Explain the magnetic susceptibility $(\chi)$ of a material. From it,explain the relative magnetic permeability and magnetic permeability of the material. Obtain the relation between them.

(N/A) The magnetization $M$ of a material is proportional to the magnetic intensity $(H)$.
$\overrightarrow{M} = \chi \overrightarrow{H}$....$(1)$
Here,$\chi$ is a dimensionless quantity called magnetic susceptibility. It measures how a magnetic material responds to an external field.
For paramagnetic materials,$\chi$ is small and positive.
For diamagnetic materials,$\chi$ is small and negative because $\overrightarrow{M}$ and $\overrightarrow{H}$ are in opposite directions.
Consider a magnetic material placed inside a solenoid. The total magnetic field $B$ when a current $I$ flows through the solenoid is:
$\overrightarrow{B} = \mu_{0}(\overrightarrow{H} + \overrightarrow{M})$....$(2)$
Substituting equation $(1)$ into $(2)$:
$\overrightarrow{B} = \mu_{0}(\overrightarrow{H} + \chi \overrightarrow{H}) = \mu_{0}(1 + \chi) \overrightarrow{H}$....$(3)$
Here,$\mu = \mu_{0}(1 + \chi)$ is defined as the magnetic permeability of the material.
Thus,$\mu = \mu_{0}(1 + \chi)$....$(4)$
Dividing by $\mu_{0}$,we get the relative magnetic permeability $\mu_{r}$:
$\mu_{r} = \frac{\mu}{\mu_{0}} = 1 + \chi$....$(5)$
Therefore,the relation is $\mu = \mu_{0} \mu_{r}$....$(6)$
Substituting this into equation $(3)$,we get:
$\overrightarrow{B} = \mu \overrightarrow{H}$....$(7)$
$\mu_{r}$ is a dimensionless quantity,analogous to the dielectric constant in electrostatics.