Answer

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) \[\dfrac{{3{p_1}{p_2}}}{{4\pi {\varepsilon _ \circ }{x^4}}}\]

**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.

Recently Updated Pages

To get a maximum current in an external resistance class 1 physics JEE_Main

f a body travels with constant acceleration which of class 1 physics JEE_Main

A hollow sphere of mass M and radius R is rotating class 1 physics JEE_Main

If the beams of electrons and protons move parallel class 1 physics JEE_Main

Two radioactive nuclei P and Q in a given sample decay class 1 physics JEE_Main

silver wire has diameter 04mm and resistivity 16 times class 12 physics JEE_Main

Other Pages

Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main

Explain the construction and working of a GeigerMuller class 12 physics JEE_Main

Which of the following is not a redox reaction A CaCO3 class 11 chemistry JEE_Main

Electric field due to uniformly charged sphere class 12 physics JEE_Main

when an object Is placed at a distance of 60 cm from class 12 physics JEE_Main

Two identical particles each having a charge of 20 class 12 physics JEE_Main