +++ b/lo_dispersion.html @@ -0,0 +1,92 @@ + + + + + + + Light and Optics - Dispersion - Physics 299 + + +
+

+
+
+

Dispersion

+
+

+
+"To err is human but to really foul things up +requires a computer"
+Anonymous
+(Farmers Almanac 1978)
+
+
+ +
    +
  • Without explicitly making the statement we have assumed that the +refractive index of a medium is independent of the frequency of the +light transmitted.  That is we have assumed for example that red +light and blue light have the same speed when passing through glass (or +any other medium).  This is not exactly true, refractive index is +frequency dependent such that for glass, nblue > nred +(blue light is bent more than red light when passing from air to glass) +and the speed of blue light in glass is slightly smaller than the speed +of red light.
  • +
    +
  • This fact may be demonstrated by the production of a spectrum of +colours when white light is incident on a glass prism. [White +light is comprised of all the wavelengths of visible light] Each +constituent of the white light has a slightly different refractive +index and is deflected (according to Snell's law) through a slightly +different angle.
    +
    +
    Prism
    +
  • +
    +
  • This phenomenon is known as DISPERSION. + White light is dispersed into its constituent +colours.
  • +
    +
  • A rainbow is created by a combination of dispersion and total +internal reflection within water droplets in the atmosphere.
    +
    Rainbow
    +
  • +
    +
  • An object appears a particular colour under white light +illumination because it reflects that colour strongly.
    +
    Colours
    +
    +For example a red object reflects most of the red wavelength +incident and absorbs most of the other wavelengths.  If +such an object is illuminated with blue light it will appear black, +the blue light is absorbed, but there is no red to be reflected.
  • +
+
+
+
"There +is no Dark Side of the Moon really.  Matter of fact it's all dark"
+Pink Floyd
+
+
+  +

Dr. C. L. Davis
+Physics Department
+University of Louisville
+email: c.l.davis@louisville.edu +

+

+
+


+

+ + diff --git a/lo_dsdiffraction.html b/lo_dsdiffraction.html new file mode 100644 index 0000000..8269c75 --- /dev/null +++ b/lo_dsdiffraction.html @@ -0,0 +1,194 @@ + + + + + + + Light and Optics - Double Slit Diffraction/Interference - + Physics 299 + + +
+

+

+
+
+

Double Slit Diffraction/Interference
+

+
+

+
+ " + + All of physics is either impossible or trivial. It is + impossible until you understand it, and then it becomes + trivial. + + "
+ + + + + + + Ernest Rutherford
+
+ +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • Intensity pattern for single slit diffraction:
  • +
+
eqn1    +     where        eqn2
+
+
    +
+
    +
  • Intensity pattern for double slit interference:
  • +
+
eqn1    +         where    +     eqn4
+
+
+
Note that φ' = φ/2, where φ is defined in the + double slit interference analysis.
+
+
+
+
+
    +
  • Remember that in the double slit interference analysis we + (implicitly) assumed that the two slits were "point" wave + sources.  Of course in reality this is impossible, no + sources are true "point" sources.  An accurate + description of double slit interference must include the + effect of the finite width of each slit - that is, the + diffraction phenomena from each slit.  A detailed + mathematical analysis leads to an intensity pattern at point + P given by,
  • +
+
eqn5
+
+
Qualitatively, the interference pattern is + modulated by the diffraction pattern, as indicated below.
+
fig1
+
+
+
+
+
exclamatiom Note + that the slit width "a" must be smaller than the slit + separation "d" (center to center).
+
+
+
+
exclamation The + number of interference maximum underneath the central + maximum of the diffraction envelope depends on the + relative values of a and d.
+
+ exclamation Diffraction minima occur at θ = + λ/a , 2λ/a , 3λ/a...      [ asinθ + = nλ ]
+          + Interference minima occur at  θ = λ/2d , 3λ/2d , + 5λ/2d...     [ dsinθ = (n+1/2)λ ]
+
+
+
+
    +
  • In the examples below the first intensity pattern + has  d = 3a, whereas the second pattern has d = + (15/2)a
    +
  • +
+
+
+

+
fig2  fig3
+
+
+
+
+
+
+
+
+
    +
+
    +
+
+
+
+
+
+
+
+
  + + + + A year after almost failing + her high school physics class, a girl told her older brother, +
+ "You know, my physics teacher was right about + the optical Doppler effect. You see those cars. The lights of + the ones approaching us are white, but the lights of the ones + moving away from us are red."

+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/lo_dsdiffraction_eqn1.jpg b/lo_dsdiffraction_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_dsdiffraction_eqn4.jpg b/lo_dsdiffraction_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_dsdiffraction_eqn5.jpg b/lo_dsdiffraction_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_dsdiffraction_fig1.gif b/lo_dsdiffraction_fig1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_dsdiffraction_fig2.jpg b/lo_dsdiffraction_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_dsdiffraction_fig3.jpg b/lo_dsdiffraction_fig3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_e_mag.gif b/lo_e_mag.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_emwave.gif b/lo_emwave.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_emwaves.html b/lo_emwaves.html new file mode 100644 index 0000000..c197b16 --- /dev/null +++ b/lo_emwaves.html @@ -0,0 +1,130 @@ + + + + + + + Light and Optics - Electromagnetic Waves - Physics 299 + + +
+

+
Electromagnetic Waves
+

+
+

+
+ "Basic research is what I am doing when I + don't know what I am doing"
+ Wernher von Braun
+
+
+
+
    +
  • At the end of the nineteenth century + James Clerk Maxwell was responsible for bringing together + the work of Coulomb, Ampere, Faraday and others, to provide an + elegant theoretical description of electricity and magnetism. +  Classical electromagnetism was described  by four "simple" + equations known as Maxwell's equations + .  These equations provided the "explanation" for + electromagnetic waves.
    +
    +
  • +
  • An electromagnetic wave is a transverse wave comprised of + oscillating Electric and Magnetic fields.  Mathematically + the analysis of EM waves is rather complex.  A plane + polarised electromagnetic wave traveling from left to right can + be represented as indicated below.
  • +
    emwave
    +
    The "traveling nature of the wave is + represented thus,emwave2 where E represents + + + the electric field and H the magnetic field.
    +
    +
    +
    +
  • Visible light is a very small part of the electromagnetic + spectrum.  The full EM spectrum extends continuously from + the very low frequency E and B oscillations of + radio waves through microwaves, infra-red, visible light, + ultra-violet, X-rays  to very high frequency gamma rays.
    +
    +
    EM spectrum
    +
  • +
    +
  • The speed of electromagnetic waves (of all frequencies and + wavelengths) in a vacuum is equal to the speed of light, c = 3 x + 108  m/s.  Speed, frequency (f) and + wavelength () are related by the usual + relationship
    +
    +
    loemwaveseqn2
    +
  • +
  •  Using Maxwell's equations it can be shown that the speed + of light is given by
    +
  • +
+
loemwaveseqn1
+
    +
    +
    +
  • Using Maxwell's decription of electromagnetic waves it is + possible to explain many of the properties of the interactions + of visible light with matter.  The details of such + explanations are technically quite difficult  and are + typically covered in senior level Physics courses.  In the + following discussion we will investigate some of the + consequences of these properites, for example, the law of + reflection, law of refraction (Snell's law), total internal + reflection, dispersion, interference and diffraction.
  • +
+
    +
  • The electromagnetic spectrum NASA video.
  • +
+
    +
  • Maxwell brief biography video.
    +
  • +
+
    +
  • Maxwell's equations and electromagnetic waves video.
    +
  • +
+
    +
+
+
+

"I've heard that the government wants to put + a tax on the mathematically ignorant. Funny, I thought that's + what the lottery was!"
+ Gallagher

+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/lo_emwaves_eqn1.jpg b/lo_emwaves_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_emwaves_eqn2.gif b/lo_emwaves_eqn2.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_emwaves_eqn2.jpg b/lo_emwaves_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_fiberopticcladdingandfiberbundle.jpg b/lo_fiberopticcladdingandfiberbundle.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference.html b/lo_interference.html new file mode 100644 index 0000000..dcc6f77 --- /dev/null +++ b/lo_interference.html @@ -0,0 +1,591 @@ + + + + + + + Light and Optics - Double Slit Interference - Physics 299 + + +
+

