(i)Obtain the expression for the torque `vecτ` experienced by an electric dipole of dipole moment `vecP` in a uniform electric field, `vecE` .
(ii) What will happen if the field were not uniform?
Solution
(i) Consider an electric dipole consisting of charges −q and +q and of length 2a placed in a uniform electric field making an angle θ with electric field.
Force on charge −q at `A=-qvecE`(opposite to`vecE`)
Force on charge +q at `B=qvecE`(along`vecE`)
Electric dipole is under the action of two equal and unlike parallel forces, which give rise to a torque on the dipole.
τ = Force × Perpendicular distance between the two forces
τ = qE (AN) = qE (2a sin θ)
τ = q(2a) E sinθ
τ = pE sinθ
`therefore vectau=vecPxxvecE`
(ii) If the electric field in not uniform then both the charges of the dipole will experience a different force (not equal and opposite) at slightly different positions in the field and hence a net force acts on the dipole in a nonuniform electric field. Also, a net torque acts on the dipole which depends on the location of the dipole in the nonuniform electric field.
