Faraday's Law of Induction Explained: Generators to Wireless Chargers

Faraday's law of induction says that the electromotive force (EMF) induced around a closed loop equals minus the rate of change of the magnetic flux through that loop: ε = −N dΦ/dt, where the flux is Φ = B A cos θ. Lenz's law supplies the minus sign: the induced current always opposes the change that caused it. Michael Faraday first produced an induced current on 29 August 1831, and the same rule now runs every generator, transformer and wireless phone charger.

This post goes one level deeper than our tour of Maxwell's equations from Faraday to radio, Wi-Fi and fiber, where Faraday's law is one of four lines. Here it gets the whole page: the history, the two different ways a flux can change, the generator equation, the transformer, eddy currents, inductive charging and NFC, and five worked examples with real numbers. If you want the circuit arithmetic that sits around every induced EMF, read Ohm's law and Kirchhoff's laws explained alongside it, and for the force on a moving conductor, Newton's three laws of motion. The wider set lives in famous equations that changed the world.

A note on sources. The physics follows OpenStax University Physics Volume 2 and Georgia State University's HyperPhysics; the history comes from the Royal Institution's collection record and Faraday's own first paper; wireless charging and NFC facts come from the Wireless Power Consortium and the NFC Forum. Coil sizes, speeds and currents in the worked examples are illustrative choices, labelled as such, and every result was recomputed with the Python script in the walkthrough.

Key takeaways

• The law: ε = −N dΦ/dt. The induced EMF depends on how fast the flux changes, not on how large the flux is. A huge steady field induces nothing.
• Flux: Φ = B A cos θ for a uniform field B through a flat area A, with θ the angle between the field and the normal to the surface. The SI unit is the weber, 1 Wb = 1 T·m² = 1 V·s.
• Three ways to change flux: change B (transformers, chargers), change A (sliding rods) or change θ (rotating generators).
• Lenz's law: the induced current creates a magnetic field that opposes the change in flux. It is energy conservation in disguise.
• Generator equation: a coil of N turns and area A spinning at angular speed ω in a field B gives ε = N B A ω sin(ωt), with peak N B A ω.
• Transformer rule: for an ideal transformer V_s/V_p = N_s/N_p, and since power is conserved, I_s/I_p = N_p/N_s.
• Eddy currents: induced currents inside solid conductors. A loss in transformer cores, useful in induction cooktops, magnetic brakes and metal detectors.
• Wireless charging: Qi is magnetic induction between a transmitter coil and a receiver coil. The Wireless Power Consortium says coupling factors of 0.3 to 0.6 are typical, and Qi2 adds magnets to hold the coils aligned.
• Maxwell form: the Maxwell-Faraday equation, ∮E·dl = −dΦ/dt or ∇×E = −∂B/∂t, says a changing magnetic field creates a circulating electric field even with no wire present.

What does Faraday's law say?

Start with the quantity the law is about: magnetic flux. For a flat loop of area A in a uniform magnetic field B, the flux through the loop is

Φ = B A cos θ

where θ is the angle between the field direction and the normal (the perpendicular) to the loop. When the field passes straight through the loop, θ = 0 and the flux is simply B A. When the loop is edge on to the field, θ = 90° and no field lines thread it, so the flux is zero. For a curved surface or a field that varies across the loop, OpenStax defines the flux as the surface integral Φ = ∫ B·n dA, which reduces to B A cos θ in the simple case.

Faraday's law then states that the EMF induced in a loop is set by the rate of change of that flux:

ε = −dΦ/dt          (one loop)
ε = −N dΦ/dt        (a coil of N identical turns)

HyperPhysics puts it in one sentence: the induced EMF in a coil "is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil." The number of turns multiplies the result because each turn sees the same change and the turns are wired in series, so their EMFs add.

Two features trip people up. First, only change counts: a coil sitting still inside a strong magnet has no EMF, and pulling it out ten times faster gives ten times the EMF for a tenth of the time. Second, HyperPhysics stresses that how the flux changes does not matter: changing the field strength, moving a magnet or the coil, or rotating the coil all induce an EMF, and the law gives the right magnitude for each. In a circuit with resistance R the induced current is then I = ε/R, which is where Ohm's law joins the story.

Units: the weber, the tesla and the volt

The SI unit of flux is the weber (Wb). OpenStax defines it directly: 1 Wb = 1 T·m², and notes that the tesla is occasionally written as webers per square metre for the same reason. Because EMF is flux per second, 1 Wb/s = 1 V, so the weber is also a volt second. That makes unit checks easy: any expression for EMF must reduce to tesla times square metres per second.

The three knobs: B, A and θ

Because Φ = B A cos θ, the derivative has three parts. By the product rule,

dΦ/dt = A cos θ · dB/dt  +  B cos θ · dA/dt  −  B A sin θ · dθ/dt

Each term is a family of machines.

The split into "transformer EMF" and "motional EMF" matters later, when we get to Maxwell's equations, because the two have different microscopic causes even though the flux rule gives the right answer for both.

Worked example 1: a coil in a changing field (illustrative numbers)

The most direct use of ε = −N dΦ/dt. A square coil of 200 turns, 5.0 cm on a side (A = 2.5 × 10⁻³ m²), sits with its face perpendicular to a magnetic field that ramps steadily from 0 to 0.40 T in 0.10 s. The coil's total resistance is 4.0 Ω.

• Change in flux per turn: ΔΦ = ΔB × A = 0.40 × 2.5 × 10⁻³ = 1.0 × 10⁻³ Wb.
• EMF: ε = N ΔΦ / Δt = 200 × 1.0 × 10⁻³ / 0.10 = 2.0 V.
• Current: I = ε / R = 2.0 / 4.0 = 0.50 A.
• Charge that flows during the ramp: Q = I Δt = 0.50 × 0.10 = 0.050 C. Notice that Q = N ΔΦ / R, independent of how fast the ramp is. A faster ramp gives a larger current for a shorter time and moves the same charge. Ballistic flux meters use this to measure flux.

