Magnetic Flux and Flux Linkage

Magnetic flux (Φ) — the flux density multiplied by the area it passes through perpendicularly:

Φ = B A cos θ

measured in webers (Wb), where one weber is one tesla metre squared.

The angle catches people out. Here θ is the angle between the field and the normal to the area, not the area itself. So:

field perpendicular to the plane of the coil, meaning along the normal — cos θ = 1, maximum flux;

field in the plane of the coil — cos θ = 0, zero flux.

Compare this with F = BIL sin θ, where the angle is measured differently and the sine appears instead. Mixing the two up is easy; it helps to ask each time whether the quantity should be largest or zero in the geometry in front of you, and let that select the function.

Flux linkage (NΦ) — for a coil of N turns, each turn links the same flux, so the total linkage is

N Φ = B A N cos θ

measured in weber-turns. This is the quantity that matters for induction: what an e.m.f. responds to is the rate of change of flux linkage, not flux alone, which is why coils have many turns.

Example: "flux linkage" sounds like jargon, but it is literal. Picture the field lines threading through the loop like thread through the eye of a needle. Add more loops and the same thread passes through more eyes. Nothing about the field has changed; the circuit simply engages with more of it.

A rotating coil — as a coil turns in a uniform field, θ changes continuously, so the flux linkage varies as cos θ and therefore sinusoidally with time. The induced e.m.f., being its rate of change, varies as sin θ — a quarter cycle out of step. This is why the e.m.f. is zero when the flux linkage is maximum, which feels wrong until you notice that a maximum is exactly where a curve is momentarily flat.