Question
An electric dipole of moment $$p$$ is placed in an electric field of intensity $$E.$$ The dipole acquires a position such that the axis of the dipole makes an angle $$\theta $$ with the direction of the field. Assuming that the potential energy of the dipole to be zero when $$\theta = {90^ \circ },$$ the torque and the potential energy of the dipole will respectively be
A.
$$pE\sin \theta , - pE\cos \theta $$
B.
$$pE\sin \theta , - 2pE\cos \theta $$
C.
$$pE\sin \theta ,2pE\cos \theta $$
D.
$$pE\cos \theta , - pE\sin \theta $$
Answer :
$$pE\sin \theta , - pE\cos \theta $$
Solution :
Here, torque, $$\tau = pE\sin \theta $$
Potential energy of the dipole,
$$\eqalign{ & U = \int {\tau d\theta } = \int_{\frac{\pi }{2}}^0 {pE\sin \theta d\theta = - pE\left[ {\cos \theta } \right]_{\frac{\pi }{2}}^0} \cr & = - pE\cos \theta \cr} $$
Here, torque, $$\tau = pE\sin \theta $$
Potential energy of the dipole,
$$\eqalign{ & U = \int {\tau d\theta } = \int_{\frac{\pi }{2}}^0 {pE\sin \theta d\theta = - pE\left[ {\cos \theta } \right]_{\frac{\pi }{2}}^0} \cr & = - pE\cos \theta \cr} $$