Tilt the coil so its normal is at 60° to the field and the effective area halves (cos 60° = 0.5): the EMF falls to 1.0 V. Turn it edge on (90°) and the EMF is zero, however fast the field changes.

Direction, by Lenz's law (next section): the flux is increasing, so the induced current circulates in the sense that makes a field opposing B inside the coil.

What does Lenz's law add?

Faraday's law gives the size of the EMF. Lenz's law gives its direction, and the minus sign in ε = −N dΦ/dt is how the equation encodes it. HyperPhysics states it this way: the polarity of the induced EMF is such that it produces a current whose magnetic field opposes the change which produces it. If the flux through a loop is increasing, the induced current makes a field pointing the other way, trying to hold the flux down. If the flux is decreasing, the induced current makes a field in the same direction, trying to prop it up.

A practical recipe, following OpenStax's approach (which uses its right hand rule 2 for the last step):

1. Decide which way the external field points through the loop.
2. Decide whether the flux through the loop is increasing or decreasing.
3. The induced field points opposite to the external field if the flux is increasing, and along it if the flux is decreasing.
4. Use the right hand rule to turn that induced field direction into a current direction: curl your fingers in the direction of the current and your thumb points along the field inside the loop.

Why the sign has to be negative

Imagine the opposite. If the induced current reinforced the change, pushing a magnet into a coil would pull it in harder, increasing the change and the current without limit: free energy from a nudge. Lenz's law forbids that. Whenever an induced current flows, something supplies the energy, whether your hand, a turbine or a charger's power supply, and the opposing force or EMF is how the bill is presented.

The same opposition shows up in motors. OpenStax calls it back EMF: a spinning motor is also a generator, and its induced EMF opposes the supply voltage, which is why a motor draws a large current at start up and much less once it reaches speed.

What did Michael Faraday show in 1831?

During the 1820s Faraday, working at the Royal Institution in London, had been trying to make electricity from magnetism. Hans Christian Ørsted had already shown that an electric current deflects a compass needle, so a current produces magnetism; OpenStax lists Ørsted with Coulomb, Gauss and Faraday as the experimenters whose laws Maxwell later unified. The natural question was whether magnetism could produce a current. His paper names the hope of "obtaining electricity from ordinary magnetism" as his motive.

The Royal Institution's collection record for the apparatus says Faraday achieved success on 29 August 1831, with a device now called the induction ring or ring-coil apparatus. It is an iron ring 170 mm across and 32 mm high, with two separate coils of copper wire wound on opposite sides and insulated with cotton. The record notes that the wire was the kind made for bonnets, that the iron ring seems to have been specially made, and that winding and insulating the coils was a tedious process. The Royal Institution describes it as the first ever electric transformer.

What Faraday saw was the key to the whole law. With one coil connected to a battery and the other to a galvanometer (a sensitive current meter), a steady current in the first coil produced no current in the second. In his paper he reports that the slight deflection of the needle at the moment of completing the connection "was always in one direction," and the deflection when the contact was broken was in the other. Induction was a transient effect of change, not a steady effect of magnetism.

Over the following weeks Faraday showed the same thing without a battery: moving a permanent magnet into or out of a coil, or moving a coil near a magnet, produced a current while the motion lasted. He then turned a copper plate between the poles of a magnet and drew a continuous current from it; the paper describes this as the production of "a permanent current of electricity by ordinary magnets." The device is now known as the Faraday disc, or homopolar generator.

Faraday was not alone. OpenStax notes that the American scientist Joseph Henry (1799 to 1878) discovered induced EMF independently, which is why the SI unit of inductance, the henry, carries his name. The rule for the direction of the induced current is named for Heinrich Lenz (1804 to 1865).

He presented the results to the Royal Society in the first series of his Experimental Researches in Electricity, which the printed collection records as "Read November 24, 1831," with sections titled On the Induction of Electric Currents and On the Evolution of Electricity from Magnetism. The collected volume describes the papers as reprinted from the Philosophical Transactions of 1831 to 1838.

Dated timeline

How does motional EMF work?

Motional EMF is the EMF produced when a conductor moves through a magnetic field. The simplest picture, the one OpenStax uses, is a metal rod of length l sliding with speed v along two parallel rails in a uniform field B perpendicular to the rails. The rod, the rails and a resistor R at the far end form a loop whose area grows by l v every second. So

dΦ/dt = B · dA/dt = B l v
ε = B l v

There is a second way to get the same answer, and it shows what is happening inside the metal. Each free electron in the rod moves with the rod at speed v, so it feels a magnetic force F = q v × B along the rod. That force pushes electrons toward one end, which charges the ends until the resulting electric field balances the magnetic force. The work done per unit charge in pushing charge along the rod is v B, times the length l, giving ε = B l v again. Two pictures, one answer: the flux rule and the Lorentz force agree.

Where the energy comes from

Once current flows, I = B l v / R. A current carrying rod in a magnetic field feels a force F = I l B, and by Lenz's law that force points backward, opposing the motion. To keep the rod moving at constant speed you must push with that same force, delivering mechanical power F v. Substitute and you get

P_mechanical = F v = (B l v / R) · l B · v = (B l v)² / R = ε² / R = P_electrical

Every watt dissipated in the resistor is a watt of mechanical work you put in. A generator is, at bottom, a machine for doing this conversion continuously.

