Induced Magnetic Fields — Question 9

Question 9

A steady electric field contributes no magnetic circulation through its time derivative

Inside a circular contour, the uniform axial field is $E_z(t)=E_b+\alpha t^2$ over area A. There is no conduction current through the disk. Take its boundary radius r, so $A=\pi r^2$.

1. Find the magnetic circulation and Bphi.

2. Compare the result when an arbitrary constant is added to Eb, and when time changes sign.

Question 9 - Solution

A steady electric field contributes no magnetic circulation through its time derivative

(a) Method and calculation

$$\partial_t\Phi_E=2\alpha t A\Rightarrow\oint\mathbf B\cdot d\mathbf l=2\alpha tA/c^2.$$$$\boxed{B_\phi=\alpha tr/c^2}.$$

(b) Interpretation and further calculation

$$E_b\to E_b+E_{\mathrm{offset}}:\quad\partial_tE_z\ \text{unchanged}.$$$$\boxed{B_\phi(-t)=-B_\phi(t),\quad B_\phi(0)=0}.$$

Physical check

Circulation responds to the rate of electric flux, even when the electric-field magnitude is large at t=0.