Induced Electric Fields — Question 1

Question 1

A changing cylindrical field induces an electric field beyond the magnetic-field region

An axially symmetric field $B(t)\hat z$ is uniform for r<a and zero for r>a. No electrostatic field is added. Let $\dot B=\beta>0$ and positive azimuth be counterclockwise viewed from +z.

1. Find the azimuthal electric field inside and outside.

2. Find its largest magnitude and evaluate it for a=.0400 m and beta=3.00 T/s.

Question 1 - Solution

A changing cylindrical field induces an electric field beyond the magnetic-field region

(a) Method and calculation

$$2\pi rE_\phi=-\frac d{dt}(\pi r^2B)\Rightarrow\boxed{E_\phi=-\beta r/2\quad(r<a)}.$$$$2\pi rE_\phi=-\pi a^2\beta\Rightarrow\boxed{E_\phi=-\beta a^2/(2r)\quad(r>a)}.$$

(b) Interpretation and further calculation

$$|E|\text{ rises inside and falls outside}\Rightarrow r_{\max}=a.$$$$\boxed{|E|_{\max}=\beta a/2=.0600\,\mathrm{V/m}}.$$

Physical check

Zero local magnetic field outside the cylinder does not imply zero induced electric field there.