Worked example 2: a sliding rod (illustrative numbers)

A 0.50 m rod slides at 4.0 m/s along rails in a 0.30 T field, with a 2.0 Ω resistor closing the circuit.

• EMF: ε = B l v = 0.30 × 0.50 × 4.0 = 0.60 V.
• Current: I = ε / R = 0.60 / 2.0 = 0.30 A.
• Retarding force: F = I l B = 0.30 × 0.50 × 0.30 = 0.045 N.
• Mechanical power to keep it moving: F v = 0.045 × 4.0 = 0.18 W.
• Electrical power in the resistor: ε I = 0.60 × 0.30 = 0.18 W. The books balance.

Double the speed and the EMF, current and force all double while the power quadruples to 0.72 W. That square law is why fast conductors in strong fields make effective brakes, which we return to under eddy currents.

How does a rotating coil generator produce EMF = N B A ω sin(ωt)?

Real generators do not slide rods along rails forever; they rotate coils. Take a rectangular coil of N turns, sides l and w, so area A = l w, spinning at constant angular speed ω about an axis perpendicular to a uniform field B. The angle between the field and the coil's normal is θ = ωt, so

Φ = B A cos(ωt)
ε = −N dΦ/dt = N B A ω sin(ωt)

OpenStax derives the same result the long way, by treating the two long sides of the coil as rods moving through the field with speed v = (w/2) ω and adding their motional EMFs, then multiplying by N turns. Either way the peak EMF is

ε₀ = N B A ω

and the output is a sine wave that swings from +ε₀ to −ε₀ once per revolution. This is alternating current, and it is why the mains supply is a sine wave: the machines that make it are rotating coils in magnetic fields (in large power station alternators, the magnet rotates and the coils stay still, which changes the engineering but not the physics).

Two things to notice in ε₀ = N B A ω. It is linear in every factor: double the turns, field, area or speed and the peak doubles. And the EMF is largest when the flux is zero: at the instant the coil is edge on to the field, Φ = 0 but it is changing fastest. When the coil faces the field squarely the flux is at its maximum and momentarily not changing, so the EMF is zero. In a simple generator, frequency and voltage are also tied together, since speed sets both.

The root mean square value of a sine wave is its peak divided by √2, so ε_rms = N B A ω / √2. Mains voltages quoted as 230 V or 120 V are rms values.

Worked example 3: a small 60 Hz generator (illustrative numbers)

A coil of 100 turns, each 10 cm by 10 cm (A = 0.010 m²), rotates at 60 revolutions per second in a 0.20 T field.

• Angular speed: ω = 2π × 60 = 377.0 rad/s.
• Peak EMF: ε₀ = N B A ω = 100 × 0.20 × 0.010 × 377.0 = 75.4 V.
• RMS EMF: 75.4 / √2 = 53.3 V.
• Peak flux linkage: N B A = 100 × 0.20 × 0.010 = 0.20 Wb turns.

To reach 120 V rms at the same 60 Hz you would need about 2.25 times as much N B A, for example 225 turns instead of 100. Note what you cannot do: you cannot keep 60 Hz and get more voltage by spinning faster, because the speed is what sets the frequency.

Worked example 4: a bicycle dynamo (illustrative numbers)

A hub dynamo is a small multipole alternator: a ring of permanent magnets turns with the wheel around a fixed coil. We will assume a design with 14 pole pairs, so the electrical frequency is 14 times the wheel's rotation rate, and a peak flux linkage N Φ₀ of 0.035 Wb turns (for example 500 turns each linking a peak flux of 7.0 × 10⁻⁵ Wb). These are illustrative values, not a specific product's specification.

A typical road wheel has a circumference of about 2.1 m. At 20 km/h:

• Road speed: 20 / 3.6 = 5.56 m/s.
• Wheel rotation rate: 5.56 / 2.1 = 2.65 rev/s.
• Electrical frequency: 14 × 2.65 = 37.0 Hz, so ω = 2π × 37.0 = 232.7 rad/s.
• Open circuit peak EMF: N Φ₀ ω = 0.035 × 232.7 = 8.1 V, about 5.8 V rms.

At 10 km/h the frequency halves and so does the EMF, to about 4.1 V peak, which is why a dynamo lamp dims at walking pace.

Under load, the output stops climbing with speed. The coil's own inductance, another consequence of Faraday's law, has a reactance X = ωL that rises with frequency, so at high speed more of the EMF is dropped inside the coil and the current to the lamp levels off.

How do transformers follow from induction?

A transformer is Faraday's induction ring, refined. Two coils share an iron core. An alternating current in the primary coil makes an alternating flux in the core. If the core guides essentially all of that flux through the secondary coil as well, then every turn of both coils sees the same dΦ/dt. Faraday's law applied to each coil gives

V_p = −N_p dΦ/dt
V_s = −N_s dΦ/dt

Divide one by the other and the flux cancels:

V_s / V_p = N_s / N_p

This is the transformer equation from OpenStax: the ratio of secondary to primary voltage equals the ratio of secondary to primary turns. A transformer with more turns on the secondary is a step up transformer; fewer turns, step down.

For an ideal transformer with no losses, power in equals power out, I_p V_p = I_s V_s, so

I_s / I_p = N_p / N_s

Voltage goes up by the turns ratio, current goes down by the same ratio. That is the whole reason long distance power lines run at high voltage. OpenStax works through the numbers: for a fixed delivered power, the ohmic loss in a line of resistance R is I²R, so cutting the current by a factor of 100 cuts the line loss by a factor of 10,000. Its worked example compares sending power at 240 V without a transformer, which would waste 2 × 10⁶ W in a 200 Ω line at 100 A, with sending it at high voltage and stepping it down near the houses.

