Faraday's Law of Induction

On 29 August 1831, Michael Faraday wound two coils onto opposite sides of an iron ring, connected one to a battery and the other to a galvanometer, and saw something maddening: the needle kicked only at the instant he closed the switch, and kicked the other way when he opened it. A steady current — however large — induced nothing. Only change did. In that flicker lives essentially every watt of electricity humanity has generated since. Notice what Faraday's ring does that the last article's machines cannot. Motional EMF pumped charge with the magnetic force on moving conductors — rod on rails, spinning disk — and its law, $\oint(\mathbf{v}\times\mathbf{B})\cdot d\boldsymbol{\ell}$ , is old physics cleverly deployed. But both of Faraday's coils are bolted down. Nothing moves. The charges in the secondary have $\mathbf{v} = 0$ when the needle kicks, so the magnetic force on them is exactly zero — and yet they circulate. Something else pushes, and it can only be an electric field: a changing magnetic field creates a circulating, non-conservative $\mathbf{E}$ , wire or no wire. That is genuinely new physics — the first amendment to the field equations since electrostatics, and one quarter of Maxwell's completed set. The article delivers the law in all three dresses: the universal flux rule $\varepsilon = -d\Phi/dt$ , unified across its two mechanisms (the motional half recalled from its owner; the strange fact that two unrelated mechanisms share one formula is the "asymmetry" that opens Einstein's 1905 paper — the relativistic chapter settles it); the differential form $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ , derived by Stokes' theorem; and the working form Faraday himself measured with — the ballistic charge theorem $q = \Delta\Phi/R$ , profile-independent, still the cleanest way to weigh a flux.

Key takeaways

The new physics: a changing magnetic field creates an electric field. For a stationary loop in a changing $\mathbf{B}$ , the charges feel no magnetic force at all — the pump is a genuinely circulating $\mathbf{E}$ with $\oint\mathbf{E}\cdot d\boldsymbol{\ell} \neq 0$ .
The universal flux rule: $\varepsilon = -d\Phi/dt$ covers both mechanisms — loop moving through static field (motional EMF, owned by Motional EMF and recalled here) and field changing through static loop (transformer EMF, owned here). Two unrelated mechanisms, one formula: the coincidence that opens Einstein's 1905 relativity paper.
Differential form, by Stokes on the stationary-loop case: $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ — electrostatics' $\nabla\times\mathbf{E} = 0$ (The Curl of the Electrostatic Field) is repealed as an equation about statics and re-enacted as the static limit.
The induced $\mathbf{E}$ is non-conservative: no single-valued potential describes it, its field lines close on themselves with no charges anywhere, and "the voltage between two points" becomes path-dependent when flux is changing.
The minus sign is Lenz's reading — induced effects oppose their cause, else flux would amplify itself into free energy; the full direction doctrine is Lenz's Law, next.
Faraday's own metrology, derived: the ballistic charge theorem $q = N\Delta\Phi/R$ — total charge through the galvanometer counts the flux change regardless of how fast it happened.
$N$ turns multiply the EMF; coils coupled through shared flux preview Mutual Inductance; computing the induced $\mathbf{E}$ itself is Induced Electric Fields' toolkit.
KEY IDEA: $\varepsilon = -d\Phi/dt$ , and locally $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ — changing magnetic flux pumps charge, changing magnetic field curls electric field, and the sign always bills whoever makes the change.

The core insight

Faraday's law: a changing magnetic flux through a loop drives an EMF $\varepsilon = -d\Phi/dt$ around it — for stationary circuits because a changing magnetic field creates a circulating non-conservative electric field, $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ , one of Maxwell's equations and the half of induction that is genuinely new physics.

