current flow and resistance

Physics II

Class Notes, 3/01/99

We have noted that when we crank a small electrical generator with the ends of the leads separated by air, the generator is easy to crank, but that when the ends of the leads are clamped to one another the generator is much harder to crank.

We conclude that whatever is going on in the wire to require our energy, more of it is going on when the leads are clamped together.

When we crank the source of electrical power in the circuit shown below, electrons are forced through the circuit.

In the circuit is a thin wire within a sealed bulb from which oxygen has been removed (i.e., a light bulb).
We note that when the source is cranked fast enough, the wire glows.
We note also that we must exert a moderate but significant force to crank the source.

We note also that if we connect the leads of the generator to a capacitor, as in the figure at top left below, the generator gets progressively easier to crank.

The lower part of the figure below depicts a wire with electrons flowing to the right. At some point the wire becomes narrower.

As we will see later, due to the strong repulsion between electrons they cannot accumulate at any point within a conductor.
The same number of electrons must therefore flow in both wires.
If we assume a constant electron volume density in both wires, which will be the case if both wires are made of the same material, we see that a continuity equation therefore holds and that the electrons in the thin wire must move faster than the electrons in the thick wire.

Electrons are accelerated by an electric field created by the generator and continue to accelerate until they interact with the molecules or atoms in the wire. In these interactions they tend to lose most of their kinetic energy, which is then transferred to the molecules or atoms in the wire.

The result of this transfer of kinetic energy is that the particles in the wire gain thermal energy.

Since the electrons in the thin wire have more kinetic energy to exchange than those in the thick wire, the thin wire becomes much warmer than the thick wire.

If the thin wire is thin enough, electrons will move really fast and exchange a great deal of thermal energy, causing the thin wire to glow.

Video File #1

The kinetic energy ratio is proportional to the square of the velocity ratio, which is inversely proportional to the area ratio and hence inversely proportional to the ratio of wire diameters.

For example if the average velocity of the electrons in the thicker wire is 1 cm / sec, and if the thinner wire is 1/10 the diameter of the thicker, then the area ratio will be 1 / 100 and the velocity ratio will be 100 / 1, while the kinetic energy ratio will be 10,000 / 1 (all assuming the same material in both wires, which is not the case for tungsten light bulb filaments in a copper wire circuit).

Video File #2

If we put two thin wires in a circuit, one thinner than the other and crank the handle of the generator, we therefore expect that the thinner wire will carry the faster electrons and so shine brighter.

If two bulb filaments of equal length have different thicknesses, which should be harder to crank?

The thinner wire will offer more resistance to the flow of electrons so that fewer electrons will flow in this wire, which will tend to make the generator easier to crank.

It is important not to confuse the resistance to the flow of electrons with the difficulty of cranking the generator. The generator is more difficult crank when the resistance of the circuit is low and more electrons are flowing.

The following concepts can be used to explain much of what we feel and do when we crank the generator.

A current of 1 ampere corresponds to the flow of 1 Coulomb of electrons per second; a Coulomb of electrons is approximately 6 * 10^18 charges per second.

A volt is a Joule per Coulomb, meaning that when a Coulomb of electrons flows through a one-volt circuit, one Joule of work is done.

The number of volts created by the generator is proportional to the rate at which the handle is cranked.

The number of amps flowing in a given circuit is proportional to the voltage.

The number of amps flowing when the leads are separated by air is effectively zero, so the only work done in cranking the generator is to overcome the friction in the gears of the generator.

The number of amps flowing when the leads are clamped together is the highest possible, with the only significant resistance to current flow being that of the generator itself.

The number of amps flowing when a light bulb is inserted into the circuit is less than the highest possible, due to the added resistance of the light bulb.

Video File #3