+

+
+
+

Double Slit Interference

+
+

+
+ "We only have + to look at ourselves to see how intelligent life might + develop into something we wouldn't want to meet."
+ + + Stephen Hawking
+
+
+
+
+
    +
  • We now begin a discussion of wave (physical) optics in which - + in contrast to geometric optics - we explicitly consider the + wave nature of light.  Remember, in geometric optics light + traveled in straight lines (light rays).  This is a valid + approximation so long as we do not consider apertures with + dimensions similar to the wavelength of the light or look too + closely at the edges of objects.
  • +
+
    +
  • So what happens when a wave passes through apertures whose + size is similar to the wavelength of the wave ?  First + we'll consider the case of double slit interference, in which a + parallel beam of incident monochromatic (containing a specific + wavelength) light from the left strikes a screen with two slits, + S1 and S2 , as below.
  • +
+
interference fig1
+
+ divider
+
+ CONDITIONS for MAXIMA and MINIMA + in DOUBLE SLIT INTERFERENCE
+
+
+
    +
  • With L >> d  geometric optics predicts that two + bright spots would be observed on the right hand screen + immediately opposite S1 and S2 .  + The rest of this screen would be in shadow.  What is + actually observed on the right hand screen is an + "interference pattern" as indicated below,
  • +
+
interference fig2     + + + + + + + + + + + + + + double slit interference
+
+
    +
  • The explanation is that each slit acts as a source of + spherical waves, which "interfere" as they move from + left to right as shown above.
  • +
+
    +
  • In the diagram at the top of the page, light reaching + P from S1 and S2 will travel + different distances.  Assuming that the light from + the two sources S1 and S2 are + initially in phase, then due to the path difference S1P + - S2P , at P the two waves will be out of + phase.  If the path difference is equal to an + integral number of wavelengths the waves will interfere + constructively, leading to a bright spot on the + screen.  Mathematically we can write this condition + + + + + + + + + + + + + + for maximum intensity as,
  • +
+
interference eqn1
+
+
where n can take on integer values, n + = 0, 1, 2, 3... and we have assumed that θ = θ' or in + other words the width of each slit is small compared + to their separation (D >> a above).
+
+ Similarly, the condition for minimum intensity at + + + + + + + + + + + + + + P, when the path difference is a multiple of half + wavelengths, is given by,
+
+
interference eqn2
+
where n can again take on integer + value, n = 0, 1, 2, 3...
+
+
+
+
+
+
+
+
    +
  • When the distance from slits to screen is much + larger than the distance on the screen, D + >> y above (or L >> x in first + diagram), then the angle θ is "small" and we may + assume tanθ is approximately equal to sinθ which + is approximately equal to θ (in radians) and we + may write,
  • +
+
interference eqn3
+
+
for the position on the screen + for maximum intensity.
+
+ divider
+
+
INTENSITY + DISTRIBUTION in DOUBLE SLIT + INTERFERENCE
+
+
+
+
+
+
    +
  • The interference pattern shown above was + first observed for visible light in 1801 by + Thomas + + + + + + + + + + + + Young Young, the experiment + is still sometimes called Young's slit + experiment.
  • +
+
exclamation In the above description we have + assumed the incident light is + monochromatic.  If white light + (containing all the wavelengths in the visible + spectrum) is used, the maxima for the + different wavelengths will occur at slightly + different positions (y) on the screen.  + In this case an interference pattern will only + be observed if the maximum - minimum + separation is much larger than the separation + between the maxima of the extreme wavelengths + in white light (red and violet) for the same + "n".
+
+
+
    +
  • In the above description we have shown + that at certain locations on the screen + there will be bright spots whereas at other + locations there will be no light - the + interference pattern.  But exactly how + does the light intensity vary as a function + of position on the screen ?
  • +
+
In the diagram at the top of this + page the electric field from light originating + at each of the slits S1 and S2  + + + + + + + + + + + + can be written,
+
+
interference eqn
+                              4
+
+
where each slit has the + same maximum E field, E0 + and φ is the phase difference due to the + path difference S1P - S2P.  + + + + + + + + + + +
+ Therefore the E field at P can be + written,
+
+
interference
+                                  eqn6
+
+
+
the product of an + amplitude and a sinusoidal time + varying wave.  In the case of + light waves the frequency of the time + varying part is so large that our eyes + and most instruments "see" only the + ampliutde part.  Actually, what + we observe is the intensity, + which is the square of the + amplitude.  The intensity + observed at P is then given by,
+
+
interference eqn7
+
+
as shown in the + red shading in the diagram + above.  Note that maxima of + the above cosine squared function + occur when φ = 2π n; this leads to + bright spots on the screen.
+
+
+
+
+
+
+
+
+
+
+
    +
  • As we have seen, when the path + difference is an integer multiple of + wavelengths, the waves from the two + sources interefer constructively.  + That is they are in phase, and as we + have seen above,  φ must be an + integer multiple of 2π,
    +
  • +
+
+
+
+
+
+
interference
+                                  eqn5
+
+
Thus for small values + of θ (sinθ = θ), φ and θ are + proportional to each other.
+
+ divider
+
+
COHERENCE
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • Throughout the above + description we have implicitly + assumed that the light waves + from the two slits S1 + and S2 are in phase + with each other.  This + ensures that any phase + difference in the light from the + two slits is due entirely to the + different path lengths the two + waves travel.  If the + relative phase of the + sources  S1 and + S2 is unknown and + randomly varying with time then + the interference pattern will + also randomly change with a + frequency similar to the + frequency of the light.  + Practically this means there + will be no observed interference + pattern.
  • +
+
    +
  • The difficulty of obtaining + two sources (S1 and S2) + emitting waves coherently + (in phase) depends on the + wavelength of the + electromagnetic wave. 
    +
  • +
+
For long wave radiation + (e.g. radio waves) a single source + illuminating the two slits results + in two coherent sources, since + this type of radiation is + typically produced in a continuous + waveform, as shown at left below.
+
+ For short wavelength radiation + (e.g. light) "waves" are typically + emitted by multiple individual + atoms in a random incoherent + manner (no definite phase between + these multiple "sources".  + The radiation is emitted in + "packets" rather than as a + continuous wave.  Thus a two + slit configuration, at left below, + will not produce an interference + pattern.  In order to observe + an interference pattern with light + the configuration at right below + must be employed.  The single + slit to the left of the two slits + ensures that light reaching the + two slits is from the same part of + the source and therefore in phase.
+ hot  Note that a + laser beam produces a coherent + light source and can be used to + create an interference pattern in + the left configuration.
+
+
interference fig4   +           +           +           +           +         +     interference fig5
+
+ divider
+
+ QUANTUM LIMIT - DOUBLE + SLIT INTERFERENCE
+
+
+
    +
  • interference
+                                              fig2Consider + double slit + interference.  In the + figure at right there are + locations on the screen + which have zero light + intensity.
  • +
  • Now gradually reduce the + intensity of the incident + light so that instead of a + "continuous" wave impacting + the slits we have individual + "photons" incident.  + This is the "quantum limit", + where we treat a source of + light as a source of + discrete "wave packets" or + photons.  with + individual photons incident + the same interference + pattern is observed.
    +
  • +
  • But with photons incident + one at a time it makes sense + to ask, "which slit did the + photon pass through ?"
  • +
  • Close one of the slits, + but continue illumination + with individual + photons.  The intensity + pattern observed on the + screen changes with only one + slit open.  But if the + photon passes through the + lower slit, how does it know + whether the upper slit is + open or closed ?  For + the interference pattern to + change it must know.
  • +
  • Explanation....  The + photon is an extended + object, so it never really + passes though one + slit.  Due to its + extended nature it can + "feel" whether the other + slit is open.
  • +
  • However, perhaps most + intriguing, is the fact that + if the two slits are + illuminated with a beam of + particles, for example + electrons, the same + interference phenomena is + observed.  You'd + naturally expect electrons + to go through one slit or + the other, but with both + slits open an interference + pattern similar to that at + left is observed.  The + electron is displaying "wave + properties".
  • +
  • This is an example of "wave-particle duality", + an important consequence of + the quantum theory of + matter.
  • +
  • For an accessible + description of what's going + on in the double slit + experiment see what Dr. + Photon has to say....
    +
  • +
+
interference fig6
+
+
+
+
interference fig7 +     +       +         +         +         +     interefernce cartoon
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
+
+
+
+
+
+
+
+
+
+
+
"One morning I shot an elephant in + my pajamas. How he got into my pajamas I'll never know."
+ + Groucho Marx
+ (in the film Animal Crackers)
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/lo_interference_eqn1.jpg b/lo_interference_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_eqn2.jpg b/lo_interference_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_eqn3.jpg b/lo_interference_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_eqn4.jpg b/lo_interference_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_eqn5.jpg b/lo_interference_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_eqn6.jpg b/lo_interference_eqn6.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_eqn7.jpg b/lo_interference_eqn7.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_fig1.jpg b/lo_interference_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_fig2.jpg b/lo_interference_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_fig3_inf.gif b/lo_interference_fig3_inf.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_fig4.jpg b/lo_interference_fig4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_fig5.jpg b/lo_interference_fig5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_fig6.jpg b/lo_interference_fig6.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_interference_fig7.jpg b/lo_interference_fig7.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_internalreflectiontwodiagrams.jpg b/lo_internalreflectiontwodiagrams.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_intthinfilm.html b/lo_intthinfilm.html new file mode 100644 index 0000000..dc17c5d --- /dev/null +++ b/lo_intthinfilm.html @@ -0,0 +1,326 @@ + + + + + + + Light and Optics - Interference from Thin Films - Physics 299 + + +
+

+

+
+
+

Interference From Thin Films
+

+
+

+
+ "Everything + + + + we call real is made of things that cannot be regarded as real"
+ + + + + Niels Bohr
+
+
+
    +
  • Double slit interference, described on the previous page, is + rarely observed in nature.  On the other hand, interference + due to thin films is quite frequently observed - swirling + colours on an oil slick, colours on a soap bubble, the purple + tinge on an expensive camera lens - are all examples of thin + film interference.
  • +
+
thin films fig1       + + + + thin films fig3      thin
+        films fig2
+
+
+
    +
  • thin films fig4Consider a thin film, + separating two regions of the same refractive index, for example + a soap bubble.  Incident light undergoes reflection and + refraction at every interface as shown at right.  There + will be multiple "rays" reflected back into the air (only the + first two are shown in the diagram) and multiple "rays" + transmitted through the film (only the first of these is shown).
  • +
+
    +
  • The reflected rays will interfere with each other, their phase + difference being determined by their path difference.
  • +
  • Similarly the transmitted rays will interfere with each other.
  • +
+
    +
  • exclamation Note that coherence of the interfering rays is + ensured since there is only one source - the incident ray + coloured blue in the diagram at right.
  • +
+
Assuming a monochromatic source and normal incidence, we + can expect that the two reflected rays will interfere + constructively if their path difference is an integer multiple of + wavelengths,
+
+
thin films eqn1
+
+
where d is the thickness of the film.
+
+
+
+
+
+
    +
  • First Correction
  • +
+
The path difference 2d is measured inside the film, + but the wavelength is the wavelength in air...  Therefore + the condition for a maximum should be,
+
+
thin films eqn4
+
+ However, we know that,
+
+
thin films eqn2
+
+
and using the definition of refractive + index,
+
+
thin films eqn3
+
+
This means the condition for + constructive interference can be written,
+
+
thin films eqn5
+
+
where n is the refractive index of + the film and λ is the wavelength of the light in + air.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • Second Correction
  • +
+
This result implies that when the film + becomes very thin (d approximately zero), we + should expect constructive interference + (equivalent to m = 0).  That is, just before + a soap bubble bursts, as its thickness gets + smaller and smaller, we should expect the swirling + colours to become brighter.  Actually the + opposite is observed - bubbles becomes dull just + before they burst - in other words destructive + interference must be taking place.confused
+
+ The explanation to this apparent conflict is that + the first reflected ray (green in the above + diagram) is phase changed by 1800 + by the act of reflection.  In + + + the above diagram, only this ray undergoes + such a phase change.  In fact the + rule (which can be verified theoretically from + Maxwell's equations) is,
+
+
"A 1800 + phase change on reflection + occurs when light incident from a less + dense medium (smaller n) reflects off the + boundary with a more dense (larger n) + medium"
+
+

+ Reflection from more dense to less dense + (large n to small n) causes no phase change + and transmitted rays never undergo phase + changes.
+
+ Incorporating this phase change leads to the + condition for constructive interference, +
+
+
thin films eqn6
+
and for destructive + interference,
+
+
thin films
+                                  eqn7
+
+
+
+
+
+
+
+
    +
  • IMPORTANT !!
  • +
+
thin films fig5The second correction + applies specifically to the air-soap-air + configuration, or more generally to the + circumstance of a film with greater refractive + index than the media on either side.  This + correction may not be needed for other thin film + configurations;  for example, a film of water + on a glass surface, shown at right.  In this + case, since nair < nwater + < nglass , there is a 1800 + phase change for the reflected rays from both + the air-water and water-glass interface.  + That is the condition for constructive + interference reverts to
+
+
thin films eqn5
+
+
+
+
+
+
+
+
and for destructive interference,
+
+
+
+
+
+
+
+
+
+
+
+
thin films eqn6
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • All of the above analysis assumes + a single wavelength of light.  + With incident white light on a thin + film maxima for different + wavelengths will occur at different + film thicknesses.  Thus, in + nature, where a thin film will not + typically have a fixed thickness, we + often observe swirling colours, as + the thickness of the film changes.
  • +
+
    +
  • Finally, why thin + films ?  A thin film implies + path differences (thicknesses) of no + more than a few wavelengths.  + Films thicker than this will not + necessarily exhibit this type of + interference due to re-emission and + scattering of light within the film.
    +
  • +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+

+
+
" + + Military intelligence is a contradiction in terms."
+ + Groucho Marx
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/lo_lenses.html b/lo_lenses.html new file mode 100644 index 0000000..e69de29 diff --git a/lo_lenses_eqn1.gif b/lo_lenses_eqn1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_malus.jpg b/lo_malus.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_maxeqn.gif b/lo_maxeqn.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_msdiffraction_eqn1.jpg b/lo_msdiffraction_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_msdiffraction_eqn2.jpg b/lo_msdiffraction_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_msdiffraction_fig1.jpg b/lo_msdiffraction_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_msdiffraction_fig2.jpg b/lo_msdiffraction_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_msdiffraction_fig3.jpg b/lo_msdiffraction_fig3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_msdiffraction_fig4.jpg b/lo_msdiffraction_fig4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_msgratings.html b/lo_msgratings.html new file mode 100644 index 0000000..183b86d --- /dev/null +++ b/lo_msgratings.html @@ -0,0 +1,234 @@ + + + + + + + Light and Optics - Multiple Slit Diffraction/Interference - + Diffraction Gratings - Physics 299 + + +
+

+

+
+
+

Multiple Slit Diffraction/Interference - Diffraction Gratings
+

+
+

+
+ " + + + + Physics is actually too hard for physicists"
+ + + + + + + David Hilbert (Mathematician)
+
+ +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • IMPORTANT: Don't be confused, diffraction + and interference are different aspects of the same phenomenon - + the superposition + of waves.
  • +
+
+

Interference:  Superposition of waves + from a finite number of "point" sources.
+

+

Diffraction:   Superposition + of waves from an infinite number of "point" sources, + comprising a single large source.
+

+
+ fig1 +
    +
  • Having discussed double slit interference/diffraction, the + natural extension is to 3, 4, 5...  multiple slits.
  • +
+
+

The diagram at right illustrates a 5 slit configuration, where + the path difference at point P between waves from each of the + slits is dsinθ with a slit separation of "d".  Clearly, if + the waves from all five slits are in phase, maximum intensity + will be observed at P when
+