Why must the current alternate? Because Faraday's law only responds to change. Steady DC makes a constant core flux, so the secondary EMF is zero, exactly what Faraday saw on his galvanometer in 1831, while the primary, opposed only by its wire resistance, draws a large current and heats. Transformers need AC, or DC chopped into pulses, which is what a switch mode power supply does.

Where transformer losses come from

OpenStax lists the losses that the ideal equations neglect: magnetic hysteresis in the core, ohmic heating in the windings, and ohmic heating of induced eddy currents in the core. It adds that a good transformer can have losses as low as 1% of the transmitted power. Here is what each one means and how designers fight it.

Leakage matters most for wireless charging. A closed iron core links nearly all the flux through both coils; a phone charger has no shared core and a gap of several millimetres, so only part of the flux reaches the receiver. That fraction is the coupling factor, covered below.

Worked example 5: a 230 V to 12 V transformer (illustrative numbers)

A transformer steps 230 V rms mains down to 12 V rms for a lighting circuit. The primary has 1,150 turns.

• Turns ratio: N_s / N_p = 12 / 230, so N_s = 1,150 × 12 / 230 = 60 turns.
• Load: the secondary feeds 2.0 A, so P_out = 12 × 2.0 = 24 W.
• Ideal primary current: I_p = 24 / 230 = 0.104 A, about one nineteenth of the secondary current, matching N_s/N_p = 60/1150.
• With 1% loss (OpenStax's figure for a good transformer): P_in = 24 / 0.99 = 24.24 W, so about 0.24 W becomes heat.
• Volts per turn: 230 / 1,150 = 0.20 V per turn, the same on both windings, because each turn links the same changing flux.

That volts per turn figure is a direct read of Faraday's law. In a sinusoidal transformer the rms EMF per turn is 4.44 f Φ_peak (from ω/√2 = 2πf/√2 ≈ 4.44f). At 50 Hz, 0.20 V per turn means a peak core flux of 0.20 / (4.44 × 50) ≈ 9.0 × 10⁻⁴ Wb. With an illustrative core cross section of 7.5 cm² that is a peak flux density of about 1.2 T. The core size is our choice; the 4.44 factor is pure Faraday. It also tells a designer something useful: halve the frequency at the same voltage and the core must carry twice the flux, which is why a transformer built for 60 Hz can overheat on 50 Hz at the same voltage, and why high frequency switch mode supplies can use tiny cores.

What are eddy currents, and how do induction cooktops and magnetic braking use them?

Faraday's law does not need a wire. Any conductor in a changing flux has EMFs induced around every closed path inside it, and because the conductor is solid, currents flow around those paths in swirls. OpenStax calls them eddy currents, by analogy with eddies in water. By Lenz's law they always oppose the change that causes them, and because the conductor has resistance, they turn energy into heat.

That gives two families of applications.

Eddy currents as a brake. Move a conductor through a magnetic field and the eddy currents produce a drag force that opposes the motion. OpenStax describes the classic demonstration: a metal plate swinging through a magnet's gap stops quickly, while the same plate with slots cut in it swings almost freely, because the slots break up the current loops. It also notes that eddy current braking is used in trains and roller coasters, where it is smoother and is not affected by rain, but that the braking force falls as the speed falls. In OpenStax's words, speed can be reduced from, say, 20 m/s to 5 m/s, but another form of braking is needed to completely stop the vehicle.

Why does the force fade? Because the EMF driving the eddy currents is proportional to the speed (ε = B l v), the current is proportional to the speed, and the force on the current is proportional to the current. So the drag force is proportional to v, like a viscous fluid. A force proportional to speed gives an exponential decay of speed with time: it never quite reaches zero.

OpenStax calls this effect magnetic damping and illustrates it with a metal pendulum bob swinging between the poles of a magnet: the bob is quickly slowed as it enters and leaves the field, while a bob made of an insulator swings on undamped.

Eddy currents as a heater. OpenStax notes that induction cooktops have electromagnets under their surface. The coil carries alternating current, the changing field passes into the base of the pan, and the eddy currents induced there heat the pan directly. The cooking surface is an insulator, so it is not heated by induction; it only gets warm from contact with the hot pan. OpenStax adds that induction cooktops have high efficiencies and good response times when the base of the pot is a conductor such as iron or steel.

Why iron and steel in particular? Two effects add up in a ferromagnetic base. The material concentrates the field, so more flux enters the metal near its surface, and the material's own resistance turns the eddy currents into heat effectively. Repeatedly flipping the magnetisation also adds some hysteresis heating, the same mechanism that counts as a loss in a transformer core. A practical home check that many cooks use is whether a fridge magnet sticks firmly to the base.

Eddy currents as a sensor. A metal detector drives a coil with alternating current. A metal object nearby carries eddy currents whose own field changes the impedance of the coil or induces a signal in a second receive coil. OpenStax lists metal detectors alongside braking systems as the major applications. Industrial eddy current testing uses the same principle to find cracks in metal parts: a crack interrupts the current paths and changes the signal.

Eddy currents as a nuisance. Inside a transformer or motor core they waste energy as heat. The fix is lamination: thin, insulated steel sheets allow only small current loops. At the higher frequencies of switch mode supplies and wireless chargers, designers use ferrite, a magnetic ceramic with very high resistivity.

Eddy current applications at a glance

That last row leads straight into wireless charging.

How do inductive wireless charging, NFC and RFID use induction?

Qi wireless charging

The Wireless Power Consortium describes how Qi works in terms any physics student would recognise: you need two coils, a transmitter coil and a receiver coil. An alternating current in the transmitter coil generates a magnetic field which induces a voltage in the receiver coil, and that voltage can power a device or charge a battery. It is a transformer with the core removed and the two halves separated by a desk and a phone case.

