Question

Define electric dipole moment Derive an expression for total work done in rotating the dipole through an angle in a uniform electric field.

Answer 1

Dipole moment ($\overrightarrow{P}$) is a measure of the strength of electric dipole. It is a vector quantity whose magnitude is equal to product of the magnitude of either charge and the distance between them.

Expression of total work done

Potential energy of dipole is the energy possessed by the dipole by virtue of its particular position in the electric field.
Suppose an electric dipole of moment $\overrightarrow{P}$ is oriented at an angle $\theta$ with the direction of uniform external electric field $\overrightarrow{E}$
we know, the torque $\overrightarrow{\tau }$ acting on the dipole is
$\tau =pE\sin \left(\theta \right)$
It tries to rotate the dipole .
Small amount of work done in rotating the dipole through a small angle $d\theta$ against the torque is
$$\begin{gathered} dW=\tau d\theta \\ =pE\sin \left(\theta \right)d\theta \end{gathered}$$
Total work done in rotating the dipole from orientation ${\theta }_{1}to{\theta }_2$ is

$$\begin{gathered} W={\int }_{{\theta }_1}^{{\theta }_2}pE\sin \left(\theta \right)d\theta \\ =pE{\left[-\cos \left(\theta \right)\right]}_{{\theta }_1}^{{\theta }_2} \\ W=-pE\left[\cos \left({\theta }_2\right)-\cos \left({\theta }_1\right)\right] \end{gathered}$$