Induced Magnetic Fields — Question 8

Question 8

Opposite electric-flux rates can hide an internal induced magnetic field

A vacuum field has rate +beta inside radius a and rate -gamma in the annulus $a<r<b$. Outside b the rate is zero. No conduction current is present. Choose gamma so the total electric-flux rate vanishes.

1. Find gamma and the magnetic field outside b.

2. Find B inside a and explain why zero external circulation does not imply zero internal field.

Question 8 - Solution

Opposite electric-flux rates can hide an internal induced magnetic field

(a) Method and calculation

$$\dot\Phi_E=\pi a^2\beta-\pi(b^2-a^2)\gamma=0.$$$$\boxed{\gamma=\beta a^2/(b^2-a^2),\quad B_\phi(r>b)=0}.$$

(b) Interpretation and further calculation

$$r<a:\quad2\pi rB_\phi=c^{-2}\pi r^2\beta.$$$$\boxed{B_\phi=\beta r/(2c^2)\ne0}.$$

Physical check

Cancellation of total enclosed flux rate applies only to contours enclosing both regions.