Faraday's Law and the Flux Rule

Derivation

Faraday's Law and the Flux Rule

D-160 Home PU-204 Threads fields · force · energy Depends on lorentz-force-law, gauss-law-for-magnetism

Statement

For any circuit C bounding a surface S, the electromotive force equals minus the rate of change of magnetic flux, ℰ = −dΦ_B/dt, and this single "flux rule" resolves into two physically distinct contributions — the induced curl of the electric field, giving the differential law ∇ × E = −∂B/∂t, and the motional term ∮ (v × B)·dl arising from the Lorentz force on charges carried by a moving or deforming conductor.

Why it matters

Faraday's law is the electromagnetic induction principle underlying every generator, transformer, inductor, and betatron, and it is the third of Maxwell's equations — the one that couples a time-varying magnetic field to a circulating electric field and thereby makes electromagnetic waves possible.

Its most subtle feature is that the deceptively simple flux rule ℰ = −dΦ_B/dt secretly bundles together two unrelated pieces of physics: a genuine field effect (a changing B creating a non-conservative E) and a mechanical effect (charges pushed by v × B). Understanding when the flux rule holds — and the celebrated cases where it fails — is a genuine test of physical maturity.

Assumptions

The Lorentz force law holds for the force per unit charge — if the force on a charge were not f = E + v × B, the motional EMF would take a different form and the two contributions would not combine into a single flux derivative.

The magnetic field is divergence-free, ∇·B = 0 — the flux transport theorem contains a term v(∇·B); if magnetic monopoles existed this term would survive and the clean result ℰ = −dΦ_B/dt would acquire a monopole correction.

The surface S is a well-defined material or geometric surface whose boundary is a single closed curve — if the "circuit" cannot be spanned by a surface deforming continuously with it (a sliding contact that switches branches, a broken loop), Φ_B is ambiguous and the flux rule breaks.

Fields and the loop velocity are non-relativistic, v « c — at relativistic speeds E and B mix under Lorentz transformation and the split into "field-induced" and "motional" EMF becomes frame-dependent; only the covariant statement ∂_[μF_νλ] = 0 is invariant.

The surface integral and its time derivative may be interchanged (fields are smooth enough for the Leibniz rule), — at a moving material discontinuity or shock the differentiation under the integral sign is invalid and boundary terms must be added by hand.

Derivation

ℰ = ∮_C f·dl, f = E + v × B

Definition of EMF as the work done per unit charge around the loop, using the assumed Lorentz force per unit charge on a charge that is convected with the wire at velocity v. A

ℰ = ∮_C E·dl + ∮_C (v × B)·dl

Split the line integral by linearity into a purely field term and a motional term; nothing yet assumes anything about how the fields evolve. A

∮_C E·dl = −d/dt ∫_S B·dA (loop at rest)

Experimental Faraday result for a stationary loop: a changing flux drives a circulating E. Here v = 0 so only this "transformer" term acts. This is the empirical input that promotes electrostatics (∮E·dl = 0) to electrodynamics. B

∫_S (∇ × E)·dA = −∫_S ∂B/∂t ·dA

Apply Stokes' theorem to the left side of Step 3, and move the time derivative inside the fixed-surface integral (legal because S is stationary, so only the explicit time dependence of B contributes). B

∇ × E = −∂B/∂t

The equality of surface integrals in Step 4 holds for every surface S and its bounding curve; a field whose flux integral vanishes over all surfaces must vanish pointwise. This is the differential (local) form of Faraday's law. C

d/dt ∫_S(t) B·dA = ∫_S ∂B/∂t ·dA + ∫_S v(∇·B)·dA − ∮_C (v × B)·dl

Flux transport (Helmholtz/Leibniz) theorem for a surface whose boundary moves with velocity v: the flux changes because B itself changes in time (first term), because sources are swept through (second term), and because the loop sweeps new area (third term). C

dΦ_B/dt = ∫_S ∂B/∂t ·dA − ∮_C (v × B)·dl

Drop the middle term of Step 6 using the prior result ∇·B = 0 (Gauss's law for magnetism). This is why no-monopoles is essential to the flux rule. C

∮_C E·dl = −∫_S ∂B/∂t ·dA

Re-integrate the differential law of Step 5 over the (now moving) surface via Stokes; note the right side involves the explicit partial ∂B/∂t only, evaluated at the instantaneous position of S. B

