Describe the physical meaning of the various components of the stress-energy tensor $T^{a b}$ of the electromagnetic field.

Suppose that one is given an electric field $\mathbf{E}(\mathbf{x})$ and a magnetic field $\mathbf{B}(\mathbf{x})$. Show that the angular momentum about the origin of these fields is

$$\mathbf{J}=\frac{1}{\mu_{0}} \int \mathbf{x} \times(\mathbf{E} \times \mathbf{B}) d^{3} \mathbf{x}$$

where the integral is taken over all space.

A point electric charge $Q$ is at the origin, and has electric field

$$\mathbf{E}=\frac{Q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}}{|\mathbf{x}|^{3}}$$

A point magnetic monopole of strength $P$ is at $\mathbf{y}$ and has magnetic field

$$\mathbf{B}=\frac{\mu_{0} P}{4 \pi} \frac{\mathbf{x}-\mathbf{y}}{|\mathbf{x}-\mathbf{y}|^{3}}$$

Find the component, along the axis between the electric charge and the magnetic monopole, of the angular momentum of the electromagnetic field about the origin.

[Hint: You may find it helpful to express both $\mathbf{E}$ and $\mathbf{B}$ as gradients of scalar potentials.]