Compute the scalar potential of a dipole

Question

Question: Consider two opposite charges, of magnitudes $q$ and $−q$, separated by the vector $\vec{d}$.
Compute the scalar potential at all points in space, in the limit in which $q$ becomes very large and d very small, with $\vec{p}\equiv q\vec{d}$ kept constant.

My attempt: So I have picked a general point $\vec{r} = (x,y,z)^T$ and labelled the origin at the point charge $q$.

Then I have taken the formula for an electric potential:

$$ \phi(\mathbb{r})= \frac{1}{4\pi\epsilon_{0}} \left(\frac{q} {\mathbb{|\vec{r}|}} -\frac{q}{|\mathbb{\vec{r}} - \mathbb{\vec{d}}|}\right) $$

However I'm not sure how to manipulate this further to reach a point where I can use $\vec{p}\equiv q\vec{d}$ is constant. Any help would be greatly appreciated.

Answer 1

Without loss of generality, we can assume that (w.l.o.g.w.c.a.t.) $\vec{d} = d\hat{k}$. You can then write $$\frac{1}{|\vec{r} - \vec{d}|} = \frac{1}{\sqrt{r^2 - 2rd\cos\theta + d^2}}.$$ The further expansion is an application of the multinomial series and binomial series in the small parameter $d/r$. The dipole term is the one linear in $d/r$, the quadrupole terms are the ones of order $(d/r)^2$, etc. If you prefer, you don't have to assume that $\vec{d}$ points in the $z$-direction, you can replace $rd\cos\theta = \vec{r}\cdot\vec{d}$.

The final step is to replace $q = p/d$ and take the limit.