Question

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

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,
\[\]$\begin{align}
  & \chi =3\times {{10}^{-4}} \\
 & {{\mu }_{r}}=1+\chi \\
 & \Rightarrow {{\mu }_{r}}=1+3\times {{10}^{-4}} \\
 & \Rightarrow {{\mu }_{r}}=1+0.0003 \\
 & \therefore {{\mu }_{r}}=1.0003 \\
\end{align}$

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 $H=\dfrac{1}{{{\mu }_{0}}}B-M$.
By rearranging the equation we get,
$\begin{align}
  & H+M=\dfrac{1}{{{\mu }_{0}}}B \\
 & \Rightarrow B={{\mu }_{0}}\left( M+H \right) \\
\end{align}$
Substituting equation (2) in the above equation,
$\begin{align}
  & B={{\mu }_{0}}\left( {{\chi }_{m}}H+H \right) \\
 & \Rightarrow B={{\mu }_{0}}\left( 1+{{\chi }_{m}} \right)H \\
\end{align}$
Thus here B is also proportional to H. Hence,
$B=\mu H$
where,
$\mu ={{\mu }_{0}}\left( 1+{{\chi }_{m}} \right)$
$\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.