Derive an expression for the torque experienced by a dipole due to a uniform electric field.

Explanation

(i) Consider an electric dipole of dipole moment `vec(p)` placed in a uniform electric field `vec(E)`.
(ii) The charge +q will experience a force `qvec(E)` in the direction of the field and
(iii) charge -q will experience a force `-qvec(E)` in a direction opposite to the field.
(iv) Since the external field `vec(E)` is uniform, the total force acting on the dipole is zero.
(v) These two forces acting at different points will constitute a couple and the dipole experience a torque This torque tends to rotate the dipole.

(vi) The total torque on the dipole about the point O
`vec(tau)=Ovec(A)xx(-qvec(E))+Ovec(B)xxqvec(E)" ".......(1)`
Using right-hand corkscrew rule it is found that total torque is perpendicular to the plane of the paper and is directed into it.
The magnitude of the total torque
`tau=|Ovec(A)||(-qvec(E))|sintheta+|vec(OB)||vec(E)|sintheta`
`tau=qE.2asintheta" "......(2)`
(vii) Where `theta` is the anlge made by `vec(p)` with `vec(E)` since p=2aq, the torque is written in terms of the vector product as
`vec(tau)=vec(p)xxvec(E)" ".......(3)`
`tau=pEsintheta`
(a) When `theta=0`
`tau=pEsintheta`
=0
The dipole moment `vec(p)` of the dipole is parallel to electric field `vec(E)` no torque acts on it
(b) `theta=90,tau_(max)=pE`
`vec(E)" is "botr` to E
The magnitude of this torque is `tau=pEsintheta` and is maximum when `theta=90^(@)`. This torque tends to rotate the dipole and align it with the electric field `vec(E)` once `vec(p)` is aligned with `vec(E)`, the total torque on the dipole becomes zero.
(c) `theta=180,tau=0`
`vec(p)` is antiparallel to `vec(E)` no torque acts. i.e., the electric dipole is in unstable equilibrum.