Question

Show that the energy of an ideal dipole p in an electric field E isgiven by

$\mathbf{U}\mathbf{=}\mathbf{-}\mathbf{p}\mathbf{\cdot }\mathbf{E}\mathbf{\in }$

Short answer

Thepotential energy for the dipole moment is$-p\cdot E$ .

Step-by-step solution

Determine the formulas

Consider the formula for the torque on the dipole moment as

$\mathbf{\tau }\mathbf{=}\mathbf{pEsin\theta }$

Here, $p$is the dipole moment and E is the electric field.

Determine the formula for the energy of the dipole moment 

Consider the formula for work done in the rotating dipole moment as:

$dw=\tau d\theta$

Substitute the values and solve as

$\begin{array}{l}dw=pE\sin \theta d\theta \\ w={\int }_{a_1}^{a_2}pE\sin \theta d\theta \\ w=pE(\cos {\theta }_1-\cos {\theta }_2)\end{array}$

Consider the change in the potential for the two dipole positions and the corresponding working done as

$\begin{array}{c}w=U\left({\theta }_2\right)-U\left({\theta }_1\right)\\ =-pE(\cos {\theta }_2-{\theta }_1)\\ =-pE\cos \theta \\ =-p\cdot E\end{array}$

Therefore, the potential energy for the dipole moment is $-p\cdot E$.