Question

What is the potential energy of an electric dipole in a uniform electric field?

Answer 1

Step 1:Torque of the dipole a in uniform electric field.

$\stackrel{\rightharpoonup }{\tau }=\stackrel{\rightharpoonup }{p}\times \stackrel{\rightharpoonup }{E}$

Here, $\stackrel{\rightharpoonup }{p}$ is dipole and $\stackrel{\rightharpoonup }{E}$ is electric field.

Dipole:

The magnetic dipole moment is the product of pole strength and magnetic length.

Step 2:Work done by a small angle $d\theta$ in a rotating dipole.

$$\begin{gathered} dw=\tau d\theta \\ =-pE\sin \theta d\theta \end{gathered}$$

Step 3:The change in potential energy.

$$\begin{gathered} du=-dw \\ =pE\sin \theta d\theta \end{gathered}$$

Total potential energy.

$$\begin{gathered} u={\int }_{90°}^{0°}pE\sin \theta d\theta \\ =-pE\cos \theta \\ =-\stackrel{\rightharpoonup }{p}.\stackrel{\rightharpoonup }{E} \end{gathered}$$