Courses
Courses for Kids
Free study material
Offline Centres
More
Store

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

Last updated date: 23rd May 2024
Total views: 45.9k
Views today: 1.45k
Verified
45.9k+ views
Hint: 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) $\dfrac{{3{p_1}{p_2}}}{{4\pi {\varepsilon _ \circ }{x^4}}}$

$E = \dfrac{1}{{4\pi {\varepsilon _ \circ }}}\dfrac{p}{{{r^3}}}\sqrt {3{{\cos }^2}\theta + 1}$ , $\theta$ =angle between the distance vector and dipole.
$V = \dfrac{{p\cos \theta }}{{4\pi {\varepsilon _ \circ }{r^2}}}$