Two pages ago a moving conductor made a voltage. Last page we defined exactly what was being “cut” — flux linkage. Now we join them into the single most important rule in this whole topic. Faraday spotted that the induced e.m.f. doesn’t care how much flux there is — it cares how fast the flux is changing. Change it quickly and you get a big voltage; change it slowly and you get a small one; hold it steady and you get nothing at all.
📘 What you need to know
Faraday’s law: the magnitude of the induced e.m.f. is directly proportional to the rate of change of flux linkage
In symbols: ε = NΔΦ / Δt — the change in flux linkage divided by the time taken
NΔΦ is the change in flux linkage (Wb turns); Δt is the time interval (s)
A faster change of flux linkage gives a bigger e.m.f.
An e.m.f. only appears when the flux linkage is changing — a steady flux gives no e.m.f.
For a rotating coil, flux linkage changes as NΦ = BAN cos θ
The e.m.f. is zero when the coil is face-on to the field (flux maximum, but not changing at that instant)
The e.m.f. is maximum when the coil is edge-on (flux zero, but changing fastest)
The induced e.m.f. is 90° out of phase with the flux linkage
The law in one line
Faraday’s law says it all in a sentence:
The magnitude of an induced e.m.f. is directly proportional to the rate of change of magnetic flux linkage.
Turn that sentence into an equation and you have the tool for every calculation on this page:
Faraday’s lawε = NΔΦ / Δtε = induced e.m.f. (V) • NΔΦ = change in flux linkage (Wb turns) • Δt = time (s)
Read it carefully: NΔΦ is a change, and it’s divided by time. So the e.m.f. is a rate — it measures how quickly the flux linkage moves, not how big it is.
Flux linkage NΦ = BAN cos θ
how fast it changes
e.m.f. ε = NΔΦ/Δt
Think of flux linkage as your position and e.m.f. as your speed. Standing at the top of a hill (lots of flux) doesn’t move you anywhere — speed zero. It’s how fast your position changes that matters. That’s why a coil sitting still in the strongest field on Earth produces no e.m.f.: nothing is changing. Movement, and only movement, makes the voltage.
WE 1
The magnetic flux through a 200-turn coil falls steadily from 8.0 mWb to 2.0 mWb in 0.15 s. (a) Calculate the change in flux linkage. (b) Calculate the average e.m.f. induced. (c) State what the e.m.f. would be if the same change happened in half the time.
(a) Step 1 — change in flux linkage = N × change in fluxΔΦ = 8.0 − 2.0 = 6.0 mWb = 6.0 × 10⁻³ WbNΔΦ = 200 × (6.0 × 10⁻³)NΔΦ = 1.2 Wb turns(b) Step 2 — Faraday’s lawε = NΔΦ / Δt = 1.2 / 0.15ε = 8.0 V(c) same change, half the timeε ∝ 1/Δt, so halving Δt doubles εε = 16 VNotice the flux fell, but Faraday’s law asks only for the magnitude of the change, so we take the size, 6.0 mWb, and don’t fuss over the sign here. The direction of the e.m.f. is Lenz’s law’s job — that’s the next page.
Why a spinning coil gives an alternating voltage
Spin a coil in a magnetic field and its flux linkage rises and falls as it turns, following NΦ = BAN cos θ. Faraday’s law then says the e.m.f. depends on how fast that flux linkage is changing at each instant — and here comes the twist that trips up almost everyone.
Face-on, the coil catches the most flux — but for that split second the flux isn’t changing, so the e.m.f. is zero. Edge-on, the flux is zero but sweeping through fastest, so the e.m.f. is at its maximum. It’s the rate that matters, not the amount.
Coil position
Flux linkage
Rate of change
Induced e.m.f.
Face-on (plane ⊥ field)
Maximum
Zero (at a turning point)
Zero
Edge-on (plane ∥ field)
Zero
Maximum
Maximum
The 90° phase shift
Plot the flux linkage and the e.m.f. against time and the reason jumps out. The e.m.f. is the rate of change (the gradient) of the flux-linkage curve. Where flux linkage is flat at a peak, its gradient is zero — so the e.m.f. is zero. Where flux linkage races through zero, its gradient is steepest — so the e.m.f. peaks. The two curves are 90° out of step.
The e.m.f. curve (red) is the gradient of the flux-linkage curve (blue). Peaks in one line up with zeros in the other — the classic 90° phase shift between flux linkage and induced e.m.f.
WE 2
A rectangular coil of 350 turns has sides 4.0 cm and 2.5 cm. It sits between the poles of a magnet giving a uniform field of flux density 80 mT. Starting from horizontal (its plane parallel to the field), the coil is turned through 40° in a time of 0.18 s. Calculate the magnitude of the average e.m.f. induced.
