Magnetisation and magnetic intensity

Orbiting electrons give atoms magnetic moments, and in bulk matter these add as vectors. The magnetisation M of a sample is its net magnetic moment per unit volume, M = m_net/V: a vector with dimensions L⁻¹ A, measured in A m⁻¹.

In an empty long solenoid B₀ = μ₀nI. Filling it with a magnetised material adds a field B_m = μ₀M, so B = B₀ + B_m.

The magnetic intensity is defined as H = B/μ₀ − M, also in A m⁻¹. Then B = μ₀(H + M): H stands for the external cause (in a solenoid H = nI, whatever the core), M for the material's response.

For most materials M is proportional to H: M = χH, where the magnetic susceptibility χ is dimensionless. χ is small and positive for paramagnetic materials, small and negative for diamagnetic ones (M opposite to H).

So B = μ₀(1 + χ)H = μ₀μᵣH = μH, with relative permeability μᵣ = 1 + χ (the magnetic twin of the dielectric constant) and permeability μ = μ₀μᵣ, in the same units as μ₀.

χ, μᵣ and μ are linked, so knowing any one gives the other two.

Example 5.5: a core of μᵣ = 400 in windings of 1000 turns per metre carrying 2 A gives H = nI = 2 × 10³ A/m, B = μᵣμ₀H = 1.0 T, and M = (μᵣ − 1)H = 399H ≈ 8 × 10⁵ A/m.

Example 5.5: the magnetising current is the extra current that would give the same B with no core, from B = μ₀n(I + I_M); it comes to I_M = 794 A, showing how much the core adds.