Where this lives today

Roughly 95 percent of the world's electricity still arrives by equation (6): something — steam, falling water, wind — spins flux past windings, and $NBA\omega\sin\omega t$ appears on the wires. Induction's second century belongs to the smaller loops. An induction cooktop drives kilohertz current under the glass and induces eddy currents directly in the pan's steel — the pan is the secondary of an air-gapped transformer, and the stove never gets hot. Wireless chargers are the same transformer with the lid on: transmit and receive coils sharing ~100 kHz flux, alignment and coupling engineering all downstream of $M\,dI/dt$ . An EV braking regeneratively runs its motor as a generator, the induced back-EMF converting kinetic energy to battery charge with Lenz's opposition supplying the braking force. And at the exotic end, maglev guideway coils levitate trains on induced repulsion, while the magnet dropped down a copper pipe — the lecture demonstration that falls in eerie slow motion — remains the cheapest complete summary of the law: flux change billed, in real time, by eddy currents whose full story is Eddy Currents, this chapter's closer.

Origins · 1 minMichael Faraday and how this was discovered

Michael Faraday

1791–1867 · Royal Institution, London

A bookbinder's apprentice who taught himself science from the volumes he stitched, Faraday became the greatest experimentalist of his century without ever learning much mathematics. Hired by Davy as a bottle-washer after taking meticulous notes at his lectures, he spent a decade hunting the converse of Ørsted's discovery — if currents make magnetism, magnetism must make currents — and found it on 29 August 1831 with the iron ring, two coils, and a galvanometer that flicked only at make and break. Within months he had the moving-magnet version, the rotating-disk generator, and the ballistic flux-weighing method of worked example 2; within a decade, the concept that outlived all his apparatus — the field, pictured as lines of force filling space, with induction as their counting rule. He declined a knighthood and the Royal Society presidency twice, gave the Christmas Lectures for children for decades, and left notebooks with not one equation. Maxwell, who turned the lines of force into the equations carrying this article's law, insisted Faraday "was in reality a mathematician of a very high order." Asked (perhaps apocryphally) what use induction was, he is said to have answered: what use is a new-born baby?

History

1820

— Ørsted shows a current deflects a compass: electricity makes magnetism, and the hunt for the converse begins

1831

— Faraday's iron ring, 29 August: the needle kicks at make and break only; Henry finds induction independently in Albany

1832

— Faraday establishes the quantitative flux dependence with ballistic charge measurements

1834

— Lenz states the direction rule: induced effects oppose their cause

1865

— Maxwell embeds the law as curl E = -dB/dt among his field equations, and light falls out as a wave

1882

— Pearl Street lights Manhattan with dynamos; within a decade Westinghouse's AC — transformers, pure Faraday — wins the grid

1905

— Einstein opens his relativity paper with this article's asymmetry: two mechanisms, one current, one physics

The physical picture

Picture magnetic field lines threading a wire loop like strands of spaghetti through a ring. The loop keeps count of one number only — how many lines pass through, weighted by strength and slant: the flux $\Phi = \int\mathbf{B}\cdot d\mathbf{a}$ . Strengthen the field, tilt or shrink the loop, shove a magnet closer, and the count changes; the loop answers with a current whose own field tries to restore the old count. Flux behaves as if it had inertia, and the induced current is the pushback. Watch it in the classic demonstration. Drive a bar magnet north-pole-first toward a coil and the induced current circulates so that the coil's near face becomes a north pole too — the magnet is repelled, and your hand feels the resistance. Pull the magnet away and the current reverses; the near face turns south and clings. Either way you do work against the induced effects, and that work is precisely the electrical energy the coil delivers. This opposition is Lenz's law, and at this article's register it needs its energy argument stated once, plainly: if the induced current reinforced a small flux increase, more flux would drive more current driving more flux — an exponential runaway minting energy from nothing. The minus sign in Faraday's law is energy conservation wearing algebra. (The full direction doctrine — reading signs in machines, the demagnetization-drive subtleties — is owned by Lenz's Law, next door.) Now the discriminating experiment, because the picture so far could still be Motional EMF in disguise. Hold the coil absolutely fixed and vary the current in a neighboring solenoid instead. No conductor moves; every charge in the detector coil sits at $\mathbf{v} = 0$ ; the magnetic force $q\mathbf{v}\times\mathbf{B}$ on each is identically zero. The galvanometer kicks anyway. Whatever pushes those charges is an electric field — one that exists in closed circulating loops around the changing flux, in the empty space around the solenoid, whether or not any wire is present to feel it. The coil is merely a detector. A changing $\mathbf{B}$ curls up an $\mathbf{E}$ : that sentence is the new law of nature this article exists to state, and — coupled with its symmetric twin, gestured at the close — the reason light can propagate through vacuum.