+
+

eqn1
+

+
+
where n = 0, 1, 2,...
+
+
+
+
+
    +
  • A detailed mathematical analysis yields an intensity pattern + for N slits each of width d given by
  • +
+
 eqn1
+
+
where β is the same β as in the single slit + diffraction and 
+
+
+
+
eqn2
+
+
Note that this pattern is comprised of a + diffraction envelope together with the (sin2Nγ)/sin2γ + term.  This latter term leads to the existence of principal + maxima predicted by dsinθ = nλ together with much + weaker secondary maxima between the principal + maxima as shown below for a five slit example.
+
fig2
+
+ divider
+

DIFFRACTION GRATINGS

+
+
+
+
+
+
+
    +
  • The diffraction + grating is an important experimental application + of the multiple slit maximum/minimum phenomenon.  + In its simplest form a diffraction grating consistes of + a metal or glass plate with many very finely spaced + grooves or slits.
  • +
+
    +
  • For a metal grating interference occurs in the + reflected light.  For a glass grating reflected or + transmitted light will interfere.
  • +
+
    +
  • The "slit" spacing, d,  is typically defined by + the number of grooves per cm (or inch).
  • +
+
    +
  • Principal maxima are located at angles θ given by sinθ + = nλ/d.  Therefore, the smaller "d" (or the more + grooves per cm) the larger the angle θ.
  • +
+
    +
  • By examining the exact intensity formula it can be + shown that the smaller "d" the brighter the principal + maxima are compared to the secondary maxima.
  • +
+
    +
  • When an atom is "excited" the spectrum of light it + emits de-exciting back to its ground state is + characteristic of that  particular atom.  + Thus, observing the spectrum of light emitted by a star + gives an accurate measure of the elemental compostion of + the star.  Since the angle θ depends on the + wavelength λ we can use a diffraction grating to obtain + the spectrum of the light emitted by the star.
  • +
+
    +
  • For a "white" light source a diffraction grating may + lead to several "orders", corresponding to n = 1, 2, + 3,... Each order will contain the complete spectrum of + colors (see below).
    +
  • +
+
fig3
+
+
exclamation + Depending on the value of "d" the various "orders" may + overlap.  In other words red light (long + wavelength) in the first order (n=1) could have the + same value of θ as blue light (smaller wavelength) in + the second order (n=2).
+
+ exclamation Note that in a diffraction + grating red light is diffracted through a larger angle + than blue light, in contrast to the way in which a + prism separates the rainbow of colors (below).
+
fig4
+
+
+
+
+
+
+
+
+
+
+
    +
+
+
+
+
+
+
+
+
  + + A little boy refused to run anymore. When his mother asked him + why, he replied, "I heard that the faster you go, the shorter + you become
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/lo_polarisation.html b/lo_polarisation.html new file mode 100644 index 0000000..08004cb --- /dev/null +++ b/lo_polarisation.html @@ -0,0 +1,148 @@ + + + + + + + Light and Optics - Polarisation - Physics 299 + + +
+

+

+
+
+

Polarisation

+
+

+
+ " + + +
A physicist is just an + atom's way of looking at itself.
+ Read more at http://www.brainyquote.com/quotes/quotes/n/nielsbohr382329.html#urRTQl9BfaJk3yj3.99
+ + + +
A physicist is just an + atom's way of looking at itself.
+ Read more at http://www.brainyquote.com/quotes/quotes/n/nielsbohr382329.html#urRTQl9BfaJk3yj3.99
+ + + +
A physicist is just an + atom's way of looking at itself.
+ Read more at http://www.brainyquote.com/quotes/quotes/n/nielsbohr382329.html#urRTQl9BfaJk3yj3.99
+ A physicist is just an atom's way of + looking at itself"
+ Niels Bohr
+
+
+
+
    +
  • We have seen that electromagnetic waves may be described by + transverse waves of E and B (at right angles to + each other).  However, there is nothing special about any + particular direction for the E or B vectors to + oscillate.  In fact an unpolarised wave contains + waves with E vectors in all possible directions (for + each E wave there is a B wave at right + angles).  If the wave propagates in the z direction this + wave has E components in all possible directions.
  • +
+
    +
  • In a polarised wave the E vector oscillates in a + specific direction - the direction of polarisation.
  • +
+
+
    +
  • Polarised light does not appear any different to unpolarised + light to our eyes.  The difference only becomes apparent + when observing through transparent polarising material,
  • +
+
polarisation fig1                  + + + polarisation fig2
+
+
+
    +
  • The intensity of initially unpolarised light which + passes through two polarisers is given by malusMalus' + Law,
  • +
+
polarisation eqn1
+
+
where I0 is the incident + intensity and θ is the angle between the directions of + the two polarisers.  Note that for "cross + polarisers" there is no transmitted wave.
+
+
+
+
    +
  • hot  Analog TV and radio + signals are transmitted as polarised waves.  When + rotating the direction of an antenna you are + effectively rotating the second polariser, thus + varying the angle θ.  Maximum and minimum signal + reception will be obtained with orientations of the + antenna at ninety degrees to each other.
    +
  • +
+
+
+
+
+
+
+
+
+
+
+
"Alimony + + is like buying hay for a dead horse"
+ Groucho Marx
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/lo_polarisation1.jpg b/lo_polarisation1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_polarisation2.gif b/lo_polarisation2.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_polarisation_eqn1.jpg b/lo_polarisation_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_rainbowgirltwodrops.jpg b/lo_rainbowgirltwodrops.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_refl1.gif b/lo_refl1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_refl4.gif b/lo_refl4.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_reflection.html b/lo_reflection.html new file mode 100644 index 0000000..6583f4e --- /dev/null +++ b/lo_reflection.html @@ -0,0 +1,130 @@ + + + + + + + Light and Optics - Reflection - Physics 299 + + +
+

+
+
+

Reflection

+
+

+
+"We believe a scientist because he can +substantiate his remarks, not because he is eloquent and forcible in +his enunciation.  In +fact we distrust him when he seems to be influencing us by his manner"
+I. A Richards
+
+
+
reflection picture
+Edouard +Manet - A bar at the Folies_Bergere
+Spot the deliberate(?) mistakes
+
+
    +
  • A general description of the reflection of electromagnetic waves +is obtained by considering the wave nature of the radiation. + However, when the wavelength of the radiation is small compared +to any apertures or obstacles nearby we may treat the wave as a stream +of "particles" traveling in a straight line.  This is the +situation when considering the formation of images by visible light in +mirrors and lenses. In this case the straight lines +in which the visible light waves travel are known as light rays +.  The study of the behaviour of  light under these conditons +is known as Geometric Optics.
  • +
    +
  • Light rays undergo one of  two types of  reflection - +diffuse or specular.
    +
    +
      +
    • Diffuse reflection is more common for almost all objects. + The reflected light is emitted in all directions due to +irregularities on the surface of the material.
    • +
    • Specular reflection occurs if the irregularities of the +surface are small compared to the wavelength of the light.  In +this case reflection occurs at a single angle, for example from the +surface of a plane mirror or water.
    • +
    +
    specular /diffuse refelction
    +
    +We will consider only specular reflection in +this course
    +
  • +
    +
  • The law of reflection.(which can be derived from +Maxwell's equations)
    +
    +
    Angle of incidence = angle of reflection
    +
    +Angles are measured with respect to the normal to the surface.
  • +
    reflection1         +                    +                    + reflection2
    +
    +
  • Ray Tracing and Image Formation in a plane mirror
    +
    +Light rays are reflected from a plane mirror according to +the law of reflection. An image is formed on the "other" side of the +mirror, where the reflected rays appear to originate.
  • +
+
reflection from plane mirror
+
    +
    +
  • Images formed in plane mirrors have the +following properties,
    +
    +
      +
    • Located the same distance behind the mirror as the object is +in front of the mirror.
    • +
    • Same size as the object.
    • +
    • Right and left inversion
      +
    • +
    • Upright
    • +
    • Virtual
    • +
    +
    +An image is virtual if light rays do not physically pass +through it.  Real images are formed if light rays actually pass +through the image.
  • +
+ +

+"The +tides are a fight between the Earth and Moon. All water tends +towards the moon, because there is no water in the moon, and nature +abhors a +vacuum. I forget where the sun joins in this fight."
+Middle School Science Test Answer
+
+
+  +

Dr. C. L. Davis
+Physics Department
+University of Louisville
+email: c.l.davis@louisville.edu +

+

+
+


+

+ + diff --git a/lo_reflpic.jpg b/lo_reflpic.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_refracpic.jpg b/lo_refracpic.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_refraction.html b/lo_refraction.html new file mode 100644 index 0000000..da81b99 --- /dev/null +++ b/lo_refraction.html @@ -0,0 +1,137 @@ + + + + + + + Light and Optics - Refraction - Physics 299 + + +
+

+
+
+

Refraction

+
+

+
+ "Technology...the knack of so arranging + the world so that we need not experience it"
+ Max Frisch
+
+
+
+
    +
  • Refraction may be defined as the "bending" of light + (or any EM wave) as it passes from one medium to another. +  The bending is caused by differing speeds of the wave in + the two media.
  • +
    +
  • Refractive Index
    + The refractive index, n, of a medium is defined as the ratio of + the speed of light in a vacuum to the speed of light in the + medium
    +
    +

    +
    +
    +
  • +
  • Snell picture Snell + 's Law (Law of Refraction), describes exactly how much + the light is bent when passing from one medium to another.
    +
  • +
+
Snells law
+
Refraction
+          picture
+

+
    +
  • Important points
  • +
+ +
    +
      +
    • Light is bent towards the normal when moving into a + medium of larger refractive index.
    • +
    +
+
    +
      +
    • Angles of incidence, reflection and refraction are all + measured with respect to the normal.
      +
    • +
    +
+
    +
      +
    • As a general rule the refractive index increases with + the density of  the medium.  For example, the + refractive index of glass (1.5) is greater than that of + water (1.33) - glass is more dense than water.
    • +
    +
+
    +
      +
    • Light incident normally is not deflected.
    • +
    +
+
    +
      +
    • Light rays are "reversible".  In other + words the refracted ray in the above diagram can act as + an incident ray and the incident ray will represent the + direction of the refracted ray.
    • +
    +
+
    +
      +
    • The refractive index of a medium depends on the + wavelength of light, e.g. in glass, blue light travels + more slowly than red light.  This phenomena is + called "dispersion".
      +
    • +
    +
+
+
+
+ +
+

What is π?
+ Mathematician: "π is + the ratio of the circumference of a circle to its diameter."
+ Engineer: "π + is about 22/7."
+ Physicist: "π is 3.14159 plus or minus 0.000005."
+ Computer Programmer: "π is 3.141592653589 in double + precision."
+ Nutritionist: "You one track math-minded fellows, Pie is + a healthy and delicious dessert!"

