Chapter 25 Problem 92

Problem

The resistor fantasy

For the wire shown, whose diameter varies uniformly from a to b, suppose a current I = 2.0 A enters at a. If a = 2.5 mm and b = 4.0 mm, what is the current density (assume uniform) at each end?

solution

We assume that all of the current that enters at a leaves at b, so that the current is the same at each end.

$j_{a}={\frac {I}{A_{a}}}={\frac {I}{\pi \left({\frac {1}{2}}a\right)^{2}}}={\frac {4I}{\pi a^{2}}}=4.1\times 10^{5}{\textrm {A}}/{\textrm {m}}^{2}$

$j_{b}={\frac {I}{A_{b}}}={\frac {I}{\pi \left({\frac {1}{2}}b\right)^{2}}}={\frac {4I}{\pi b^{2}}}=1.6\times 10^{5}{\textrm {A}}/{\textrm {m}}^{2}$