Question

The force of interaction of two dipoles, if the two dipoles are parallel to each other and placed at distance x apart.

1. $\dfrac{{3{p_1}{p_2}}}{{4\pi {\varepsilon _ \circ }{x^4}}}$
2. $\dfrac{{{p_1}{p_2}}}{{4\pi {\varepsilon _ \circ }{x^4}}}$
3. $\dfrac{{{p_1}{p_2}}}{{4\pi {\varepsilon _ \circ }{x^4}}}$
4. $\dfrac{{{p_1}{p_2}}}{{3\pi {\varepsilon _ \circ }{x^4}}}$

Answer 1

First, we will need to find the electrostatic field of dipole ${p_2}$ at ${p_1}$ . Then we will find the potential energy of two dipoles. In the final step we will differentiate the potential energy to get the Force of interaction between two dipoles.

Complete step-by-step Solution
A dipole is separation of two opposite charges and it is quantified by electric dipole moment and is denoted by p.
As we know electric field of dipole along perpendicular bisector of the axis,
$\overrightarrow E = - \dfrac{{\overrightarrow p }}{{4\pi {\varepsilon _ \circ }{r^3}}}$ , where r= distance
${\varepsilon _ \circ }$ = permittivity of free space
${E_{21}}$ is the field due to dipole ${p_1}$ at dipole ${p_2}$
${E_{21}} = \dfrac{{{p_1}}}{{4\pi {\varepsilon _ \circ }{x^3}}}$
Potential energy of dipole system
$U = - \overrightarrow {{p_2}} .\overrightarrow {{E_{21}}}$
$U = - {p_2}\dfrac{{{p_1}}}{{4\pi {\varepsilon _ \circ }{x^3}}}\cos (\pi )$
Angle between the dipole and electric field is 180 degrees.
$U = \dfrac{{{p_1}{p_2}}}{{4\pi {\varepsilon _ \circ }{x^3}}}$
Now, to find the force
$F = - \dfrac{{dU}}{{dx}} = \dfrac{3}{{4\pi {\varepsilon _ \circ }}}\dfrac{{{p_1}{p_2}}}{{{x^4}}}$
F is positive, so it is a repulsive force.
Option (1) $\dfrac{{3{p_1}{p_2}}}{{4\pi {\varepsilon _ \circ }{x^4}}}$

Additional Information
Electric field due to dipole at a general point
$E = \dfrac{1}{{4\pi {\varepsilon _ \circ }}}\dfrac{p}{{{r^3}}}\sqrt {3{{\cos }^2}\theta + 1}$ , $\theta$ =angle between the distance vector and dipole.
Potential due to dipole at a general point
$V = \dfrac{{p\cos \theta }}{{4\pi {\varepsilon _ \circ }{r^2}}}$

Note
1. You need to keep in mind the direction of the electric field and dipole.
2. While using the formula of potential energy of dipole, you need to find the angle between field and dipole otherwise you will get the wrong force direction.
3. While finding electric fields, Approximation is made that the length of the dipole is negligible as compared to the distance of the point from the dipole.