Question

What is the relation between magnetic susceptibility $X_m$ and relative permeability ${\mu }_r$?
(A) $X_m={\mu }_r$
(B) $X_m-1={\mu }_r$
(C) ${\mu }_r=1+X_m$
(D) ${\mu }_r=1-X_m$

Answer 1

Hint: The relation between the magnetic susceptibility and relative permeability can be determined by understanding the three equations. They are the magnetic susceptibility equation, magnetic permeability equation, and relative permeability equation. By using these three equations the relation between magnetic susceptibility and relative permeability can be determined.

Complete step by step answer:
1. Equation of magnetic susceptibility:
$X_m={\displaystyle \frac{M}{H}}.................\left(1\right)$
Where,
$X_m$ is the magnetic susceptibility
$M$ is the magnetization
$H$ is the magnetic field strength

2. Equation of Magnetic permeability:
$\mu ={\displaystyle \frac{B}{H}}......................\left(2\right)$
Where,
$\mu$ is the permeability
$B$ is the magnetic induction
$H$ is the magnetic field strength

3. Equation of relative permeability:
${\mu }_r={\displaystyle \frac{\mu }{{\mu }_0}}..................\left(3\right)$
Where,
${\mu }_r$ is the relative permeability
$\mu$ is the permeability
${\mu }_0$ is the absolute permeability

The relation between $H$, $B$ and $M$ is,
$B={\mu }_0\left(M+H\right)............\left(4\right)$
Substituting the value of $\left(M\right)$, taking from the equation (1),
$\Rightarrow B={\mu }_0\left(X_{m}H+H\right)$
By taking the common term $\left(H\right)$ out from the bracket, then the above equation is written as,
$\Rightarrow B={\mu }_0\left(X_m+1\right)H$
By rearranging the terms in the above equation,
$\Rightarrow B={\mu }_0\left(1+X_m\right)H.................\left(5\right)$
Now, the relation between the relative permeability and the susceptibility is,
$\Rightarrow B=\mu H$
Substituting the value of $\left(\mu \right)$ which is taken from the equation (3), then,
$\Rightarrow B={\mu }_0{\mu }_{r}H..................\left(6\right)$
By equating the equation (5) and equation (6), then,
$\Rightarrow {\mu }_0\left(1+X_m\right)H={\mu }_0{\mu }_{r}H$
Now, cancelling the same terms on each side, the above equation can be written as,
$\Rightarrow \left(1+X_m\right)={\mu }_r....................\left(7\right)$
From equation (7), it clearly shows the relationship between relative permeability $\left({\mu }_r\right)$ and magnetic susceptibility $\left(X_m\right)$.

Hence, option (C) is the correct answer.

Note:
The relation between the relative permeability and the magnetic susceptibility can be determined in another way also. The magnetic induction $\left(B\right)$ is equal to the sum of number of lines of force $\left(B_0\right)$ and the number of lines due to magnetisation of specimen $\left({\mu }_{0}I\right)$, by using this statement the relation between relative permeability and the magnetic susceptibility can be determined.