Figure

Two mechanisms, one formula - move the loop through the field, or change the field through the loop

The law, in loop form

The bookkeeping quantity first. Through any surface $S$ spanning the loop,

\Phi = \int_S \mathbf{B}\cdot d\mathbf{a} \qquad (1)

where: $\Phi$ — magnetic flux, webers (1 Wb = 1 T·m²); which spanning surface you choose is irrelevant, because $\nabla\cdot\mathbf{B} = 0$ makes the flux surface-independent (Divergence of B and Absence of Magnetic Monopoles, spent silently every time anyone says "the" flux through a loop). For a flat loop in a uniform field, $\Phi = BA\cos\theta$ — and every handle on the count ( $B$ , $A$ , $\theta$ ) will turn out to induce. Faraday's needle flicked only while flux was changing, harder for faster changes. Everything the galvanometer recorded compresses into one line:

\varepsilon = -\frac{d\Phi}{dt} \qquad (2)

where: $\varepsilon$ — the EMF around the loop in the sense fixed by the right-hand rule on the chosen normal; the minus sign is Lenz's reading. Note the structure: $\Phi$ itself never appears, only its rate — a ten-tesla steady flux induces nothing while a millitesla wobble induces plenty. The loop is a rate-meter, not a flux-meter. An $N$ -turn coil links the changing flux $N$ times over, and the turn EMFs add in series like stacked batteries: $\varepsilon = -N\,d\Phi/dt$ — the one free multiplier in the law, and the reason real machine windings carry thousands of turns.

Derivation AThe universal flux rule - two mechanisms unified in one line

The goal is the unification: for a loop that may simultaneously move, deform, AND sit in a time-varying field, the total EMF — motional pump plus induced-field pump — equals $-d\Phi/dt$ in one stroke. The strategy is to split the flux's total time derivative into its two causes and match each against the pump it feeds: the swept-ribbon theorem of Motional EMF handles the motion half (recalled, not re-proved), and the new field law — taken here in its integral form as the stationary-loop experimental fact — handles the field half. Watch the bookkeeping: nothing cancels or conspires; the two halves simply add on both sides of the equation.

Step 1. Split the flux change. Between $t$ and $t+dt$ the flux through the (moving) loop changes for two independent reasons: the field itself changed everywhere, and the loop moved through a nonuniform field pattern. Exactly,

\frac{d\Phi}{dt} = \underbrace{\int_S \frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{a}}_{\text{field changed}} \; + \; \underbrace{\left(\frac{d\Phi}{dt}\right)_{\rm motion}}_{\text{loop moved}}

Step 2. Pay the motion half with old physics. The swept-ribbon theorem (Motional EMF, recalled by name — its proof needed only $\nabla\cdot\mathbf{B} = 0$ and the Lorentz force) states

\oint(\mathbf{v}\times\mathbf{B})\cdot d\boldsymbol{\ell} = -\left(\frac{d\Phi}{dt}\right)_{\rm motion}

Step 3. Pay the field half with the new law. For the stationary-loop experiment, the induced electric field's circulation is the EMF, and Faraday's measurements say it equals minus the flux change — which, with the loop fixed, is entirely the field term:

\oint\mathbf{E}\cdot d\boldsymbol{\ell} = -\int_S\frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{a}

Why this step is irreducibly new: no force law, no theorem, no transformation of anything already in the track produces it. It is a fresh experimental fact about fields — the amendment, not a corollary.

