Ampère's Law — Question 4

Question 4

A radial current-density profile changes the interior field without changing the exterior law

An infinite cylindrical conductor of radius a has steady axial current density $J_z(r)=J_0r/a$ for r between zero and a, with J0 positive. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find the total current and interior magnetic field.

2. Express B(r) in terms of the measured total current I and compare the interior field at r=a/2 with that of a uniform-current conductor having the same I and a.

Question 4 - Solution

A radial current-density profile changes the interior field without changing the exterior law

(a) Method and calculation

$$I_{\mathrm{enc}}(r)=\int_0^r\frac{J_0s}{a}2\pi s\,ds=\frac{2\pi J_0r^3}{3a},\quad I=\frac{2\pi J_0a^2}{3}.$$$$B(r)=\frac{\mu_0I_{\mathrm{enc}}}{2\pi r}=\boxed{\frac{\mu_0J_0r^2}{3a}}\quad(r<a).$$

(b) Interpretation and further calculation

$$B(r)=\frac{\mu_0Ir^2}{2\pi a^3},\quad B(a/2)=\frac14B(a).$$$$B_{\mathrm{uniform}}(a/2)=\frac12B(a)\Rightarrow\boxed{B/B_{\mathrm{uniform}}=1/2}\text{ at }r=a/2.$$

Physical check

Both distributions produce the same exterior field because they have the same total current and cylindrical symmetry.