Induced Magnetic Fields — Question 4

Question 4

Induced magnetic field is in quadrature with a sinusoidal electric field

A uniform axial field $E_z=E_0\cos\omega t$ occupies a cylindrical region of radius a. Consider r<a and the quasistatic limit with no conduction current.

1. Find Bphi and its peak magnitude.

2. Compare B at an electric-field maximum with B at the first electric-field zero.

Question 4 - Solution

Induced magnetic field is in quadrature with a sinusoidal electric field

(a) Method and calculation

$$B_\phi=\frac r{2c^2}\partial_tE_z=-\frac{r\omega E_0}{2c^2}\sin\omega t.$$$$\boxed{B_{\mathrm{peak}}=r\omega E_0/(2c^2)}.$$

(b) Interpretation and further calculation

$$t=0:\ E_z=E_0,\quad B_\phi=0.$$$$\boxed{t=\pi/(2\omega):\ E_z=0,\quad B_\phi=-B_{\mathrm{peak}}}.$$

Physical check

The largest electric field and the largest induced magnetic field occur at different phases.