The WPC's technical notes, written by Philips Research engineers, explain what determines efficiency. The transmitter coil L1 and receiver coil L2 form a system of magnetically coupled inductors, and the efficiency of the transfer depends on the coupling factor k between them and on the quality factor Q of each coil. The coupling factor is the fraction of the flux from the transmitter that passes through the receiver. It runs from 0 (independent coils) to 1 (all flux linked). According to the WPC, coupling factors in the range of 0.3 to 0.6 are typical, and k depends on the distance between the coils, their relative size, their shape and the angle between them. If coils are axially aligned, a sideways displacement lowers k.

In the language of circuits, two coupled coils have a mutual inductance M = k √(L1 L2), and the EMF induced in the receiver when the transmitter current changes is

ε_2 = −M dI_1/dt

For a sinusoidal transmitter current of peak I₁ at angular frequency ω, the peak induced EMF is ω M I₁. That is Faraday's law with the flux linkage written as M I₁ (OpenStax's chapter on mutual inductance develops the same relation). Because ω appears directly, inductive chargers drive their coils far above mains frequency: with the illustrative 130 kHz used in worked example 6 below, the same small coils produce about 2,600 times the EMF they would at 50 Hz. The exact operating frequencies are set in the Qi specification, which is available to WPC members, so we use an illustrative value rather than quoting one.

The WPC also explains why the system needs more than coils: for energy efficient transfer, the phone must be able to shut down the transmitter when the battery is full, so the phone needs to send control signals back to the charging station. The WPC's public overview does not detail the signalling method. A natural way to do it in any inductive link, and the way NFC tags reply to readers (below), is load modulation: the receiver switches a small load on and off, and by Lenz's law the transmitter sees that as changes in its own coil current. Either way the charger and phone are running a control loop, and a control loop is software.

Qi, Qi2 and Qi2 25W

According to the WPC's specification history page, version 1.0 of the Qi Specification was released in 2010, version 1.1 in 2012, version 1.2 in 2015, version 1.3 in 2021 and version 2.0 in 2023. The same page says the latest version, v2.3, is available to WPC members.

On its Qi page the WPC says Qi2 brought 15 W charging to mobile devices certified for interoperability and safety, and that Qi2 devices feature magnetic attachment technology which aligns devices and chargers for improved energy efficiency, faster charging and easier use. In Faraday's law terms, the magnets fix the receiver coil over the transmitter coil, holding the coupling factor high instead of letting it drop with a sideways slip. The WPC says Qi2 25W delivers nearly 70% more charging power than Qi2 and can charge a smartphone battery from 0 to 50% in about 30 minutes. To display the Qi or Qi2 logo, the WPC says products must be tested for safety and interoperability in one of its independent authorised test laboratories.

The WPC's public pages do not list the features added in each revision, so the table shows only what they state.

Foreign object detection

Recall the last row of the eddy current table. A coin, key or foil wrapper between the coils sits in the charger's alternating field, carries eddy currents and heats up. Inductive chargers therefore need some form of foreign object detection. The general idea is a power balance: if the transmitter knows how much power it is sending and the receiver reports how much it is receiving, a large unexplained gap means something else is absorbing energy, and the charger should back off. The exact methods Qi requires are in its specification, which we did not have access to for this post.

NFC and RFID

Near Field Communication uses the same coupled coil physics for data, and a small amount of power. According to the NFC Forum, NFC is a short range wireless technology for devices a few centimetres apart, operating at 13.56 MHz. The Forum says the Certified Compliant range of NFC Forum connections is 5 mm, and that NFC devices can talk to cards and readers built to ISO/IEC 14443 Type A and Type B, ISO/IEC 15693, ISO/IEC 18092 and JIS X 6319-4, as well as NFC Forum Tags. The Forum also lists a mode for transmitting up to 1 W of power for wireless charging.

How does a passive tag, with no battery, answer a phone? The reader's coil drives a 13.56 MHz alternating field. The tag's antenna is a flat coil, and by Faraday's law the changing flux through it induces an EMF. A rectifier turns that into DC to run the tag's chip. To reply, the tag switches a load across its coil on and off, which changes how much energy it draws from the field. By Lenz's law the tag's current produces a field that acts back on the reader's coil, so the reader sees small changes in its own coil current and decodes them as data. This is called load modulation, and the NFC Forum page refers to it when it explains that different contactless protocols use different coding for signal and load modulation.

Contactless cards built to the ISO/IEC standards the NFC Forum lists work this way: inductively coupled, in the near field, over centimetres. Some longer range RFID systems instead communicate by radio waves in the far field, so "RFID" as a label does not by itself mean induction; check the frequency and the standard.

Induction based wireless technologies compared

Worked example 6: a phone charger coil (illustrative numbers)

Take two flat coils, each with inductance 10 µH, driven at an illustrative 130 kHz with a transmitter current of 1.5 A peak. Assume a coupling factor of 0.5, the middle of the WPC's typical 0.3 to 0.6.

• Mutual inductance: M = k √(L1 L2) = 0.5 × √(10 × 10) µH = 5.0 µH.
• Angular frequency: ω = 2π × 130,000 = 816,800 rad/s.
• Peak open circuit EMF in the receiver: ω M I₁ = 816,800 × 5.0 × 10⁻⁶ × 1.5 = 6.13 V.

Now slide the phone sideways until k falls to 0.3, the low end of the WPC's range:

• M = 0.3 × 10 = 3.0 µH, EMF = 816,800 × 3.0 × 10⁻⁶ × 1.5 = 3.68 V.

The induced EMF drops by 40%. To deliver the same power, the transmitter must drive more current, which raises copper losses in both coils as the square of the current and puts more stray field into any nearby metal. That is the physics behind Qi2's magnets: by holding the coils in place they keep k up.

