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Question: If the dipole moment of a short bar magnet is \[1.25\;{\rm{A}}{{\rm{m}}^{\rm{2}}}\], the magnetic fi...

If the dipole moment of a short bar magnet is 1.25  Am21.25\;{\rm{A}}{{\rm{m}}^{\rm{2}}}, the magnetic field on its axis at a distance of 0.5 m from the centre of the magnet is
A. 1×10−4  NA−1m−11 \times {10^{ - 4}}\;{\rm{N}}{{\rm{A}}^{{\rm{ - 1}}}}{{\rm{m}}^{{\rm{ - 1}}}}
B. 2×10−6  NA−1m−12 \times {10^{ - 6}}\;{\rm{N}}{{\rm{A}}^{{\rm{ - 1}}}}{{\rm{m}}^{{\rm{ - 1}}}}
C. 4×10−4  NA−1m−14 \times {10^{ - 4}}\;{\rm{N}}{{\rm{A}}^{{\rm{ - 1}}}}{{\rm{m}}^{{\rm{ - 1}}}}
D. 6.64×10−4  NA−1m−16.64 \times {10^{ - 4}}\;{\rm{N}}{{\rm{A}}^{{\rm{ - 1}}}}{{\rm{m}}^{{\rm{ - 1}}}}

Explanation

Solution

The above problem can be resolved using the concepts and the fundamental relation for the magnetic field at the axis of the bar magnet, basically at the centre. Moreover, in the given formula, the magnetic moment is given. The distance is also given, along with some constants' values like the magnetic permeability of free space, these values are generally known. On final substituting the corresponding variables' values, one must obtain the desired magnetic field's value at the desired point

Complete step by step answer:
Given:
The dipole moment of the magnet is, M=1.25  Am2M = 1.25\;{\rm{A}}{{\rm{m}}^{\rm{2}}}.
The distance of separation is, d=0.5  md = 0.5\;{\rm{m}}.
The magnetic field at the point on the axis of a bar magnet is given as,
B=μ04π×2Md3B = \dfrac{{{\mu _0}}}{{4\pi }} \times \dfrac{{2M}}{{{d^3}}}
Here, μ04π\dfrac{{{\mu _0}}}{{4\pi }} is the magnetic constant and its value is 10−7{10^{ - 7}} μ0{\mu _0} denotes the magnetic permeability of free space.
Solve by substituting the values as,

B = \dfrac{{{\mu _0}}}{{4\pi }} \times \dfrac{{2M}}{{{d^3}}}\\\ B = \left( {{{10}^{ - 7}}} \right) \times \left( {\dfrac{{2 \times 1.25\;{\rm{A}}{{\rm{m}}^2}}}{{{{\left( {0.5\;{\rm{m}}} \right)}^3}}}} \right)\\\ B = 2 \times {10^{ - 6}}\;{\rm{N}}{{\rm{A}}^{{\rm{ - 1}}}}{{\rm{m}}^{{\rm{ - 1}}}} \end{array}$$ Therefore, the required value of the magnetic field on its axis at a distance of 0.5 m from the centre of the magnet is $$2 \times {10^{ - 6}}\;{\rm{N}}{{\rm{A}}^{{\rm{ - 1}}}}{{\rm{m}}^{{\rm{ - 1}}}}$$ **So, the correct answer is “Option B”.** **Note:** To resolve the given problem, one must understand the fundamentals of the terms like the magnetic field, magnetic moment, and their significance to find their values in a given condition. Moreover, the magnetic field is that particular region where one can sense the effect of the magnetic force and its magnitude if considered in mathematical calculations, and the magnetic moment is a special couple of magnetic forces that can generate the turning moment of the magnetic dipole. Moreover, the applications of such variables had a wide range of modern physics.