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# The relation between magnetic susceptibility ${{\chi }_{m}}$ and relative permeability ${{\mu }_{r}}$ isA.${{\chi }_{m}}={{\mu }_{r}}$B.${{\chi }_{m}}-1={{\mu }_{r}}$C.${{\mu }_{r}}=1+{{\chi }_{m}}$D.${{\mu }_{r}}=1-{{\chi }_{m}}$

Last updated date: 23rd Feb 2024
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Hint: To derive relation between magnetic susceptibility and relative permeability first we will discuss how permeability measures resistance of material against magnetic field and magnetic susceptibility to indicate whether a material is attracted or repelled by magnetic field.

Formula used:
${{\mu }_{r}}=\dfrac{\mu }{{{\mu }_{\circ }}}$
$\overrightarrow{B}=\overrightarrow{{{B}_{\circ }}}+\overrightarrow{{{B}_{m}}}$

Permeability measures the magnetization material can obtain in the applied magnetic field and can be represented by $\mu$. Relative permeability is the ratio of permeability of a specific medium $\mu$ and permeability of free space ${{\mu }_{\circ }}$.
$\Rightarrow {{\mu }_{r}}=\dfrac{\mu }{{{\mu }_{\circ }}}$
Magnetic susceptibility measures whether a material will be attracted or repelled by a magnetic field, for example paramagnetic materials align with the magnetic field present in the area and diamagnetic material will be pushed away in the lower magnetic field regions. Magnetic susceptibility is denoted by ${{\chi }_{m}}$.
Now, magnetic field inside the substance is given by,
$\Rightarrow \overrightarrow{B}=\overrightarrow{{{B}_{\circ }}}+\overrightarrow{{{B}_{m}}}$
$\Rightarrow \overrightarrow{B}={{\mu }_{\circ }}\overrightarrow{H}+{{\mu }_{\circ }}\overrightarrow{I}$
$\Rightarrow \overrightarrow{B}={{\mu }_{\circ }}(\overrightarrow{H}{{+}_{\circ }}\overrightarrow{I})$
$\Rightarrow \overrightarrow{B}={{\mu }_{\circ }}\overrightarrow{H}\left( 1+\dfrac{\overrightarrow{I}}{\overrightarrow{H}} \right)$
$\Rightarrow \chi =\dfrac{\overrightarrow{I}}{\overrightarrow{H}}$
Here,$\overrightarrow{I}$(intensity of magnetization) and $\overrightarrow{H}$(magnetic field strength)
$\Rightarrow \dfrac{\overrightarrow{B}}{\overrightarrow{H}}={{\mu }_{\circ }}\left( 1+\chi \right)$
Now, we know that,
$\Rightarrow B=\mu H$
So,
$\Rightarrow \mu ={{\mu }_{\circ }}\left( 1+\chi \right)$
Or by rearranging the equation we have,
$\Rightarrow \dfrac{\mu }{{{\mu }_{\circ }}}=\left( 1+\chi \right)$
$\Rightarrow {{\mu }_{r}}=\left( 1+\chi \right)$
$\therefore$ Relative permeability of medium will be equal to one added to magnetic susceptibility which is given by,
$\Rightarrow {{\mu }_{r}}=\left( 1+\chi \right)$

Hence, option (C) is correct.

Note:
Permeability is not a constant value, it can vary with position in the medium, humidity, temperature or frequency of the applied magnetic field. SI unit of permeability is measured in henries per meter (H/m). The permeability of a free space ${{\mu }_{\circ }}$ is constant and its value is $4\pi \times {{10}^{-7}}H/m$. Magnetic susceptibility is a dimensionless property which indicates degree of magnetization on a substance in an applied magnetic field. If ${{\chi }_{m}}$is positive material can be paramagnetic or magnetic field in material will strengthened by induced magnetization and get weakened if ${{\chi }_{m}}$is negative.