Run it backward: to induce 9 V peak at k = 0.5 you need I₁ = 9 / (816,800 × 5.0 × 10⁻⁶) ≈ 2.20 A peak in the transmitter coil. Real chargers resonate their coils with capacitors so a modest drive voltage produces that current, and the control loop trims the drive to hold the phone's requested output.

How does Faraday's law appear as one of Maxwell's equations?

Faraday's law as written so far is about EMF in a loop. Maxwell's insight was to treat it as a statement about fields that holds whether or not a wire is present. OpenStax's section on induced electric fields makes the step: since EMF is the work done per unit charge around a closed path, and that work comes from an electric field, the law can be written as

∮ E · dl = −dΦ_B/dt = −d/dt ∫ B · dA

In words: a changing magnetic flux through any surface creates an electric field that circulates around the surface's boundary. Using Stokes' theorem on a fixed surface gives the differential form, the Maxwell-Faraday equation:

∇ × E = −∂B/∂t

The curl of the electric field at a point equals minus the rate of change of the magnetic field at that point.

Why this electric field is different

OpenStax points out that the induced electric field is nonconservative: it does net work moving a charge around a closed path, whereas the electrostatic field from charges (Coulomb's law) is conservative and does no net work around a closed loop. That is why you cannot define a single valued electric potential for an induced field, and why the reading of a voltmeter connected near a transformer winding can depend on how its leads are routed through the changing flux. It is also the fine print on Kirchhoff's voltage law: the rule assumes a conservative field, so it holds only where you can neglect changing flux through the circuit loop, or where you treat each induced EMF as a lumped source, as circuit diagrams do when they draw an inductor or a transformer winding as a component.

The flux rule and the Maxwell-Faraday equation are not identical

The Maxwell-Faraday equation ∇ × E = −∂B/∂t accounts for the "transformer EMF" part of the flux rule, the part where B changes in time. The motional part, where a conductor moves through a static field, is not in that equation at all; it comes from the magnetic part of the Lorentz force, q v × B, acting on the moving charges. The flux rule ε = −dΦ/dt happens to give the right total for both cases in ordinary circuits, which is why it is so useful. But there are setups, the Faraday disc among them, where you must go back to the fields and the Lorentz force to get the right answer, because it is not obvious which "loop" the flux should be counted through.

This asymmetry is famous. Einstein's paper On the Electrodynamics of Moving Bodies, dated 30 June 1905, opens with it, taking as its example "the reciprocal electrodynamic action of a magnet and a conductor." If the magnet moves and the conductor is still, the changing field produces an electric field in the conductor's region, the transformer EMF. If the conductor moves and the magnet is still, there is no electric field; the charges are pushed by the magnetic force on moving charges, the motional EMF. The observed current is the same in both cases. Special relativity resolved it: electric and magnetic fields are two aspects of one field, and which part you see depends on how you are moving.

Faraday's law among Maxwell's four equations

Pair the third line with the fourth and you get electromagnetic waves: a changing B makes a changing E, which makes a changing B, and the disturbance propagates at speed 1/√(μ₀ε₀), the speed of light. Our Maxwell's equations post follows that thread to radio, Wi-Fi and fibre. NIST's CODATA 2022 value for the vacuum magnetic permeability μ₀ is 1.256 637 061 27 × 10⁻⁶ N A⁻², listed with a relative standard uncertainty of 1.6 × 10⁻¹⁰, which tells you it is now a measured quantity rather than an exactly defined one.

Walkthrough: reproduce every number in Python

Every figure in the worked examples above came from this script. It uses only the Python standard library. Run it, then change a parameter (turns, field, speed, coupling) and watch the result move.

Save it as faraday_examples.py:

import math

def coil_ramp(N, side_m, B_final, t_s, R_ohm, theta_deg=0.0):
    A = side_m ** 2
    dphi = B_final * A * math.cos(math.radians(theta_deg))
    emf = N * dphi / t_s
    I = emf / R_ohm
    return dict(A=A, dphi=dphi, emf=emf, I=I, Q=I * t_s)

def sliding_rod(B, l, v, R):
    emf = B * l * v
    I = emf / R
    F = I * l * B
    return dict(emf=emf, I=I, F=F, P_mech=F * v, P_elec=emf * I)

def generator_peak(N, B, A, f_hz):
    w = 2 * math.pi * f_hz
    peak = N * B * A * w
    return dict(omega=w, peak=peak, rms=peak / math.sqrt(2))

def dynamo(speed_kmh, circumference_m, pole_pairs, flux_linkage):
    v = speed_kmh / 3.6
    rev_s = v / circumference_m
    f = pole_pairs * rev_s
    w = 2 * math.pi * f
    peak = flux_linkage * w
    return dict(v=v, rev_s=rev_s, f=f, omega=w, peak=peak, rms=peak / math.sqrt(2))

def transformer(Vp, Vs, Np, I_load, loss_frac=0.01, f_hz=50, core_area_m2=7.5e-4):
    Ns = Np * Vs / Vp
    P_out = Vs * I_load
    Ip_ideal = P_out / Vp
    P_in = P_out / (1 - loss_frac)
    v_per_turn = Vp / Np
    phi_peak = v_per_turn / (4.44 * f_hz)
    return dict(Ns=Ns, P_out=P_out, Ip=Ip_ideal, P_in=P_in,
                v_per_turn=v_per_turn, phi_peak=phi_peak,
                B_peak=phi_peak / core_area_m2)

def charger(L1_uH, L2_uH, k, f_hz, I1_peak):
    M = k * math.sqrt(L1_uH * L2_uH) * 1e-6
    w = 2 * math.pi * f_hz
    return dict(M_uH=M * 1e6, omega=w, emf_peak=w * M * I1_peak)

