Question

A short dipole is placed along the x-mds at x =x (Fig. 3.120).

a. Find the force acting on the dipole due to a point charge q placed at the origin.

b. Find the force on the dipole if the dipole is rotated by ${180}^{\circ }$ about the z-axis.

c. Find the force on dipole if the dipole is rotated by ${90}^{\circ }$ anticlockwise about z-axis, i.e., it becomes parallel to the y-axis.

Answer 1

$E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{x^2}$

$U=-pEcos{0}^{\circ }=-\frac{qp}{4\pi {\epsilon }_0{x}^2}$

$=-\frac{\partial U}{\partial x}=\frac{-pq}{2\pi {\epsilon }_0{x}^3}$

Negative sign indicates that force on dipole is toward the positive x-direction or the force is attractive.

b. $U=-pEcos{180}^{\circ }=\frac{qp}{4\pi {\epsilon }_0{x}^2}$

$=-\frac{\partial U}{\partial x}=\frac{-pq}{2\pi {\epsilon }_0{x}^3}$

Positive sign indicates that force on dipole is toward the positive x-direction or the force is repulsive.

C. $E=\frac{1}{4\pi {\epsilon }_0}\frac{p}{x^3}$

Let us first find force on q due to p.

$F=qE=\frac{qp}{4\pi {\epsilon }_0{x}^3}$

Charge q will also apply the same force on dipole but in an opposite direction, so the force on dipole is

$F=\frac{qp}{4\pi {\epsilon }_0{x}^3}$ along $\overrightarrow{p}$ or parallel to y-axis.