Obtain the relation between magnetisation `overset(to)((m) )` and magnetic intensity `overset(to) ((H))` for a solenoid.

Explanation

Consider a long solenoid of n turns per unit length and carrying a current I.
The magnetic field in the interior of the solenoid `overset(to) (B_0) = mu_(0) nI " "…(1)`
If the interior of the solenoid is filled with a material with non-zero magnetization, the field inside the solenoid will be greater than `overset(to) (B_0)`. The net `overset(to) (B) ` field in the interior of the solenoid may be expressed as,
`overset(to) (B) = overset(to)(B_0) + overset(to) (B_m) " "...(2)`
where `overset(to) (B_m)` is the field contributed by the material core. It turns out that this additional field `B_m` is proportional to the magnetisation `overset(to) (M)` of the material and is expressed as
`overset(to) (B_m) prop overset(to) (M)`
`overset(to) (B_m) = mu_(0) overset(to) (M) " "...(3)`
where `mu_0` is same constant (permeability of vacuum) that appear in Biot-Savart.s law.
It is convenient to introduce another vector field H called the magnetic intensity which is defined by,
`overset(to) (H) = (overset(to) (B) )/( mu_0) - overset(to) (M) " "...(4)`
where `overset(to) (H)` has the same dimension as `overset(to) (M)` and is measured in units of `Am^(-1)`. Thus, the total magnetic field `overset(to) (B)` is written as,
`overset(to) (B) = mu_(0) ( overset(to) (H) + overset(to) (M) )" "...(5)`
The total magnetic field inside the sample is divided into two parts :
(1) Due to external factors such as the current `overset(to) (H)` in the solenoid.
(2) The other is due to the specific nature of the magnetic material namely `overset(to) (M)`.