Exercises

1. Demonstrate that the energy contained in the magnetic field generated by a stationary current distribution ${\bf j}({\bf r})$ in vacuum is given by
   $$W = \frac{\mu_0}{8\pi}\int\int \frac{{\bf j}({\bf r})\cdot{\bf j}({\bf r}')}{\vert{\bf r} - {\bf r}'\vert}\,dV\,dV'.$$

2. A transverse plane wave is incident normally in vacuum on a perfectly absorbing flat screen. Show that the pressure exerted on the screen is equal to the electromagnetic energy density of the wave.

3. Consider an infinite parallel-plate capacitor. Let the lower plate lie at $z=-d/2$ , and carry the charge density $-\sigma$ . Likewise, let the upper plate lie at $z=+d/2$ , and carry the charge density $+\sigma$ . Calculate the electromagnetic momentum flux across the $y$ -$z$ plane. Hence, determine the direction and magnitude of the force per unit area that the plates exert on one another.

4. The equation of electromagnetic angular momentum conservation takes the general form
   $$\frac{\partial {\bf L}}{\partial t} + \nabla\cdot{\bf M} = - {\bf r}\times (\rho\,{\bf E} + {\bf j}\times {\bf B}),$$
   where ${\bf L}$ is the electromagnetic angular momentum density, and the tensor ${\bf M}$ is the electromagnetic angular momentum flux. Demonstrate that
   $${\bf L} = {\bf r}\times {\bf g},$$
   and
   $${\bf M} = {\bf r}\times {\bf G},$$
   where ${\bf g}$ is the electromagnetic momentum density, and ${\bf G}$ the electromagnetic momentum flux tensor.

5. A long solenoid of radius $R$ , with $N$ turn per unit length, carries a steady current $I$ . Two hollow cylinders of length $l$ are fixed coaxially such that they are free to rotate. The first cylinder, whose radius is $a<R$ , carries the uniformly distributed electric charge $Q$ . The second cylinder, whose radius is $b>R$ , carries the uniformly distributed electric charge $-Q$ . Both cylinders are initially stationary. When the current is switched off the cylinders start to rotate. Find the final angular momenta of the two cylinders, and demonstrate that the total angular momentum of the system is the same before and after the current is switched off.

6. Consider a system consisting of an electric charge $e$ and a magnetic monopole $g$ separated by a distance $d$ . Demonstrate that the total angular momentum stored in the resulting electromagnetic fields is
   $$L= \frac{\mu_0}{4\pi}\,e\,g.$$
   [Hint: The radial magnetic field generated a distance $r$ from a magnetic monopole of strength $g$ is of magnitude $(\mu_0/4\pi)\,(g/r^{\,2})$ .]