+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/lo_refraction_eqn1.gif b/lo_refraction_eqn1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_anim.gif b/lo_sm_anim.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_anim2.gif b/lo_sm_anim2.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_anim3.gif b/lo_sm_anim3.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_anim4.gif b/lo_sm_anim4.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_anim5.gif b/lo_sm_anim5.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_cc.gif b/lo_sm_cc.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_convex.gif b/lo_sm_convex.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_defs.gif b/lo_sm_defs.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_objatF.gif b/lo_sm_objatF.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_sm_parallel.gif b/lo_sm_parallel.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_snellslaw.gif b/lo_snellslaw.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_spearfishing.jpg b/lo_spearfishing.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_speculardiffuse.jpg b/lo_speculardiffuse.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_spmirror.html b/lo_spmirror.html new file mode 100644 index 0000000..d3fbcb0 --- /dev/null +++ b/lo_spmirror.html @@ -0,0 +1,473 @@ + + + + + + + Light and Optics - Spherical Mirrors - Physics 299 + + +
+

+
+
+

Spherical Mirrors

+
+
+
+
+"A modern compter hovers between the +obsolescent and the non existent"
+Sydney Brenner
+
+ +
+
    +
  • There are two kinds of spherical mirrors, concave +and convex.Concave convex
  • +
  • Concave parallel The focal point (F) of a concave mirror is +the point at which a parallel beam of light is "focussed" after +reflection in the mirror.  For a convex mirror the focal point is +the point from which light appears to have originated after reflection +from the mirror.  The centre of curvature (C) is the centre of the +circle (sphere) of which the mirror is an arc.
    +
    +
  • +
  • Concave definitions The focal length (f) and radius of curvature +(R) are defined in the diagram at the right. It can be shown that R = +2f.  "A" in the diagram is known as the "vertex" (often +labeled V).
    +
    +
  • +
    +
  • Image formation in spherical mirrors is defined by +certain +"characteristic" rays whose behaviour is governed by the law +of reflection.
  • +
      +
    • Rays parallel to the principal axis are reflected through the +focal point - concave (or as if they came from the focal point +- convex ).
    • +
    • Rays passing through the focal point are reflected parallel to +the principal axis - concave (for convex mirrors a ray +that would have passed through the focal point is reflected parallel to +the axis).
    • +
    • Rays passing through the centre of curvature are reflected back +along their original path -concave (or a ray which would have +passed through the centre of curvature is reflected back along itself - + convex )
    • +
    +
    + Concave animation +
  • Concave mirror.  In +the animation the first two rays from the object are examples of the +first two characteristic rays described above.  Only two rays are +needed to define the position of the image.  The paths of the +other rays in the animation are +defined since the image position is already known.
    +The general characteristics of the image depend on the location of the +object with respect to the centre of +curvature and the focal point.  Animations of image formation +in a concave mirror for the five possible object positions can be +observed by choosing from the following options.
    + +
  • +
  • Convex mirror.  The general characteristics +of images in convex mirrors are independent of the location of the +object.  Three examples are shown below. +
    Convex images
    +
  • +
    +
  • Note that for the convex mirror the reflected rays DIVERGE +(this is also the case for the concave mirror when the object is closer +than the focal point to the mirror).  In these cases the image +formed is virtual - light rays do not pass though it, but to an +observer " appear" to come from it.
  • +
+
    +
  • Perhaps the most common everyday experience of a convex mirror is +the passenger side rear-view mirror in your car.  As can be seen +from the above ray diagrams, the image you see  in a convex mirror +is always smaller than the object.  You "know" the typical size of +a car or truck, so much so that the "size" it appears tells you how far +away it is (if you see a tiny car in the mirror you "know" it is far +away).  Thus, seeing a vehicle "looking smaller" due to the convex +mirror, will make you think the vehicle is further away from the mirror +than it actually is, hence the warning, "objects mirror are +closer than they appear".
    +
  • +
    +
  • Mirror equation.  For objects placed close to +the +principle axis the distance of the object from the vertex of the mirror +(p), the distance of the image from the vertex of the mirror (q) and +the focal length (f) are related by the following equation,
    +
    +

    +
    +

    +
    +
    +
  • +
  • Magnification.  The magnification (m) is +defined by,
    +
    +

    +
    +
  • +
    +
  • Sign Conventions.  In order to make use of the +above formulae the following sign conventions must be followed.
    +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    SIGN
    +
    +
    +
    +
    +
    +
    +
    -
    +
    +
    +
    f - focal length
    +
    +
    +
    Concave
    +
    +
    +
    Convex
    +
    +
    +
    p - object distance
    +
    +
    +
    Real
    +
    +
    +
    Virtual
    +
    +
    +
    q - image distance
    +
    +
    +
    Real
    +
    +
    +
    Virtual
    +
    +
    +
    m - magnification
    +
    +
    +
    Upright image
    +
    +
    +
    Inverted image
    +
    +
    +
    +
  • +
    +
  • Image properties. There are four basic properties, +dependent on the position of the object, as indicated in the table +below.  These properties can be verified either graphically or by +using the mirror +equation and the definition of magnification.
    +

    +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    Mirror
    +
    +
    +
    Object location
    +
    +
    +
    Image location
    +
    +
    +
    Type
    +
    +
    +
    Orientation
    +
    +
    +
    Relative size
    +
    +
    +
    CONCAVE
    +
    +
    +
    At infinity
    +
    +
    +
    At F
    +
    +
    +
    Real
    +
    +
    +
    Inverted
    +
    +
    +
    Smaller
    +
    +
    +
    CONCAVE
    +
    +
    +
    Beyond C
    +
    +
    +
    Between F and C
    +
    +
    +
    Real
    +
    +
    +
    Inverted
    +
    +
    +
    Smaller
    +
    +
    +
    CONCAVE
    +
    +
    +
    At C
    +
    +
    +
    At C
    +
    +
    +
    Real
    +
    +
    +
    Inverted
    +
    +
    +
    Same size
    +
    +
    +
    CONCAVE
    +
    +
    +
    Between C and F
    +
    +
    +
    Beyond C
    +
    +
    +
    Real
    +
    +
    +
    Inverted
    +
    +
    +
    Larger
    +
    +
    +
    CONCAVE
    +
    +
    +
    At F
    +
    +
    +
    At infinity
    +
    +
    +
    No image
    +
    +
    +
    No image
    +
    +
    +
    No image
    +
    +
    +
    CONCAVE
    +
    +
    +
    Closer than F
    +
    +
    +
    Behind the mirror
    +
    +
    +
    Virtual
    +
    +
    +
    Upright
    +
    +
    +
    Larger
    +
    +
    +

    +
    +
    +
    CONVEX
    +
    +
    +
    Anywhere
    +
    +
    +
    Behind the mirror
    +
    +
    +
    Virtual
    +
    +
    +
    Upright
    +
    +
    +
    Smaller
    +
    +
    +
    +
  • +
+ +
What's +a light-year?
+One-third less +calories than a regular year.
+(Very Punny)
+
+
+  +

Dr. C. L. Davis
+Physics Department
+University of Louisville
+email: c.l.davis@louisville.edu +

+

+

+
+


+

+ + diff --git a/lo_spmirror_eqn1.gif b/lo_spmirror_eqn1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_spmirror_eqn2.gif b/lo_spmirror_eqn2.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction.html b/lo_ssdiffraction.html new file mode 100644 index 0000000..9263a5f --- /dev/null +++ b/lo_ssdiffraction.html @@ -0,0 +1,251 @@ + + + + + + + Light and Optics - Single Slit Diffraction - Physics 299 + + +
+

+

+
+
+

Single Slit Diffraction
+

+
+

+
+ " + + + + Physics is really nothing + more than a search for ultimate simplicity, but so far all we + have is a kind of elegant messiness.”
+
Bill Bryson, A Short History of Nearly Everything

+
+ +
+
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+
+
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+
+
    +
  • Diffraction may be thought of as the "spreading out" of waves + as they pass through or by an aperture or edge.  We now + investigate this phenomenon more closely for a single slit of + width "a".
  • +
+
    +
  • By considering point sources in pairs across the width of the + slit we can use the ideas of double slit interference to show + that minimum intensity is observed at point P on a screen when
  • +
+
eqn1
+
+
+
fig1                               + + + fig4
+
+
    +
  • fig2However, in order + to obtain the intensity profile as a function of θ we must + perform a more complex analysis.  Considering each + point in the slit as a point source, for electromagnetic + waves the electric field at a point P on the screen at right + can be written,
  • +
+
eqn4
+

+
where ω is the angular frequency of the wave and + Δφ is the phase angle.  The relationship between + phase angle and path difference Δs is given by
+
eqn2
+
+
From the diagram at right the path + difference Δs, relative to the center, is given by + ysinθ.  In this case
+
eqn3
+
+
so that the total electric field + at a point on the screen is obtained by + integration over the slit
+
+
eqn5
+
+
Performing this integration we + obtain
+
eqn6
+
where eqn7
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
+
+
+
    +
  • The intensity observed is proportional to the square of the + electric field
    +
  • +
+
eqn8
+
+

+ ssdifffig5
+

+
    +
  • Using the intensity function above the minima condition is + given by sinβ = 0  ( β ≠ 0 ), which means β = nπ where + n = 1,2,3...   Using the definition of β above + gives the condition for minima
  • +
+
eqn1
+
+
as expected.
+
+
+
+
+
    +
  • exclamation For n = 0 above, θ = 0, but + the intensity function leads to the central maximum.
  • +
  • exclamation Note that the width of the + central maximum - 2λ/a - is double that of secondary + maxima - λ/a.  This is in contrast to the double + slit interference pattern where all maxima have the same + width.
  • +
  • exclamation The location of the secondary + maxima are given approximately by 
  • +
+
eqn9
+
+
The exact position of these maxima is + shifted slightly towards smaller θ.
+
+
+
+
    +
  • In the above analysis we have (implicitly) assumed + that the source and observation screen are + infinitely far from the single slit (on opposite + sides of the slit).  This allows us to use the + plane wave approximation leading to the intensity + expression above.  This is described as Fraunhofer fraunhoferdiffraction.  + Practically we can approximate a Fraunhofer + situation by using converging lenses to produce + parallel rays.
  • +
+ +
+
+
    +
+
+
+
    +
    +
    +
+
+
+
+
+
+
  + + + + + + + + The Official Unabashed Scientific Dictionary defines a + transistor as a nun who's had a sex change
+
+
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/lo_ssdiffraction_eqn1.jpg b/lo_ssdiffraction_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_eqn2.jpg b/lo_ssdiffraction_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_eqn3.jpg b/lo_ssdiffraction_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_eqn4.jpg b/lo_ssdiffraction_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_eqn5.jpg b/lo_ssdiffraction_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_eqn6.jpg b/lo_ssdiffraction_eqn6.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_eqn7.jpg b/lo_ssdiffraction_eqn7.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_eqn8.jpg b/lo_ssdiffraction_eqn8.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_eqn9.jpg b/lo_ssdiffraction_eqn9.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_fig1.gif b/lo_ssdiffraction_fig1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_fig2.jpg b/lo_ssdiffraction_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_fig4.gif b/lo_ssdiffraction_fig4.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_ssdiffraction_fig5.jpg b/lo_ssdiffraction_fig5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilm_fig1.jpg b/lo_thinfilm_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_eqn1.png b/lo_thinfilms_eqn1.png new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_eqn2.png b/lo_thinfilms_eqn2.png new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_eqn3.png b/lo_thinfilms_eqn3.png new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_eqn4.png b/lo_thinfilms_eqn4.png new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_eqn5.png b/lo_thinfilms_eqn5.png new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_eqn6.jpg b/lo_thinfilms_eqn6.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_eqn7.jpg b/lo_thinfilms_eqn7.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_fig2.jpg b/lo_thinfilms_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_fig3.jpg b/lo_thinfilms_fig3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_fig4.jpg b/lo_thinfilms_fig4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_thinfilms_fig5.jpg b/lo_thinfilms_fig5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_tir.html b/lo_tir.html new file mode 100644 index 0000000..351163d --- /dev/null +++ b/lo_tir.html @@ -0,0 +1,120 @@ + + + + + + + Light and Optics - Total Internal Reflection - Physics 299 + + +
+

+
+
+

Total Internal Reflection

+
+

+
+ "One machine can do the work of fifty + ordinary men.  No machine can do the work of one + extraordinary man"
+ Elbert Hubbard
+
+
+
    +
  • When light passes from a medium with refractive index n1 + to a medium with refractive index n2, where n 1 + > n2, if the incident angle is greater than some + critical angle , the light will undergo + total internal reflection (TIR).
  • +
    +
  • The diagram below represents a light source underwater with + light rays striking the water-air interface.  Rays 1, 2 and + 3 are refracted at the interface according to Snell's law, the + refracted ray bending away from the normal such that    > . When = 90o + the refracted ray is directed along the surface of the water and + is called the critical angle .  For incident angles greater than + there is no refracted ray, the light + undergoes total internal reflection back into the water.
  • +
+
exclamation Note that even when TIR does not take + place there will always be a reflected wave (back into the water), + in this case, of low intensity (about 4% of the total energy is + reflected).  In other words, at an interface where the + incident angle is less than the critical angle, part of the + incident energy is transmitted and part is reflected.
+
+
    +
    + confused smiley  Even though all the light energy undergoes + total internal reflection, there is a "disturbance" in the other + medium (air in the example above) called the evanescent wave which + travels along the interface between the two media...
    +
    TIR
    +
    +
  • At the critical angle
    +

