Formulas/physics/Electric Charges Fields/Work Done in Rotating a Dipole

Work Done in Rotating a Dipole

Work done by external agent rotating dipole from angle θ₁ to θ₂ in a uniform electric field E.

Class 11Class 12

Derivation

From the potential energy expression $U = -pE\cos\theta$ , the work done by the external agent in rotating the dipole from $\theta_1$ to $\theta_2$ equals the change in potential energy:

W = U(\theta_2) - U(\theta_1) = (-pE\cos\theta_2) - (-pE\cos\theta_1)

\boxed{W = pE(\cos\theta_1 - \cos\theta_2)}

Interpreting the sign

$W > 0$ : external agent does positive work (angle increases, system gains energy)
$W < 0$ : field does work (angle decreases toward alignment)

Special cases

From perpendicular to aligned ( $\theta_1 = 90° \to \theta_2 = 0°$ ):

W = pE(\cos 90° - \cos 0°) = pE(0 - 1) = -pE

The field does work $pE$ ; the external agent does $-pE$ .

From aligned to anti-aligned ( $\theta_1 = 0° \to \theta_2 = 180°$ ):

W = pE(\cos 0° - \cos 180°) = pE(1 - (-1)) = 2pE

Note

Work done depends only on the initial and final angles — not on the path of rotation. This confirms $U = -pE\cos\theta$ is a genuine potential energy.

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