Magnetic vs electric circuits, work law and Ohm's law

A magnetic circuit is the closed path followed by magnetic flux, e.g. the iron core of a transformer with its air gaps. It behaves much like an electric circuit, with mmf driving flux against reluctance.

Similarities

Differences (in brief): flux does not actually flow and needs no energy to keep it up, while current does; $\mu$ changes with B (saturation), while R is nearly constant.

Work law (Ampere's circuital law)

The work done in moving a unit magnetic pole once round a closed path equals the ampere-turns enclosed:

$$\oint \vec H \cdot d\vec l = NI$$

For a path made of sections in which H is constant,

$$H_1 l_1 + H_2 l_2 + \dots = NI = F$$

So the total mmf equals the sum of the "magnetic drops" $Hl$ round the circuit, just as KVL says the emf equals the sum of IR drops.

Ohm's law for a magnetic circuit

Take a ring of mean length $l$, area $A$, permeability $\mu = \mu_0\mu_r$, with $N$ turns carrying $I$. From the work law $H = NI/l$. Then

$$\begin{aligned} B &= \mu H = \frac{\mu NI}{l} \\ \phi &= BA = \frac{\mu A NI}{l} = \frac{NI}{l/(\mu A)} \end{aligned}$$ $$\phi = \frac{F}{S}, \quad S = \frac{l}{\mu_0 \mu_r A}$$

i.e. flux = mmf / reluctance, the magnetic form of $I = V/R$.