    +
    +
    + For light  passing into air from water the critical angle + is equal to about 49o
  • +
    +
  • Thus a fish-eye's view is of the world in a 98o + cone.  Outside this cone all is darkness...[Actually this + is not true due to the scattering of light in water]
    +
    Fish view
    +
    +
  • +
    +
  • Total Internal Reflection is the mechanism by which fibre + optic cables work.  Signal losses along such cables is + extremely small since ALL the incident energy is + reflected.
    +
    +
    Fibre
+            Optic
    +
  • +
+ +
"I know that this defies the law of gravity, + but, you see, I never studied law."
+
Bugs Bunny
+
+
+
 
+ Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu
+   +

+
+


+

+ + diff --git a/lo_tir_eqn1.gif b/lo_tir_eqn1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tir_eqn2.gif b/lo_tir_eqn2.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tir_eqn3.gif b/lo_tir_eqn3.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tir_eqn4.gif b/lo_tir_eqn4.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tl_conimg1.gif b/lo_tl_conimg1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tl_conimg2.gif b/lo_tl_conimg2.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tl_conv2.gif b/lo_tl_conv2.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tl_conv3.gif b/lo_tl_conv3.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tl_div2.gif b/lo_tl_div2.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tl_div3.gif b/lo_tl_div3.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tl_divimg1.gif b/lo_tl_divimg1.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tl_lens_anim.gif b/lo_tl_lens_anim.gif new file mode 100644 index 0000000..e69de29 diff --git a/lo_tl_typelens.jpg b/lo_tl_typelens.jpg new file mode 100644 index 0000000..e69de29 diff --git a/lo_young.jpg b/lo_young.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_LR.html b/mag_LR.html new file mode 100644 index 0000000..a650724 --- /dev/null +++ b/mag_LR.html @@ -0,0 +1,208 @@ + + + + + + + Magnetism - LR Circuits - Physics 299 + + + +
+

+
+
+

LR Circuits
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • As we have seen, due to Faraday's Law, an induced emf will + appear in any loop/coil/circuit in which the current (and + therefore the magnetic field) changes with time.  Certain + circuit elements have a larger (self) inductance than others - + the solenoid is the most obvious example of a circuit element + with a large inductance.
  • +
+
    +
  • An ideal inductance (L) has no resistance and is represented + by the following symbol  magLRfig1 + .
  • +
+
+
    +
  • magLRfig2Placing an inductance (L) and a + resistance (R) in a simple circuit gives us the "LR" circuit, + at right.  When the switch is closed to include the power + supply in the circuit we can apply Kirchoff's loop theorem to + obtain the following equation,
  • +
+
magLReqn1
+
+
exclamation Note + that in applying the loop theorem we make use of the fact + that the induced emf across the inductance opposes the + changing emf that causes it.
+
+
+
+
    +
  • magLRfig3The solution of this + differential equation is given by,
  • +
+
magLReqn2
+
+
shown at right, where Imax is + ε/R  .  Also shown is the time dependence of + the voltage across the inductance.
+
+
+
+
    +
  • The time constant of the inductance, given by  + magLReqn3 is a measure of the time + needed for the current in the circuit to reach 63% of + its maximum value.
  • +
+


+

+
    +
+
    +
  • magLRfig4Taking the + power supply out of the circuit leads to an even + simpler differential equation
    +
  • +
+
+
+
+
+
+
magLReqn4
+
+
the solution of which is
+
magLReqn5
+
shown at right.  In this figure I0 + = ε/R.
+
+
+
+
+
    +
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+
+
+
+
+
+

+ +

+ This girl said she recognized me from + the vegetarian club, but I'd never met herbivore.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+
+ + diff --git a/mag_LR_eqn1.jpg b/mag_LR_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_LR_eqn2.jpg b/mag_LR_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_LR_eqn3.jpg b/mag_LR_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_LR_eqn4.jpg b/mag_LR_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_LR_eqn5.jpg b/mag_LR_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_LR_fig1.jpg b/mag_LR_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_LR_fig2.jpg b/mag_LR_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_LR_fig3.jpg b/mag_LR_fig3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_LR_fig4.jpg b/mag_LR_fig4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere.html b/mag_ampere.html new file mode 100644 index 0000000..bcea834 --- /dev/null +++ b/mag_ampere.html @@ -0,0 +1,243 @@ + + + + + + + Magnetism - Ampere's Law - Physics 299 + + + +
+

+
+
+

Ampere's Law
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • magamperefig3We have just seen that + the Biot-Savart Law is in some sense the magnetic equivalent of + Coulomb's Law.  Is there a magnetic equivalent of Gauss's + Law ?  The answer is (of course) yes - Ampere's Law.
    +
  • +
+
magampereeqn1
+
+
where ds is an element of length + around an arbitrary closed loop "C", called an Amperian loop + and the summation is over all currents passing through + the loop.
+
+ exclamation Currents passing "out of" the loop are defined + as positive, currents passing into the loop are negative, + whereas current which do not pass through the loop are not + included in the summation.
+
+ exclamation Ampere's Law forms part of the second of + Maxwell's equations.   We will shortly adjust it + slightly (following Maxwell), to complete the second of + Maxwell's equations.  Remember, Gauss's Law was the first + of Maxwell's equations.
+
+
+
+
    +
  • As a reminder, Gauss's Law appears below,
    +
  • +
+
+ elecgausseqn3
+
+
exclamation Note that + Gauss's Law involves a surface integral + of E over a closed Gaussian surface "S".  + Ampere's Law involves a line integral + around a closed Amperian loop "C".
+
+ exclamation Gauss's Law is valid for any arbitrary closed + Gaussian surface.  Similarly Ampere's Law is valid for + any closed (Amperian) loop.
+
+
+
+
    +
  • Although Ampere's Law is true for any closed loop "C", it + is only useful to calculate B for some very + symmetric cases, where we already know (from symmetry) some + of the properties of B.
  • +
+
divider
+
+

Simple Applications

+
+
    +
  • B due to an infinite straight current carrying + wire.
    +
      +
      +
    • magamperefig4Symmetry argument: +

      Since the wire is infinite, we know from the + Biot-Savart Law that B is perpendicular to dl + and r and thus lines of B must form + concentric circles around the current.  Also, B + can, at most, depend only on the distance from the + wire, r.
      +

      +
    • +
    • Choice of Amperian Loop: +

      The Amperian loop is chosen so that B is + constant on the loop and in the same direction as ds + - that is a circle whose plane is perpendicular to the + wire and centered on the wire.  This allows us to + take B "out of the integral".

      +
    • +
    • Evaluation of B: +

      With the Amperian loop above we have
      +

      +
      magampereeqn2
      +
      so that
      +
      magampereeqn3
      +
      +
      directed "circumferentially" + around the loop, with the sense given by the + right-hand-rule described under the Biot-Savart + law.
      +
      +
      +
      +
    • +
    +
    magamperefig5
    +
    +
      +
    +
  • +
    +
+
+
    +
    +
  • magamperefig1B due to + an infinite solenoid +
      +
      +
    • Symmetry argument: +

      Since the solenoid is infinite, we conclude that B + is directed along the axis of the solenoid.  + Also, B can, at most, depend only on the + distance from the axis of the solenoid.  For an + infinite solenoid B = 0 outside the solenoid.
      +

      +
    • +
    • magamperefig2Choice of Amperian Loop: +

      The rectangular Amperian loop (at right) is chosen + so that B is constant on the two sides + parallel to the solenoid axis and perpendicular to the + ds on the other two sides. This allows us to + take B "out of the integral".

      +
    • +
    • Evaluation of B: +

      With the Amperian loop above we have
      +

      +
      magampereeqn4
      +
      so that
      +
      magampereeqn5
      +
      +
      where n is the number of turns + per unit length and the field is directed along + the solenoid axis as shown. (Use + right-hand-rule) 
      +
      +
      +
      +
    • +
    + exclamation Note that, the above analysis is + true only for an infinite solenoid.  For + a real solenoid it is a good approximation inside, away from + the ends.  The B field outside is not zero, but + much smaller than the field inside.
  • +
+
exclamation The shape of the B field due + to a solenoid is the same as that of a bar magnet and a + magnetic dipole.
+
magamperefig6
+
+
+
    +
+
+
+
+

+ +

+ What does a clock do when it's hungry ? + It goes back four seconds.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+
+ + diff --git a/mag_ampere_eqn1.jpg b/mag_ampere_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_eqn2.jpg b/mag_ampere_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_eqn3.jpg b/mag_ampere_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_eqn4.jpg b/mag_ampere_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_eqn5.jpg b/mag_ampere_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_fig1.jpg b/mag_ampere_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_fig2.jpg b/mag_ampere_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_fig3.gif b/mag_ampere_fig3.gif new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_fig4.gif b/mag_ampere_fig4.gif new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_fig5.jpg b/mag_ampere_fig5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_ampere_fig6.gif b/mag_ampere_fig6.gif new file mode 100644 index 0000000..e69de29 diff --git a/mag_biotsavart.html b/mag_biotsavart.html new file mode 100644 index 0000000..2a87074 --- /dev/null +++ b/mag_biotsavart.html @@ -0,0 +1,259 @@ + + + + + + + Magnetism - Biot-Savart Law - Physics 299 + + + +
+

+
+
+

The Law of Biot-Savart
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
+
+
+
+
+
+
+
+
+
+
    +
  • We have seen that a magnetic field exerts + a force on a moving charge or current, just + as an electric field exerts a force on a + charge.
  • +
+
magforcechargeeqn1 +          and   +      magforcecurrenteqn3
+
+ magbiotsavarteqn1
+
+
    +
  • In the electric case Coulomb's Law + gives us a method of determining E + from an arbitrary distribution of + charges.  For a point charge we + have
    +
  • +
+
magbiotsavarteqn3
+
+
where magbiotsavarteqn3is a unit + vector from the point charge to + location at which the E field + is to be determined.
+
+
+
+
+
    +
  • magbiotsavartfig1The Biot-Savart Law + provides a general method of + determining the B field from + an arbitrary current distribution.
  • +
+
magbiotsavarteqn4
+
+
+
where μ0 + is the permeability of the vacuum + (free space) = 4π x 10-7 + T.m/A, dl is a "current element" + directed along the current in the + wire and magbiotsavarteqn3is a unit + vector from dl to where + the B field is to be + calculated as in the diagram at + right.
+
+
+
+
    +
  • There are a number of + important points to be aware of:
  • +
+
exclamation The law + must be written in differential + form since there is no such thing + as a "point current".  Note + that a moving point charge is + equivalent to a current and a + stationary point charge does not + create a magnetic field.
+
+ exclamation  + Due to the cross product dB + is at right angles to both dl + and magbiotsavarteqn3.  + You can't avoid considering 3 + dimensions.
+
+ exclamation  magbiotsavartfig2The "sense" of dB + (into or out of the plane in the + above diagram) is determined by + the basic definition of a vector + cross product or equivalently + another "right-hand-rule" + shown at right.
+
+ exclamation  (μ0/4π) + + + plays a similar role to the + Coulomb constant, k (= 1/4πε0).  + + But compare the values of μ0 + and k; this is an indication of + the relative strengths of the + electric and magnetic interactions + of charges.
+
+ exclamation  + Similar to our initial statement + of Coulomb's Law, the above + expression of the Biot-Savart Law + gives dB due to a current + element in a vacuum.  For + real world applications wheres the + current is not in a vacuum a + slight adjustment must be made to + take into account the magnetic + properties of the medium.  + This can be done in a similar + manner to the adjustment of + Coulomb's Law in a dielectric + + + medium, but is beyond the + scope of this course.
+
+
+
    +
  • Needless to say determining + the magnetic field is typically + significantly more complicated + than the electric field. sadface
    +
  • +
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+
+

+ +

+ Energizer bunny arrested.  Charged + with battery. 
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+ + diff --git a/mag_biotsavart_eqn1.jpg b/mag_biotsavart_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_biotsavart_eqn2.jpg b/mag_biotsavart_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_biotsavart_eqn3.jpg b/mag_biotsavart_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_biotsavart_eqn4.jpg b/mag_biotsavart_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_biotsavart_fig1.jpg b/mag_biotsavart_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_biotsavart_fig2.jpg b/mag_biotsavart_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_dipole.html b/mag_dipole.html new file mode 100644 index 0000000..f19a1bc --- /dev/null +++ b/mag_dipole.html @@ -0,0 +1,171 @@ + + + + + + + Magnetism - Dipoles - Physics 299 + + + +
+

