CBSE Class 12 › Unit III: Magnetic Effects

Magnetism & Matter

NCERT Chapter 5 • The Bar Magnet, Gauss’s Law & Magnetic Properties

NCERT 2025–26 Theoretical Concepts

1. The Bar Magnet

A bar magnet has two poles: North and South. The magnetic field lines are a visual representation of the magnetic field.

Comparison of magnetic field lines between a bar magnet and a finite solenoid. — The magnetic field lines of a solenoid resemble those of a bar magnet.

Properties of Magnetic Field Lines:

They form continuous closed loops (unlike electric field lines which start from positive and end on negative).
The tangent to the field line at any point gives the direction of the net magnetic field $\vec{B}$ .
The larger the number of field lines crossing per unit area, the stronger the magnitude of $\vec{B}$ .
Magnetic field lines do not intersect.

Solenoid as Bar Magnet:

A finite solenoid produces an axial field similar to a bar magnet. The field at a large distance $r$ from the center is:
$B = \dfrac{\mu_0}{4\pi} \dfrac{2m}{r^3}$ .

2. The Dipole in a Uniform Magnetic Field

When a magnetic needle of magnetic moment $\vec{m}$ is placed in a uniform magnetic field $\vec{B}$ , it experiences a torque.

Magnetic needle oscillating in a uniform magnetic field experiencing torque. — A magnetic dipole experiences a torque $\tau = m \times B$ .

Derivation: Torque & Potential Energy 3 Marks

1. Torque
The torque acting on the needle is given by the cross product:
$\vec{\tau} = \vec{m} \times \vec{B}$ or $\tau = mB \sin \theta$ .
Here $\theta$ is the angle between $\vec{m}$ and $\vec{B}$ .

2. Potential Energy
The work done in rotating the dipole is stored as potential energy $U_m$ .
$U_m = \int \tau(\theta) d\theta = \int mB \sin \theta \, d\theta = -mB \cos \theta$ .
$U_m = -\vec{m} \cdot \vec{B}$ .

3. Stability

Stable Equilibrium: $\theta = 0^\circ$ (PE is minimum, $-mB$ ).
Unstable Equilibrium: $\theta = 180^\circ$ (PE is maximum, $+mB$ ).

3. The Electrostatic Analog

We can obtain magnetic field formulas from electric field formulas by making the following replacements:

$\vec{E} \rightarrow \vec{B}$
$\vec{p} \rightarrow \vec{m}$
$\frac{1}{4\pi\epsilon_0} \rightarrow \frac{\mu_0}{4\pi}$

Property	Electrostatics	Magnetism
Equatorial Field	$-p / 4\pi\epsilon_0 r^3$	$-\mu_0 m / 4\pi r^3$
Axial Field	$2p / 4\pi\epsilon_0 r^3$	$\mu_0 2m / 4\pi r^3$
Torque	$\vec{p} \times \vec{E}$	$\vec{m} \times \vec{B}$

4. Gauss’s Law for Magnetism

The net magnetic flux through any closed surface is always zero. This implies that isolated magnetic poles (monopoles) do not exist.

Diagram showing zero net magnetic flux through a closed surface enclosing a magnet. — The number of magnetic field lines leaving a surface is equal to the number entering it.

\phi_B = \oint \vec{B} \cdot d\vec{S} = 0

Difference from electrostatics: $\oint \vec{E} \cdot d\vec{S} = q/\epsilon_0$

5. Magnetisation and Magnetic Intensity

To classify magnetic properties of materials, we define the following terms:

1. Magnetisation ( $\vec{M}$ )
Net magnetic dipole moment per unit volume.
$M = \frac{m_{net}}{V}$ . (Unit: A/m)

2. Magnetic Intensity ( $\vec{H}$ )
Considering a long solenoid with current $I$ , the field in the interior is $B_0 = \mu_0 n I$ .
We define vector field $H$ such that: $H = \frac{B}{\mu_0} - M$ .
Thus, total field: $B = \mu_0 (H + M)$ .

3. Magnetic Susceptibility ( $\chi$ )
A measure of how a material responds to an external field.
$M = \chi H$ .

\mu_r = 1 + \chi

Relative Permeability: $\mu = \mu_0 \mu_r$

6. Magnetic Properties of Materials

Materials are classified based on their susceptibility $\chi$ and relative permeability $\mu_r$ .

Illustration of magnetic domains and field line behavior in different magnetic materials. — Behavior of field lines in Diamagnetic (repelled) vs Paramagnetic (attracted) materials.

Type	Susceptibility ( $\chi$ )	Relative Permeability ( $\mu_r$ )	Behavior
Diamagnetic	Small, Negative ( $-1 \le \chi < 0$ )	$0 \le \mu_r < 1$	Repelled by magnets. Field lines expelled. Example: Water, Copper.
Paramagnetic	Small, Positive ( $\chi > 0$ )	$\mu_r > 1$	Weakly attracted. Field lines concentrated inside. Example: Aluminum, Oxygen.
Ferromagnetic	Large, Positive ( $\chi \gg 1$ )	$\mu_r \gg 1$	Strongly attracted. Domains align with field. Example: Iron, Cobalt.

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