The magnetic potential V at a certain point on the axial line of a bar magnet of dipole moment M. What is the magnetic potential V due to a bar magnet of magnetic dipole moment $\dfrac{M}{4}$ at the same point?
(A) $4V$
(B) $2V$
(C) $\dfrac{V}{2}$
(D) $\dfrac{V}{4}$

Hint: To find the answer of this question, we’ll need to know the relation between the magnetic dipole and magnetic potential. There are two types of position, one is axial and the other is equatorial, so, we need to remember the relation between potential on the axial line and the magnetic dipole moment.

Complete step by step solution:
Let the point on the axial line of the bar magnet be P. The magnetic potential at point P of a bar magnet with dipole moment M is V. And is given by following formula:
$V = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{M}{{{r^2}}}$ (equation 1)
Where, r is the distance of the point P from the bar magnet.

Now, when the magnetic dipole moment of the bar magnet becomes $\dfrac{M}{4}$ . Then the magnetic potential formula will be as follows:
${V_1} = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{\dfrac{M}{4}}}{{{r^2}}}$ (equation 2)

Now, equating the equation 1 and 2 for value of M, we get;
${V_1} = \dfrac{V}{4}$

Hence the correct answer is Option(D).

Note:
Find the magnetic potential at the given point on the axial line of the bar magnet in both the cases when the magnetic dipole moment of the bar magnet is $M\,and\,\dfrac{M}{4}$ and finally equate both the equation and get the final answer. Here the exact value of the potential was not given hence we get the final potential in terms of initial potential.