ℰ = −∫_S ∂B/∂t ·dA + ∮_C (v × B)·dl

Substitute Step 8 into Step 2. The first term is the field-induced (transformer) EMF; the second is the motional EMF. Both are now expressed with the same fields and the same loop. B

ℰ = −(∫_S ∂B/∂t ·dA − ∮_C (v × B)·dl) = −dΦ_B/dt

Recognise the bracket as exactly the right-hand side of the flux transport theorem, Step 7. The two physically distinct EMFs have merged into a single total flux derivative — the flux rule. C

Result

∇ × E = −∂B/∂t ⇔ ∮_C E·dl = −∫_S ∂B/∂t ·dA ⇒ ℰ = −dΦ_B/dt

Reading. The local law says a magnetic field changing in time is inseparable from an electric field that curls around it; the electric field lines close on themselves rather than terminating on charges. The global flux rule then states that the EMF around any loop is minus the total rate of change of the magnetic flux threading it — whether that change comes from the field varying in time (transformer action) or from the loop moving and reshaping in a static field (motional action). The minus sign is Lenz's law: the induced EMF opposes the change that produced it, guaranteeing energy conservation.

Units check. Φ_B has units of tesla·metre² = weber = volt·second, so dΦ_B/dt is in volts, matching ℰ. For the local law, ∇×E has units (V/m)/m = V/m², while ∂B/∂t is T/s = (V·s/m²)/s = V/m² — both sides agree.

Limiting cases

Static fields (∂B/∂t = 0) and stationary loop: both terms vanish, ∮E·dl = 0 — electrostatics is recovered, E is conservative.
Uniform static B, moving rigid loop: pure motional EMF ℰ = ∮(v×B)·dl; the induced E in the lab frame is zero and all the EMF is mechanical Lorentz work.
Fixed loop, time-varying uniform B: pure transformer EMF ℰ = −A dB/dt; there is a real circulating E even in empty space where no wire exists.
Slowly varying (quasi-static) regime: displacement-current feedback is negligible, Φ_B is well defined instantaneously, and lumped-circuit inductance ℰ = −L dI/dt emerges.

Breaks when

The circuit topology changes discontinuously. In Faraday's copper-disc (homopolar) generator and in sliding-contact "paradoxes", the material path of the current is not a fixed loop deforming continuously; the flux rule ℰ = −dΦ_B/dt can give the wrong answer while the fundamental law ℰ = ∮(E + v×B)·dl stays correct. Always trust the force law, not the flux bookkeeping, at switching contacts.
Magnetic monopoles are present (∇·B ≠ 0). The v(∇·B) term in the transport theorem no longer vanishes, so a moving loop in a monopole field acquires an extra EMF not captured by −dΦ_B/dt alone.
Relativistic loop or field speeds. When v ∼ c the separation into "transformer" and "motional" EMF is frame-dependent, retardation matters, and only the covariant Faraday tensor equation is unambiguous.
The surface cannot be defined / the loop is not closed. With a broken circuit, a spark gap, or a fractally deforming boundary, Φ_B has no single value and the rule is meaningless; only the local ∇×E = −∂B/∂t survives.

Failure modes

Sign errors from Lenz's law: choosing the surface normal and the loop circulation direction inconsistently (they must obey the right-hand rule together), then getting the polarity of ℰ backwards.
Confusing total and partial derivatives: writing ∮E·dl = −dΦ/dt for a moving loop — the line integral of E equals only the ∫∂B/∂t·dA piece; the full dΦ/dt also carries the motional term.
Double-counting the motional EMF: including v×B work and the full flux change for a moving loop, effectively counting the same physics twice.
Forgetting the turns factor N: using ℰ = −dΦ/dt for one turn when the coil has N turns; the correct flux linkage is NΦ.
Assuming E = 0 inside the region because "there's no charge": a changing B produces a non-conservative E with no source charges at all.
Applying the flux rule to a homopolar disc without checking that the conducting path is a continuously deforming loop — the classic disc-generator trap.

Discussion

The deepest lesson of Faraday's law is that the flux rule is a coincidence of two laws, not a fundamental principle in its own right. The field-induced part comes from Maxwell's equation ∇×E = −∂B/∂t; the motional part comes from the Lorentz force v×B. Feynman emphasised that "in general the two effects contribute" and that no other place in physics has such a simple and accurate principle needing two distinct phenomena to be understood. The transport theorem shows algebraically why they conspire to give a single flux derivative, but the conspiracy relies on ∇·B = 0 — remove the monopole assumption and the coincidence dissolves.

