Induced Magnetic Fields — Question 6

Question 6

A magnetic pulse integral records an electric-field change

At radius r<a in a uniform axial electric-field region, $B_\phi(t)=B_*\sin(\pi t/T)$ for $0<t<T$ and zero outside. No conduction current crosses the contour. Use the quasistatic relation.

1. Find the net electric-field change during the pulse.

2. If E initially vanishes, find E(t) during the pulse and its final value.

Question 6 - Solution

A magnetic pulse integral records an electric-field change

(a) Method and calculation

$$\dot E=2c^2B_\phi/r\Rightarrow\Delta E=\frac{2c^2B_*}{r}\int_0^T\sin(\pi t/T)dt.$$$$\boxed{\Delta E=4c^2B_*T/(\pi r)}.$$

(b) Interpretation and further calculation

$$E(t)=\frac{2c^2B_*T}{\pi r}[1-\cos(\pi t/T)].$$$$\boxed{E(T)=\Delta E\ne0\quad\text{although }B_\phi(T)=0}.$$

Physical check

Vanishing magnetic circulation after the pulse implies no further electric-flux change, not zero electric flux.