Faraday's law

Faraday's experiments settled one question: a circuit feels the motion of flux, not its size. A 500-turn coil of area 20 cm² held still in a 0.80 T field links Φ = 0.80 T × 20 × 10⁻⁴ m² = 1.6 × 10⁻³ Wb per turn, a flux linkage of 0.80 Wb, and produces no emf whatsoever. Pull the same coil clear of the field in 40 ms and the average emf is NΔΦ/Δt = 0.80 Wb ÷ 0.040 s = 20 V. Nothing about the coil changed; only the clock entered. So the quantity to compute is always a derivative, and the units say so: one weber per second is one volt. A useful habit follows. Before reaching for a formula, ask what is changing and how fast, because if the answer is nothing, the emf is zero no matter how impressive the field.

Faraday's law has a sign, and a sign needs a convention. Choose a direction to walk around the loop. Curl the right hand that way and the thumb gives the positive normal, which fixes the area vector and therefore the sign of Φ = ∫B⋅dA. Then ε = −dΦ/dt is positive when it drives current in the direction you chose. Choose the opposite sense and both Φ and ε flip together, so the physical current is unchanged — the convention never affects the answer, only its label. Two practical consequences: keep the same normal for the whole calculation, since flipping it midway is the commonest sign error, and remember that the flux may be negative to start with, in which case a flux becoming less negative is still a positive dΦ/dt. Turning that sign into the statement that the induced effect opposes the change is the job of Lenz's law.

For a flat loop in a uniform field, Φ = BA cos θ is a product of three things that can each move in time. The product rule gives dΦ/dt = A cos θ (dB/dt) + B cos θ (dA/dt) − BA sin θ (dθ/dt): a changing field, a changing area, a turning loop. Most problems switch on one term. A fixed 0.040 m² loop facing a field rising at 1.5 T s⁻¹ has dΦ/dt = 0.040 m² × 1.5 T s⁻¹ = 0.060 Wb s⁻¹, so one turn develops 60 mV. But the terms are signed and they can fight: a loop being squeezed while the field grows has a positive first term and a negative second, and at the right squeeze rate the emf is exactly zero while both the field and the area are changing fast. When the field is not uniform over the loop, go back to Φ = ∫B⋅dA, integrate first with the geometry frozen, then differentiate the result.

A coil of N identical turns is N loops in series, each threaded by the same flux Φ, so their emfs add: ε = −N dΦ/dt. The bookkeeping is tidier in terms of the flux linkage λ = NΦ, since then ε = −dλ/dt for any coil. Two cautions. The word identical is doing work: near the ends of a short solenoid, or in a coil wider than the magnet that feeds it, the outer turns link less flux and the sum must be done turn by turn. And a coil with more turns is not automatically a better current source. Winding twice as many turns of the same wire round the same former doubles the emf but also doubles the length of wire, hence the resistance, so I = ε/R barely moves. More turns buy voltage; a lower-resistance circuit buys current.

Since ε = −N dΦ/dt, the emf waveform is the negative slope of the flux waveform, scaled by N. A flat stretch of flux gives zero emf however high it sits. A straight ramp gives a constant emf, so a triangular flux gives a square-wave emf, and the sharp corner where the ramp reverses appears as an instantaneous jump in the emf. A steeper ramp in half the time gives twice the emf, which is why short, fast changes are the violent ones. The sinusoidal case runs the whole course: Φ = Φ₀ sin ωt gives ε = −NΦ₀ω cos ωt, an emf a quarter cycle behind the flux with amplitude NΦ₀ω. With N = 250 turns, Φ₀ = 2.0 mWb and f = 10 Hz, ω = 2π(10) = 62.8 rad s⁻¹ and the peak emf is 250 × 2.0 × 10⁻³ Wb × 62.8 s⁻¹ = 31 V. Double the frequency and the peak doubles, though the flux curve keeps its height.

Faraday's law can also be run in reverse. With a total circuit resistance R the current is I = −N(dΦ/dt)/R, and integrating over a flux change gives the charge q = ∫I dt = N|ΔΦ|/R. Every trace of the timing has gone: the same charge circulates whether the change takes a millisecond or an hour, which is exactly why a fluxmeter measures charge in order to get ΔΦ. Calculus handles the smooth cases the same way. A search coil of 350 turns and area 4.0 cm² in a field decaying as B = (0.60 T) e^(−t/τ) with τ = 12 ms sees ε = NAB₀ e^(−t/τ)/τ, which starts at 350 × 4.0 × 10⁻⁴ m² × 0.60 T ÷ 0.012 s = 7.0 V and then decays with that same τ. Integrating the emf over all time recovers NAB₀, so with R = 22 Ω the total charge is 0.14 m² × 0.60 T ÷ 22 Ω = 3.8 mC.