Induced Magnetic Fields — Question 2

Question 2

An annular electric-flux change leaves a field-free central hole

A quasistatic vacuum field changes at uniform rate beta along +z only in the annulus $a<r<b$. No conduction current is present. Use $\oint B\,dl=c^{-2}\dot\Phi_E$.

1. Find Bphi in all three radial regions.

2. Locate its greatest magnitude for beta positive.

Question 2 - Solution

An annular electric-flux change leaves a field-free central hole

(a) Method and calculation

$$r<a:\quad\boxed{B_\phi=0}.$$$$a<r<b:\quad\boxed{B_\phi=\frac{\beta(r^2-a^2)}{2c^2r}};\quad r>b:\quad\boxed{B_\phi=\frac{\beta(b^2-a^2)}{2c^2r}}.$$

(b) Interpretation and further calculation

$$\frac d{dr}(r-a^2/r)=1+a^2/r^2>0,\quad\frac d{dr}(1/r)<0.$$$$\boxed{B_{\max}=\frac{\beta(b^2-a^2)}{2c^2b}\quad\text{at }r=b}.$$

Physical check

The field is continuous at both radial boundaries even though the changing electric-field region has sharp edges.