Question

Deduce the formula of torque on an electric dipole placed in a uniform electric field. 
 

Hint:

The torque on an electric dipole in a uniform electric field is given by \( T = pE \sin \theta \), where \( p \) is the dipole moment, \( E \) is the electric field, and \( \theta \) is the angle between \( \mathbf{p} \) and \( \mathbf{E} \).

Answer 1

Consider an electric dipole consisting of two charges \( +q \) and \( -q \), separated by a distance \( d \). The dipole moment \( \mathbf{p} \) is given by: \[ \mathbf{p} = q \cdot \mathbf{d}, \] where \( \mathbf{d} \) is the vector pointing from the negative to the positive charge. When the dipole is placed in a uniform electric field \( \mathbf{E} \), it experiences a torque that tends to align the dipole with the electric field. The torque \( \mathbf{T} \) is given by: \[ \mathbf{T} = \mathbf{p} \times \mathbf{E}, \] where the cross product ensures that the torque is perpendicular to both the dipole moment and the electric field. The magnitude of the torque is: \[ T = pE \sin \theta, \] where \( \theta \) is the angle between the dipole moment \( \mathbf{p} \) and the electric field \( \mathbf{E} \).