Physically, the local law tells us that electric field lines can form closed loops. In electrostatics every field line begins and ends on charge, so ∮E·dl = 0 and a scalar potential exists. Once ∂B/∂t ≠ 0, part of E becomes solenoidal (curling), the line integral around a closed path is nonzero, and no single-valued potential describes it. This is why we introduce the vector potential and write E = −∇φ − ∂A/∂t: the induced field is precisely the −∂A/∂t piece, and ∇×E = −∂(∇×A)/∂t = −∂B/∂t follows identically.

Faraday's law is also the energy-thread partner of Ampère's law. Together with the displacement current they make E and B mutually regenerating, which is the origin of electromagnetic radiation: a changing B curls up an E, whose change curls up a B, and the disturbance propagates at c = 1/√(μ₀ε₀). The minus sign (Lenz's law) is not a separate postulate but a thermodynamic necessity: a plus sign would make induced currents reinforce their cause, giving runaway energy growth and violating conservation.

At the covariant level all of this collapses into one line. Faraday's law and Gauss's law for magnetism are the four components of the homogeneous Maxwell equation ∂_[αF_βγ] = 0, equivalently dF = 0 where F is the electromagnetic 2-form. The statement dF = 0 is a pure geometric identity (the field 2-form is closed), so the vector potential exists globally on contractible regions, F = dA. The split of F into E and B, and hence the split of EMF into transformer and motional parts, is merely the observer-dependent way a boost slices the invariant 2-form — which is exactly why the two "different" phenomena were destined to unify.

Common misconceptions. The flux rule is often taught as the definition of Faraday's law, but it is a derived convenience with genuine exceptions (homopolar disc, switching contacts). The fundamental laws are the local ∇×E = −∂B/∂t together with the Lorentz force; when in doubt, integrate the force per charge directly rather than counting flux.

Worked examples

AC generator (rotating coil): transformer-type flux change from a rotating loop in a uniform field.

A flat coil of N = 200 turns and area A = 0.010 m² spins at f = 50 Hz in a uniform field B = 0.50 T. Find the peak EMF. B

Φ_B(t) = B·A cos θ = B A cos(ωt), ω = 2πf

Flux through one turn as the normal makes angle θ = ωt with B; symbols first. A

ℰ = −N dΦ_B/dt = N B A ω sin(ωt) ⇒ ℰ_peak = N B A ω

Differentiate and multiply by turns; the peak occurs when sin(ωt) = 1. B

ω = 2π(50) = 314.2 rad/s ⇒ ℰ_peak = 200 × 0.50 × 0.010 × 314.2

Insert numbers only now, all SI. A

ℰ_peak ≈ 314 V (≈ 0.31 kV)

Reading. The output is a sinusoid of amplitude 314 V at 50 Hz; the EMF is largest when the coil plane is parallel to B (flux zero, changing fastest) and zero when the plane is perpendicular (flux maximal, momentarily stationary). Units check. [1][T][m²][s−¹] = T·m²/s = Wb/s = V. ✓

Sliding rod (motional EMF and reaction force): pure Lorentz-force induction on a moving conductor.

A rod of length L = 0.20 m slides at v = 3.0 m/s along rails in a uniform field B = 0.40 T perpendicular to the plane; the circuit resistance is R = 0.50 Ω. Find the EMF, current, and the force needed to keep the rod moving steadily. B

ℰ = ∮(v × B)·dl = B L v (v ⊥ B ⊥ L)

Only the rod contributes; v×B is uniform along the rod and parallel to it, so the line integral is just the product. B

I = ℰ/R, F_ext = B I L = B²L²v/R (steady state)

Ohm's law for the loop; at constant velocity the applied force balances the magnetic braking force BIL (Lenz's law opposes the motion). B

ℰ = 0.40 × 0.20 × 3.0; I = 0.24/0.50; F = (0.40)²(0.20)²(3.0)/0.50

Numbers in SI. A

ℰ = 0.24 V, I = 0.48 A, F_ext = 0.038 N

Reading. Mechanical power delivered F_extv = 0.038 × 3.0 = 0.115 W exactly matches electrical dissipation I²R = (0.48)²(0.50) = 0.115 W — the flux rule here is entirely motional, yet energy balances perfectly. Units check. [T][m][m/s] = T·m²/s = V; [T][A][m] = N. ✓

