Derivation : Electric Potential Energy of an Electric Dipole in an Electric Field

Electric Potential Energy of an Electric Dipole in an Electric Field

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

Let an electric dipole of dipole moment $\vec{p}$ be placed in an electric field $\vec{E}$ making an angle $\theta$ with the direction of electric field intensity $\vec{E}$. The torque acting on the dipole is given by,

$ \tau = pE\sin\theta \qquad \cdots (i)$

Work done to rotate the dipole through an angle $d\theta$ is given by,

$dW = \tau d\theta = pE\sin\theta d\theta$

Work done in rotating the dipole from an angle $\theta_1$ to $\theta_2$ is given by,

$W = \int dW = \int_{\theta_1}^{\theta_2} pE\sin\theta d\theta$

$W= pE\int_{\theta_1}^{\theta_2} \sin\theta d\theta$

If $\theta_1 = 90^\circ$ and $\theta_2 = \theta$, then

    $W= pE [-\cos \theta]_{\theta_1}^{\theta_2} = -pE [\cos \theta_2 - \cos \theta_1]$

    $= -pE [\cos \theta - \cos 90^\circ]$

    $= -pE \cos \theta \qquad \cdots (ii)$

This work done is stored as the electric potential energy (U) of a dipole in an electric field.

That is,

$U = -pE \cos \theta = -\vec{p} \cdot \vec{E} \qquad \cdots (iii)$

Eqn. (iii) represents the expression of the electric potential energy of an electric dipole in an electric field.

Special Cases : 

(i) When $\theta = 0^\circ$ (i.e., dipole is parallel to direction of electric field), $U = -pE \cos 0^\circ = -pE$

Thus, electric potential energy of an electric dipole in an electric field is minimum (-pE), when the dipole is parallel to the direction of electric field. The dipole in this position is in STABLE EQUILIBRIUM.

(ii) When $\theta = 90^\circ$ (i.e., dipole is perpendicular to the direction of electric field), $U = -pE \cos 90^\circ = 0$

(iii) When $\theta = 180^\circ$ (i.e., dipole is anti-parallel to electric field), $U = -pE \cos 180^\circ$, i.e., $U = pE$

Thus, electric potential energy of a dipole is maximum (pE), when it is anti-parallel to the direction of the electric field.

The dipole in this position is in UNSTABLE EQUILIBRIUM.