Question

A molecule with a dipole moment p is placed in the electric field of strength E. Initially the dipole is aligned parallel to the field. If the dipole is to be rotated to be anti-parallel to the field, the work required to be done by an external agency is
A. \[ - 2pE\]
B. \[ - pE\]
C. \[pE\]
D. \[2pE\]

Answer 1

Hint:A dipole is the combination of two charges of equal in magnitude and opposite in nature. The work done by the external agency is stored as the potential energy of the system. So by finding the change in electric potential energy of the dipole, we can find the work done by the external agency.

Formula used:
\[U = - pE\cos \theta \]
where U is the electric potential energy stored in the dipole system, p is the dipole moment, E is the electric field and \[\theta \] is the angle between the dipole moment vector and the electric field vector.
\[W = \Delta U\]
where W is the work done by the external agency and \[\Delta U\] is the change in potential energy.

Complete step by step solution:

Image: Dipole in electric field
Initially, the dipole moment is parallel to the external electric field vector. When two vectors are parallel to each other, then the angle between the vectors is 0°. So, the initial potential energy of the dipole system is,
\[{U_i} = - pE\cos {\theta _i}\]
\[\Rightarrow {U_i} = - pE\cos 0^\circ \]
\[\Rightarrow {U_i} = - pE\]
Finally, the dipole moment is anti-parallel to the external electric field vector. When two vectors are antiparallel to each other, then the angle between the vectors is 180°.

So, the final potential energy of the dipole system is,
\[{U_f} = - pE\cos {\theta _f}\]
\[\Rightarrow {U_f} = - pE\cos 180^\circ \]
\[\Rightarrow {U_i} = pE\]
The work done by the external agency is the change in electric potential energy of the dipole system.

So, the work done can be calculated as,
\[W = \Delta U\]
\[\Rightarrow W = {U_f} - {U_i}\]
\[\Rightarrow W = pE - \left( { - pE} \right)\]
\[\therefore W = 2pE\]
Hence, the work done by the external agency to rotate the given dipole from the initial position of being parallel to the electric field to the final position of being anti-parallel to the electric field.

Therefore, the correct option is D.

Note: The work done by the restoring force is negative to the change in potential energy of the system and the work done by the external force is equal to the change in potential energy of the system.