Exercises

1. Given that the bound charge density associated with a polarization field ${\bf P}({\bf r})$ is $\sigma_b=-\nabla\cdot{\bf P}$ , use charge conservation to deduce that the current density due to bound charges is
   $${\bf j}_p = \frac{\partial{\bf P}}{\partial t}.$$

2. Given that $\nabla\times {\bf H}={\bf0}$ in the absence of true currents, and ${\bf H}={\bf B}/\mu_0-{\bf M}$ , demonstrate that the current density due to magnetization currents is
   $${\bf j}_m = \nabla\times{\bf M}.$$

3. A cylindrical hole of radius $a$ is bored parallel to the axis of a cylindrical conductor of radius $b>a$ which carries a uniformly distributed current of density $j$ running parallel to its axis. The distance between the center of the conductor and the center of the hole is $x_0$ . Find the ${\bf B}$ field in the hole.

4. A sphere of radius $a$ carries a uniform surface charge density $\sigma$ . The sphere is rotated about a diameter with constant angular velocity $\omega$ . Find the vector potential and the ${\bf B}$ field both inside and outside the sphere.

5. Find the ${\bf B}$ and ${\bf H}$ fields inside and outside a spherical shell of inner radius $a$ and outer radius $b$ which is magnetized permanently to a constant magnetization ${\bf M}$ .

6. A long hollow, right cylinder of inner radius $a$ and outer radius $b$ , and of relative permeability $\mu$ , is placed in a region of initially uniform magnetic flux density ${\bf B}$ at right-angles to the field. Find the flux density at all points in space. Neglect end effects.

7. A transformer consists of a thin uniform ring of ferromagnetic material of radius $a$ , cross-sectional area $A$ , and magnetic permeability $\mu$ . The primary circuit is wrapped $N_1$ times around one side of the ring, and the secondary $N_2$ times around the other side. Show that the mutual inductance between the two circuits is
   $$M = \frac{\mu\,N_1\,N_2\,A}{2\pi\,a}.$$
   Suppose that a thin gap of thickness $d\ll a$ is cut in a part of the ring in which there are no windings. What is the new mutual inductance of the two circuits? Suppose that the gap is filled with ferromagnetic material of permeability $\mu'$ . What, now, is the mutual inductance of the circuits? You may neglect flux-leakage (i.e., you may assume that magnetic field-lines do not leak out of the transformer core into the surrounding vacuum, except in the gap).