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Question
Answer the following question.
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.
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