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

Explanation

The magnetisation of material M proportional to the magnetic intensity (H).
`overset(to) (M) prop overset(to) (H)`
`therefore overset(to) (M) = chi overset(to) (H) " "…(1)`
Here `chi` is dimensionless quantity called it as the magnetic susceptibility. It is a measure of how a magnetic material responds to an external field.
For paramagnetic material it is small and positive.
For diamagnetic material it is small and negative because this material `overset(to) (M) and overset(to)(H)` are opposite in direction.
Think the material which can be magnetize are filled in interior of solenoid. Total magnetic field obtain when current I in solenoid pass through is,
`overset(to) (B) = mu_(0) (overset(to) (H) + overset(to) (M) )" "...(2)`
From equation (1) and (2),
`overset(to) (B) = mu_(0) (overset(to) (H) + chi overset(to) (H) )`
`therefore overset(to) (B) = mu_(0) (1+ chi) overset(to) (H)" "...(3)`
Here `mu_(0) (1 + chi) ` is the permeability `mu` of the material.
Hence, `mu= mu_(0) (1 + chi) " "...(4)`
`(mu) /( mu_0) = (1 + chi ) = mu_r" "...(5)`
`therefore mu= mu_(0) (1 + chi) = mu_(0) mu_r " "...(6)`
Here,
(1) `mu_r` is called the relative magnetic permeability of the substance. It is a dimensionless quantity. It is the analog of the dielectric constant in electrostatics.
(2) Equation (6) indicates the interrelation between `mu, mu_r and H` and only one of them is independent. Given one the other two may be easily determined.
Putting equation (6) in equation (3),
`overset(to) (B) = mu_(0) mu_r overset(to) (H)" "...(7)`
`overset(to) (B) = muoverset(to) (H) ( because mu_(0) mu_(r) = H) " "...(8)`