Question

The torque acting on a dipole of momentum p​ in an electric field \[\overrightarrow E \] :
A. \[\overrightarrow p \times \overrightarrow E \]
B. \[\overrightarrow p \cdot \overrightarrow E \]
C. zero
D. \[\overrightarrow E \times \overrightarrow p \]

Answer 1

Hint: Torque can be measured as the force that causes a body to rotate about an axis of rotation. Dipole moment can be defined as the product of charge and the dipole length. When a dipole is placed in a uniform electric field then it will experience a torque which aligns to the direction of the electric field having zero net force. The torque acting on the dipole is calculated as the cross product of dipole moment \[(\overrightarrow p )\] and the electric field \[(\overrightarrow E )\].

Formula used:
Force acting on dipole due to electric field:
\[\overrightarrow F = q\overrightarrow E \]
Where q is the charge and \[\overrightarrow E \] is the electric field.
Torque \[\tau \] on a dipole in an electric field is,
\[\tau = F \times d = Fd\sin \theta \]
Where F represents the force act on the dipole, d represents the length of the moment arm and \[\theta \] is the angle between the force and the moment arm.

Complete step by step solution:
The force acting on the dipole due to the electric field is:
\[\overrightarrow F = q\overrightarrow E \]
According torque on a dipole in an electric field is:
\[\tau = F \times d\]
\[\Rightarrow \tau = Fd\sin \theta \]
Now using the value of force in above equation, we get
\[\tau = q\overrightarrow E d\sin \theta \]
\[\Rightarrow \tau = qd\overrightarrow E \sin \theta \]

As we know that dipole moment,
\[\overrightarrow p = qd\]
Substituting this value in the above equation,
\[\begin{array}{l}\tau = qd\overrightarrow E \sin \theta \\ \Rightarrow \tau{\rm{ = }}\overrightarrow p \overrightarrow E \sin \theta \\ \Rightarrow \tau {\rm{ = }}\overrightarrow p \times \overrightarrow E \end{array}\]
Therefore the torque acting on a dipole of momentum p​ in an electric field \[\overrightarrow E \] is \[\overrightarrow p \times \overrightarrow E \].

Hence option A is the correct answer.

Note: Torque, dipole moment, force and distance between the dipoles all are vector quantities because all have magnitude and direction. Torque can be measured as the cross product of dipole moments and the force acting on the dipole. A dipole consists of two charges having equal magnitude but opposite charges.