Question

Obtain an expression for the potential energy of a dipole in an external field.

Answer 1

1. Consider a dipole with charges –q and +q separated by a finite distance 2l, placed in a uniform electric field `vec"E"`.
2. It experiences a torque `vectau` which tends to rotate it as shown in figure below, `vec tau = vec"p" xx vec"E"` = pE sinθ
   
   Couple acting on a dipole
3. To neutralize this torque, let us assume an external torque `vectau_"ext"` be applied, which rotates it in the plane of the paper from angle θ₀ to angle θ, without angular acceleration and at an infinitesimal angular speed. 
4. Work done by the external torque
   W = `int_{θ_0}^θ tau_"ext"(θ)"d"θ = int_{θ_0}^θ"pE" sinθ "d"θ`
   = pE`[-cosθ]_{θ_0}^θ`
   = pE`[-cosθ - (-cosθ_0)]`
   = pE`[-cosθ + cosθ_0]`
   = pE`[cosθ_0 - cosθ]`
   This work done is stored as the potential energy of the system in the position when the dipole makes an angle θ with the electric field.
5. Thus, potential energy of electric dipole in external electric field is, U(θ) – U(θ₀) = pE(cosθ₀– cosθ)