4.2.2 Fundamental equations of magnetostatics in a medium

With $H$ defined so that only the free current is its source, the two fundamental equations of Section 4.1 keep their form in every medium.

Fundamental equations of magnetostatics

First fundamental equation	Second fundamental equation
$\nabla \times H = J$	$\nabla \cdot B = 0$
$\oint_{C} H \cdot d l = \int_{S} J \cdot d S$	$\oint_{S} B \cdot d S = 0$

(differential form above, integral form below)

\nabla \times H = J, \nabla \cdot B = 0, B = μ H (μ = μ_{0} μ_{r}) . (4.2.9)

In plain words

Both equations hold in any medium. The medium enters only through the constitutive relation $B = μ H$ . Ampère's law with the free current gives $H$ , the material turns it into $B$ , and the flux of $B$ through any closed surface is zero: magnetic field lines have no ends, in iron just as in air.

The integral forms are obtained from the differential ones exactly as in Section 4.1, by Stokes' theorem and the divergence theorem (Section 2.4). They have a property the differential forms lack: they remain valid when the contour $C$ or the surface $S$ crosses a surface on which $μ$ jumps, and this is what we exploit next.

Recipe · Fields in a simple magnetic medium

Find $H$ from the free currents with Ampère's law (or Biot–Savart with $μ_{0} \to 1$ in front), exactly as in free space. The material does not enter.
Multiply by $μ = μ_{0} μ_{r}$ to get $B = μ H$ .
If wanted, $M = B - μ_{0} H = μ_{0} χ_{m} H$ , and the bound currents from Eq. (4.2.5).