Magnetized material consists of a large number of tiny magnetic dipoles, and these magnetic dipoles then generate their own magnetic field.

$\vec M$ : Magnetic dipole moment per unit volume.

Equivalent to a surface current per unit length.

$\oint \vec{B} \cdot \overrightarrow{d l}=\mu_0 I_{enc}$

$I_ {enc} = N I l + M l$

$\oint \vec{B} \cdot \overrightarrow{d l}=\mu_0 (N I l + M l)$

$\oint \vec{M} \cdot \vec{d l} = M l$

$\oint \frac{\vec{B}}{\mu_0} \cdot \vec{d l} = N I l+\oint \vec{M} \cdot \overrightarrow{d l}$

N : Number of turns per unit length

$\oint\left(\frac{\vec{B}}{\mu_0}-\vec{M}\right) \cdot \vec{d l}=N I l$

$\vec{H}=\frac{\vec{B}}{\mu_0}-\vec{M}$

$\oint \vec{H} \cdot \vec{d l} = N I l =I_{f,_{enc}}$

$I_{f._{enc}}$ = Free current enclosed by the loop

$\oint \vec{H} \cdot \vec{d l} = I_{f,_{enc}}$

For a large class of materials

$\vec{M}=\chi_m \vec{H}$

$\chi_m$ : Magnetic susceptibility

Linear media

Diamagnetic : $\chi_m<0$

Paramagnetic : $\chi_m>0$

$|\chi_m|$ <<< 1

$\vec{H} =\frac{\vec{B}}{\mu_0}-\vec{M}$

$\vec{B} =\mu_0(\vec{H}+\vec{M})$

$\vec{M} =\chi_m \vec{H}$

$\vec{B} =\mu_0\left(1+\chi_m\right) \vec{H} =\mu {\vec{H}}$

$\mu =\mu_0\left(1+\chi_m\right)$

$\mu_0$ : Permeability of free space

$\mu$ : Permeability of the medium

For diamagnetic and paramagnetic material, $\left|\chi_m\right| \lll 1$

$\mu \simeq \mu_0$

Diamagnetic: $\chi_m<0 \Rightarrow \mu \leqslant \mu_0$

Paramagnetic: $\chi_m>0 \Rightarrow \mu \geq \mu_0$

Relative permeability

$K_m=\frac{\mu}{\mu_0}=1+\chi_m$

$\mu_0$ = Permeability of free space.

$\mu$ = Permeability of the medium.

When you place a medium in a magnetic field, external magnetic field magnetizes the medium then generates its magnetic field and the total magnetic field gets altered because of magnetization.

N : Number of turns per unit length

I : Current through the wire

$\oint \vec H. \vec {d l}$ = $I_ {f_{enc}}$

$\oint \vec{H} \cdot \overrightarrow{d L}=I_{f,_{enc}}$

$\vec{H}=\frac{\vec{B}}{\mu_0}-\vec{M}$

$\vec{B}=\mu_0\left(1+\chi_m\right) \vec{H}$

For path $C_1$

$H \cdot l = N.I.l$

$H = N I$

$\vec{H} = N I \hat{k}$

For path $C_2$

$\oint \vec{H} \cdot \overrightarrow{d \ell}=I_{f,_{enc}}$

$H^{\prime} \ell=N I \ell$

$\Rightarrow \quad H^{\prime}=N I$

$\overrightarrow{H^{\prime}}=N I \hat{k}=\vec{H}$

$\vec{H}=N I \hat{k}$ Everywhere within the solenoid.

$\vec{B}=\mu_0\left(1+\chi_m\right) \vec{H}$

Reason I

${\chi_m}=0$

$\vec{B}=\mu_0 \vec{H} =\mu_0 N I \hat{k}$

Reason II

$\vec{B} =\mu_0\left(1+\chi_m\right) \vec{H}$

$=\mu_0\left(1+\chi_m\right) N I \widehat{k} =\mu N I \hat k$

$\vec M = \chi_m \vec H$

= $\chi_m N I \hat k$

Diamagnetic core

Paramagnetic core

In a diamagnetic materials magnetization is downward, because of bound carrying conductor and so the magnetic field inside the material is slightly less than the magnetic field outside.

In paramagnetic materials the magnetization has the same direction and hence produces the magnetic field in the same direction as the coil.

The magnetic field inside the paramagnetic material is slightly more than the magnetic field outside.

Primary class of magnetic materials

Diamagnetic

Paramagnetic

Ferromagnetic

No intrensic dipole moment of constituent atoms.

Dipoles induceds by the external magnetic field.

Induced dipoles directent opposite to external applied magnetic field.

Magnetization disappears when the external field is removed.

Pushed from region of high field to smaller magnetic field in an inhomogenous field.