## The Electric Field of a Dipole

The electric field of a dipole can be seen as the result of two charges approaching each other. Learn in this problem how to use the Taylor expansion of $$1/\left|\mathbf{r}-\mathbf{r}^{\prime}\right|$$ to calculate this field. Find out how this series expansion yields multipole moments, a very powerful description of the electric field.

## Problem Statement

Find the electric field of an electric dipole using the following steps:

• Find the electrostatic potential of two point charges $$q_{1/2}=\pm q$$ at $$\mathbf{r}_{1/2}=\pm\frac{1}{2}\mathbf{d}$$.
• Take the limit $$\left|\mathbf{d}\right|\rightarrow0$$ of the potential holding $$\mathbf{p}:=q\mathbf{d}$$ constant.
• Calculate the electric field from the electrostatic potential.
• additional: Verify that the charge distribution $$\rho\left(\mathbf{r}\right)=-\mathbf{p}\cdot\nabla\delta\left(\mathbf{r}\right)$$ yields the correct dipole potential.

## Background: The Dipole as Limiting Process

As we have found in The Electric Field of two Point Charges, the electric field of two opposite charges is fundamentally different from that of a single charge decreasing with $$1/r^{3}$$ at large distances, not only with $$1/r^{2}$$. This is not just a coincidence but stems from the so-called multipole expansion as we have outlined earlier.

For now, let us try to find the dipole term by the limiting process of the given problem!