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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
$pE\cos\theta$, $-pE\sin\theta$
B
$pE\sin\theta$, $-pE\cos\theta$
C
$pE\sin\theta$, $-2pE\cos\theta$
D
$pE\sin\theta$, $2pE\cos\theta$
Detailed Solution
Torque: $\vec\tau = \vec p\times\vec E$, so $\tau = pE\sin\theta$
Potential energy = work done in rotating the dipole from $90^\circ$ to $\theta$: $U = \int_{90^\circ}^{\theta}pE\sin\theta\,d\theta = -pE\cos\theta$
i.e. $U = -\vec p\cdot\vec E = -pE\cos\theta$
So the torque is $pE\sin\theta$ and the potential energy is $-pE\cos\theta$.
Potential energy = work done in rotating the dipole from $90^\circ$ to $\theta$: $U = \int_{90^\circ}^{\theta}pE\sin\theta\,d\theta = -pE\cos\theta$
i.e. $U = -\vec p\cdot\vec E = -pE\cos\theta$
So the torque is $pE\sin\theta$ and the potential energy is $-pE\cos\theta$.