+
+
+

Magnetic Dipoles
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • Shown below is a rectangular current loop sides a and b placed + in a region of space with uniform magnetic field (B).
    +
  • +
+
magdipfig1          +                magdipfig2
+
+
+
    +
  • Application of the expression for the force on a current + carrying wire to the top and bottom sides of the rectangle + leads to cancellation of forces as shown.
  • +
+
    +
  • The forces on the other two sides (length a) are also + equal and opposite, as shown in (b) above, magdipeqn3.  Although the net force + in this case is zero, because the forces are not co-linear, + there is a net torque on the current loop, + given by
  • +
+
magdipeqn1
+
+
The magnitude of this torque is given by
+
magdipeqn2
+
+
where A = ab is the cross section area + of the coil.
+
+
+
+
+
+
+
+
    +
  • The magnitude of the magnetic dipole moment of the + loop with N turns is defined by,
  • +
+
magdipeqn4
+
+
magdipfig3μ is a vector quantity + whose direction is perpendicular to the loop with + a sense defined by (another) Right Hand Rule.
+
+
+
+
    +
  • With this definition of μ the torque on the + current loop is given by
  • +
+
 magdipeqn5
+
+
    +
  • Without proof, the potential energy of a + magnetic dipole in an external magnetic + field is given by
  • +
+
magdipeqn6
+
+
hot Note that the + above expressions for the torque and + potential energy of a magnetic dipole in an + external magnetic field are identical in + form to those obtained for an electric + dipole in an external electric field.
+
+
elecdipext      + + and      elecdipextf2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ +
+

+ +

+ I stayed up all night to see where the + sun went.  Then it dawned on me.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/mag_dipole_eqn1.jpg b/mag_dipole_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_dipole_eqn2.jpg b/mag_dipole_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_dipole_eqn3.jpg b/mag_dipole_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_dipole_eqn4.jpg b/mag_dipole_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_dipole_eqn5.jpg b/mag_dipole_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_dipole_eqn6.jpg b/mag_dipole_eqn6.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_dipole_fig1.jpg b/mag_dipole_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_dipole_fig2.jpg b/mag_dipole_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_dipole_fig3.jpg b/mag_dipole_fig3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement.html b/mag_displacement.html new file mode 100644 index 0000000..2f2103f --- /dev/null +++ b/mag_displacement.html @@ -0,0 +1,287 @@ + + + + + + + Magnetism - Magnetic Energy - Physics 299 + + + +
+

+
+
+

Displacement Current
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • Having discussed Gauss' Law for Magnetism, we now have four + equations describing electromagnetism,
  • +
+
Gauss' Law:   elecgausseqn3
+
+
+
Ampere's Law:  magampereeqn1
+
+
+
Faraday's Law:  magfaradayeqn6
+
+ Nobody's Law:  magmonopolefig5
+
+
+
    +
  • Looking carefully at these equations, the two flux + equations are now consistent with our physical understanding + of electric and magnetic charges. 
    +
  • +
+
    +
  • The line integrals are a different matter.  The right + hand side of Ampere's Law has a summation over electric + currents.  Naively we would expect a similar sum over + "magnetic currents" on the right hand side of Faraday's + Law.  But since there are no magnetic charges, + "magnetic currents" do not exist and the necessity for such + an additional term disappears.
  • +
+
    +
  • However, in words, Faraday's Law states that
  • +
+
+
+

"A changing magnetic field ( magdisplacementeqn1 ) gives rise to an + electric field ( magdisplacementeqn2 )"
+

+

For a more complete correspondence between E + and B we would expect a term added to the right + hand side of Ampere's Law which indicates,
+

+

 "A changing electric field ( magdisplacementeqn3)  gives rise to a + magnetic field ( magdisplacementeqn4 )"
+

+
+
+
    +
  • Maxwell proposed this addition, the existence of which + is now physically verified.  Furthermore, Maxwell + showed that this additional term, known as the + displacement current, is essential for the description + of electromagnetic waves.
  • +
+
    +
  • The form of the displacement current term can be + obtained by the argument described below.
  • +
+
    +
  • We have seen that current and current density are + related by the following equation
  • +
+
eleccurrenteqn3
+
+
Suppose we consider a closed surface, + then charge conservation tells us that this flux + integral must be zero.  Charge cannot be created + or destroyed inside the closed surface, therefore,
+
magdisplacementeqn5
+
exclamation This equation can be considered as a + formal statement of conservation of charge.
+
+
+
+
+
+
    +
  • magdisplacementfig1Consider a parallel + plate capacitor (at right).  The closed surface + over which we will apply the above integral is S1 + and S2.  Current I passes in though + S1, but since S2 is between + the capacitor plates, no current passes out of the + closed surface.  As it stands, we have violated + conservation of charge... What to do  ?  + Propose that there is an equal current passing out + of the closed surface through S2 - the + "displacement current".
  • +
+
    +
  • Now
  • +
+
magdisplacementeqn6
+
+
+
Therefore, if we define
+
magdisplacementeqn7
+
as the displacement current, + by writing
+
magdisplacementeqn8
+
we can ensure charge is + conserved.
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • Therefore, including the displacement + current, Ampere's Law becomes,
  • +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+

+ +

+ This girl said she recognized me from + the vegetarian club, but I'd never met herbivore.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+
+ + diff --git a/mag_displacement_eqn1.jpg b/mag_displacement_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement_eqn2.jpg b/mag_displacement_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement_eqn3.jpg b/mag_displacement_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement_eqn4.jpg b/mag_displacement_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement_eqn5.jpg b/mag_displacement_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement_eqn6.jpg b/mag_displacement_eqn6.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement_eqn7.jpg b/mag_displacement_eqn7.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement_eqn8.jpg b/mag_displacement_eqn8.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement_eqn9.jpg b/mag_displacement_eqn9.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_displacement_fig1.jpg b/mag_displacement_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_energy.html b/mag_energy.html new file mode 100644 index 0000000..3e552a2 --- /dev/null +++ b/mag_energy.html @@ -0,0 +1,203 @@ + + + + + + + Magnetism - Magnetic Energy - Physics 299 + + + +
+

+
+
+

Magnetic Energy
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • magenergyfig1Let's take a second + look at the differential equation for the simple LR circuit
  • +
+
magenergyeqn1
+
+
Multiplying throughout by the current I gives
+
magenergyeqn2
+
This is now an energy conservation + equation,
+
+
Rate of thermal heating + (I2R)  + +  Rate at which energy is stored in the + inductance (LI dI/dt)  =  Rate at which + energy is supplied  (εI)
+
+
+
+
+
+
+
+
+
+
+
    +
  • The energy stored in the inductance is associated + with the magnetic field of the inductance, UB.  + Therefore,
  • +
+
magenergyeqn3
+
+
which upon integration gives
+
magenergyeqn4
+
+
equivalent to the electrical + case of energy stored in a capacitor UE + = q2/2C = (1/2)CV2.
+
+
+
+
+
+
+
+
    +
  • Exactly as in the electric field case, we + can determine a general expression for the + energy density (uB) stored in a + magnetic field,
  • +
+
magenergyeqn5
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+

+ +

+ This girl said she recognized me from + the vegetarian club, but I'd never met herbivore.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+
+ + diff --git a/mag_energy_eqn1.jpg b/mag_energy_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_energy_eqn2.jpg b/mag_energy_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_energy_eqn3.jpg b/mag_energy_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_energy_eqn4.jpg b/mag_energy_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_energy_eqn5.jpg b/mag_energy_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_energy_fig1.gif b/mag_energy_fig1.gif new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday.html b/mag_faraday.html new file mode 100644 index 0000000..82e70ec --- /dev/null +++ b/mag_faraday.html @@ -0,0 +1,440 @@ + + + + + + + Magnetism - Faraday's Law of Induction - Physics 299 + + + +
+

+
+
+

Faraday's Law of Induction
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • So far we have treated electricity and magnetism as almost + separate subjects.  We now begin to discuss phenomena which + show that electricity and magnetism are inextricably connected, + hence the term electromagnetism.  The first + of these properties is known as Faraday's Law of + Induction.
  • +
+
exclamation  Formally, time independent + electrical and magnetic properties can be described by considering + electricity and magnetism as largely separate phenomena.  + However, when time dependence becomes part of the "equation"  + we find that electrical and magnetic properties become + inextricably linked - electromagnetism.
+
+
    +
+
    +
  • This law is conveniently written in terms of magnetic flux, + which is defined in the same way as electric flux.
  • +
+
magfaradayeqn1
+
+
where S is the surface over which the flux is + evaluated.
+
+ For constant B, perpendicular to the surface, ΦB + = BA where A is the surface area of S.
+
+ exclamation  The magnetic flux, ΦB, is so + important it has its own unit the Weber  -  1 Weber + = 1 T.m2 .  In the early days of + electromagnetism it was common to measure the magnetic (B) + field in Weber/m2 .
+
+
+
+
    +
  • In term of the magnetic flux Faraday's Law of Induction is + given by,
  • +
+
magfaradayeqn2
+
+
The induced electromotive force (emf) + in a circuit is equal to the rate of change of magnetic + flux through the circuit.
+
+ exclamation An emf is not a force, + rather it can be considered as the voltage induced + in a closed circuit.
+
+ exclamation Faraday experimentally + determined his law in the form presented above.
+
+

+
+
+
+
    +
  • One of the easiest ways to change the magnetic flux + through a circuit is to move a permanent (bar) magnet + towards or away from the circuit as shown in the + diagrams below.
  • +
+
magfaradayfig1
+
+
(a)  magfaradayfig2Magnetic flux passes through the + circuit, but does not change with time, so there is no + induced emf and so no induced current.
+
+ (b)  The flux through the circuit increases with + time causing an induced emf and current.
+
+ (c)  As the magnet moves faster the rate of + change of flux with time is increased causing a larger + emf and current.
+
+ (d)  When the magnet moves away from the circuit + the flux decreases with time so the induced emf + and current are reversed.
+
+

+
+
+
+
+
    +
  • The origin of the changing magnetic flux (field) + is not limited to permanent magnets.  The + magnetic field due to a second circuit can produce a + similar effect, as described in the examples + below. 
    +
  • +
+
+
+
magfaradayfig3In the diagram at + right the current in the left circuit is constant, + but the flux through the other circuit increases + as the two circuits get closer.
+
+
+
+
+ magfaradayfig4In the situation at + left both circuits are stationary.  The + current in the left circuit is initially zero, but + rapidly increases to a constant value when the + switch is closed.  As the current reaches its + final (constant) value the flux through the right + circuit is increasing with time, thus by Faraday's + Law,  causing a brief pulse of induced + current in the second circuit.  When the + switch is opened the flux in the right circuit + rapidly decreases causing a short induced current + pulse in the opposite direction.
+
+
+
+
+
    +
  • exclamation Important !  In both the + above examples a magnetic field (B) + changing with time results in a changing + magnetic flux.  But it is possible to have + a changing flux with a constant B if the + cross sectional area - dA - can be made + to change with time as in the case of the + (electric) generator.
    +
  • +
+
magfaradayfig5
+
    +
+
divider
+
+
+
+

LENZ'S LAW

+
+
+
+
+
    +
  • Mathematically the negative sign in + Faraday's Law tells us about the + direction of the induced (emf) + current.  Practically we use Lenz's + Law to determine the direction in + specific cases.
    +
  • +
+
+
+
+
confused "The induced + current will appear in such a + direction that it opposes the change + that produced it"  + confused
+
+ Confusing ?  Yes !
+
+
+
+
+
+
+                           + + + magfaradayfig7
+
+
    +
+
    +
  • exclamation The induced (flux) + magnetic field (associated with the + induced current) does not necessarily + oppose the field which causes the change + in flux, rather it opposes the CHANGE + in this field.
    +
  • +
+
    +
  • exclamation Lenz's Law always ensures + that there is a force resisting the + motion of the magnet.  It is the + work done against this force which + appears as the energy of the moving + charges of the induced current.
    +
  • +
+

magfaradayfig6

+
    +
+
divider
+
+ FARADAY'S + LAW = MAXWELL + EQUATION
+
+
    +
  • Whenever there is an induced + electric current there must also be + an induced electric field, E.  + + + + + + The work dW done by this induced + field moving a charge q0 + a distance ds around a loop + is given by,
  • +
+
magfaradayeqn3
+
+
where dε is the + potential difference in ds.
+
+ Therefore,
+
magfaradayeqn4
+
So that the emf + + + + + around the whole loop is
+
magfaradayeqn5
+
+
Equating + this emf to that + given by Faraday's Law we + obtain the integral form + of Faraday's Law, the + third of Maxwell's + equations we have + encountered so far,
+
magfaradayeqn6
+
+
exclamation  + Note that the line + integral of E + must be round a closed + loop (circuit).
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    +
  • Written in the + above form the + relationship between + the E and B + fields is clear
  • +
+

