# 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.