Question

A dipole of dipole moment ‘p’ is placed I a non-uniform electric field along the x-axis. Electric field is increasing at the rate of 1 V/m. The force on dipole is:
A. 0
B. 2p
C. p/2
D. p

Answer 1

Force is defined as negative of change of potential energy with displacement and potential energy for an electric dipole is negative of dot product dipole moment and electric field.
Formula used:
F=$- \dfrac{{dU}}{{dx}}$ and U=$- \overrightarrow p .\overrightarrow E$
Where, F=force on dipole
U=potential energy of dipole
E=electric field of the surrounding.

Complete step by step answer:
We know that force is given as the negative gradient of potential energy. F=$- \dfrac{{dU}}{{dx}}$
Potential energy for an electric dipole is U=$- \overrightarrow p .\overrightarrow E$
So, we get the force as,
F=$\dfrac{{d(\overrightarrow p .\overrightarrow E )}}{{dx}}$
=$\dfrac{{d(pE\cos 0^\circ )}}{{dx}}$
As cos$0^\circ$=1 and p is constant, we get
F=p$\dfrac{{dE}}{{dx}}$
Given, rate of increase of electric field=1V/m.
Therefore, $\dfrac{{dE}}{{dx}}$=1
So,
F=p×1
=p

So, the correct answer is “Option D”.

Additional Information:
Dipole is a pair of equal and opposite charged or magnetised poles separated by a small distance and dipole moment is the product of either of the charge and the length of the dipole. In a uniform electric field the net force on an electric dipole is zero. But if the dipole is not aligned with the electric field, then a torque acts on the dipole trying to align the dipole with the electric field. The work done by this torque is stored in the form of potential energy, U=-pEcosθ.

Note:
Here the dipole is aligned in the direction of the electric field, therefore $\theta = 0^\circ$. This angle may vary in other questions and should be checked carefully. The direction of dipole is from negative to positive charge.