Electric Potential Energy of an Electric Dipole in an Electric Field - Param Himalaya
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.