Maharashtra HSC · Class XII · Physics

Chapter 11: Magnetic Materials

Full revision notes for Chapter 11 - Magnetic Materials.

Part 1: Key Definitions & Formulas

The core quantities you need before the material comparison below: what magnetization and magnetic intensity mean, how susceptibility ties them together, and the master relations connecting $B$ , $H$ and $M$ .

M

Magnetization

The net magnetic dipole moment $m_{\text{net}}$ per unit volume $V$ of the material - how strongly the material is magnetized.

SI unit ampere per metre (A/m) · dimensions $[L^{-1}I]$

M = \frac{m_{\text{net}}}{V}

H

Magnetic intensity

The capability of an external magnetic field to magnetize a material. It depends on the field source alone, not on the material.

SI unit ampere per metre (A/m)

chi

Magnetic susceptibility

The ratio of the induced intensity of magnetization $M$ to the applied magnetic intensity $H$ .

\chi = \frac{M}{H}

A pure, dimensionless number (scalar)

mu_r

Relative permeability

The ratio of the material's permeability $\mu$ to that of free space $\mu_0$ ; it equals $1 + \chi$ .

\mu_r = \frac{\mu}{\mu_0} = 1 + \chi

Dimensionless

Core master relations

B = \mu_0\left(H + M\right)

B = \mu_0 H\left(1 + \chi\right)

B = \mu H

$\mu_0 = 4\pi \times 10^{-7} \; \text{N/A}^2$ is the permeability of free space.

Diamagnetic, paramagnetic and ferromagnetic

The three classes of magnetic material, compared against the properties you are asked about in the board exam.

Property	Diamagnetic	Paramagnetic	Ferromagnetic
Behavior in a non-uniform field	Weakly repelled - drawn away from the strong-field region.	Weakly attracted - drawn toward the strong-field region.	Strongly attracted - pulled into the strongest-field region.
Atomic dipole moment	No permanent atomic dipoles; field-induced moments cancel (net zero).	Permanent dipoles, randomly oriented - net moment near zero without a field.	Permanent dipoles aligned within domains - large net moment that persists.
Susceptibility $\chi$	Small and negative, $\chi \approx -10^{-5}$ .	Small and positive, $\chi \approx 10^{-5}\,\text{to}\,10^{-3}$ .	Large and positive, $\chi \approx 10^{2}\,\text{to}\,10^{5}$ .
Relative permeability $\mu_r$	$\mu_r$ slightly less than 1.	$\mu_r$ slightly more than 1.	$\mu_r$ very large - much greater than 1.
Temperature effect	Independent of temperature - a small, constant susceptibility.	Obeys Curie's law: $\chi \propto 1/T$ , so susceptibility falls as temperature rises.	Ferromagnetic below the Curie temperature; above it, becomes paramagnetic.
Examples	Cu, Bi, water, gold.	Al, Pt, sodium, oxygen.	Fe, Ni, Co and their alloys.

Torque on a Magnetic Dipole in a Uniform Magnetic Field (Balbharati Sec 11.2)

Physical concept - a pure torque

A magnetic dipole (for example a bar magnet) has a magnetic dipole moment vector $\vec{m}$ . In a uniform magnetic field $\vec{B}$ the two poles experience equal and opposite forces: the net (resultant) force on the dipole is zero, so there is no translation. But the two equal, opposite and offset forces form a couple that tends to rotate the dipole until $\vec{m}$ aligns with $\vec{B}$ .

ec{ au}

Torque on the dipole

The magnitude of the torque is $\tau = m B \sin\theta$ , where $\theta$ is the angle between $\vec{m}$ and $\vec{B}$ .

Maximum at $\theta = 90^{\circ}$ : $\tau_{\max} = mB$ . Minimum (zero) torque at $\theta = 0^{\circ}$ or $\theta = 180^{\circ}$ .

\vec{\tau} = \vec{m} \times \vec{B}

U

Potential energy

The dipole is most stable pointing along the field and least stable pointing against it.

Stable equilibrium at $\theta = 0^{\circ}$ with $U_{\min} = -mB$ . Unstable equilibrium at $\theta = 180^{\circ}$ with $U_{\max} = +mB$ .

U = -\vec{m}\cdot\vec{B} = -mB\cos\theta

Time period of small angular oscillations

About the stable equilibrium ( $\theta=0^\circ$ ), the dipole performs small angular simple harmonic oscillations with time period

T = 2\pi\sqrt{\frac{I}{mB}}

$I$ is the moment of inertia of the magnet about its axis of rotation, $m$ its magnetic dipole moment, and $B$ the magnitude of the magnetic field. This expression holds only for small oscillations about the stable equilibrium position.

Domain theory of ferromagnetism

The unmagnetized state

A ferromagnetic material is divided into many small regions called domains, each already magnetized to saturation in some direction. In an unmagnetized piece the domains point at random, so their magnetic moments largely cancel one another and the net magnetization is near zero.

The magnetized state

With an external field applied, the domains whose moments point along the field grow at the expense of the less favourably oriented ones (the domain walls move), and the rest rotate into the field direction. The material acquires a large net magnetization aligned with the applied field.

When the field is removed

The alignment is not completely lost. Domains that have grown or rotated stay partly aligned, so the material keeps a residual (permanent) magnetization - this is why a magnet retains its magnetism. Pushing the magnetization back to zero takes a reverse field, which is the origin of the hysteresis loop.

Curie's law and the Curie-Weiss law

Curie's law - paramagnets

For a paramagnetic material the susceptibility falls in inverse proportion to the absolute temperature,

\chi = \frac{C}{T}

Here $C$ is the Curie constant, a property of the material proportional to the density of its atoms and the square of their magnetic moment, and $T$ is the absolute temperature in kelvin. As $T$ rises, thermal agitation randomizes the dipoles and the susceptibility falls.

Curie-Weiss law - ferromagnets above the Curie temperature

Below its Curie temperature $T_c$ a ferromagnetic material shows spontaneous domain magnetization. Above $T_c$ that order breaks down and the material behaves as a paramagnet, obeying the Curie-Weiss law,

\chi = \frac{C}{T - T_c}

As before $C$ is a constant of the material and $T_c$ is the Curie temperature. The law is valid only for $T > T_c$ : at $T = T_c$ the susceptibility tends to diverge as ordering sets in, and for $T < T_c$ ferromagnetic order takes over with a spontaneous magnetization, so the formula must not be used there.

Magnetic intensity H, magnetization M and B

Magnetic intensity H

The applied magnetizing field - a measure of the external field the coil or magnet imposes on the specimen. It depends only on the current and geometry of the field source, not on the material. Its SI unit is the ampere per metre (A/m).

Magnetization M

The magnetic dipole moment per unit volume that the material develops in response to the field. How readily $M$ responds to $H$ is the susceptibility, given by $\chi = M/H$ . Its SI unit is also the ampere per metre (A/m).

Relation between B, H and M

Inside a material the total magnetic flux density $B$ is the sum of the applied field's contribution and that of the material's own magnetization,

B = \mu_0\left(H + M\right)

$\mu_0 = 4\pi \times 10^{-7}\,\text{T}\cdot\text{m/A}$ is the permeability of free space.

$B$ is the total field inside the material, in tesla (T). $H$ and $M$ are both in ampere per metre (A/m).