Question

Derive the formula for the electric potential energy of an electric dipole in a uniform electric field.

Answer 1

Potential energy of dipole in uniform electric field.

$\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}$

work done in rotating dipole by small angle

$dw=\tau d\theta$

$=-p\epsilon \sin \theta d\theta$

The change in potential energy

$dU=-dW=pE\sin \theta d\theta$

If $\theta \to {90}^o$ to ${\theta }^o$

$U\left(\theta \right)-U\left({90}^o\right)={\int }_{{90}^o}^{0^o}P\epsilon \sin \theta d\theta$

$=P\epsilon [-\cos \theta {]}_{90}^0$

$=-\overrightarrow{P}.\overrightarrow{E}$

So

$U\left(\theta \right)=-\overrightarrow{P}.\overrightarrow{E}$