Write down the electric potential due to a point charge $Q$ at the origin.

A dipole consists of a charge $Q$ at the origin, and a charge $-Q$ at position $-\mathbf{d}$. Show that, at large distances, the electric potential due to such a dipole is given by

$$\Phi(\mathbf{x})=\frac{1}{4 \pi \epsilon_{0}} \frac{\mathbf{p} \cdot \mathbf{x}}{|\mathbf{x}|^{3}}$$

where $\mathbf{p}=Q \mathbf{d}$ is the dipole moment. Hence show that the potential energy between two dipoles $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$, with separation $\mathbf{r}$, where $|\mathbf{r}| \gg|\mathbf{d}|$, is

$$U=\frac{1}{8 \pi \epsilon_{0}}\left(\frac{\mathbf{p}_{1} \cdot \mathbf{p}_{2}}{r^{3}}-\frac{3\left(\mathbf{p}_{1} \cdot \mathbf{r}\right)\left(\mathbf{p}_{2} \cdot \mathbf{r}\right)}{r^{5}}\right)$$

Dipoles are arranged on an infinite chessboard so that they make an angle $\theta$ with the horizontal in an alternating pattern as shown in the figure. Compute the energy between a given dipole and its four nearest neighbours, and show that this is independent of $\theta$.