The infinite plane $z=0$ is earthed and the infinite plane $z=d$ carries a charge of $\sigma$ per unit area. Find the electrostatic potential between the planes.

Show that the electrostatic energy per unit area (of the planes $z=$ constant) between the planes can be written as either $\frac{1}{2} \sigma^{2} d / \epsilon_{0}$ or $\frac{1}{2} \epsilon_{0} V^{2} / d$, where $V$ is the potential at $z=d$.

The distance between the planes is now increased by $\alpha d$, where $\alpha$ is small. Show that the change in the energy per unit area is $\frac{1}{2} \sigma V \alpha$ if the upper plane $(z=d)$ is electrically isolated, and is approximately $-\frac{1}{2} \sigma V \alpha$ if instead the potential on the upper plane is maintained at $V$. Explain briefly how this difference can be accounted for.