Question

The ratio of magnetic field due to short bar magnet on the equatorial line and to the magnetic field on the axis line at the same distance will be
A. ${2^{ - 3}}$
B. ${2^3}$
C. ${2^{\dfrac{1}{3}}}$
D. ${2^{ - 1}}$

Answer 1

Hint: To calculate the ratio of magnetic field due to the short bar magnet on the equatorial line and to the magnetic field on the axis line at the same distance, we need to know the magnetic field at that position due to a bar magnet.

Complete step by step answer:
We know that magnetic field due to short bar magnet on the equatorial line is given by
${B_{equi}} = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{M}{{{r^3}}} - - - - - - - - - (1)$
While, magnetic field due to short bar magnet on the axis line is
${B_{axis}} = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{2M}}{{{r^3}}} - - - - - - - - (2)$
Where, ${\mu _0}$ - permeability of the free space, $M$ - Magnetic dipole moment and $r$ - distance between the point and the center of the bar magnet.

Now, ratio of magnetic field due to short bar magnet on the equatorial line and to the magnetic field on the axis line is
\[\dfrac{{{B_{equi}}}}{{{B_{axis}}}} = \dfrac{{\dfrac{{{\mu _0}}}{{4\pi }}\dfrac{M}{{{r^3}}}}}{{\dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{2M}}{{{r^3}}}}}\]
Here, ${\mu _0}$ is constant and we are given that we have to take the same distance, therefore $r$ is also the same for both cases.
\[\therefore \dfrac{{{B_{equi}}}}{{{B_{axis}}}} = \dfrac{1}{2}\]
We get , \[\dfrac{{{B_{equi}}}}{{{B_{axis}}}} = {2^{ - 1}}\]

Hence option D is correct.

Note: We should know the formulae for magnetic fields due to the short bar magnet on the equatorial line and on the axis line. We should analyze the constant terms in the given data and make the calculations to get the desired answer.A magnetic field is produced by moving electric charges and intrinsic magnetic moments of elementary particles associated with a fundamental quantum property known as the spin.