EM-18 · Magnetostatics field

Magnetic Materials and Magnetization

Put a magnetic material in a field and its atomic magnets line up, strengthening the net field. Raise the relative permeability μ_r to see the moments align and the field grow by a factor μ_r — the opposite of how a dielectric weakened the electric field.

Raise μ_r and watch the field grow

A bar of magnetic material sits in an external field. Raise the relative permeability μ_r. How far do the atomic magnets inside line up? What happens to the net field (the gold arrow)?

Relative permeability μ_rμ_r = 5.0

Drag to orbit. The slider sets the relative permeability μ_r.

The net field inside the material

B = μ_r B₀ = μH · the net field grows by a factor μ_r

M = (μ_r − 1)H · atomic magnets line up

Net field (vs external) · ×5.0

Middle

What a magnetic material is

Every atom carries a small magnetic moment from the motion and spin of its electrons — an atom-sized magnet. Normally these point every which way and cancel, so there is no net field. But apply an external field and they begin to line up with it. Rather than a current, it is the existing atomic magnets being brought into line. This capacity to become magnetized is magnetism.

Magnetization M, moments align

Apply an external field and countless atomic magnets line up with it. The degree of alignment, as magnetic moment per unit volume, is the magnetization M. It closely mirrors the dielectric polarization P of EM-10. The difference: polarization was charge shifting slightly, while magnetization is existing magnets falling into line. When the aligned atomic magnets all point one way, their fields add in that same direction.

ObserveB = μ_r B₀

Inside a magnetic material the net field is μ_r times stronger.

The net field grows

Here it parts ways with the dielectric decisively. A dielectric's bound charge made a field opposing the external one, weakening it. Aligned atomic magnets instead add a field in the same direction as the external one, strengthening it. So the net field is stronger than outside. The factor by which it grows is the relative permeability μ_r: B = μ_r B₀. Paramagnets have μ_r just above 1; ferromagnets (iron, nickel) reach hundreds to thousands, amplifying the field enormously. Diamagnets have μ_r just below 1 and weakly push back.

ChooseM = ?

The magnetization is (μ_r − 1) times H.

Permeability, H and B

To handle magnetization separately, we use the field intensity H. The flux density B combines the applied H with the material’s added magnetization M: B = μ₀(H + M). In a linear material M is proportional to H, so B = μH cleanly, with permeability μ = μ_r μ₀. Think of H as the part made by free currents and B as the actual field including the magnetization. Ferromagnets are nonlinear: once magnetized they resist returning, showing hysteresis — the basis of permanent magnets and recording media.

Fill inB = μ₀(H + ?)

The flux density is the sum of the applied H and the magnetization M.

On your ownμ = ?

Permeability is relative permeability times the vacuum value.

Back to the first screen

On the first screen, raising μ_r aligned the atomic magnets inside ever more sharply, and the gold arrow for the net field grew longer and longer, because the aligned magnets added their field in the same direction as the external one. The factor of growth is μ_r itself: B = μ_r B₀. Exactly opposite to how a dielectric tamed the electric field, a magnetic material amplifies the magnetic field. That is why putting an iron core in a coil makes its inductance jump by μ_r (EM-17).

Put a magnetic material in a field and its atomic magnetic moments align (magnetization M), adding to the external field. The net field thus grows by a factor μ_r: B = μ_r B₀ = μH (μ = μ_r μ₀); precisely, B = μ₀(H + M). Opposite to how a dielectric weakened the electric field, paramagnetic and ferromagnetic materials strengthen the magnetic field. Ferromagnets (iron and the like) have μ_r in the hundreds to thousands, so they serve as cores in coils and transformers.

What comes next

The last piece of magnetostatics is energy. The energy stored in coils and magnetic materials in fact resides in the magnetic field itself. The next unit (EM-19, magnetic energy) sees it as a field energy density u = B²/(2μ). It is the magnetic counterpart of the electrostatic u = ½εE², and the coil’s ½LI² comes from here too. That completes the seven pieces of magnetostatics, and we move on to time-varying fields and Maxwell.