1.II.18B

Inside a volume $D$ there is an electrostatic charge density $\rho(\mathbf{r})$, which induces an electric field $\mathbf{E}(\mathbf{r})$ with associated electrostatic potential $\phi(\mathbf{r})$. The potential vanishes on the boundary of $D$. The electrostatic energy is

$W=\frac{1}{2} \int_{D} \rho \phi d^{3} \mathbf{r}$

Derive the alternative form

$W=\frac{\epsilon_{0}}{2} \int_{D} E^{2} d^{3} \mathbf{r}$

A capacitor consists of three identical and parallel thin metal circular plates of area $A$ positioned in the planes $z=-H, z=a$ and $z=H$, with $-H<a<H$, with centres on the $z$ axis, and at potentials $0, V$ and 0 respectively. Find the electrostatic energy stored, verifying that expressions (1) and (2) give the same results. Why is the energy minimal when $a=0$ ?

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