Question

A paramagnetic substance is the form of a cube with sides $1cm$ has a magnetic dipole moment of $20 \times {10^{ - 6}}J/T$ when a magnetic intensity of $60 \times {10^3}A/m$ is applied. Its magnetic susceptibility is?
(A) $2.3 \times {10^{ - 2}}$
(B) $3.3 \times {10^{ - 2}}$
(C) $3.3 \times {10^{ - 4}}$
(D) $4.3 \times {10^{ - 4}}$

Answer 1

To solve this question, we have to use the basic formula of the magnetic susceptibility, which relates it with the magnetic dipole moment and the magnetic intensity. Then substituting the values given in the question, we will get the answer.
Formula used: The formula used to solve this question is given by
- $\chi = \dfrac{M}{H}$
- $M = \dfrac{m}{V}$
$M$ is the magnetisation,$H$ is the intensity of the magnetic field, $\chi$ is the magnetic susceptibility, $m$ is the magnetic moment, and $V$ is the volume of the substance.

Complete step by step answer:
We know that the magnetic susceptibility $\chi$ of a substance is defined as the ratio of the magnetisation $M$ and the magnetic intensity $H$
That is, $\chi = \dfrac{M}{H}$ (1)
We also know that the magnetisation is defined as the net magnetic dipole moment $m$ per unit volume $V$ of a material.
That is, $M = \dfrac{m}{V}$ (2)
Substituting (2) in (1), we have
So that, $\chi = \dfrac{m}{{VH}}$ (3)
According to the question, we have $m = 20 \times {10^{ - 6}}J/T$ and $H = 60 \times {10^3}A/m$
Also the material is a cube with side, $a = 1cm = 0.01m$
So, the volume of the cube $V = {a^3}$
$V = {0.01^3} = {10^{ - 6}}{m^3}$
Substituting these values in (3), we get
$\chi = \dfrac{{20 \times {{10}^{ - 6}}}}{{{{10}^{ - 6}} \times 60 \times {{10}^3}}}$
$\chi = \dfrac{{{{10}^{ - 3}}}}{3}$
On solving, we get
$\chi = 3.3 \times {10^{ - 4}}$
So, the magnetic susceptibility of the given material is $3.3 \times {10^{ - 4}}$
Hence, the correct answer is option (C), $3.3 \times {10^{ - 4}}$.

Note:
Do not forget to convert all the given quantities into the SI units. This is very much important in these types of entirely numerical based problems, as here only mistakes can be committed. For example in this question, the side of the square was given in centimetres, which is not an SI unit. So, first we converted it into the corresponding SI unit, that is meters, and then we made the substitution into the formula. If we forget to make the conversion, we will get an incorrect answer.