Magnetic Intensity, Permeability and Susceptibility

Magnetic Materials

Some materials are strongly attracted, some are weakly attracted and some are actually repelled. Magnetic materials are diamagnetic, paramagnetic and ferromagnetic materials. In an atom, various orbital and spin magnetic moments of electrons exactly cancel so that the atom is not magnetic. The magnetic moments of these atoms are randomly oriented so that there is no net magnetic moment in any volume of the material. When such material is placed in a magnetic field, the field exerts a torque on each magnetic moment which tends to align the magnetic moment along the field.

Intensity of Magnetization

Consider a magnetic material composed of atoms having magnetic dipole moment. TheIntensity of magnetisation (I) is defined as the net dipole moment (M) per unit volume (V).

$$ I = \frac MV$$

The unit of intensity of magnetisation is ampere per metre(A/m). for a bar magnet of length 2l and cross-sectional area A, we have

$$ I = \frac {m \times 2l}{A \times 2l} = \frac {m}{A} $$

where m is the pole strength of the magnet which is a vector quantity.

so, the intensity of magnetisation may also be defined as the pole strength developed per unit area of the substances when subjected to a uniform magnetic field. It may be positive or negative depending upon either the magnetisation taking place in the direction or in the opposite direction of the magnetising field.

Magnetic Intensity, Permeability and Susceptibility

Suppose a material placed in a uniform magnetic field B₀, such as inside long solenoid. The applied field magnetizes the material and aligns the dipoles which produce a magnetic field B_M of their own. At any point, the magnetic field B is the sum of the applied field B₀ and the field produced by the B_M.

The magnetization field B_M is related to the intensity of magnetization I.

$$\begin{align*} B_M &= \mu_0I \\ \text {where}\: \mu_0 \:\text {is the permeability of free space}\\ \text {Both}\: B_0 \: \text {and I are parallel with axis and so,} \\ B &= B_0 + \mu_0I \\ \end{align*}$$

The magnetic permeability \(\mu \) of a magnetic material is the ratio of total induced magnetic field B to the external magnetic field H. Actually, magnetic permeability is the measure of the degree of penetration of magnetic field through the substance kept in an external magnetic field. The relative permeability is defined as the ratio of the permeability of substance to the permeability of vacuum. It is denoted by \(\mu_r\).

The permeability is expressed as

$$\begin{align*} \mu &= \frac {\vec B}{\vec H} \\ \text {or,}\: \vec B &=\mu \vec H \\ \And mu _r &= \frac {\mu }{\mu _r} \\ \text {or,}\: \mu &= \mu_r\mu_o \\ \end{align*}$$

The magnetic susceptibility, \(\chi \) of the material is the ratio of intensity of magnetization to the magnetic intensity, H

$$\chi = \frac {1}{H} $$

It is a measure of how susceptible or easily and strongly a material be magnetized by H. \(\chi \) is unitless as unit of I and H is same.

Relation between \(\mu \) and \(\chi \)

The magnetic field intensity in a material is

$$\begin{align*} B &= B_0 + B_M \\&= \mu_0(H + I) \\ \text {Since} \chi = \frac 1H,\: \text{so} \\ B &= \mu_0(H + \chi H) \\ \text {or,}\: B &= \mu_0 H (1 + \chi )\\ \text {or,}\: \frac BH &= \mu _0(1 + \chi) \\ \text {But}\: B/H = \mu \text {permeability of medium, then} \\ \mu &= \mu_0 (1 + \chi) \\ \text {as the relative permeability of medium, then} \mu_r = \mu/\mu_0, \text{so} \\ \frac {\mu}{\mu_0} &= (1 + \chi )\\ \text {or,}\:\mu_r &= (1 + \chi )\\ \text {or,}\: \chi &= (\mu_r – 1) \\\end{align*}$$