By considering the force per unit volume $\mathbf{f}=\rho \mathbf{E}+\mathbf{J} \times \mathbf{B}$ on a charge density $\rho$ and current density $\mathbf{J}$ due to an electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$, show that

$$\frac{\partial g_{i}}{\partial t}+\frac{\partial \sigma_{i j}}{\partial x_{j}}=-f_{i}$$

where $\mathbf{g}=\epsilon_{0} \mathbf{E} \times \mathbf{B}$ and the symmetric tensor $\sigma_{i j}$ should be specified.

Give the physical interpretation of $\mathbf{g}$ and $\sigma_{i j}$ and explain how $\sigma_{i j}$ can be used to calculate the net electromagnetic force exerted on the charges and currents within some region of space in static situations.

The plane $x=0$ carries a uniform charge $\sigma$ per unit area and a current $K$ per unit length along the $z$-direction. The plane $x=d$ carries the opposite charge and current. Show that between these planes

$$\sigma_{i j}=\frac{\sigma^{2}}{2 \epsilon_{0}}\left(\begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)+\frac{\mu_{0} K^{2}}{2}\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array}\right)$$

and $\sigma_{i j}=0$ for $x<0$ and $x>d$.

Use $(*)$ to find the electromagnetic force per unit area exerted on the charges and currents in the $x=0$ plane. Show that your result agrees with direct calculation of the force per unit area based on the Lorentz force law.

If the current $K$ is due to the motion of the charge $\sigma$ with speed $v$, is it possible for the force between the planes to be repulsive?