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

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

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,

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

$ \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.


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