H-field

The H-field is the "currents-we-control" part of the magnetic field. Inside a magnetised material, the microscopic magnetic field B is produced by two populations of current: free currents, which are the ones you drive through wires with a power supply, and bound currents, which are the tiny atomic circulations that add up to the material's magnetization M. Writing B as the sum of these two contributions and rearranging gives H = B/μ₀ − M — the auxiliary field whose curl is driven purely by free current density, ∇×H = J_free.

Mathematically, this is the exact analogue of what the displacement field D does in dielectrics: D = ε₀E + P tracks the free-charge part of electrostatics, and H = B/μ₀ − M tracks the free-current part of magnetostatics. Ampère's law, restated for H, reads ∮H·dℓ = I_free — no factor of μ₀ and no explicit bound-current term, because M has already absorbed the bound currents into its definition. In SI units H is measured in amperes per metre.

The H-field is an experimentalist's tool. If you wind a solenoid around a chunk of iron and run 1 A through N turns per metre, the H-field inside the iron is exactly N·I = N amperes per metre, regardless of what the iron is doing. The iron's response shows up separately as M — an enormous M, in fact, often thousands of times larger than H itself. The B-field inside is then B = μ₀(H + M), dominated by the M term. H cleanly separates "what you put in" (current through the coil) from "how the material responded" (magnetization), which is exactly what you want when characterising magnetic materials.