Classical Electrodynamics · Problem 6.8

Plane Electromagnetic Waves and Wave Propagation — John David Jackson

Physics — E&M · rotating-dielectric-sphere-magnetic-field

Problem

A dielectric sphere of dielectric constant

\epsilon

and radius

a

is located at the origin. There is a uniform applied electric field

E_0

in the

x

direction. The sphere rotates with an angular velocity

\omega

about the

z

axis. Show that there is a magnetic field

\mathbf{H} = -\nabla\Phi_M

, where

\Phi_M = \frac{3}{5}\left(\frac{\epsilon - \epsilon_0}{\epsilon + 2\epsilon_0}\right)\epsilon_0 E_0 \omega \left(\frac{a}{r_>}\right)^5 \cdot xz

where

r_>

is the larger of

r

and

a

. The motion is nonrelativistic. You may use the results of Section 4.4 for the dielectric sphere in an applied field.