I understand that this is a complex question, however, I am stumped. Please help! Thank you!

Problem 4: Let's build a simplified model of the fields inside a resistor. Imagine you have a perfectly cylindrical resistor of length \ell and radius a, carrying total current I. If the total resistance is R, then there is a voltage drop V=I R down the length of the resistor. This means that the resistor contains an electric field

inside it. What is the magnitude |

| and direction of that field E ? Give your answer in terms of R,

, a, I.

Now that

field must be created by charges; where are those charges? What charge densities must exist in this problem, and where? Imagine that the ends of the resistor are conducting circular plates of radius a and the

interior of the resistor is some kind of resistive material. Does it make sense now that any resistor must also have capacitance? Roughly how much charge +Q,-Q must be on those plates? Give your answer in terms of R,

, a, I.

In order to solve this problem, assume that the

field is entirely contained inside the resistor. That's not true! But this problem is intended to be conceptual not exact.

Now imagine (incorrectly, it also turns out) that the current is uniformly distributed throughout the resistor. What is the current density

inside the resistor (magnitude and direction). You might have to look up the definition of current density in your textbook. The resistivity \rho we encountered in a previous Problem Set should be related to these by

= \rho

. This is the "differential form" of Ohm's Law. Check that this works out in terms of units.

Now, finally, what is the direction and magnitude of the magnetic field

at the surface of the resistor? Compute the "Poynting vector" quantity

at the surface of the resistor. Integrate the flux of

into the resistor by the standard way we do integrals of vectors over the surface of the resistor. Compare this integral to the power

R dissipated by the resistor. Are the units the same? Is the value the same?