Current Density — Question 2

Question 2

A parabolic radial profile concentrates current in the central core

A cylindrical wire of radius R carries $J_z(r)=J_0[1-(r/R)^2]$ with $J_0>0$.

1. Find total current and average current density.

2. Find the radius containing half the current and compare it with the half-area radius.

Question 2 - Solution

A parabolic radial profile concentrates current in the central core

(a) Method and calculation

$$I=2\pi J_0\int_0^R(1-r^2/R^2)r\,dr=\tfrac12\pi J_0R^2.$$$$J_{\mathrm{avg}}=I/(\pi R^2)=J_0/2.$$

(b) Interpretation and further calculation

$$I(r)/I=2u-u^2,\quad u=r^2/R^2.$$$$2u-u^2=1/2\Rightarrow r_{1/2}=R\sqrt{1-1/\sqrt2}=.541196R<R/\sqrt2.$$

Physical check

The denser central current makes the half-current core smaller than the half-area core.