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

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

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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.
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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