Question

An electric dipole of dipole moment p is placed at the origin of the coordinate system along the z-axis. The amount of work required to move a charge 'q' from the point (a,0,0) to the point (0,0,a) is

• A. \(\frac{pq}{4\pi\epsilon_0a}\)
• B. 0
• C. \(-\frac{pq}{4\pi\epsilon_0a^2}\)
• D. \(\frac{pq}{4\pi\epsilon_0a^2}\)

Answer 1

The Correct Option is D

Given:
An electric dipole with dipole moment $\vec{p}$ produces an electric field at a point on the x-axis, making an angle $\theta$ with the axis.

Using the identity: 
$\tan \alpha = \frac{1}{2} \tan \theta$, and $\theta = 60^\circ + \alpha$

Electric Field Components:
Resolving the electric field $\vec{E}$ into components:
$E \cos \alpha = \frac{1}{4\pi\epsilon_0} \cdot \frac{2p \cos 60^\circ}{r^3}$ ⟶ (1)
$E \sin \alpha = \frac{1}{4\pi\epsilon_0} \cdot \frac{p \sin 60^\circ}{r^3}$ ⟶ (2)

Dividing (2) by (1):
$\tan \alpha = \frac{1}{2} \tan \theta$
Also given: $\tan \alpha = \tan(\theta + \alpha)$

After simplification and solving, the resulting expression for potential energy is:
Answer: $\frac{pq}{4\pi\epsilon_0 a^2}$

Correct Option: (D)