Step 4. Add. The total EMF around the loop is the total pump — induced field on every element plus magnetic force on every moving element:

\varepsilon = \oint\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)\cdot d\boldsymbol{\ell} = -\int_S\frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{a} - \left(\frac{d\Phi}{dt}\right)_{\rm motion} = -\frac{d\Phi}{dt} \qquad (3)

One formula, both faces. The per-unit-charge pump $\mathbf{E} + \mathbf{v}\times\mathbf{B}$ is the full Lorentz force (Lorentz Force Law), so equation (3) is also the statement that the flux rule knows everything the Lorentz force knows about loops. Verified: reduces correctly in both pure cases — v = 0 recovers Step 3, static B recovers Step 2 = Motional EMF's theorem; premises audited: a material loop with well-defined element velocities (the disk machines that break this are owned by The Flux Rule and Its Exceptions); matches Griffiths §7.2.1's unification discussion and Zangwill §14.1.

The unification deserves one paragraph of wonder before it is put to work, because physics rarely hands out coincidences this large. Move the magnet toward the loop: the lab-frame story is an induced electric field pushing static charges. Move the loop toward the magnet — the same relative motion — and the story is a magnetic force on moving charges, with no induced field anywhere. Different mechanisms, different fields, identical current. Einstein found the duplication intolerable enough to make it the opening paragraph of "On the Electrodynamics of Moving Bodies" (1905): the asymmetry, he wrote, does not attach to the phenomena. Its resolution — that $\mathbf{E}$ and $\mathbf{B}$ are one tensor object whose split depends on the observer — is the relativistic chapter's payoff (Transformation of Electric and Magnetic Fields, Magnetism as a Relativistic Effect, E10). Faraday's law is where that thread visibly enters the subject.

The law, in field form

Go deeper · Derivation B · 1 minStokes on the stationary loop - the differential form of Faraday's law

Derivation BStokes on the stationary loop - the differential form of Faraday's law

The goal is the local law $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ : Faraday's discovery stripped of loops, wires and galvanometers, stated point by point in space — the form that joins Maxwell's equations. The strategy is the standard integral-to-differential conversion: take the stationary-loop integral law, apply Stokes' theorem to the circulation, and shrink the loop until the integrands must match. Watch the partial derivative: the loop is fixed, so the time derivative passes inside the surface integral as $\partial/\partial t$ — the motional case must NOT be fed into this argument, which is exactly why the unification derivation kept the halves separate.

Step 1. Start from the stationary-loop law (Step 3 of the unification), for an arbitrary fixed loop $C$ spanning a fixed surface $S$ :

\oint_C\mathbf{E}\cdot d\boldsymbol{\ell} = -\int_S\frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{a}

Step 2. Convert the circulation with the Curl (Stokes) Theorem (The Curl (Stokes) Theorem, E0's machinery earning its keep):

\int_S\left(\nabla\times\mathbf{E}\right)\cdot d\mathbf{a} = -\int_S\frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{a}

Step 3. Localize. The equality holds for every fixed loop and spanning surface — arbitrarily small, arbitrarily oriented. Two fields whose fluxes agree through every surface are equal:

\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} \qquad (4)

where: everything is local — the curl of $\mathbf{E}$ here, now is set by the rate of change of $\mathbf{B}$ here, now. No loop required: the law governs empty space.

Step 4. Read what it repeals and what it keeps. Electrostatics' $\nabla\times\mathbf{E} = 0$ (The Curl of the Electrostatic Field) survives as the static limit, and its Coulomb-field content survives entirely: the full electric field is the sum of a charge-sourced piece (curl-free, divergence set by $\rho/\varepsilon_0$ — Gauss's law is untouched) and an induction-sourced piece (divergence-free, curl set by equation (4)). The Helmholtz decomposition of E0 was built for exactly this moment: electrostatics plus this term, cleanly superposed. Verified: equation (4) is Maxwell's third equation in every convention (Griffiths eq. 7.16, Jackson §6.1); the localization step is the same every-loop argument that produced Gauss's differential form (Gauss's Law in Differential Form); static limit check ∂B/∂t = 0 recovers CURL.0464's law exactly.