if __name__ == "__main__":
    print("coil ramp       ", coil_ramp(200, 0.05, 0.40, 0.10, 4.0))
    print("coil ramp 60deg ", coil_ramp(200, 0.05, 0.40, 0.10, 4.0, 60))
    print("sliding rod     ", sliding_rod(0.30, 0.50, 4.0, 2.0))
    print("generator 60 Hz ", generator_peak(100, 0.20, 0.010, 60))
    print("dynamo 20 km/h  ", dynamo(20, 2.1, 14, 0.035))
    print("dynamo 10 km/h  ", dynamo(10, 2.1, 14, 0.035))
    print("transformer     ", transformer(230, 12, 1150, 2.0))
    print("charger k=0.5   ", charger(10, 10, 0.5, 130e3, 1.5))
    print("charger k=0.3   ", charger(10, 10, 0.3, 130e3, 1.5))

Run it in a fresh virtual environment so it cannot pick up anything unexpected from your global site packages:

python3 -m venv .venv
source .venv/bin/activate
python -I faraday_examples.py

Expected output (rounded):

coil ramp        emf 2.0 V, I 0.5 A, Q 0.05 C
coil ramp 60deg  emf 1.0 V
sliding rod      emf 0.6 V, I 0.3 A, F 0.045 N, P 0.18 W both ways
generator 60 Hz  omega 377.0 rad/s, peak 75.4 V, rms 53.3 V
dynamo 20 km/h   f 37.0 Hz, peak 8.14 V, rms 5.76 V
dynamo 10 km/h   f 18.5 Hz, peak 4.07 V
transformer      Ns 60, P_out 24 W, Ip 0.104 A, P_in 24.24 W, 0.2 V/turn, B_peak 1.2 T
charger k=0.5    M 5.0 uH, emf_peak 6.13 V
charger k=0.3    M 3.0 uH, emf_peak 3.68 V

See induction at work on your own devices

You can watch the software side of wireless charging on hardware you already own. These commands only read status; they change nothing.

On a Linux laptop, the kernel's power supply class exposes each charger and battery to user space; the kernel documentation says its attributes are available through sysfs and uevent interfaces:

ls /sys/class/power_supply/
cat /sys/class/power_supply/*/uevent

Each entry is a driver reporting what the charging hardware tells it: whether a supply is online, the battery's voltage and current, and its status. Plug in and unplug the charger while watching the online attribute change.

On an Android phone with USB debugging enabled, Android's dumpsys tool dumps the state of system services; Android's documentation uses it for battery diagnostics among other tasks. List the services, then dump the battery service:

adb shell dumpsys -l | grep -i battery
adb shell dumpsys battery

The battery service output typically includes separate lines for AC, USB and wireless power; check what your device reports. Put the phone on a Qi pad and run it again to see which path is active. Nudge the phone off centre and, on many phones, charging pauses or slows. You have just changed the coupling factor by moving one coil relative to the other, and the charger's control loop has responded.

Where does software enter?

Almost every device that relies on Faraday's law today has a processor in the loop. At the level the public sources support:

• Wireless charger firmware. The WPC is explicit that a working system needs more than coils: the phone must signal the charger to reduce or stop power. Both ends run firmware that decodes those messages, sets the drive level, watches for foreign objects and handles faults.
• Phone power management. On the receiving side, a charger chip and the phone's operating system decide how much power to request, when to slow down as the battery fills, and what to report to the user. The Android dumpsys battery output above is that software's view of the world.
• Certification. The WPC says products must pass safety and interoperability testing at independent authorised labs before using the Qi or Qi2 logo. Whether and how a phone can cryptographically check a charger's certification is a matter for the Qi specification itself; we could not confirm the details from the WPC's public pages, so read the specification before relying on specifics.
• NFC stacks. The NFC Forum describes modes in which a device emulates a contactless card, reads or writes tags and cards, or transmits up to 1 W of power for charging, and says card emulation is the most common today. Each mode is implemented in a controller's firmware plus an operating system stack.
• Generators, cooktops and the grid. Excitation controllers, protection relays, inverters and the controllers inside induction hobs are digital systems wrapped around induction.

The firmware, companion apps, cloud dashboards and lab test scripts around these devices all pull in open source components.

Common mistakes

1. Confusing flux with rate of change of flux. A coil in a strong, steady field has no EMF. Only change induces EMF. If your answer depends on B but not on dB/dt or on motion, check it.
2. Forgetting the number of turns. ε = −N dΦ/dt. Missing N is the most common factor of 100 or 1,000 error in coil problems.
3. Using the wrong angle. θ in Φ = B A cos θ is measured between B and the normal to the loop, not between B and the plane of the loop. Mix them up and you swap sine and cosine.
4. Expecting peak EMF at peak flux. In a rotating generator the EMF peaks when the flux is zero and changing fastest.
5. Applying the transformer equation to DC. Transformers need changing current. Steady DC gives zero secondary voltage and a hot primary.
6. Treating the turns ratio as a power ratio. Voltage scales with N_s/N_p and current with N_p/N_s. Power stays the same (less losses).
7. Losing the minus sign, then losing the direction. Lenz's law is how you find the current direction. Do it as a separate step: external field direction, increasing or decreasing, opposing or supporting, then the right hand rule.
8. Mixing peak and rms. ε₀ = N B A ω is a peak value. Mains voltages are rms. Divide by √2 to convert for sinusoids.

FAQ

What is Faraday's law in simple terms?

