Question

Establish a relation between magnetic permeability and magnetic susceptibility.

Answer 1

When a magnetic material is placed in a magnetic field the magnetic field inside the magnetic material changes use the relationship how the magnetic field changes inside the magnetic material to find the relation between permeability and magnetic susceptibility.

Complete step by step answer:
When a magnetic material is kept in a magnetising field (H).
Then the total number of magnetic lines of force inside the material =magnetic lines of force due to magnetising field +magnetic lines of force due to magnetization of specimens.
i.e. Magnetic induction(B) = ${B_0}$(no of lines of force sur to H)+${\mu _0}I$(no of lines due to magnetization of specimen)
$B = {B_0} + {\mu _0}I = {\mu _0}H + {\mu _0}I = {\mu _0}(H + I) \\\ \Rightarrow B = {\mu _0}H(1 + {\chi _m})as{\chi _m} = \dfrac{1}{H} \\\$
Now $B = \mu H = {\mu _0}H(1 + {\chi _m}) \Rightarrow \mu = {\mu _0}(1 + {\chi _m})$
${\mu _r} = \dfrac{\mu }{{{\mu _0}}} = 1 + {\chi _m}$
Hence relationship between magnetic permeability and magnetic susceptibility is
${\mu _r} = \dfrac{\mu }{{{\mu _0}}} = 1 + {\chi _m}$.

Note:
As we can see ${\mu _r} = 1 + {\chi _m}$ so if ${\chi _m}$ is positive relative permeability will be greater than 1 and if ${\chi _m}$ is negative then relative permeability will be less than one which is the case of the diamagnetic materials.