Question

The magnetic susceptibility of a magnetic material is $3\times {10}^{-4}$. Its relative permeability will be
$\begin{array}{rl}& \left(A\right)31\times {10}^{-4}\\ & \left(B\right)1.003\\ & \left(C\right)1.0003\\ & \left(D\right)29\times {10}^{-4}\end{array}$

Answer 1

Hint: Use the equation connecting the relative permeability to the magnetic susceptibility of a material. That, one plus the value of magnetic susceptibility gives the relative permeability of a material. Magnetic susceptibility is described as a dimensionless quantity that varies from one substance to another. Magnetic susceptibility is usually positive for paramagnets and negative value for diamagnets.

Formula used:
${\mu }_r=1+\chi$
where, $\chi$ is the magnetic susceptibility
${\mu }_r$ is the relative permeability.

Complete step by step solution:
Given that magnetic susceptibility is,
$$$$$\begin{array}{rl}& \chi =3\times {10}^{-4}\\ & {\mu }_r=1+\chi \\ & \Rightarrow {\mu }_r=1+3\times {10}^{-4}\\ & \Rightarrow {\mu }_r=1+0.0003\\ & \therefore {\mu }_r=1.0003\end{array}$

So, the correct answer is “Option C”.

Additional Information: In paramagnetic and diamagnetic materials, the magnetization is sustained by the field; when the magnetic field is removed, magnetization disappears. For most of the substances the magnitude of magnetization is proportional to its magnetic field.
That is,
$M={\chi }_{m}H$ ………(2)
The constant of proportionality ${\chi }_m$ is the magnetic susceptibility.
Magnetic susceptibility is described as a dimensionless quantity that varies from one substance to another. Magnetic susceptibility is usually positive for paramagnets and negative value for diamagnets.
The material which obeys the equation $M={\chi }_{m}H$is called linear media.
Let's consider the equation $H={\displaystyle \frac{1}{{\mu }_0}}B-M$.
By rearranging the equation we get,
$\begin{array}{rl}& H+M={\displaystyle \frac{1}{{\mu }_0}}B\\ & \Rightarrow B={\mu }_0(M+H)\end{array}$
Substituting equation (2) in the above equation,
$\begin{array}{rl}& B={\mu }_0({\chi }_{m}H+H)\\ & \Rightarrow B={\mu }_0(1+{\chi }_m)H\end{array}$
Thus here B is also proportional to H. Hence,
$B=\mu H$
where,
$\mu ={\mu }_0(1+{\chi }_m)$
$\mu$ is called the permeability of the material and ${\mu }_0$ is called the permeability of free space.
${\mu }_r=1+{\chi }_m$
where, ${\mu }_r$ is the relative permeability.

Note: In paramagnetic and diamagnetic materials, the magnetization is sustained by the field; when the magnetic field is removed, magnetization disappears. For most of the substances the magnitude of magnetization is proportional to its magnetic field. Magnetic susceptibility is a dimensionless quantity that varies from one substance to another. Magnetic susceptibility is usually positive for paramagnets and negative value for diamagnets.