Induced Magnetic Fields — Question 3

Question 3

A tapered electric-field profile moves the magnetic maximum inside the edge

For r<a, a prescribed quasistatic vacuum field has $\partial_tE_z=\beta(1-r^2/a^2)$; outside, its time derivative vanishes. There is no conduction current.

1. Integrate the electric-flux rate to find Bphi inside and outside.

2. Find the radius of maximum B for beta positive.

Question 3 - Solution

A tapered electric-field profile moves the magnetic maximum inside the edge

(a) Method and calculation

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

(b) Interpretation and further calculation

$$\partial_rB_\phi=(\beta/c^2)(1/2-3r^2/(4a^2))=0.$$$$\boxed{r_{\max}=a\sqrt{2/3},\quad B_{\max}=\frac{\beta a}{3c^2}\sqrt{2/3}}.$$

Physical check

Using the center value of the electric-field rate everywhere would overcount flux.