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# 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.
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,

where e is the charge on the electron and vd is the drift velocity.

- Current Density, J (A/m2) is defined by,

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
The current is the flux of J through a surface.
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.

- 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.

> ## Resistance

- In metallic conductors the electric field and current density are in the same direction and are found to be proportional to each other,

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
The resistance of the conductor R, is defined by,
Resistance is measured in ohms (Ω), then resistivity has units ohm.metre and conductivity (ohm.metre)-1

- Important: The relationship V = IR is NOT Ohm's Law !

Ohm'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. A and B form part of a complete circuit.

## 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

> so that the rate at which energy is transferred (power), P, is given by,
> 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

> ## Electro-motive Force - "emf"

- In 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.

> ## Internal Resistance

- All emfs - batteries, generators etc - and electrical measuring devices - ammeters, voltmeters etc - have an "internal resistance".

- As 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.

- 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