# Derive the Expression for the Torque Acting on an Electric Dipole, When It is Held in a Uniform Electric Field. Identify the Orientation of the Dipole in the Electric Field, in Which It - Physics

Derive the expression for the torque acting on an electric dipole, when it is held in a uniform electric field. identify the orientation of the dipole in the electric field, in which it attains a stable equilibrium.

#### Solution

Dipole in a Uniform External Field

Consider an electric dipole consisting of charges − q and + q and of length 2a placed in a uniform electric field vec"E"  making an angle θ with the electric field.

Force on charge - q at "A" = -q vec"E" ("opposite to"_ vec"E")

Force on charge + q at "B" = qvec"E" ("along"_ vec"E")

The Electric dipole is under the action of two equal and unlike parallel forces, which give rise to a torque on the dipole.
T = Force x Perpendicular distance between the two forces

T = qE (AN) = qE (2a sin θ)

T = q(2a) E sinθ

T = pE sinθ

∴ vec"t" = vec"p" xx vec"E"

in a uniform electric field, the net force on dipole will always be zero but torque is zero for theta = 0° and theta = 180°

Now Potential Energy of a dipole in a uniform external electric field is given by the expression "P"."E" = - vec"p" .vec"E"

1. For theta = 0°, U = - pE  (minimum), the equilibrium will be stable and if the dipole is slightly displaced, it performs oscillations.
2. For theta = 180°, U = +pE(maximum), it will be an unstable equillibrium.

Concept: Dipole in a Uniform External Field
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