Question

derive necessary formula for couple acting on electric dipole in a electric field​

Answer 1

Consider a dipole with charges +q and –q forming a dipole since they are a distance d away from each other.

Let it be placed in a uniform electric field of strength E such that the axis of the dipole forms an angle θ with the electric field.

force on the charges is

F+−→ = +qE→

F−−→ = −qE→

The components of force perpendicular to the dipole are:

F⊥+ = +qE sinθ

F⊥− = −qE sinθ

Since the force magnitudes are equal and are separated by a distance d, the torque on the dipole is given by:

Torque (τ) = Force × distance seperating forces

τ = d qE sin θ

Since dipole moment is given by

p = qd

And the direction of the dipole moment is from the positive to the negative charge; it can seen from the above equation that the torque is the cross product of the dipole moment and electric field.

Notice that the torque is in the clockwise direction (hence negative)  if the direction of Electric field is positive.

Thus,

τ = −pE sin θ

Or,

τ→ = p→ × E→

|τ→| = |p→ × E→| = pE sin θ