Answer

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**Hint**To solve this question, we have to use the formula for the magnetic field at the axial position due to a magnet. The parameters of the magnets are given in the question, which can be used to get the final answer.

**Formula Used:**The formulae used to solve this question are given by

$ \Rightarrow {B_A} = \dfrac{{{\mu _0}\left( {2m} \right)}}{{4\pi {r^3}}} $

Here $ m $ is the magnetic dipole moment, $ r $ is the distance from the centre of the magnet, and $ {\mu _0} $ is the magnetic permeability in vacuum.

$ \Rightarrow F = - \dfrac{{dU}}{{dr}} $

Here $ U $ is the potential energy corresponding to the conservative force $ F $, and $ r $ is the distance.

**Complete step by step answer**

As both the dipoles are placed axially, they will exert force on each other due to the axial magnetic field.

We know that the axial magnetic field is given by

$ \Rightarrow {B_A} = \dfrac{{{\mu _0}\left( {2m} \right)}}{{4\pi {r^3}}} $

Here, according to the question, the dipoles are of length $ d $ and their mid points are separated by a distance $ x $. But it is also given that \[\left( {x > > d} \right)\]. So the length of the dipoles can be neglected and approximately we take $ r = x $. So, we have the magnetic field

$ \Rightarrow {B_A} = \dfrac{{{\mu _0}\left( {2m} \right)}}{{4\pi {x^3}}} $

$ \Rightarrow {B_A} = \dfrac{{{\mu _0}m}}{{2\pi {x^3}}} $ ……...(i)

Now, the magnetic potential energy is given by

$ \Rightarrow U = - mB $

Substituting (i), we get

$ \Rightarrow U = - \dfrac{{{\mu _0}{m^2}}}{{2\pi {x^3}}} $ ……...(ii)

Also, as the magnetic force is conservative in nature, so it is related to the potential energy by $ F = - \dfrac{{dU}}{{dr}} $

Here $ r = x $

$ \therefore F = - \dfrac{{dU}}{{dx}} $

Substituting $ U $ from (ii), we have

$ \Rightarrow F = - \dfrac{{d\left[ { - \dfrac{{{\mu _0}{m^2}}}{{2\pi {x^3}}}} \right]}}{{dx}} $

$ \Rightarrow F = - \dfrac{{3{\mu _0}{m^2}}}{{2\pi {x^4}}} $

As we can see from the final expression of the force that it is proportional to $ \dfrac{1}{{{x^4}}} $ or $ {x^{ - 4}} $ .

Comparing $ {x^{ - n}} $, we get $ n = 4 $ .

**Hence, the correct answer is option D.**

**Note**

If we do not remember the formula for the equatorial magnetic field, then we can take the help of the electrostatic analogy. We know that the electric field due to a dipole at its equatorial position is given by $ E = \dfrac{1}{{4\pi {\varepsilon _0}}}\dfrac{{2p}}{{{r^3}}} $. Replacing the electric field $ E $ with the magnetic field $ B $, the electric dipole moment $ p $ with the magnetic dipole moment $ m $ and the constant $ \dfrac{1}{{4\pi {\varepsilon _0}}} $ with the constant $ \dfrac{{{\mu _0}}}{{4\pi }} $ we can get the corresponding expression for the axial magnetic field.

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