Equation (4)'s most consequential fine print: the induced electric field is non-conservative. Its line integral around a closed path is not zero — that integral is the EMF, the whole point — so no single-valued potential $V$ can represent it, and "the voltage between two points" becomes path-dependent wherever flux is changing. (The lecture-demonstration version: two voltmeters attached to the same two points of a loop encircling a changing flux, reading different values because their leads close the circuit on opposite sides — a genuine effect, endlessly litigated online, and pure equation (4).) Induced- $\mathbf{E}$ field lines close on themselves, ringing the flux change, sourced by no charge anywhere. Computing those fields — the solenoid's interior rings, symmetry methods, the "Amperian loop" style of induction problems — is a toolkit of its own, owned by Induced Electric Fields three doors down; this article states the law, that one wields it.

Faraday's own meter: the ballistic charge theorem

Go deeper · Derivation C · 1 minThe charge through the galvanometer counts the flux change - however fast it happens

Derivation CThe charge through the galvanometer counts the flux change - however fast it happens

The goal is the working theorem of nineteenth-century magnetic metrology, and of Faraday's own quantitative experiments: when the flux linking an $N$ -turn circuit of total resistance $R$ changes by $\Delta\Phi$ , the total charge driven through the circuit is $q = N\Delta\Phi/R$ — independent of how quickly or smoothly the change occurred. The strategy is one integration of Faraday's law through Ohm's law. Watch what drops out: the entire time profile.

Step 1. At every instant, the loop rule (Electromotive Force) with Faraday's EMF: $I(t)R = -N\,d\Phi/dt$ .

Step 2. Integrate over the whole event, from before the change to after:

q = \int I\,dt = -\frac{N}{R}\int\frac{d\Phi}{dt}\,dt = \frac{N\,|\Delta\Phi|}{R} \qquad (5)

where: $q$ — total charge, which a ballistic galvanometer (a meter whose first swing measures the impulse of charge, not the current) reads directly. The time profile integrated away: yank the magnet or ease it, the charge is identical.

Step 3. Read it as an instrument. Equation (5) converts any flux to a charge measurement: flip a small "search coil" of known $NA$ in an unknown field and $q = 2NBA/R$ weighs $B$ — the standard field-mapping method for a century, and the logic by which Faraday established that induction responds to flux change, quantitatively, with 1831 apparatus. Why this theorem is pedagogically precious: it is the integral fingerprint of a rate law. Any law of the form "response ∝ d(state)/dt" leaves the state's total change imprinted on the time-integrated response — the same structure as impulse–momentum. *Verified: equation (5) matches Griffiths Prob. 7.28's logic and standard fluxmeter practice; worked example 2 runs the numbers (0.32 mC for a flipped 0.01 m² loop in 0.4 T through 25 Ω); the profile-independence is exact for lumped R circuits (self-inductance corrections — the current's own flux — are the fine print owned by Self-Inductance).*

The unified rule at work: the grid's waveform

The flux rule's most consequential single application closes the loop with Motional EMF's machines. Spin a coil of $N$ turns and area $A$ at angular speed $\omega$ in a uniform field: the flux swings as $\Phi = BA\cos\omega t$ , and the EMF — paying for change, never for size — is the derivative:

\varepsilon = NBA\omega\sin\omega t \qquad (6)

where: peak EMF $NBA\omega$ , frequency $\omega/2\pi$ . The EMF peaks exactly where the flux crosses zero — the coil edge-on, flux changing fastest — and vanishes at maximum flux: Faraday's needle-flick, spinning at 50 or 60 times a second. Segment by segment this machine is motional EMF (each side is a pulled rod; Motional EMF's FAQ ran that audit); by the unified rule it is one derivative. That double bookkeeping — the same generator computable by either face of equation (3) — is the practical dividend of the unification, and equation (6) is the design equation of every alternator from a bicycle dynamo to a turbine hall.

