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

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Solution

Potential energy of dipole in uniform electric field.
$\to τ = \to p \times \to E$
work done in rotating dipole by small angle
$d w = τ d θ$
$= - p ε sin θ d θ$
The change in potential energy
$d U = - d W = p E sin θ d θ$
If $θ \to 90^{o}$ to $θ^{o}$
$U (θ) - U (90^{o}) = \int_{90^{o}}^{0^{o}} P ε sin θ d θ$
$= P ε [- cos θ]_{90}^{0}$
$= - \to P . \to E$
So
$U (θ) = - \to P . \to E$

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