Exercise 3.3.2: Two point charges: exact potential and dipole limit

By superposition of the two point-charge potentials,

where the distances from $\pm q$ to P follow from the law of cosines (the angle between $r$ and the $+z$-axis is $\theta$, that between $r$ and the $-z$-axis is $\pi -\theta$).

For $r\gg d$, neglect $d^2/4$ against $r^2$ and use $(1+x{)}^{-1/2}\simeq 1-x/2$ for $|x|\ll 1$:

The field follows from the gradient in spherical coordinates,

with $\partial V/\partial r=-2qd\cos \theta /(4\pi {\epsilon }_0{r}^3)$, $\partial V/\partial \theta =-qd\sin \theta /(4\pi {\epsilon }_0{r}^2)$ and $\partial V/\partial \varphi =0$:

$V$ changes sign when $\theta \to \pi -\theta$ (swapping the roles of $+q$ and $-q$), as it must. The unit of $p/({\epsilon }_0{r}^2)$ is $\mathrm{C}\mathrm{m}/[(F/m){\mathrm{m}}^2]=\mathrm{V}$, and $E\propto 1/r^3$, one power faster than a single charge.