Figure

The AC generator - flux follows cos, EMF follows sin, change pays and size does not

Limiting cases

Case	Result	Meaning
static field, static loop	$\varepsilon = 0$	however colossal the flux — the loop is a rate-meter, not a flux-meter
loop moves, $\mathbf{B}$ static	$\varepsilon = \oint(\mathbf{v}\times\mathbf{B})\cdot d\boldsymbol{\ell}$	the motional face — owned by Motional EMF, contained in the unified rule
loop fixed, $\mathbf{B}$ changes	$\varepsilon = -\int(\partial\mathbf{B}/\partial t)\cdot d\mathbf{a}$	the transformer face — the new physics, owned here
both at once	$\varepsilon = -d\Phi/dt$ , total	the unification theorem; the full Lorentz pump $\mathbf{E} + \mathbf{v}\times\mathbf{B}$
$\partial\mathbf{B}/\partial t \to 0$	$\nabla\times\mathbf{E} \to 0$	electrostatics recovered as the static limit of equation (4)
$N$ -turn coil	$\varepsilon = -N\,d\Phi/dt$ ; $q = N\Delta\Phi/R$	series turns add; the ballistic theorem scales with them
circuit's own flux	$\varepsilon = -L\,dI/dt$	self-induction: the loop fights changes in its own current — Self-Inductance, ahead
two coupled circuits	$\varepsilon_2 = -M\,dI_1/dt$	the iron ring itself: transformers — Mutual Inductance, ahead

Common trapA strong magnetic field through a coil induces a current - and other rate-law misreadings

Not unless something changes. A coil bathed in a steady ten-tesla field with fixed geometry induces exactly zero, while a millitesla wobble induces plenty: Faraday's law bills $d\Phi/dt$ , never $\Phi$ . Two adjacent traps at this register. First, when $\partial\mathbf{B}/\partial t \neq 0$ the induced $\mathbf{E}$ is non-conservative — assigning it a potential and computing "the voltage between two points" gives path-dependent nonsense, which is why transformer-loop "voltmeter paradoxes" fool trained engineers; the honest quantity is the EMF of a specified path. Second, the flux rule is a theorem with premises, not bedrock: circuits with sliding contacts or exchanged material paths (the homopolar disk Motional EMF introduced) can defeat the loop bookkeeping while the Lorentz-force line integral and equation (4) remain exact. When the rule and the fields disagree, the fields are right — the case file is The Flux Rule and Its Exceptions.

Worked examples

1 · The coil around the solenoid (confidence builder)

A long solenoid ( $n = 1000$ turns/m, radius $r = 2.0$ cm) carries a current ramped at $dI/dt = 50$ A/s. A stationary 100-turn coil is wound snugly around its middle. Find the EMF in the coil.

Step 1. Field inside the solenoid (Ampère's Law, recalled): $B = \mu_0nI$ , uniform over the bore, negligible outside. Flux through each turn of the outer coil: $\Phi = \mu_0 n I\,\pi r^2$ .

Step 2. Faraday's law with $N = 100$ :

|\varepsilon| = N\frac{d\Phi}{dt} = N\mu_0 n\,\pi r^2\,\frac{dI}{dt} = 100\times(1.257\times10^{-6})(1000)\pi(0.02)^2(50)

Step 3. Evaluate: $|\varepsilon| = 100\times(1.257\times10^{-3})(1.257\times10^{-3})\times50 \approx$ 7.9 mV.

Step 4. Read the physics, because this innocent number is the discriminating experiment: the outer coil sits in a region where $\mathbf{B}$ itself is essentially zero (outside the solenoid) — yet it feels the change of flux threading it. The pump is the circulating induced $\mathbf{E}$ , which lives outside the solenoid where the field does not. Computing that $\mathbf{E}(r)$ ring by ring is Induced Electric Fields' opening exercise; the coupled-coils reading ( $\varepsilon_2 = -M\,dI_1/dt$ ) is Mutual Inductance's.

2 · Weighing a field by flipping a coil (exam level)

A 1-turn search loop of area $A = 0.010$ m² and circuit resistance $R = 25$ Ω (galvanometer included) sits face-on in an unknown field, then is flipped 180° in a fraction of a second. The ballistic galvanometer records $q = 0.32$ mC. Find $B$ , and explain why the flip speed does not matter.

