Q.Derive the expression $\tau = pE\sin\theta$ for the torque on an electric dipole of moment $p$ placed in a uniform external electric field $E$ making angle $\theta$ with the field, and write the corresponding vector relation $\vec{\tau}=\vec{p}\times\vec{E}$.

In a uniform field $\vec{E}$, the force on $+q$ is $qE$ (along $\vec{E}$) and on $-q$ is $qE$ (opposite to $\vec{E}$) -- equal and opposite, so net force is zero, but since they act at different points (the two ends of the dipole), they form a couple.

If $\vec{p}$ makes angle $\theta$ with $\vec{E}$, the perpendicular distance between the two parallel lines of action of the two forces is $2a\sin\theta$ (the component of the charge separation perpendicular to $\vec{E}$). The torque of a couple is (force)$\times$(perpendicular distance):

$\tau = (qE)(2a\sin\theta) = (q\cdot2a)E\sin\theta = pE\sin\theta$ …

Identify the two forces as a couple, find the perpendicular distance between their lines of action ($2a\sin\theta$), then a …

• Using the full separation $2a$ instead of the perpendicular distance $2a\sin\theta$, which would incorrectly give $\tau=pE$ for all angles. …