Induced Electric Fields — Question 2

Question 2

An annular changing field leaves the central hole free of induced circulation

A changing axial field has uniform derivative beta in the annulus a<r<b and zero derivative elsewhere. The system is axially symmetric and has no additional electric field.

1. Find Ephi in all three radial regions.

2. Show continuity at both boundaries and locate the maximum magnitude.

Question 2 - Solution

An annular changing field leaves the central hole free of induced circulation

(a) Method and calculation

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

(b) Interpretation and further calculation

$$E_\phi(a)=0,\quad E_\phi(b)=-\beta(b^2-a^2)/(2b).$$$$\frac d{dr}\left(r-\frac{a^2}r\right)=1+\frac{a^2}{r^2}>0\Rightarrow\boxed{|E|_{\max}=\frac{|\beta|(b^2-a^2)}{2b}}.$$

Physical check

The hole result uses symmetry in addition to zero circulation; an unrelated electrostatic field could change the local field.