Question 5.4: (a) A current I is uniformly distributed over a wire of circ...

Introduction to Electrodynamics 4th. Edition [EXP-2861]

(a) A current I is uniformly distributed over a wire of circular cross section, with radius a (Fig. 5.15). Find the volume current density J .

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The area (perpendicular to the flow) is $\pi a^{2}$ , so

J=\frac{I}{\pi a^{2}}.

This was trivial because the current density was uniform.
(b) Suppose the current density in the wire is proportional to the distance from the axis,

J=ks

(for some constant k). Find the total current in the wire.

Solution
Because J varies with s, we must integrate Eq. 5.25. The current through the shaded patch (Fig. 5.16) is $Jda_{\bot }$ . and $da_{\bot}=sdsd\phi .$ so

I=\int{(ks)(sdsd\phi)}=2\pi k\int_{0}^{a}{s^{2}ds}=\frac{2\pi ka^{3}}{3}.

$J=\frac{dI}{da_{\bot }} .$ (5.25)

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