Question

What is the relation between relative permeability and magnetic susceptibility?

Answer 1

Formula used:

Magnetic susceptibility, ${\mathrm{X}}_{\mathrm{m}}=\frac{\mathrm{I}}{\mathrm{H}}$

I is the intensity of magnetization.

H is the intensity of the magnetizing field.

Relative permeability, ${\mathrm{\mu }}_{\mathrm{r}}=\frac{\mathrm{\mu }}{{\mathrm{\mu }}_0}$

$\mathrm{\mu }$ the permeability of the material.

${\mathrm{\mu }}_0$ free space.

Relative permeability in terms of susceptibility:

${\mathrm{\mu }}_{\mathrm{r}}=(1+{\mathrm{X}}_{\mathrm{m}})$

Calculation:

Let's compute the net field within a magnetic substance.

$\mathrm{B}={\mathrm{B}}_0+{\mathrm{B}}_{\mathrm{i}}$

Here,

${\mathrm{B}}_0$ is the net field will be a resultant of magnetization due to the applied field.

${\mathrm{B}}_{\mathrm{i}}$ magnetization due to induced magnetism.

and,

$$\begin{gathered} {\mathrm{B}}_0={\mathrm{\mu }}_0\mathrm{H} \\ {\mathrm{B}}_{\mathrm{i}}={\mathrm{\mu }}_0\mathrm{I} \end{gathered}$$

By substituting $\mathrm{\mu }=\frac{\mathrm{B}}{\mathrm{H}}$ and ${\mathrm{X}}_{\mathrm{m}}=\frac{1}{\mathrm{H}}$ in the above equation, we get

$$\begin{gathered} \Rightarrow \mathrm{\mu }={\mathrm{\mu }}_0(1+{\mathrm{X}}_{\mathrm{m}}) \\ \frac{\mathrm{\mu }}{{\mathrm{\mu }}_0}=(1+{\mathrm{X}}_{\mathrm{m}}) \\ {\mathrm{\mu }}_{\mathrm{r}}=(1+{\mathrm{X}}_{\mathrm{m}}) \end{gathered}$$

Therefore, the relationship between relative permeability and magnetic susceptibility is ${\mathrm{\mu }}_{\mathrm{r}}=(1+{\mathrm{X}}_{\mathrm{m}})$.