Induced Electric Fields — Question 3

Question 3

A nonuniform magnetic profile moves the strongest induced electric field inside its edge

For r<a an axial field is $B_z(r,t)=\beta t(1-r^2/a^2)$, and it is zero outside. Assume axial symmetry and no added electrostatic field.

1. Find Ephi inside and outside.

2. Locate the interior maximum of its magnitude for beta positive.

Question 3 - Solution

A nonuniform magnetic profile moves the strongest induced electric field inside its edge

(a) Method and calculation

$$\dot\Phi(r)=2\pi\beta\int_0^r(1-s^2/a^2)s\,ds=\pi\beta(r^2-r^4/(2a^2)).$$$$\boxed{E_\phi=-\beta(r/2-r^3/(4a^2))\ (r<a);\quad E_\phi=-\beta a^2/(4r)\ (r>a)}.$$

(b) Interpretation and further calculation

$$\frac d{dr}|E_\phi|=\beta\left(\frac12-\frac{3r^2}{4a^2}\right)=0\Rightarrow\boxed{r=a\sqrt{2/3}}.$$$$\boxed{|E|_{\max}=\frac{\beta a}{3}\sqrt{2/3}}>\beta a/4=|E(a)|.$$

Physical check

The radius of largest induced field is determined by the integrated flux profile, not solely by the field-region boundary.