Question

The relation between magnetic susceptibility $\chi_m$ and the relative permeability $\mu_r$ is:
A). $\chi_m = \mu_r$
B). $\chi_m-1 = \mu_r$
C). $\mu_r = 1+\chi_m $
D). $\mu_r = 1-\chi_m $

Answer 1

Hint: Recall that $\chi_m$ is defined as the ratio of the intensity of magnetization to the intensity(force) of the magnetizing field. And permeability, in general, is given as the ratio of the applied magnetic field to the intensity(force) of the magnetizing field. Using these relations, as well as deriving an expression for the net field inside a magnetic material arising from the applied field as well as induced magnetism, you should be able to arrive at the solution just by rearranging a few terms in the net field expression.

Formula Used:
Magnetic susceptibility $\chi_m = \dfrac{I}{H}$, where I is the intensity of magnetization and H is the intensity of the magnetizing field (magnetizing force of applied field)
Relative permeability $\mu_r = \dfrac{\mu}{\mu_0}$, where the permeability of the material is $\mu$, and that of free space $\mu_0$
Relative permeability in terms of susceptibility: $\mu_r = (1+\chi_m)$

Complete step-by-step solution:
Let us begin by establishing an understanding of what magnetic susceptibility and relative permeability mean.
Magnetic susceptibility $\chi_m$ indicates the intensity or the degree of magnetization (I) of a material in response to the intensity of the magnetizing force (H) of the applied magnetic field (B). In other words, it describes the magnetizability of a material. This is a dimensionless proportionality constant. A higher susceptibility implies that the material is more susceptible to the field and hence can be easily magnetized.
It is given as: $\chi_m = \dfrac{I}{H}$
Magnetic permeability is a property of the material that characterizes the amount of magnetizing force (H) that a material experiences under the influence of an applied magnetic field (B).
It is given as: $\mu = \dfrac{B}{H}$
Relative permeability is the ratio of the permeability of the material($\mu$) to that of free space($\mu_0$).
It is given as: $\mu_r = \dfrac{\mu}{\mu_0}$
Now, let us calculate the net field inside a magnetic material.
The net field will be a resultant of magnetization due to applied field ($B_0$) and magnetization due to induced magnetism ($B_i$),
$B = B_0 +B_i$
Where, $ B_0 = \mu_0 H$ and $B_i = \mu_0 I$.
$\Rightarrow B = \mu_0 (H+I)$
$\Rightarrow \dfrac{B}{H} = \mu_0 \left(1+\dfrac{I}{H}\right)$
Substituting $\mu = \dfrac{B}{H}$ and $\chi_m = \dfrac{I}{H}$ in the above equation, we get:
$\mu =\mu_0(1+\chi_m) \Rightarrow \dfrac{\mu}{\mu_0} = (1+\chi_m) \Rightarrow \mu_r = (1+\chi_m)$
Therefore, the correct choice would be: C. $\mu_r = (1+\chi_m)$.

Note: Do not get confused between the intensity of magnetization I, and the intensity of the magnetizing field H. ‘I’ is a material-dependent property whereas H is the property of the field applied. The symbiosis between the two gives the magnetic susceptibility that we’ve already seen.
Also, remember that permeability is usually the term associated with magnetism whereas permittivity is associated with electrostatics and electrodynamics. Permittivity is the obstruction produced by the material in the formation of the electric field whereas permeability is the ability of the material to allow magnetic field lines to pass through it. The free space permittivity is given as $8.85 Fm^{-1}$ whereas the free space permeability is $1.26 Hm^{-1}$, where F and H denote electric force and magnetic force respectively.