Question

an electric dipole is placed in a uniform electric field E shaw that torque= p ×E and potential energy U = -p×E

Answer 1

Torque:

Torque = Force * perpendicular dist.

here, Forces on dipole form couple ( due to presence of +ve and -ve charges ).

let dist. b/w the charges = 2a
amount of each charge = q
dipole moment = p = 2qa (-q ---> +q)

Force = qE ( along the field on +ve charge and opposite to field in -ve charge)

angle b/w dipole and E = @

Torque = qE * 2a * sin(@) = pE sin(@)

so, in vector form,

Torque = p x E (cross product)