Question

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

Answer 1

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,

& \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{aligned}$ **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, $\begin{aligned} & H+M=\dfrac{1}{{{\mu }_{0}}}B \\\ & \Rightarrow B={{\mu }_{0}}\left( M+H \right) \\\ \end{aligned}$ Substituting equation (2) in the above equation, $\begin{aligned} & B={{\mu }_{0}}\left( {{\chi }_{m}}H+H \right) \\\ & \Rightarrow B={{\mu }_{0}}\left( 1+{{\chi }_{m}} \right)H \\\ \end{aligned}$ 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.