Key Points

A bar magnet has two poles, a north and a south pole. Like poles repel and unlike poles attract, but isolated magnetic poles (monopoles) do not exist; cutting a magnet creates two smaller magnets.

Magnetic field lines are imaginary curves that form continuous closed loops, originating from the north pole and ending at the south pole outside the magnet. The tangent to a field line at any point gives the direction of the net magnetic field B at that point.

The magnetic field of a bar magnet is very similar to that of a finite current-carrying solenoid. This analogy supports Ampere's hypothesis that all magnetic phenomena arise from circulating electric currents.

This law states that the net magnetic flux through any closed surface is zero. Mathematically, $\phi_B = \sum \mathbf{B} \cdot \Delta \mathbf{S} = 0$. This implies the absence of magnetic monopoles.

A bar magnet or a current loop possesses a magnetic dipole moment, denoted by the vector $\mathbf{m}$. For a solenoid with N turns, area A, and current I, its magnitude is $m = NIA$.

In a uniform magnetic field $\mathbf{B}$, a magnetic dipole with moment $\mathbf{m}$ experiences a torque given by the vector product $\tau = \mathbf{m} \times \mathbf{B}$. The magnitude is $\tau = mB \sin\theta$.

The potential energy of a magnetic dipole in an external field is $U_m = -\mathbf{m} \cdot \mathbf{B} = -mB \cos\theta$. The system is in stable equilibrium when $\theta = 0^{\circ}$ and unstable equilibrium when $\theta = 180^{\circ}$.

The magnetic field on the axis of a short bar magnet is $\mathbf{B}_{A} = \frac{\mu_0}{4\pi} \frac{2\mathbf{m}}{r^3}$, and on the equatorial line, it is $\mathbf{B}_{E} = -\frac{\mu_0 \mathbf{m}}{4\pi r^3}$.

Magnetisation $\mathbf{M}$ is the net magnetic moment per unit volume. Magnetic intensity $\mathbf{H}$ is related to the total magnetic field $\mathbf{B}$ by the equation $\mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M})$.

Magnetic susceptibility $\chi$ is a dimensionless measure of how a material is magnetized, where $\mathbf{M} = \chi \mathbf{H}$. Relative magnetic permeability is $\mu_r = 1 + \chi$, and magnetic permeability is $\mu = \mu_0 \mu_r$.

Materials are classified as diamagnetic, paramagnetic, or ferromagnetic based on their magnetic susceptibility $\chi$.

Diamagnetic materials have a small, negative susceptibility ($\chi < 0$) and are weakly repelled by magnetic fields. This property is present in all materials but is often overshadowed. Example: Bismuth, Copper, Water.

Paramagnetic materials have a small, positive susceptibility ($\chi > 0$) and are weakly attracted to magnetic fields. Their atoms have permanent magnetic dipole moments. Example: Aluminum, Calcium, Oxygen.

Ferromagnetic materials have a large, positive susceptibility ($\chi \gg 1$) and are strongly attracted to magnetic fields. They can be permanently magnetized due to the formation of magnetic domains. Example: Iron, Cobalt, Nickel.

Materials that retain their ferromagnetic property for a long time are called hard ferromagnets and are used to make permanent magnets. Materials that lose their magnetism easily are called soft ferromagnets and are used in electromagnets.