Question

Derive the expression for torque acting on a dipole $\tau =P.E\sin \theta$

Answer 1

Consider a dipole with charges $+q$ and $-q$ farming a dipole since they are a distance $\left(d\right)$ from each other and makes an angle $\theta$ with electric field.

The force on the charge is,

i) ${\overrightarrow{F}}_{-}=-qE$

ii) ${\overrightarrow{F}}_{+}=+qE$

And the parallel component cancel out each other because of equal magnitude and opposite direction.

$\therefore$ The perpendicular component adds, which results into,

$\overrightarrow{F}={\overrightarrow{F}}_0\sin \theta +{\overrightarrow{F}}_{+}\sin \theta$

$=2qE\sin \theta$

and we know torque, $\overrightarrow{C}=F\times$ distance b/w force

$=(2qE\sin \theta \times d)$

and dipole moment, $p=2qd$

$\Rightarrow \overrightarrow{C}=P.E\sin \theta$ {Hence Proved}