4.3.1 From Ampère's law to a circuit

The circulation of H along a closed path

The first fundamental equation of magnetostatics, Ampère's law of Section 4.1, says in integral form that the circulation of $H$ around any closed path $C$ equals the current through any surface $S$ bounded by $C$:

The path $C$ may be cut into pieces. For the rectangle of Figure 4.3.1, made of the four sides $C_1$, $C_2$, $C_3$ and $C_4$ taken in order, the circulation is the sum of four line integrals,

where $I$ is the current linked by the rectangle, counted positive when it circulates in the sense given by the right-hand rule with respect to the direction of travel along $C$. This trivial-looking split is the whole idea of the magnetic circuit: when the path runs along an iron core, each side of the rectangle will be one branch of the core, and each line integral will become a simple product $H_i{l}_i$.

Flux lines do not end: the node rule

The second fundamental equation, ${\oint }_{S}B\cdot dS=0$ for every closed surface (Section 4.1), is the statement that lines of $B$ have no beginning and no end. Figure 4.3.2 shows a bundle of flux lines near a point A where the bundle splits. Wrap a closed surface $S$ around A and divide it into the three parts $S_1$, $S_2$ and $S_3$ through which the lines pass. With the outward normal, the flux entering through $S_1$ is counted negative and the fluxes leaving through $S_2$ and $S_3$ positive, so

In words: what flows in must flow out. This is the magnetic version of Kirchhoff's current law, and in a core it will apply at every point where the iron branches.