Derive the equation for couple acting on a electric dipole in uniform electric field.

Consider an electric dipole consisting charges +q and -q and of length 2a placed in uniform electric field $\overrightarrow{\mathrm{E}}$ making an angle $\mathrm{\theta }$ with it.

Its dipole moment P = (q) (2a)
There is a force qE on charge +q and -qE on charge -q. The net force on dipole is zero. But the forces acts at different points of the dipole. So these forces form a couple. Magnitude of Torque = Either force × perpendicular of distance between two forces
$\mathrm{\tau }=\left(\mathrm{qE}\right)\times 2\mathrm{asin}\mathrm{\theta };\mathrm{\tau }=(\mathrm{q}\times 2\mathrm{a})\mathrm{Esin}\mathrm{\theta }$
$\mathrm{\tau }=\mathrm{PEsin}\mathrm{\theta }\text{.The direction of }\overrightarrow{\mathrm{\tau }}\text{ is perpendicular to both }\overrightarrow{\mathrm{P}}\text{ and }\overrightarrow{\mathrm{E}};\overrightarrow{\mathrm{\tau }}=\overrightarrow{\mathrm{P}}\times \overrightarrow{\mathrm{E}}$