3.4: Potential Energy of a Dipole in an Electric Field

Refer again to Figure III.3. There is a torque on the dipole of magnitude $pE \sin θ$. In order to increase $θ \text{ by }δθ$ you would have to do an amount of work $pE \sin θ\, δθ$. The amount of work you would have to do to increase the angle between $\textbf{p} \text{ and }\textbf{E}$ from 0 to $θ$ would be the integral of this from 0 to $θ$, which is $pE(1 − \cos θ)$, and this is the potential energy of the dipole, provided one takes the potential energy to be zero when $\textbf{p} \text{ and }\textbf{E}$ are parallel. In many applications, writers find it convenient to take the potential energy (P.E.) to be zero when $\textbf{p} \text{ and }\textbf{E}$ perpendicular. In that case, the potential energy is

$$\text{P.E}=-pE\cos \theta = -\textbf{p}\cdot \textbf{E}.\label{3.4.1}$$

This is negative when $θ$ is acute and positive when $θ$ is obtuse. You should verify that the product of $p \text{ and }E$ does have the dimensions of energy.