1. A long cylindrical rod of radius R has a magnetization

in the azimuthal direction. There are no free currents. Find the magnetic field both inside and outside the cylinder due to magnetization.

The bound volume current is zero since the magnetization has constant magnitude. The surface current density is . The magnetic field inside is zero as there is no volume current inside. The current outside is given by

2. In the preceding problem, take the magnetization to be given by where r is the distance from the cylindrical axis. Find the magnetic field both inside and outside the cylinder due to magnetization.

Bound current densities are as follows and

By symmetry the fields at The total current enclosed within a length L of the cylinder is . Thus the magnetic field outside is zero.

Since the current is in azimuthal direction, in order to calculate the magnetic field, we need to take an Amperian loop which is in the form of a rectangle of height L whose one side is at a distance r from the axis and the other parallel side is outside. Since the field outside is zero, we have,

which gives .

3. Repeat Problem 2 by calculating the H field and adding the magnetization contribution.

Since there are no free currents, H=0. Thus .Since the magnetization is zero outside, the field is zero. Inside, the field is equal to .

4. A magnetized material is in the shape of a cube of side *a * as shown. The magnetization is directed along the z direction and varies linearly from its value zero on its surface on the y-z plane to a value on the plane .
Calculate the bound currents both on the surface and inside the material.