Derive a expression for the torque acting on a electric dipole in uniform electric field. Under what condition, the torque will be maximum?

Explanation

Consider an electric dipole placed in a uniform electric field `vecE` such that dipole moment `vecP` of the electric, dipole makes an angle`theta` with electric field `vecE`.
Consider two charges + q and constituting a dipole experiences equal and opposite forces `vecF= qvecE and vecF=-qvecE` respectively due to electric field.
`:." "`The net force acting on dipole is
`vecF_("net")=vecqE-qvecE=0`
Two equal and opposite forces acting on dipole constitute a couple. This couple tends to rotate the dipole in clockwise direction.
`:." "`Torque acting on the dipole `(pi) = "moment of Couple" = "Magnitude of either force X perpendicular distance between two forces"`
`pi=qE xx AC" "...(1)`
In `DeltaABC, (AC)/(AB)=sin theta`
`:." "AC=AB sin theta`
Since`" "AB=2a`
`:." AC-2 a sin theta" "...(2)`
From (1) and (2)
`pi=qE xx 2a sin theta`
`pi=(q xx 2 a) E sin theta `
Since`" "q xx 2a =p" "("dipole moment")`
`:." "pi=pE sin theta`
In vector from
`vecpi=vecp xx vecE`
The toeque will be maxiumum force, when `theta=90^(@)`
`:." "pi=p E sin theta`
`pi_(max)=pE sin 90^(@)" "{sin 90^(@)=1}`
`pi_(max)=pE(1)`
`pi_(max)=pE("maximum")`