Faraday's law of electromagnetic induction

Why it matters

Faraday's law is behind a large share of instrumentation: electromagnetic flowmeters, tachogenerators, LVDTs, current transformers, search coils, eddy-current proximity probes and induction-type energy meters. It also explains unwanted effects — pick-up in measurement loops, eddy-current losses in cores — and it is the first of Maxwell's equations that couples electric and magnetic fields.

Key ideas

The law. The EMF induced around a closed path equals the negative rate of change of the magnetic flux linking it: e = −dΦ/dt for one turn, e = −N dΦ/dt = −dλ/dt for N turns. Here Φ = ∫B·dS through any surface bounded by the path.

Three ways to change the flux:

1. Transformer EMF — a stationary loop in a time-varying B (transformers, CTs, search coils).
2. Motional EMF — a conductor moving through a steady B (generators, flowmeters, tachogenerators). Each free charge feels Q u × B, giving an EMF ∫(u × B)·dl, which for a straight rod perpendicular to B and to its velocity reduces to Bℓu.
3. Both together — a moving loop in a time-varying field; the total EMF is the sum.

Lenz's law (the minus sign). The induced current flows in the direction whose own magnetic field opposes the change in flux. This is energy conservation: if the induced current aided the change, it would grow without limit. The mechanical work needed to move a conductor against the retarding force is exactly the electrical energy delivered.

Point form. Using Stokes' theorem, ∮E·dl = −d/dt ∫B·dS becomes ∇ × E = −∂B/∂t for stationary paths. So a time-varying B produces an electric field with non-zero curl: the electric field is no longer conservative, and "voltage" between two points depends on the path — which is why the routing of measurement leads matters near AC magnetic fields.

Rotating coil. A coil of N turns and area A rotating at angular speed ω in a uniform B has λ = NBA cos ωt and e = NBAω sin ωt: the principle of the AC generator and the tachogenerator, whose output is proportional to speed.

Electromagnetic flowmeter. A conducting liquid flowing with mean velocity v through a pipe of diameter d across a field B generates e = B d v between electrodes on opposite walls. The output is linear in flow, independent of viscosity and density, but the liquid must be conductive.

Eddy currents. Changing flux in solid conductors induces circulating currents that heat them (loss roughly proportional to f²B²t² for laminations of thickness t) and create opposing fields. They are reduced by laminations or ferrites, and exploited in eddy-current brakes, induction heating, energy-meter discs and proximity sensors.

Formulas

• e = −N dΦ/dt = −dλ/dt — Faraday's law (V); Φ in Wb, λ = NΦ in Wb-turns.
• Φ = ∫ B·dS = BA cos θ — uniform B; θ between B and the loop's normal.
• ∮ E·dl = −d/dt ∫ B·dS — integral form; ∇ × E = −∂B/∂t — point form.
• e = ∮ (u × B)·dl, e = Bℓu — motional EMF; ℓ = rod length (m), u = speed (m/s), mutually perpendicular.
• e = NBAω sin ωt, E_max = NBAω, E_rms = NBAω / √2 — rotating coil; ω = 2πN_r/60 for N_r rpm.
• e = B d v — electromagnetic flowmeter; d = pipe diameter (m), v = mean velocity (m/s).
• E_rms = 4.44 f N Φ_max — transformer EMF for sinusoidal flux.

Worked examples

Example 1 (standard). A circular loop of radius 0.1 m lies perpendicular to a uniform field that falls from 0.5 T to 0 in 0.2 s. Find the average EMF.

1. A = π(0.1)² = 0.0314 m²; Φ1 = 0.5 × 0.0314 = 0.0157 Wb; Φ2 = 0.
2. e = −ΔΦ/Δt = −(0 − 0.0157)/0.2 = 78.5 mV.
3. Direction (Lenz): the induced current tries to maintain the vanishing flux, so its own field is in the same direction as the original B.

Example 2 (motional EMF). A rod 0.5 m long slides at 4 m/s along rails in a perpendicular field of 0.8 T; the rails are joined through 2 Ω (rod and rail resistance neglected). Find the EMF, current, retarding force and power.

1. e = Bℓu = 0.8 × 0.5 × 4 = 1.6 V.
2. I = e/R = 1.6/2 = 0.8 A.
3. Retarding force F = BIℓ = 0.8 × 0.8 × 0.5 = 0.32 N, opposing the motion (Lenz).
4. Mechanical power F × u = 0.32 × 4 = 1.28 W = electrical power eI = 1.6 × 0.8 = 1.28 W ✓ — energy is conserved.

Example 3 (GATE level, instruments). (a) An electromagnetic flowmeter on a 0.1 m pipe uses B = 0.05 T, and the liquid flows at 2 m/s. Find the electrode voltage and the volume flow. (b) A tachogenerator coil has 100 turns of area 0.01 m² in 0.5 T and turns at 3000 rpm. Find the peak and RMS EMF.

1. (a) e = Bdv = 0.05 × 0.1 × 2 = 10 mV. Volume flow Q = (πd²/4)v = 7.854 × 10⁻³ × 2 = 0.0157 m³/s. The sensitivity is e/Q = 0.637 V per m³/s, a constant: the meter is linear.
2. (b) ω = 2π × 3000/60 = 314.16 rad/s. E_max = NBAω = 100 × 0.5 × 0.01 × 314.16 = 157.1 V; E_rms = 157.1/√2 = 111.1 V.

Common mistakes

• Using B instead of flux: e depends on the change of BA cos θ, so a field parallel to the plane of a loop induces nothing, however fast it changes.
• Measuring θ from the plane of the loop rather than its normal.
• Forgetting N, or multiplying flux by N twice (λ already includes N).
• Saying "a constant field induces no EMF" — a conductor moving through a constant field does get an EMF.
• Treating Lenz's law as opposing the flux rather than the change in flux.
• Using rpm directly as ω; convert with ω = 2πN/60.

For GATE IN

Common items: average EMF for a changing field or area; motional EMF in sliding rods and rotating rods (e = ½Bωℓ² for a rod pivoted at one end); rotating-coil EMF; transformer EMF equation; flowmeter output and sensitivity; direction of induced current by Lenz's law; and ∇ × E = −∂B/∂t to find E from a given B(t). Practise identifying which of the three mechanisms is at work.

Quick check

1. A loop's flux changes from 0.5 Wb to 0.2 Wb in 0.1 s. What is the average EMF magnitude?
2. Does a loop moving at constant velocity through a uniform, steady B have an EMF around it?
3. What is e for a 0.5 m rod moving at 4 m/s perpendicular to 0.8 T?
4. Why are transformer cores laminated?
5. Does an electromagnetic flowmeter work with non-conducting oil?

Answers: 1. 3 V. 2. No — the flux through it does not change (the EMFs on opposite sides cancel). 3. 1.6 V. 4. To reduce eddy-current loss. 5. No — the liquid must be conductive.