Step 1 — known quantitiesA = 0.040 × 0.025 = 1.0 × 10⁻³ m²B = 80 mT = 0.080 T, N = 350, Δt = 0.18 sStep 2 — flux linkage uses the angle to the normal
Plane starts parallel to the field, so the normal is at θ = 90°.
initial: NΦ = BAN cos 90° = 0
After turning 40°, the normal is at 90 − 40 = 50°.
final: NΦ = BAN cos 50° = 350 × 0.080 × (1.0 × 10⁻³) × cos 50°final NΦ = 0.018 Wb turnsStep 3 — change in flux linkage, then Faraday’s lawΔ(NΦ) = 0.018 − 0 = 0.018 Wb turnsε = Δ(NΦ) / Δt = 0.018 / 0.18ε = 0.10 V = 100 mVThe whole trick is Step 2: “plane parallel to the field” means the normal is at 90°, so the starting flux linkage is zero, not maximum. Get the normal angle right and the rest is arithmetic.
WE 3
A flat coil of 120 turns and area 6.0 × 10−3 m2 lies face-on in a uniform field of flux density 0.25 T. It is rotated through a quarter-turn to edge-on in 0.30 s. (a) Calculate the average e.m.f. induced. (b) Explain why the e.m.f. is not steady during the turn, even though the average is this value.
(a) Step 1 — flux linkage at start and end
Face-on means the normal lines up with the field, θ = 0°.
initial: NΦ = BAN cos 0° = (0.25)(6.0 × 10⁻³)(120) = 0.18 Wb turns
Edge-on means θ = 90°.
final: NΦ = BAN cos 90° = 0Step 2 — Faraday’s lawΔ(NΦ) = 0.18 − 0 = 0.18 Wb turnsε = 0.18 / 0.30ε = 0.60 V(b) why not steady?
The flux linkage changes as cosθ, whose gradient is not constant.
the e.m.f. starts small, rises to a peak, so 0.60 V is only the averageThis is the difference between average and instantaneous e.m.f. Faraday’s law with Δ gives the average over the interval; the true e.m.f. rises from zero (face-on) to its peak (edge-on) as the coil sweeps round.
⚡ Working a Faraday’s-law question
Find the flux linkage at the start and end: NΦ = BAN cos θ.
Watch the angle.θ is from the normal. “Plane parallel to field” → θ = 90° → flux zero.
Change in flux linkage:NΔΦ = (final) − (initial), taking the magnitude.
Divide by the time:ε = NΔΦ / Δt.
Convert units first: cm → m (then square for area), mT → T.
Average vs peak: the Δ form gives the average e.m.f. over the interval.
💡 Top tips
Faraday’s law is about the rate of change of flux linkage — a steady flux gives no e.m.f.
Always start from flux linkageNΦ = BAN cos θ, then find its change.
Get the angle to the normal right — “plane parallel to field” means θ = 90°, flux = 0.
e.m.f. is maximum when the coil is edge-on (flux zero, changing fastest).
e.m.f. is zero when the coil is face-on (flux maximum, but momentarily still).
The ε and NΦ graphs are 90° out of phase.
⚠ Common mistakes
Thinking a big flux means a big e.m.f. It’s the rate of change that counts
Saying e.m.f. is maximum when the coil is face-on — it’s actually zero there
Getting the normal angle wrong: “plane parallel to field” is θ = 90°, not 0°
Forgetting the N — Faraday’s law uses flux linkage, not bare flux
Leaving area in cm2 or B in mT
Treating the average e.m.f. as if it were constant throughout the motion
Worrying about the sign here — the direction is Lenz’s law, the next page
Quick recap:Faraday’s law says the induced e.m.f. is proportional to the rate of change of flux linkage: ε = NΔΦ / Δt. A faster change gives a bigger e.m.f., and a steady flux gives none. For a rotating coil the flux linkage follows NΦ = BAN cos θ, so the e.m.f. is zero when the coil is face-on (flux maximum, not changing) and maximum when edge-on (flux zero, changing fastest) — the two are 90° out of phase. Always work from flux linkage, mind the angle to the normal, and remember this gives the average e.m.f. over the interval.
Faraday’s law tells you the size of the induced e.m.f., but not its direction. Which way does the induced current flow? There’s a beautifully simple rule, tied to conservation of energy: the induced current always fights the change that made it. That’s Lenz’s law, and it’s why the equation is really written with a minus sign, ε = −NΔΦ/Δt. Next page: Lenz’s Law.
Faraday’s law not adding up?
Book a free meeting and we’ll drill ε = NΔΦ/Δt, the flux-linkage setup, and the “max flux, zero e.m.f.” idea.