"A + + + + + magnetic field + changing with + time induces + an electric + field"
+

+
+
Shortly + + + + + we will see that the + reverse of this + statement is also + true.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+

+ +

+ They told me I had type A blood, but it + was a Type O.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+
+ + diff --git a/mag_faraday_eqn1.jpg b/mag_faraday_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_eqn2.jpg b/mag_faraday_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_eqn3.jpg b/mag_faraday_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_eqn4.jpg b/mag_faraday_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_eqn5.jpg b/mag_faraday_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_eqn6.jpg b/mag_faraday_eqn6.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_fig1.jpg b/mag_faraday_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_fig2.jpg b/mag_faraday_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_fig3.jpg b/mag_faraday_fig3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_fig4.jpg b/mag_faraday_fig4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_fig5.jpg b/mag_faraday_fig5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_fig6.jpg b/mag_faraday_fig6.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_faraday_fig7.jpg b/mag_faraday_fig7.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_2wires.html b/mag_force_2wires.html new file mode 100644 index 0000000..689e378 --- /dev/null +++ b/mag_force_2wires.html @@ -0,0 +1,181 @@ + + + + + + + Magnetism - Force Between Parallel Wires: Ampere Definition - + Physics 299 + + + +
+

+
+
+

Force Between Two Parallel Wires: Ampere Definition
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • We have just seen that a long straight wire carrying current + i1  produces a B field which forms + concentric circles around the wire whose magnitude a distance + "d" from the wire is given by
    +
  • +
+
magforce2wireseqn1
+
+
with direction given by "the" right-hand-rule.
+
+
+
+
    +
  • magforce2wiresfig1If a second parallel wire carrying a current + i2 is placed a distance "d" from the first, it + will feel a force due to the presense of B1 .  + + The force felt by a length l2  of this + second wire is
  • +
+
magforce2wireseqn2
+
+
so that
+
magforce2wireseqn3
+
+
with a direction such that i2 + is attracted towards i1 as shown at right.
+ Then the force per unit length on i2 is + given by
+
magforce2wireseqn4
+
+
+
+
+
+
+
+
+
    +
  • exclamation Note that if we reverse + the roles of i1 and i2 we + obtain the same expression for F1/l1 + .  That is, i1 exerts a force + on  i2 and  i2  + exerts an equal, but opposite, force on i1  + + + as expected from Newton's Third Law ("Action equals + Reaction").
  • +
+
    +
  • The above analysis applies for parallel currents - + the two wires are attracted to each other.  If + one of the currents is reversed, so that the + currents are "anti-parallel", the force per unit + length is unchanged, but the force acting on each + wire is reversed.  In other words the wires are + now repelled from each other.
  • +
+
exclamation "Like + + + currents are attractive, unlike currents are + repelled".  Note that this is + opposite to the behavior of point electric charges.
+
+
+
    +
  • Definition + + + of the Ampere
  • +
+
+

The unit of electric current, the Ampere, is + defined using the force between parallel wires + carrying current.
+

+

tick"The Ampere is + that current which when flowing in two + infinite parallel wires one + metre apart produces a force between them of 2 + x 10-7 + N/m"
+
+

+

Proof:
+

+
magforce2wireseqn5
+
+
+
+
+
+
+
+ +
+
+
+

+ +

+ What do you call a dinosaur with an + extensive vocabulary ?  A thesaurus.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+
+ + diff --git a/mag_force_2wires_eqn1.jpg b/mag_force_2wires_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_2wires_eqn2.jpg b/mag_force_2wires_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_2wires_eqn3.jpg b/mag_force_2wires_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_2wires_eqn4.jpg b/mag_force_2wires_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_2wires_eqn5.jpg b/mag_force_2wires_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_2wires_fig1.jpg b/mag_force_2wires_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_charge.html b/mag_force_charge.html new file mode 100644 index 0000000..c066f27 --- /dev/null +++ b/mag_force_charge.html @@ -0,0 +1,296 @@ + + + + + + + Magnetism - Force on Charges - Physics 299 + + + +
+

+
+
+

Magnetic Forces on Charged Particles
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+
+
    +
  • Proceeding under the assumption that magnetic fields exist - + created by an as yet not described mechanism - the magnetic + force on a positively charged particle is found + experimentally to be given by +
    magforcechargeeqn1
    +
  • +
+
+

where q is the charge (positive) of the particle, v its + + + + + + + + + + + + + velocity, FB the force it experiences and B + the magnetic field causing the force.
+

+
+
    +
  • +
    +
    exclamation Note + that this equation is the magnetic equivalent of the + electric expression  FE = qE
    +
    +
  • +
+
    +
    +
  • The expression for the magnetic force is written in terms of + a vector (cross) product, which means, like it or not, you + have to work in three dimensions.  Several important + facts emerge from this equation.
    +
      +
      +
    1. If v = 0 there is no force.  Electric + charges at rest in a magnetic field do not feel a magnetic + force.
    2. +
      +
    3. The magnitude of FB is given by +
      magforcechargeeqn2
      +
      + where φ is the angle between v and B. +
        +
      +
        +
      • So that, when v is parallel to B (φ + = 0) or antiparallel to B (φ = 180o) + then sinφ = 0 and FB = 0.  + Therefore, charged particles moving along magnetic + field lines experience no magnetic force.
      • +
      +
        +
      • When v and B are at 90o + FB has its maximum value, FB + = qvB.
      • +
      +
    4. +
      +
    5. FB is at right angles to v + and B.  The "sense" is given by the usual + rules for the vector (cross) product, sometimes called + (the) "Right Hand Rule". +
      magforcechargefig1
      +
      +
      + This leads to circular (or spiral) motion around the B + field lines.
      +
      +
      magforcechargefig2       magforcechargefig3
      +
      +
      +
      +
      hotDon't + + + + + forget that the direction of the magnetic force + obtained above is for a positive charge.  + For a negative charge the direction of FB  + is reversed.
      +
      +
      +
    6. +

      +
    7. The work done by the magnetic force when a charged + particle is displaced by ds is given by,
    8. +
      +
      +

      + magforcechargeeqn3 +
      +
      where θ is the angle between FB  + + + + + + + + + + + + + + and ds.
      +
      +
      The displacement ds and the + velocity of the particle, v, are in the same + direction, so θ is also the angle between FB  + + + + + + + + + + + + + + and v.  But from the form of the + magnetic force we know FB  and + + + + + + + + + + + + + v are perpendicular (θ = 90o) therefore + dW is always zero.  In other words the magnetic + + + + + + + + + + + + + force does no work, which means, unlike the + electric force, it cannot give a charged particle energy + (increase or decrease its kinteic energy). The effect of a + magnetic force is to change the direction not the kinetic + energy.
      +
    +
  • +
    +
    +
+
divider
+
+
    +
  • If a charged particle feels both a magnetic and electric + field the resultant force is given by
  • +
+
+
magforcechargeeqn4
+
+
+
this is known as the Lorentz Force Law.
+
+
divider
+
+
+
+
+
+
+
    +
  • UNITS
  • +
+
+

From the form of the magnetic force we see that the units + of B are  N/(C.m/s) = N/(A.m).  This + combination of basic units is defined as the Tesla.
+

+
magforcechargeeqn5 +
+

The Tesla is a very large unit of magnetic field.  + For this reason you may occasionally come across the + smaller unit of magnteic field, the Gauss; where  1 + Tesla = 104 Gauss.
+

+
+
+
+
+
+
+
+
+ +
+

+ +

+ Q: What did one quantum physicist say + when he wanted to fight another quantum physicist?
+ A: Let me atom.

+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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+

+
+
+

Magnetic Forces on Electric Currents
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • An electric current is simply a collection of charges moving + along a wire with velocity vd , known as the + drift velocity.  We have already seen that the drift + velocity and current density are related by
  • +
+
eleccurrent eqn2
+
+
+
Considering the magnetic force on a single + charge (e) we find
+
magforcecurrent
+
+
Now the total force on a wire comprised of + (nAl) charge carriers each with charge e is given by
+
+
magforcecurrenteqn2
+
+
Defining a vector l along the + wire in the same direction as J and using |JA| + + + = i (electric current) we find for the total force on + the wire
+
+
magforcecurrenteqn3
+
+ magforcecurrentfig1         +          +        magforcecurrentfig2
+
+
exclamation  Note that for a given current + negative and positive charge carriers move in + opposite directions.  This leads to  a + force on the wire in the same direction (see + above), whether the charge carriers are positive + or negative.
+
+
+
+
+
+
+
+
+
+
+

+ +

+ When chemists die, they barium.
+  

+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email:
c.l.davis@louisville.edu +
+  

+

+
+


+

+ + diff --git a/mag_force_current_eqn1.jpg b/mag_force_current_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_current_eqn2.jpg b/mag_force_current_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_current_eqn3.jpg b/mag_force_current_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_current_fig1.jpg b/mag_force_current_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_force_current_fig2.jpg b/mag_force_current_fig2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_intro.html b/mag_intro.html new file mode 100644 index 0000000..3de1575 --- /dev/null +++ b/mag_intro.html @@ -0,0 +1,135 @@ + + + + + + + Magnetism - Introduction - Physics 299 + + + +
+

+
+
+

Introduction to Magnetism
+

+
+

+
+ + + + + +
"Physics Is + Imagination In A Straight Jacket"
+
+ John Moffat
+
+
+
    +
+
fig1
+
    +
    +
  • As we will see shortly, Electricity and Magnetism are + different aspects of the same basic physical phenomenon - hence + the description Electromagnetism.  
  • +
+
    +
  • However, initially it is convenient to treat magnetism in a + separate, but similar, manner to electricity.  In other + words we will define, magnetic forces, magnetic fields and + magnetic dipoles exactly as we did the electric variables.
  • +
+
    +
  • After this initial description we will develop the + relationship between magnetism and electricity through Faraday's + Law of Induction and the Displacement Current, leading finally + to a complete description of (classical) electromagnetism in the + form of Maxwell's + equations.
  • +
+
    +
  • So our starting point in describing magnetism will be the + assumption of the existence of a magnetic field, B, the + origin of which will be described later.
  • +
+
    +
  • fig3Exactly as for the electric field E, + we will assume that the B field can be represented by + field lines where,
  • +
+
    +
      +
    • the tangent to a field line gives the direction of B + at that point and
    • +
    +
+
    +
      +
    • the number of field lines per unit cross sectional area is + proportional to the strength of B.
    • +
    +
+
    +
  • exclamation It is important to realize immediately that unlike + electric field lines, magnetic field lines DO NOT + represent the direction of the force acting on a charged + particle.
  • +
+
exclamation  The magnetic phenomenon most + familiar to most people is that of permanent magnetism - + refrigerator magnets etc.  As it happens, permanent magnetism + is not a simple topic to explain.  In fact a complete + description requires a detailed knowledge of the quantum behavior + of materials.  For this reason we will barely mention + permanent magnets in this course.
+
+
fig2
+
+
+
    +
+ +
+

