Question

The ratio of magnetic potential due to magnetic dipole at the end to that at the broad side on a position for the same distance from it is:
(A) Zero
(B) $\infty$
(C) 1
(D) 2

Answer 1

Hint: The magnetic dipole is defined as the pair of equal and opposite magnetic charges (monopoles) kept at a certain distance apart. It is analogous to electric dipole, the only difference being that the magnetic monopoles do not exist in nature.

Formula used:
we define the scalar magnetic potential as:
\[\psi (r) = \dfrac{{m.r}}{{4\pi {r^3}}}\]
Where,
m is the magnetic moment and,
r is the distance from the center of the magnetic dipole to the point where potential is to be defined.

Complete step by step answer:
Using electric dipole as an analogy, we define magnetic dipole as the pair of equal and opposite magnetic monopoles situated at a certain distance. In general, like electric monopoles, magnetic monopoles do not exist.
Potential is an imaginary parameter (do not exist in real) which is created to ease the way of solving problems. In magnetism, we generally define a vector potential based on the fact that the magnetic field is created by currents. From a magnetic monopole limit, we can define a scalar potential for situations not involving currents such as permanent magnets, magnetic dipoles, etc. as:
\[\psi (r) = \dfrac{{m.r}}{{4\pi {r^3}}} = \dfrac{{m\cos \theta }}{{4\pi {r^2}}}\]
Where, $\theta $ is the angle between magnetic moment and radial vector r.
So, for axial position:
\[
\theta = 0 \Rightarrow \cos \theta = 1 \\
{\psi _a} = \dfrac{m}{{4\pi {r^2}}} \\
\]
For an equatorial position,
\[
\theta = {90^ \circ } \Rightarrow \cos \theta = 0 \\
{\psi _e} = 0 \\
\]
So, the ratio of magnetic potential at ends to the potential at a broad position is:
\[\dfrac{{{\psi _a}}}{{{\psi _e}}} = \dfrac{{{\psi _a}}}{0} = \infty \]
Therefore, the correct answer is option B.

Note: There are two types of potentials in the case of magnetism. The vector potential is used when we have free currents, i.e. the magnetic field is produced due to a current density. In case of permanent magnets/dipoles we use the scalar potential. It is used when a magnetic field is produced by magnets and not currents i.e. zero current density.