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\begin{figure}\centerline{
\psfig{file=problems/prob_10_7.eps,height=2.5in}
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This is basically problem 64 from your homework. Our archetypical model for a resistor is drawn above: two circular conducting plates (metal contacts) with radius $R$, separated at a distance $d$ by a material with resistivity $\rho$.

a) In a steady state situation where a DC voltage $V$ is applied as shown, find the field $\vec{E}$ inside the resistive material.

b) Find the current density $\vec{J}$ inside the resistive material.

c) From Ampere's law, find the magnetic field as a function of $r$ in the region between the plates.

d) From your answers to a) and c), find the Poynting vector $\vec{S}$ (magnitude and direction) as a function of $r$ in the region in between the plates.

e) NOW show that:


\begin{displaymath}\oint_A \vec{S}\cdot\hat{n} dA = -I^2 R\end{displaymath}

where $A$ is the outer surface of the resistor and $\hat{n}$ is its outward-directed normal unit vector.

Thus the heat that appears in the resistor can be thought of as the electromagnetic field energy that flows in through its outer surface!

Solution Page

next up previous contents
Next: . Up: AC Circuits Previous: .   Contents
Robert G. Brown 2003-02-09