Formulas/physics/Electric Charges Fields/Potential Energy of Dipole in Uniform Field

Potential Energy of Dipole in Uniform Field

Potential energy of a dipole in uniform field E. Minimum at θ = 0 (stable equilibrium), maximum at θ = π (unstable equilibrium).

Class 11Class 12

Derivation

From torque to potential energy

The torque on a dipole in a uniform field $\vec{E}$ is $\tau = -pE\sin\theta$ (negative because it opposes increase in $\theta$ ).

Work done by the external agent rotating the dipole by $d\theta$ :

dW = \tau_{ext}\, d\theta = pE\sin\theta\, d\theta

Integrating from a reference angle $\theta_0$ to $\theta$ :

W = \int_{\theta_0}^{\theta} pE\sin\theta'\, d\theta' = pE[-\cos\theta']_{\theta_0}^{\theta} = pE(\cos\theta_0 - \cos\theta)

By convention, $U = 0$ at $\theta_0 = 90°$ (dipole perpendicular to field):

U(\theta) = -pE\cos\theta + \underbrace{pE\cos 90°}_{=0}

\boxed{U = -\vec{p} \cdot \vec{E} = -pE\cos\theta}

$\theta$	$U$	State
$0°$	$-pE$	Minimum, stable equilibrium
$90°$	$0$	Reference
$180°$	$+pE$	Maximum, unstable equilibrium

Remember

The negative sign: the system has lowest energy when $\vec{p}$ aligns with $\vec{E}$. This is why polar molecules orient along electric fields.