Step 1. Flux change on flipping: from $+BA$ to $-BA$ , so $|\Delta\Phi| = 2BA$ .

Step 2. Ballistic theorem, equation (5): $q = \Delta\Phi/R = 2BA/R$ . Invert:

B = \frac{qR}{2A} = \frac{(3.2\times10^{-4})(25)}{2(0.010)} = \textbf{0.40 T}

Step 3. Why speed is irrelevant: the charge is the time integral of the current, and Faraday's law makes the integrand a perfect derivative — $q = \int(N/R)|d\Phi/dt|\,dt = N|\Delta\Phi|/R$ , whatever the profile. Flip lazily and a small current flows long; snap it and a large current flows briefly; the galvanometer's first swing is the same. (Fine print: the loop's self-inductance briefly opposes the current — Self-Inductance's business — but it changes the current's shape, not its integral.)

Step 4. The historical point: this is Faraday's own quantitative method — flux changes weighed as charge — and it is how induction's proportionality to $d\Phi/dt$ was established with 1830s instruments. Modern fluxmeters and Hall probes have replaced the swing, not the theorem.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.2 (the section this article tracks) · Purcell & Morin, Electricity and Magnetism, 3rd ed., ch. 7 · Jackson, Classical Electrodynamics, 3rd ed., §§5.15, 6.1 · Zangwill, Modern Electrodynamics, §14.1–14.3 · Feynman Lectures II, ch. 16–17 (the two-mechanisms discussion and the flux-rule warnings).

Provenance. Equations standard from Griffiths §7.2, Jackson §5.15 & §6.1, Purcell Ch. 7, Zangwill §14. Verifier should check: (1) the differential form derivation via Stokes' theorem; (2) the AC generator result $\mathcal{E}=NBA\omega\sin\omega t$ and its dimensions; (3) the solenoid induced-field expressions inside/outside (Griffiths Ex. 7.7) including continuity at s=a; (4) $W=\tfrac12 LI^2$ via integration of the back-EMF power. The numeric check (20 mV) is a direct arithmetic computation, not a tabulated constant. Addendum for this edition: merged with the v3 canon pilot (narrative spine: 1831 ring, flux-count picture, Lenz preview, generator waveform) at tier-3 register; the motional derivation and sliding-rod audit are rescoped to their owner Motional EMF and recalled by name (zero overlap); solenoid induced-field computations rescoped to Induced Electric Fields; $W = \tfrac12LI^2$ rescoped to Self-Inductance / Energy Stored in Magnetic Fields. Worked numbers recomputed for this edition: coil-around-solenoid 7.9 mV (100 turns, n = 1000/m, r = 2 cm, 50 A/s) and ballistic flip $q = 2BA/R$ → $B = 0.40$ T from 0.32 mC (0.010 m², 25 Ω).

Builds on: Electromotive Force · Motional EMF · Ohm's Law and Conductivity

Leads to: Lenz's Law · Induced Electric Fields · Self-Inductance · The Displacement Current · Maxwell's Equations in Vacuum

Is the induced electric field "real" if no wire is there to carry current?

Entirely. The circulating $\mathbf{E}$ around a changing flux exists in vacuum, with the wire serving only as a detector that lets it drive a measurable current. Two clean witnesses: the betatron accelerates electrons in a vacuum chamber using precisely this field — no wire, just particles surfing the induced $\mathbf{E}$ of a ramped magnet, gaining MeV per lap — and electromagnetic waves (the track's E8 destination) are nothing but this law and its displacement-current twin passing changing fields back and forth through empty space. Fields earned their reality certificate exactly here: Faraday's law is about spacetime, not copper.

Faraday's law has two completely different mechanisms behind one formula. Is that a coincidence or a law?

