Question

Derive an expression for potential energy of electric-dipole placed in an uniform electric field.

Answer 1

Consider an electric dipole of dipole moment \(\vec P\) = q(2\(\vec a\)) placed at an angle 0 in the direction of uniform electric field \(\vec E\). 

The dipole experiences torque 

τ = P x E = pE sinθ 

This torque tends to rotate the dipole in the direction of the electric field. 

Suppose an external torque τ_ext is applied just to neutralise the torque (τ) and to rotate the dipole from θ₀ to θ₁ without angular acceleration then, the amount of work done by the external torque.

The work done in rotating the dipole from a position of zero potential energy (θ₀ = \(\frac{\pi}{2}\)) to any given position in the field ( θ₁= θ) then,

This quantity of work done is stored as potential energy U. 

Thus, potential energy of an electric dipole in a uniform Electric field is 

U = -pE cosθ

= -\(\vec p\).\(\vec E\)

Answer 2

Consider a dipole placed a torque when kept in a uniform electric field \(\vec E\). A dipole experiences a torque when kept in a uniform electric field \(\vec E\). This torque rotates the dipole to align it with the direction of the electric field. To rotate the dipole (at constant angular velocity) from its initial angle θ’ to another angle θ against the torque exerted by the electric field, an equal and opposite external torque must be applied on the dipole.

The dipole in a uniform electric field

The work done by the external torque to rotate the dipole from angle θ’ to θ at constant angular velocity is

\(W = \int \limits_{\theta'}^\theta \tau_{ext} \,d\theta \)    ......(1)

Since τ_ext is equal and opposite to τ_E = \(\vec P \times \vec E\), we have

\(|\vec r_{ext}|= |\vec r_E| = |\vec P \times \vec E|\)     ......(2)

Substituting equation (2) in equation (1) We get,

\(W = \int \limits_{\theta'}^\theta pE\sin \theta d\theta = pE (\cos\theta ' - \cos \theta)\)

his work done is equal to the potential energy difference between the angular positions θ and θ’.

U(θ) – (Uθ’) = AU = -pE cos θ + PE cos θ’.

If the initial angle is = θ’ = 90° and is taken as reference point, then U(θ’) + pE cos θ’ = θ.

The potential energy stored in the system of dipole kept in the uniform electric field is given by El = -pE cos θ = \(-\vec P.\vec E\) .....(3)

In addition to p and E, the potential energy also depends on the orientation θ of the electric dipole with respect to the external electric field.

The potential energy is maximum when the dipole is aligned anti-parallel (θ = π) to the external electric field and minimum when the dipole is aligned parallel (θ = 0) to the external electric field.