Question

An electric dipole is placed in uniform external electric field E. Show that the torque on the dipole is given by τ = p x E, where p is the dipole moment of the dipole. What is the net force experienced by the dipole? Identify two pairs of perpendicular vectors in the expression.

Answer 1

Consider a dipole with charges +q and –q with a distance d away from each other. The dipole is placed in a uniform electric field E such that the axis of the dipole forms an angle θ with the electric field.The force on the charges is

--> ----->

F+ = +qE

---> ------->

F− = −qE

The components of force perpendicular to the dipole are:

F+⊥=+qEsinθ

F−⊥=−qEsinθ

Since the force magnitudes are equal and are separated by a distance d, the torque on the dipole is given by:Torque(τ)=Force×Distanceτ=(qE)dsinθ

Now, the dipole moment is given byp=qd

The direction of the dipole moment is from the positive to the negative charge.

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

Thus, τ=pEsinθ

----> ---->

τ = p × E