Magnetostatics in Media

The magnetic field has no sources:

$$\nabla \cdot \mathbf{B}=0$$

No experiment has ever found a magnetic monopole. Every magnetic field line closes on itself, and every magnet has two poles no matter how finely you cut it.

The vector potential

Because $\mathbf{B}$ is divergence-free it can always be written as a curl:

$$\mathbf{B}=\nabla \times \mathbf{A}$$

$\mathbf{A}$ is not unique. Adding the gradient of any scalar leaves $\mathbf{B}$ unchanged. That freedom is gauge invariance, and it is a nuisance in electrostatics and the organising principle of all of modern field theory.

Choosing the Coulomb gauge $\nabla \cdot \mathbf{A}=0$ reduces Ampère's law to Poisson's equation again, one component at a time:

$${\nabla }^2\mathbf{A}=-{\mu }_0\mathbf{J}$$

so every technique from electrostatics carries over unchanged.

Matter responds in three ways

Inside a material, bound currents contribute and it is cleaner to work with $\mathbf{H}$:

$$\mathbf{H}=\frac{\mathbf{B}}{{\mu }_0}-\mathbf{M},\nabla \times \mathbf{H}={\mathbf{J}}_{\text{free}}$$

• Diamagnetism. An induced moment opposing the field. Universal, weak, and a direct consequence of Lenz's law at the atomic scale. Strong enough, with a strong enough field, to levitate a live frog.
• Paramagnetism. Pre-existing moments partially aligning against thermal agitation. Susceptibility follows Curie's law $\chi \propto 1/T$.
• Ferromagnetism. Moments aligning with each other. Not a magnetic effect at all: the alignment comes from the exchange interaction, which is electrostatic repulsion combined with the Pauli principle.

Hysteresis and domains

A ferromagnet does not retrace its path. The $B$–$H$ curve encloses an area, and that area is energy dissipated per cycle as heat.

The mechanism is domains. An unmagnetised piece of iron is not disordered. It is divided into regions each fully magnetised in a different direction, arranged so the external field cancels. Applying a field grows the favourable domains at the expense of the others by moving the walls between them. Those walls snag on defects and release suddenly, which is why the magnetisation of a real sample advances in audible jumps.

Those audible jumps are Barkhausen noise, and hearing them through a coil and an amplifier was the first direct evidence that domains exist at all.

The loop's shape is what a material is selected on. A hard material has high coercivity, hard to demagnetise, so it makes a permanent magnet. A soft material has a thin loop and little loss per cycle, which is why transformer cores are soft iron: at 50 Hz you traverse the loop fifty times a second, and the enclosed area is dissipated as heat on each pass.