Magnetic Circuits and Electromagnetic Induction

By the end you should be able to: Define MMF, field strength, flux density and reluctance, compare magnetic with electric circuits, classify series and parallel magnetic circuits, and state Faraday's and Lenz's laws with self and mutual inductance.

The syllabus explicitly exempts both halves of this topic from numerical problems — magnetic circuits with composite materials, and electromagnetic induction. One worked calculation is given below because the analogy is clearest with numbers attached, but the examinable content is definitions, comparisons and statements of law.

The magnetic circuit idea

Flux, like current, prefers a low-resistance path and largely stays in it. An iron core guides magnetic flux the way a copper wire guides current, and the same algebra applies with different names.

Magnetomotive force (MMF) drives flux, as EMF drives current:

\mathcal{F} = NI \qquad \text{ampere-turns (At)}

Magnetic field strength $H$ is MMF per unit length:

H = \frac{NI}{l} \qquad \text{A/m}

Flux density $B$ is the response of the material:

B = \mu H = \mu_0\mu_r H \qquad \text{tesla (T)}

with $\mu_0 = 4\pi\times10^{-7}$ H/m and $\mu_r$ the relative permeability — around 2000 for silicon steel, 1 for air.

Flux is flux density times area:

\Phi = BA \qquad \text{weber (Wb)}

Reluctance opposes flux, as resistance opposes current:

S = \frac{l}{\mu_0\mu_r A} \qquad \text{At/Wb}

and the magnetic Ohm's law:

\boxed{\,\Phi = \frac{\mathcal{F}}{S} = \frac{NI}{S}\,}

The comparison, and where it breaks

Electric	Magnetic
EMF, $E$ (V)	MMF, $\mathcal{F} = NI$ (At)
Current, $I$ (A)	Flux, $\Phi$ (Wb)
Resistance, $R = \rho l/A$	Reluctance, $S = l/\mu A$
$I = E/R$	$\Phi = \mathcal{F}/S$
Conductivity $\sigma$	Permeability $\mu$
Current density $J$	Flux density $B$

The analogy is genuinely useful, but three differences matter and are commonly examined:

Current flow dissipates energy; flux does not. A steady current in a resistor produces heat continuously. A steady flux in a core consumes no power at all — only establishing it takes energy, and that energy is stored, not lost.
Permeability is not constant. Resistivity is essentially fixed; $\mu_r$ varies enormously with $B$ and collapses at saturation, around 1.5–2 T for iron. Beyond saturation, adding MMF buys almost no extra flux. There is no electrical equivalent.
There is no magnetic insulator. Copper conducts around $10^{20}$ times better than good insulators, so current stays in the wire. Iron is only a few thousand times more permeable than air, so some flux always escapes as leakage. A magnetic circuit is never as tidy as the diagram.

Series and parallel magnetic circuits

Series — a single flux path, possibly through several materials or an air gap. Same flux throughout; reluctances add:

S_{total} = S_1 + S_2 + \cdots

A composite circuit is exactly this: iron of one length and area in series with iron of another, or with an air gap. Note how dominant an air gap is. Since $\mu_r = 1$ for air against 2000 for steel, a 1 mm gap in a 500 mm steel path has, for equal area, a reluctance $\tfrac{1}{500}\times 2000 = 4$ times that of the entire steel path. A small air gap dominates the magnetic circuit, which is why motor designers fight to keep gaps small — and also why gaps are deliberately introduced to linearise inductors, since the gap's constant $\mu$ swamps the iron's variable one.

Parallel — the flux divides between two or more paths, as in a three-limb transformer core. Total flux is the sum of branch fluxes, and the MMF is common across parallel branches:

\Phi_{total} = \Phi_1 + \Phi_2, \qquad \frac{1}{S_{eq}} = \frac{1}{S_1} + \frac{1}{S_2}

Electromagnetic induction

Faraday's first law. Whenever the flux linked with a circuit changes, an EMF is induced.

Faraday's second law. The magnitude of the induced EMF equals the rate of change of flux linkage:

e = N\frac{d\Phi}{dt}

Lenz's law. The induced EMF acts in the direction that opposes the change producing it:

e = -N\frac{d\Phi}{dt}

The minus sign is not bookkeeping — it is conservation of energy. If the induced effect aided the change, the change would grow without limit and generate energy from nothing.

Statically and dynamically induced EMF

Statically induced. Nothing moves; the flux itself changes with time, usually because the current producing it is alternating. This is the transformer, and it subdivides:

Self-induced — the EMF induced in a coil by the change of its own current.
Mutually induced — the EMF induced in one coil by the change of current in a neighbouring coil.

Dynamically induced. The flux is steady but the conductor moves through it, cutting field lines. This is the generator:

e = Blv\sin\theta

for a conductor of length $l$ moving at $v$ at angle $\theta$ to the field. Motion parallel to the field cuts no lines and induces nothing.

Self, mutual inductance and coupling

Self-inductance relates the EMF to a coil's own rate of change of current:

e = -L\frac{di}{dt}, \qquad L = \frac{N\Phi}{I} = \frac{N^{2}}{S}

Note $L \propto N^{2}$ : doubling the turns quadruples the inductance, because each of twice as many turns links twice as much flux.

Mutual inductance between two coils:

e_2 = -M\frac{di_1}{dt}, \qquad M = \frac{N_2\Phi_{12}}{I_1}

Coefficient of coupling measures what fraction of one coil's flux reaches the other:

k = \frac{M}{\sqrt{L_1L_2}}, \qquad 0 \le k \le 1

$k = 1$ is perfect coupling, approached by two windings on a common closed iron core; $k \approx 0$ for coils far apart or at right angles. Rearranged, $M = k\sqrt{L_1L_2}$ .