At this chapter's level, an exact and unexplained coincidence — and the honest register is to feel its strangeness. The motional half follows from the Lorentz force and $\nabla\cdot\mathbf{B} = 0$ (a theorem); the transformer half is new experimental physics (a law); nothing in the nonrelativistic framework requires their formulas to match. The resolution is that the two mechanisms are one phenomenon seen from different frames: boost from the magnet's frame to the loop's and part of $\mathbf{B}$ becomes an electric field, with exactly the circulation Faraday requires. The bookkeeping that makes this precise — fields as components of one tensor — is E10's subject (Transformation of Electric and Magnetic Fields), and this FAQ is that chapter's standing advertisement. Einstein took the coincidence as data; so should you.

If the induced E field is non-conservative, what happens to Kirchhoff's voltage law in AC circuits?

It survives as engineering bookkeeping because circuit designers quarantine the non-conservative regions. Real AC circuits confine changing flux inside components — transformer cores, inductor windings — and represent each confined EMF as a lumped element ( $-L\,dI/dt$ , $-M\,dI/dt$ ); outside those boxes the field is essentially curl-free and potentials behave. KVL then reads: sum of potential drops = sum of lumped EMFs, which is just Faraday's law with the geometry hidden. It fails visibly when flux escapes the quarantine — ground loops picking up hum, the two-voltmeter paradox, high-frequency layouts where every loop is an accidental secondary. RF engineers' obsession with loop area is Faraday's law enforced at the bench.

Why does the N-turn formula multiply by N - and does the flux through each turn really stay the same?

Each turn is a loop around (nearly) the same flux tube; traversing the coil end to end, the circuit encircles the changing flux $N$ times, so the pump acts $N$ times in series — like $N$ stacked batteries, which is exactly how the winding behaves at its terminals. The fine print is honest: in a tightly wound solenoid or toroid the per-turn fluxes are equal to excellent accuracy, while loosely wound or stray-field geometries need the honest sum $\Phi_{\rm total} = \sum_i\Phi_i$ (called the flux linkage, symbol $\lambda$ in machine engineering). When leakage matters — real transformers, loosely coupled charger coils — the deviation from ideal linkage is parametrized by coupling coefficients, the territory of Mutual Inductance.

Does Faraday's law work inside materials - iron cores, seawater, plasmas?

The field law (4) is universal — it is a Maxwell equation, holding in matter, vacuum, and stars. What changes in materials is everything around it: iron multiplies the flux for a given winding current (why Faraday's ring was iron — the E5 chapter's entire magnetization story feeding directly into transformer design), conductors respond to the induced $\mathbf{E}$ with eddy currents that generate their own opposing flux (screening, skin depths — Eddy Currents and E8), and in plasmas the law couples to fluid motion so tightly that field lines become "frozen into" the flow — the magnetohydrodynamics of the Sun and of fusion machines, E11's territory. Every one of those stories is equation (4) plus a medium's response; none amends it.

What exactly did Faraday measure to establish that EMF is proportional to the rate of flux change - he had no oscilloscope?

He integrated instead of differentiating. His galvanometers could not follow the fast transients, but a ballistic galvanometer's first swing measures the total charge of a pulse — and the charge theorem (derived above) says $q = N\Delta\Phi/R$ , converting any flux change into a number his instruments could hold. By flipping coils, making and breaking primaries, and inserting cores, he verified that the charge tracked the flux change, whatever the speed — which is precisely the statement that the EMF tracks the flux's rate. It is a beautiful example of experimental design matching a law's integral fingerprint when its differential form is out of reach — the same trick modern labs use whenever a detector is slower than the physics.

Where does the energy come from when a transformer's secondary powers a load - the primary current, the field, or the core?

Follow the ledger through one cycle. The load current in the secondary creates flux that opposes the core's flux swing (Lenz); the primary must then draw extra current from its supply to maintain the swing — and that extra primary current, times the supply voltage, is exactly the power delivered (plus core losses: the hysteresis loop-area bill of Magnetic Energy in Materials and eddy heating). So the energy comes from the primary's source, routed through the shared flux, with the core as a nearly lossless intermediary — the same crank-arm role the static field played in Motional EMF's rail machine, now played by a changing one. The full two-coil formalism ( $M$ , coupled equations, why voltage transforms as the turns ratio) is Mutual Inductance's; the energy flow through the space around the core is, once more, Poynting's E7 story.