Magnetic Materials and Magnetization

Raise μ_r and the field inside grows stronger

A bar of magnetic material sits in an external magnetic field pointing up. The cyan arrows are the external field, and the small arrows inside the bar are atomic magnets. Raising the relative permeability μ_r with the slider makes the atomic magnets line up more clearly in the same direction, and the gold arrow, the net field inside the bar, grows longer.

What a magnetic material is

Electrons in atoms carry tiny magnetic moments from their orbital motion and spin, and in many atoms these add up into a small atom-sized magnet. Normally these magnets point every which way and cancel, so there is no net magnetic field. But when an external magnetic field is applied, they start lining up along it. Nothing new is made to flow; the atomic magnets already present are put in line, and this ability of a material to act like a magnet is called magnetism.

Magnetization M, moments align

An external field lines up countless atomic magnets along its direction. The degree of alignment, expressed as the magnetic moment per unit volume, is the magnetization M. It closely resembles polarization in EM-10, but differs in that polarization is charge pulled slightly apart, while magnetization is existing magnets falling into line. When the aligned atomic magnets point one way, their fields all add on the same side.

The net field grows

This is where the decisive split from dielectrics comes. A dielectric's bound charge made a field opposite to the external electric field and weakened it, but aligned atomic magnets add a field in the same direction as the external magnetic field and strengthen it. So the net magnetic field is stronger than outside, and the factor of strengthening is the relative permeability μ_r. B = μ_r B₀. This formula works well when the material is elongated along the field. In paramagnets μ_r is a little above 1, while in ferromagnets such as iron or nickel it reaches hundreds to tens of thousands, enlarging the field enormously. In diamagnets μ_r is slightly below 1, so they weaken the field a little and are pushed out of strong fields.

Permeability, H and B

To handle the share of magnetization separately, the magnetic field strength H is used. The magnetic flux density B combines the externally applied H and the magnetization M the material adds. B = μ₀(H + M). In a linear magnetic material M is proportional to H, so it tidies up to B = μH, with permeability μ = μ_r μ₀. You can think of it this way: the circulation of H is set by free current alone, while B is the real field that includes the share of magnetization. Ferromagnets are not linear and show hysteresis, staying magnetized once magnetized, and this property underlies permanent magnets and recording media.

Back to the first screen

On the first screen, raising μ_r made the atomic magnets in the bar line up more clearly, and the gold arrow for the net field grew longer and longer, because the aligned magnets added field in the same direction as the external one. The factor of strengthening is exactly μ_r. B = μ_r B₀. In exact opposition to a dielectric calming the electric field, a magnetic material enlarges the magnetic field, which is why putting an iron core in a coil makes the inductance jump by a factor of μ_r (EM-17).

The last piece of magnetostatics is energy. The energy stored in coils and magnetic materials actually resides in the magnetic field itself. The next unit (EM-19 magnetic energy) views it as the field's energy density u = B²/(2μ). It pairs with electrostatic energy u = ½εE², and the ½LI² of a coil comes from here too. That rounds off the seven units of magnetostatics, and we move on to fields that change in time and Maxwell's equations.