Question

A long dielectric cylinder of radius R is statically polarized so that at all its points the polarization is equal to P = αr, where a is a positive constant, and r is the distance from the axis. The cylinder is set into rotation about its axis with an angular velocity ω. Find the magnetic induction B at the centre of the cylinder.

Answer 1

Because of polarization a space charge is present within the cylinder. It’* density is

Since the cylinder as a whole is neutral a surface charge density σp must be present on the surface of the cylinder also. This has the magnitude (algebraically)

When the cylinder rotates, currents are set up which give rise to magnetic fields. The contribution of ρp and σp can be calculated separately and then added. For the surface charge the current is (for a particular element)

Its contribution to the magnetic field at the centre is

and the total magnetic field is

As for the volume charge density consider a circle of radius r, radial thickness dr and length dx.