electric circuits - What causes a resistor to heat up?

In the following video clip at 2:10,

http://www.youtube.com/v/YslOUw5oueQ ,

Professor Walter Lewin talks about a misconception people have that the energy going through a wire to a resistor is in the form of kinetic energy of electrons. He proves this cannot be so as follows. The current density is J = I/A = Vne where V is the drift velocity (or average velocity), n is the number of electrons per volume, and e is the charge of an electron. A is a cross-sectional area of the wire.

We can make A as large as we want (keeping the current constant), and therefore V will have to become very small, and the electrons will have very little kinetic energy. Yet the resistor (say a light bulb) dissipates the same amount of power P=(I^2)*R. Therefore, it must be that the form of energy is not the kinetic energy of electrons.

My first question is, if we make A larger why does it have to be that V goes down? Perhaps n goes down - we increased the volume (by increasing the cross-sectional area), so there should be fewer electrons per volume?

My second question, my main question is, if the energy is not the kinetic energy of the electrons, what does in fact bring energy to the resistor and how does it heat up?

Answer

my main question is, if the energy is not the kinetic energy of the electrons, what does in fact bring energy to the resistor and how does it heat up?

We assume steady state operation.

The drift velocity of the electrons entering the resistor must equal the drift velocity of the electrons leaving the resistor. This follows from the fact that the current into the resistor equals the current out of the resistor.

However, the electrons leaving the resistor have less potential energy than those entering the resistor. This follows from the fact that there is an electric field through the structure of the resistor and, thus, there is a potential difference between the ends of the resistor.

The electric field within the resistor structure accelerates (does work on) the electrons increasing their kinetic energy however, this energy is quickly given up to the structure of the resistor via collisions; the resistor gets hotter.

On a more fundamental level, the energy flows from the battery to the resistor through the space around the conductors via the electromagnetic field. See, for example, William Beaty's description of energy flow in a simple circuit here.