Derivation of the electric potential energy of an electric dipole in the uniform electric field:
Let us consider an electric dipole $AB$, which is made up of two charges $q_1$ and $q_2$, which are placed at a distance of $2l$ in the electric field $E$. So force acting on each charge due to the electric field will be $qE$. If the dipole gets rotated a small-angle $d\theta$ against the torque acting on it in the uniform electric field $E$ then the small work done is
$dW=\tau. d\theta \qquad (1)$
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| Force of moment on an electric Dipole |
The torque (i.e moment of force) on an electric dipole in a uniform electric field
$ \tau=p.E\:sin\theta$
Now substitute the value of $\tau$ in equation $(1)$. So work done
$ dW=p.E\:sin\theta.d\theta$
If the dipole rotate the angle from $\theta_{1}$ to angle $\theta_{2}$ then workdone
$\int_{W_{1}}^{W_{2}}dW=p.E\int_{\theta_{1}}^{\theta_{2}}sin\theta \: d\theta$
$ W_{2}-W_{1}=p.E\left[-cos\theta \right]_{\theta_{1}}^{\theta_{2}}$
$ \Delta W= p.E \left( cos\theta_{1}-cos\theta_{2} \right)\qquad\qquad (2)$
This work is stored in the form of the electric potential energy of an electric dipole in the electric field. So
$U=\Delta W$
$U=p.E \left( cos\theta_{1}-cos\theta_{2} \right)$
If the electric dipole rotates from $0^{\circ}$ (when the direction of electric dipole moment $p$ is aligned in the direction of the electric field $E$) to an angle $\theta$ in the electric field i.e $\theta_{1}=0^{\circ}$ and $\theta_{2}=\theta$ then the electric potential energy of dipole in a uniform electric field
$U=p.E (1-cos\theta)$
Case-(I) If $\theta=0^{\circ}$ i.e It is stable equilibrium position then
$U_{min}=0$
Case-(II) If $\theta=90^{\circ}$ i.e Position of zero energy then
$U=pE$
Case-(III) If $\theta=180^{\circ}$ i.e It is unstable equilibrium position then
$U_{max}=2pE$
