Classical Electrodynamics · Problem 11.31

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · wilson-experiment-rotating-cylinder

Problem

Consider a hollow right-circular cylinder of magnetic insulator (relative permeabilities

\epsilon

and

\mu

and inner and outer radii

a

and

b

) set in rotation about its axis at angular speed

\omega

in a uniform axial magnetic field

B_0

. In 1913 the Wilsons measured the voltage difference between its inner and outer surfaces caused by a radial internal electric field. Assuming that locally the relations of Problem 11.30 hold, that the velocity

\mathbf{v} = \omega\rho\hat{\boldsymbol{\phi}}

, and that there are only the field components

E_\rho

and

B_z

, which are independent of

z

and

\phi

, solve the equations

\boldsymbol{\nabla} \cdot \mathbf{D} = 0

and

\boldsymbol{\nabla} \times \mathbf{H} = 0

within the cylinder and show that the internal fields are

E_\rho = -\frac{\mu\omega\rho B_0}{c(1 - \omega^2\rho^2/c^2)}\left(1 - \frac{1}{\mu\epsilon}\right); \quad B_z = \mu B_0\left[\frac{1 - \omega^2\rho^2/c^2\mu\epsilon}{1 - \omega^2\rho^2/c^2}\right]

and that for nonrelativistic motion (

\omega b/c \ll 1

) the voltage difference is

V = \frac{\mu\omega}{2c}B_0(b^2 - a^2)\left(1 - \frac{1}{\mu\epsilon}\right)

This experiment was an early validation of special relativity and Minkowski's electrodynamics of material media in motion. Reference: M. Wilson and H. A. Wilson, Proc. Roy. Soc. London A89, 99--106 (1913).

Browse the problem library