(i) Consider charges $-q$ at $\pm \mathbf{d}$ and $2 q$ at $(0,0,0)$. Write down the electric potential.

(ii) Take $\mathbf{d}=(0,0, d)$. A quadrupole is defined in the limit that $q \rightarrow \infty, d \rightarrow 0$ such that $q d^{2}$ tends to a constant $p$. Find the quadrupole's potential, showing that it is of the form

$$\phi(\mathbf{r})=A \frac{\left(r^{2}+C z^{D}\right)}{r^{B}}$$

where $r=|\mathbf{r}|$. Determine the constants $A, B, C$ and $D$.

(iii) The quadrupole is fixed at the origin. At time $t=0$ a particle of charge $-Q(Q$ has the same sign as $q)$ and mass $m$ is at $(1,0,0)$ travelling with velocity $d \mathbf{r} / d t=(-\kappa, 0,0)$, where

$$\kappa=\sqrt{\frac{Q p}{2 \pi \epsilon_{0} m}} .$$

Neglecting gravity, find the time taken for the particle to reach the quadrupole in terms of $\kappa$, given that the force on the particle is equal to $m d^{2} \mathbf{r} / d t^{2}$.