Question

A dipole having charges ‘±q’ and dipole length ‘2d’ is placed in a uniform external electric field at an angle ‘θ’. Deduce the expression of work done in rotating the charge from an angle θ1 = 90 to angle θ2 = 180

Answer 1

F⃗ ⋅ds⃗ =(qEsinθ)(rdθ).

Therefore we have that the work done by the external field in rotating the electric dipole through some angle is

W=∫θfθiqEsinθrdθ+∫θfθiqEsinθrdθ.

Therefore

W=2∫θfθiqrEsinθdθ.

Since r=ℓ/2 we find that the work done on the dipole by the torque provided by E⃗ is

W=∫θfθiqℓEsinθdθ=∫θfθipEsinθdθ

where the electric dipole is given by p⃗ =qℓ⃗ and has magnitude p=qℓ.

We can express the total work done by the torque in rotating the dipole as

W=∫θfθiτzdθ,

where τz=pEsinθ.

If, in the above derivation, I assumed the opposite direction for the field, i.e. E⃗ is replaced with −E⃗ , a clockwise torque would occur and the work done on the electric dipole would be

W=−∫θfθipEsinθdθ=∫θfθiτzdθ,

where τz=−pEsinθ.