Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 2 / 6B / Solution

Take the electric dipole moment along the polar axis, $\mathbf{p}=p\hat{\mathbf{z}}$. The electric field is minus the gradient of the electric potential; in spherical polar coordinates,Since $\mathbf{p}=p(\cos \theta {\hat{\mathbf{e}}}_r-\sin \theta {\hat{\mathbf{e}}}_{\theta })$ and $\mathbf{p}\cdot {\hat{\mathbf{e}}}_r=p\cos \theta$, the radial and angular components combine asThe origin is excluded because the point-dipole potential is singular there.