Question

When a ball made of uniform magnetic is introduced into an external uniform magnetic field with induction ${\overrightarrow{B}}_0$, it gets uniformly magnetized. Find the magnetic induction $\overrightarrow{B}$ inside the ball with permeability $\mu$; recall that the magnetic field inside a uniformly magnetized ball is uniform and its strength is equal to $\overrightarrow{H^{\prime }}=-\frac{\overrightarrow{J}}{3}$, where $\overrightarrow{J}$ is the magnetization.

Answer 1

We use the fact that within an isolated uniformly magnetized ball,
$\overrightarrow{H^{\prime }}=-\frac{\overrightarrow{J}}{3}$, $\overrightarrow{B^{\prime }}=\frac{2{\mu }_0\overrightarrow{J}}{3}$, where $\overrightarrow{J}$ is the magnetization vector. Then in a uniform magnetic field with induction $\overrightarrow{B_0}$, we have by superposition,
$\overrightarrow{B_{in}}=\overrightarrow{B_0}+\frac{2{\mu }_0\overrightarrow{J}}{3}$, $\overrightarrow{H_{in}}=\frac{\overrightarrow{B_0}}{{\mu }_0}-\frac{\overrightarrow{J}}{3}$
or, $\overrightarrow{B_{in}}+2{\mu }_0\overrightarrow{H_{in}}=3\overrightarrow{B_0}$
also, $\overrightarrow{B_{in}}=\mu {\mu }_0\overrightarrow{H_{in}}$
Thus, $\overrightarrow{H_{in}}=\frac{3\overrightarrow{B_0}}{{\mu }_0(\mu +2)}$ and $\overrightarrow{B_{in}}=\frac{3\mu \overrightarrow{B_0}}{\mu +2}$