Question

A dipole is placed in x-y plane such that charges \(+q\) and \(-q\) are located at \(x=a\) and \(x=b\) respectively. There exists an electric field \[ \vec E = 2\hat{i}\,\frac{N}{C} \] in the region. Calculate the force \(\vec F\) and torque \(\vec \tau\) experienced by the dipole.

Hint:

For a dipole placed in a uniform electric field: \[ \vec F = 0 \] and \[ \vec\tau = \vec p \times \vec E. \] If \(\vec p\) is parallel or antiparallel to \(\vec E\), then \[ \tau = 0. \]

Answer 1

Concept: When an electric dipole is placed in a uniform electric field,
• the net force on the dipole is zero,
• a torque acts on the dipole if its dipole moment is not parallel to the field. The force on a charge in an electric field is \[ \vec F=q\vec E. \] The torque on a dipole is \[ \vec\tau=\vec p\times\vec E. \]

Step 1: Determine the dipole moment. The charges are situated along the \(x\)-axis. Therefore the dipole axis lies along the \(x\)-direction. Hence \[ \vec p = p\,\hat{i}. \] where \[ p=q|a-b|. \]

Step 2: Calculate the net force on the dipole. Force on charge \(+q\) is \[ \vec F_1 = q\vec E. \] Since \[ \vec E=2\hat{i}, \] \[ \vec F_1 = 2q\hat{i}. \] Force on charge \(-q\) is \[ \vec F_2 = (-q)\vec E. \] Therefore, \[ \vec F_2 = -2q\hat{i}. \] Hence total force is \[ \vec F = \vec F_1+\vec F_2. \] \[ \vec F = 2q\hat{i}-2q\hat{i}. \] \[ \boxed{ \vec F=0 } \] Thus a uniform electric field exerts no net translational force on the dipole.

Step 3: Calculate the torque on the dipole. The torque is \[ \vec\tau = \vec p\times\vec E. \] Since both \(\vec p\) and \(\vec E\) are along the \(x\)-axis, \[ \vec p = p\hat{i}, \qquad \vec E = 2\hat{i}. \] Therefore, \[ \vec\tau = (p\hat{i})\times(2\hat{i}). \] Since \[ \hat{i}\times\hat{i}=0, \] we get \[ \boxed{ \vec\tau=0 } \] Final Answer: \[ \boxed{\vec F=0} \] and \[ \boxed{\vec\tau=0} \]