+ +

+ + + Q: What is the name of the first electricity detective?
+ A: Sherlock Ohms

+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

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zhiGPYp51OB6%1Z8qJ|p4ITUAYP4Ui}lAfaViw_V2wtMIg z?8qaOEwmf9^5?VG7qpAIDk3rnPtw`OI!7w?JNnM<#%>*QQu}bMlo`KEz{Wp0Q4U*)Yo+(E)%uIyc&djqvMW1@@S9IjaWZoqU$AmAxa|<>KoiI}mjJ#ObIJ?t{@Q5D7 z&Q>~2Yvskl23-7j_4g5;#1I;A-l$(oW`-B)T!Ukale2&#C$DxeEzRV=(B(55b4!X9 zQVhk{8BBcL@M1>$mGd|6VO8^@;GqoB#|U3cOSo0rZcf9CCEV`KG<*Mwgw&S$c}n5` z^s57VV$!u*5POmx&g;Rm5QfJSi^CIz5rB<(Z5 z(KBgpJn)y9Hkg_@EHAZ z=NvpJUIrjGDhKeNtTo?GNXqB>9_v?Adx;(pzv`OXym<|V{kWIK#}elw3-vU2OrI?R z3wz76=r4-6wP#=-#z@F_Oa*gY!4AcxK~1dKh86DHP>vMZ5ggIA_1f2m+Q8e||7j7^ zjEZIJ+urqbKg$Y%P4qouS8zJ8*=0s34!Aas)l8oLdhVgTEH_2`h4oFJf2md + + + + + + Magnetism - Magnetic Energy - Physics 299 + + + +

+

+
+
+

Magnetic Monopoles & Gauss' Law for Magnetism
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
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+ +
+
    +
+
+

questionmark     Magnetic + + + Monopoles    questionmark

+
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    +
  • In our initial discussion of magnetism we made the point that + we were going to treat magnetism in a similar way to + electricity.  However, whereas our discussion of + electricity began with electric charges and the electric field + associated with these charges, the magnetic discussion started + with the existence of the magnetic field.  No mention was + made of "magnetic charges", which would play the same role in + magnetism as electric charges in electricity.  This is + because individual magnetic charges - magnetic monopoles + - are apparently impossible to isolate.
  • +
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magmonopolefig2    + + + + + magmonopolefig3
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  • The bar magnetic may be considered to be a combination of two + magnetic monopoles, usually labelled North and South.  This + is similar to the electric dipole comprised of equal but + opposite electric charges.
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+


+

+
mag_monopole_fig1   +                   +                   +     magintrofig3
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    +
  • magmonopolefig4Whereas with the + electric dipole it is possible to isolate the positive and + negative charges, experimentally it is not possible to + separate the North and South poles of a bar magnet.  + Break a magnet in two and you get two magnets, each with a + North and South pole.  Continuing this splitting + process down to the atomic level we find that even + elementary particles behave as magnetic dipoles, each with a + North and South pole.  It appears that nature does not + allow us to create magnetic monopoles in this way.
  • +
+
    +
  • exclamation  However, theoreticians + developing unified quantum theories of the Universe,  + so-called "Theories of Everything", are almost unanimous in + the necessity for magnetic monopoles as elementary particles + created shortly after the birth of the Universe.
  • +
+
+

The belief is that shortly after their creation, magnetic + monopoles were "frozen out" - meaning that their + interactions with the rest of the matter in the Universe is + highly suppressed.  This does not prevent physicists + from searching for evidence for the existence of magnetic + monopoles.
+

+


+
+

+


+

+
divider
+

Gauss' Law for Magnetism

+
+
+
+
+
    +
  • So far we have discussed three basic equations + describing electromagnetic phenomena - the first three + of Maxwell's equations.
  • +
+
+
Gauss' Law:   elecgausseqn3
+
+
+
Ampere's Law:  magampereeqn1
+
+
+
Faraday's Law:  magfaradayeqn6
+
+
+
+
+
    +
  • Gauss' Law involves the flux integral for the + electric field.  To complete the correspondence + between electricity and magnetism we expect a fourth + equation involving the magnetic flux - "Gauss' Law + for Magnetism".
  • +
+
    +
  • The right hand side of Gauss' Law includes a + summation over electric charges.  Therefore, + for magnetism, we expect a summation over "magnetic + charges".  But magnetic charges, North and + South poles (equivalent to positive and negative + electric charges) always exist in pairs, the net + "magnetic charge" is thus always zero.  Gauss' + Law for Magnetism must therefore take the form,
  • +
+
magmonopolefig5
+
+
the flux of B through a + closed surface is zero.
+
+ exclamation Note that the fact that + the surface is closed is very important !  A + magnetic flux integral  appears in Faraday's + Law - in this case the surface is generally not + closed.
+
+ hot Electric field lines begin + (positive) and end (negative) on charges.  + Since there are no magnetic charges magnetic field + lines form closed loops.
+
+
+
magmonopolefig7
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+

+ +

+ This girl said she recognized me from + the vegetarian club, but I'd never met herbivore.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+
+ + diff --git a/mag_motionch.html b/mag_motionch.html new file mode 100644 index 0000000..c0b5b7c --- /dev/null +++ b/mag_motionch.html @@ -0,0 +1,239 @@ + + + + + + + Magnetism - Motion of Charged Particles in B and E - Physics + 299 + + + +
+

+
+
+

Motion of Charged Particles in Magnetic and Electric Fields
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
+
+
+
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+
+
+
MAGNETIC + + + FIELDS
+
    +
  • Return now to the case of a "point" charge + moving with velocity v in a region + of constant magnetic field (B).  + We previously stated (without proof) that + such a particle would move in a circle or + helix.  Now let's prove it.
  • +
+
    +
  •  magforcechargefig2Consider + + + the positive charge (e) at the top of the + figure at right.  With the directions + of v and B indicated the + right-hand-rule leads to a downwards force + as shown.  This force changes the + direction of v but does not change + the magnitude of v, as shown.
  • +
+
    +
  • Quantitatively we find
  • +
+
magmotionch
+
+
Assuming that this force + causes circular motion, FB + is the centripetal force so that
+
+
magmotioneqn2
+
which leads to a + radius
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+
+
magmotioncheqn3
+
+
The period (time taken to + perform one revolution) is given by
+
+
magmotioncheqn4
+
The frequency, f, is
+
magmotioncheqn5
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+
+
    +
  • exclamation Notice that this frequency + is independent of the velocity.  + That is fast particles will move in + large  radius circles, slow + particles will have smaller radii.  + This phenomenon is the basis for the + cyclotron one of the earliest charged + particle accelerators.
  • +
+
    +
  • exclamation  If the velocity of + the particle has a component along B + this will be unchanged and the resultant + trajectory will be helical.
  • +
+
divider
+
+ MAGNETIC + and ELECTRIC FIELDS
+
+
    +
  • There are many useful applications in + which B and E fields are + applied simultaneously.  Perhaps + the simplest application is the + "crossed" B and E + fields.  
  • +
+
magmotionchfig1
+
    +
+
+

For this configuration the electric and + magnetic forces are in opposite + directions.  If a charged particle + moves through this region of space with + no deflection then

+

magmotioncheqn6
+

+
so that
+
+
magmotioncheqn7
+

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    +
  • Other applications include + mass spectrometers and particle + accelerators.  (EXPAND + ?)
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  • +
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+ +

+ I'm reading a book about + anti-gravity.   I can't put it down.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+ + diff --git a/mag_motionch_eqn1.jpg b/mag_motionch_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_motionch_eqn2.jpg b/mag_motionch_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_motionch_eqn3.jpg b/mag_motionch_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_motionch_eqn4.jpg b/mag_motionch_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_motionch_eqn5.jpg b/mag_motionch_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_motionch_eqn6.jpg b/mag_motionch_eqn6.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_motionch_eqn7.jpg b/mag_motionch_eqn7.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_motionch_fig1.gif b/mag_motionch_fig1.gif new file mode 100644 index 0000000..e69de29 diff --git a/mag_mutualind.html b/mag_mutualind.html new file mode 100644 index 0000000..d8fd7b6 --- /dev/null +++ b/mag_mutualind.html @@ -0,0 +1,210 @@ + + + + + + + Magnetism - Mutual Induction - Physics 299 + + + +
+

+
+
+

Mutual Induction
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • As we have seen, Faraday's Law of Induction tells us that a + changing magnetic flux through a circuit will induce an emf and + therefore an "induced current".  Consider the situation of + two nearby circuits (below) where the flux through circuit 2 + changes due to the changing current in circuit 1.
  • +
+
magmutualindfig1
+
+
    +
  • With N2 turns in circuit 2 the emf is given by
  • +
+
magmutualindeqn1
+
+
but the total flux through circuit 2 is + proportional to the current in circuit 1, where the + proportionality constant is called the mutual inductance + of the coils, M21,
+
+
magmutualindeqn2
+
Combining these two equations gives
+
+
+
+
magmutualindeqn3
+
In other words the emf in circuit 2 is + proportional to the rate of change of current in + circuit 1.  As we will see shortly, the mutual + inductance M21 depends only on the + geometric configuration of the two circuits.
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    +
  • If the roles of the two circuits are reversed - + that is a changing current in circuit 2 induces a + current in circuit 1 - then
  • +
+
magmutualindeqn4
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+
    +
  • It is "easy" to show that  M21 + = M12 .  In other words given + two circuits in a particular configuration it + doesn't matter in which circuit the current is + induced the mutual inductance is the same.
  • +
+
    +
  • UNITS: Inductance is measured in + Henrys Henry
    +
  • +
+
magmutualindeqn5
+
+
exclamation The concept of inductance is + related to the magnetic field in a similar way + that capacitance is related to the electric + field.
+
+
+
+
    +
  • In order to calculate mutual inductance + you will typically follow the steps below:
  • +
+
+
    +
  1. Determine B due to one circuit + at the location of the other using + Ampere's Law or the Biot-Savart Law.
  2. +
  3. Using this B calculate the + magnetic flux through the 'other' circuit.
  4. +
  5. Then use the equation N2Φ2 + = MI1 to obtain M.
  6. +
+ You will always find that M depends + only on the geometric parameters of the + two circuits and the number of turns in + each circuit.
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+

+ +

+ This girl said she recognized me from + the vegetarian club, but I'd never met herbivore.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+
+ + diff --git a/mag_mutualind_eqn1.jpg b/mag_mutualind_eqn1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_mutualind_eqn2.jpg b/mag_mutualind_eqn2.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_mutualind_eqn3.jpg b/mag_mutualind_eqn3.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_mutualind_eqn4.jpg b/mag_mutualind_eqn4.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_mutualind_eqn5.jpg b/mag_mutualind_eqn5.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_mutualind_fig1.jpg b/mag_mutualind_fig1.jpg new file mode 100644 index 0000000..e69de29 diff --git a/mag_selfind.html b/mag_selfind.html new file mode 100644 index 0000000..4019dfe --- /dev/null +++ b/mag_selfind.html @@ -0,0 +1,172 @@ + + + + + + + Magnetism - Self Inductance - Physics 299 + + + +
+

+
+
+

Self Inductance
+

+
+

+
+ + + + + +
"A fact is a simple + statement that everyone believes.  It is innocent, + unless found guilty.  A hypothesis is a novel + suggestion that no one wants to believe.  It is + guilty, until found effective"
+
+ Edward Teller
+
+ +
+
    +
  • magselfindfig1Consider now a single + coil/circuit/loop with N turns.  As the current in the loop + changes the magnetic field and hence the magnetic flux through + the loop changes.  This changing "self flux" through + Faraday's Law leads to and induced emf in the circuit,
  • +
+
magselfindeqn1
+
+
    +
  • In the same manner as mutual inductance we define the self + inductance of the circuit, L, as follows
  • +
+
magselfindeqn2
+
+
so that the induced emf - sometimes called + a "back emf" - is given by
+
magselfindeqn3
+
This "back emf" always opposes the + primary emf (due to Lenz's Law), as shown in the + diagram at right.
+
+
+
+
+
hot From above,   magselfindeqn4 which can be compared to the similar + equation for capacitance  magselfindeqn5 .
+
    +
  • Calculating self inductance proceeds by the same steps + as mutual inductance - Calculate B, calculate + magnetic flux, then use the second equation above to + determine L.
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+ +

+ This girl said she recognized me from + the vegetarian club, but I'd never met herbivore.
+
+  
+   +

Dr. C. L. Davis
+ Physics Department
+ University of Louisville
+ email: c.l.davis@louisville.edu +
+  

+

+
+


+

+
+
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