Problems

(A) Solenoid ramp. A long solenoid has n = 1000 turns/m carrying a current increasing at dI/dt = 50 A/s. A single-turn loop of radius r = 2.0 cm encircles it coaxially inside the solenoid. Find the induced EMF.
Solution
Inside, B = μ₀nI, uniform, so flux through the loop uses the solenoid area only if the loop is inside; here the loop radius equals its own area A = πr². ℰ = A μ₀n dI/dt = π(0.020)²(4π×10−⁷)(1000)(50). Compute: πr² = π(4.0×10−⁴) = 1.257×10−³ m²; μ₀n dI/dt = (1.2566×10−⁶)(1000)(50) = 0.0628 T/s. So ℰ = 1.257×10−³ × 0.0628 = 7.9×10−⁵ V ≈ 79 μV.
(A) Flux linkage and turns. A 500-turn coil of area 2.0×10−³ m² sits in a field that drops uniformly from 0.80 T to 0.20 T in 0.10 s, normal to the coil. Find the average EMF.
Solution
ℰ = −N ΔΦ/Δt = −N A ΔB/Δt. ΔB = 0.20 − 0.80 = −0.60 T. ℰ = −(500)(2.0×10−³)(−0.60)/(0.10) = (500)(2.0×10−³)(6.0) = 6.0 V. Positive, and its polarity opposes the decrease (drives current to maintain flux).
(B) Rotating rod. A conducting rod of length L = 0.50 m rotates about one end at angular speed ω = 20 rad/s in a uniform field B = 0.30 T parallel to the rotation axis. Find the EMF between the ends.
Solution
Each element at radius s moves at v = ωs, contributing dℰ = (v×B)·dl = Bωs\,ds. Integrate: ℰ = ∫₀^L Bωs\,ds = ½BωL². Numerically ℰ = ½(0.30)(20)(0.50)² = ½(0.30)(20)(0.25) = 0.75 V. Note the flux rule needs care here (rotating radial "loop"); integrating the motional force directly is safest.
(B) Falling loop through a field edge. A square loop of side a = 0.10 m, resistance R = 0.20 Ω, mass m = 5.0 g, falls under gravity with its lower edge crossing the sharp boundary of a region of field B = 0.50 T (horizontal, into the loop). Find the terminal velocity.
Solution
While one edge is in the field, motional EMF ℰ = Bav, current I = Bav/R, retarding force F = BIa = B²a²v/R (upward, Lenz). Terminal velocity when F = mg: v_t = mgR/(B²a²). Numerically v_t = (5.0×10−³)(9.81)(0.20)/[(0.50)²(0.10)²] = (9.81×10−³)/(2.5×10−³) = 3.9 m/s.
(C) Transformer vs motional decomposition. A rectangular loop of width w = 0.20 m lies in the xy-plane with its right edge at x = ut (moving at u = 2.0 m/s), in a field B(x,t) = B₀(1 + kx)ẑ with B₀ = 0.40 T, k = 0.50 m−¹, that also grows as B ∝ e^αt with α = 3.0 s−¹, evaluated at t = 0, loop from x = 0 to x = 0.30 m, length ℓ = 0.15 m in y. Identify the transformer and motional parts of ℰ.
Solution
Write B(x,t) = B₀(1+kx)e^αt. Transformer part: ℰ_T = −∫∂B/∂t\,dA = −αℓ∫₀^0.30B₀(1+kx)dx. The integral = B₀[x + ½kx²]₀^0.30 = 0.40[0.30 + ½(0.50)(0.09)] = 0.40[0.30+0.0225] = 0.129 Wb/m. Times ℓ = 0.15: flux = 0.01935 Wb; times α = 3.0: ℰ_T = −0.058 V. Motional part: only the moving right edge at x = 0.30 contributes, ℰ_M = (u×B)·(edge) = u B(0.30,0) ℓ = (2.0)(0.40)(1+0.50·0.30)(0.15) at t=0 = (2.0)(0.40)(1.15)(0.15) = 0.138 V. Total ℰ = ℰ_M + ℰ_T = 0.138 − 0.058 = 0.080 V, which equals −dΦ/dt computed directly — confirming the flux rule bundles both.