Question

Answer the following Questions.
Derive an expression for the work done in rotating a dipole from the angle ${\theta }_0$ to ${\theta }_1$ in a uniform electric field $E$.

Answer 1

Let a dipole is placed in a uniform field of intensity $E$ and dipole vector makes an angle $\theta$ with field direction.

Positive charge experiences a force qE in the field direction. Negative charge experiences a force $-qE$ in opposite direction.

Though net force is zero,two forces separated by a distance making a couple.

Torque $\tau$ of the couple is product of force and perpendicular distance between them.

Torque $\tau =qE\times 2a\times sin\theta$ --(1)

but we know that $q\times 2a$ is dipole moment $p$, hence torque $\tau =p\times E\times sin\theta$ --(2)

Since $\tau ,pandE$ are vector quantities, eqn.(2) is expressed as cross product or vector poroduct, $\tau =p\times E$

Workdone W to turn the dipole from angular position ${\theta }_0$ to ${\theta }_1$ is given by,

$W={\int }_{{\theta }_0}^{{\theta }_1}\tau d\theta ={\int }_{{\theta }_0}^{{\theta }_1}pEsin\theta d\theta =pE(cos{\theta }_0-cos{\theta }_1)$

By convention, ${\theta }_0$ is taken as $\pi /2$ (reference point, point of minimum potential energy ), hence $W=-pEcos\theta$ --(3)

eqn(3) can be written in scalar product or dot product of vectors as $W=-p\cdot E$

Dipole is stable, when it is aligned in the direction of electric field