Question

derive expression for work done in turning a dipole in a uniform electric field.

Answer 1

To derive an expression for workdone in turning a dipole in a uniform electric field:
If E= electric field
F= force
q= point charges,
Thetha= angle with dipole seperation vector, therefore

F⃗ ⋅ds⃗ =(qEsinθ)(rdθ).
FdsqEsinθrdθ
Therefore we have that the work done by the external field in rotating the electric dipole through some angle is
W=∫θfθiqEsinθrdθ+∫θfθiqEsinθrdθ.
WθiθfqEsinθrdθθiθfqEsinθrdθ
Therefore
W=2∫θfθiqrEsinθdθ.
W2θiθfqrEsinθdθ
Since r=ℓ/2rℓ2 we find that the work done on the dipole by the torque provided by E⃗ E is
W=∫θfθiqℓEsinθdθ=∫θfθipEsinθdθ
WθiθfqℓEsinθdθθiθfpEsinθdθ
where the electric dipole is given by p⃗ =qℓ⃗ pqℓ and has magnitude p=qℓ.pqℓ

We can express the total work done by the torque in rotating the torque as
W=∫θfθiτzdθ,
Wθiθfτzdθ
where τz=pEsinθτzpEsinθ.

If, in the above derivation, I assumed the opposite direction for the field, i.e. E⃗ E is replaced with −E⃗ E, a clockwise torque would occur and the work done on the electric dipole would be
W=−∫θfθipEsinθdθ=∫θfθiτzdθ,
WθiθfpEsinθdθθiθfτzdθ
where τz=−pEsinθ