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Question 7.1: A cylindrical resistor of cross-sectional area A and length ...

A cylindrical resistor of cross-sectional area A and length L is made from material with conductivity σ. (See Fig. 7.1; as indicated, the cross section need not be circular, but I do assume it is the same all the way down.) If we stipulate that the potential is constant over each end, and the potential difference between the ends is V, what current flows?

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As it turns out, the electric field is uniform within the wire (I’ll prove this in a moment). It follows from Eq. 7.3 that the current density is also uniform, so

\pmb{J}=\sigma \pmb{E}.           (7.3)

I = J A = \sigma EA=\frac{\sigma A}{L}V.

 

 

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