When the magnetic field passing through a loop of wire changes, a voltage appears around the loop. The faster the change and the more turns of wire, the larger the voltage. Written as an equation, the EMF is minus the number of turns times the rate of change of magnetic flux.

What is the formula for Faraday's law?

ε = −N dΦ/dt, where ε is the induced EMF in volts, N is the number of turns, and Φ = B A cos θ is the magnetic flux through one turn in webers. For a coil rotating at angular speed ω in a uniform field, this becomes ε = N B A ω sin(ωt).

What is the difference between Faraday's law and Lenz's law?

Faraday's law gives the magnitude of the induced EMF. Lenz's law gives its direction: the induced current flows so that its magnetic field opposes the change in flux. The minus sign in Faraday's law is Lenz's law, and it is a consequence of conservation of energy.

When did Faraday discover electromagnetic induction?

According to the Royal Institution, Faraday achieved success with his induction ring on 29 August 1831, in his basement laboratory at the Royal Institution in London. His first paper on the subject was read to the Royal Society on 24 November 1831.

Why does a transformer not work with DC?

Because induction requires a changing magnetic flux. A steady direct current makes a steady flux in the core, which induces no voltage in the secondary.

How does wireless charging work?

A Qi charger drives an alternating current through a transmitter coil. The changing magnetic field passes through a receiver coil in the phone and, by Faraday's law, induces a voltage that is rectified to charge the battery. The phone sends control messages back to the charger so it can adjust or stop power.

Why do Qi2 chargers use magnets?

The Wireless Power Consortium says Qi2's magnetic attachment aligns devices and chargers for better efficiency and faster charging. In physics terms, alignment keeps the coupling factor between the two coils high, so more of the transmitter's flux passes through the receiver.

Why does my phone get warm on a wireless charger?

Losses. Not all of the transmitter's flux reaches the receiver, the coils have resistance, the power conversion circuits are not perfectly efficient, and any nearby metal carries eddy currents. All of those become heat.

What are eddy currents used for?

Induction cooktops (heating the pan), eddy current brakes on trains and roller coasters, metal detectors, and crack testing of metal parts. In transformers and motors they are an unwanted loss, reduced by laminated or ferrite cores.

Is Faraday's law one of Maxwell's equations?

Yes. In field form it is the Maxwell-Faraday equation, ∇ × E = −∂B/∂t, or in integral form, the circulation of the electric field around a loop equals minus the rate of change of magnetic flux through it. It says a changing magnetic field creates an electric field, even with no wire present.

The software that runs the hardware: an honest bridge

Safeguard does not do physics. Our models are single purpose security models, and nothing in this post was calculated by them; the numbers come from the textbook equations and the Python script above. Safeguard does not design coils, test chargers, analyse firmware images or certify hardware. The link to this topic is narrower. The chargers, phones, cooktops and grid equipment that run on Faraday's law are controlled by software, and so are the companion apps, test harnesses, simulation notebooks and monitoring dashboards that engineers build around them. Those programs pull in open source components, and those components need the same dependency hygiene as any other application.

A charger test rig, a lab data logger or a smart charger's cloud backend is usually a thin layer of your own code on top of open source libraries, and every rebuild pulls those dependencies again. Wireless stacks, charger firmware toolchains and NFC libraries ship open source components too. When one of them ships a vulnerability, the questions are the usual ones: is there a fixed version, and how quickly can you move to it.

Fix it without waiting for upstream

Whatever the vulnerability, most of the risk sits in two gaps. The first is the time between a fix being published and that fix actually running in your production builds. The second, and worse, is the time before any fix exists at all, while a maintainer works out a patch and a release schedule. Waiting through both gaps is a habit, not a requirement. Safeguard gives teams three ways to close them.

1. Autonomous remediation: pull requests, not tickets

When a fixed version exists, Safeguard's Griffin models open a tested, policy gated pull request that moves the affected dependency to the safe version. You can run it on one repository or in bulk across every project that carries the vulnerable version. Remediation categories include dependencies, licenses, security posture and findings, and each one is set separately. Reachability analysis ranks the vulnerable code paths your application actually calls first, so the first pull requests you merge are the ones that matter most. Autonomous remediation ships switched off, with every category set to Manual, so your team decides how much runs on its own and what still needs a human approval.

2. Gold Open Source: the safe version in seconds, free

gold.safeguard.sh is free and needs no login. Search any CVE or package to see affected and fixed versions, CVSS, EPSS, CISA KEV status and a recommended safe version, or use the "Am I affected?" checker. The same data is available through a free read API, README status badges and RSS threat feeds.

3. Gold Registry: start from components that are already fixed

Gold Registry provides hardened, drop in replacements for upstream packages and container images. Every Gold artifact is held continuously to zero exploitable critical or high CVEs and a completed license check, with signed build provenance (signed with Safeguard's key and verifiable with cosign, SLSA Build Level 3 as the design target) and a public evidence page you can hand to an auditor. An artifact loses its Gold badge if it regresses. Teams can point their own package directory at the registry so hardened equivalents resolve without changing how developers install anything.

When upstream has no fix yet, or will not ship one, Safeguard first works to get the fix accepted upstream. If the issue is severe or upstream will not move, Safeguard builds a fork, fixes the vulnerability there and publishes a hardened Gold build, so you are not stuck on a maintainer's release schedule. You can also request a hardened, compatibility tested build of a specific package. Ask which packages it covers before you plan around it.

One honest limit: when the vulnerable code lives in a vendor appliance, a hosted service or an operating system runtime you do not build yourself, the fix still has to come from that vendor or distribution. The steps above apply to the open source dependencies and container images you build and ship, which is where most teams carry the largest and least tracked share of their exposure.

See how Safeguard would help

None of these demos involve induction calculations; they show the software side described in the bridge above.

The first demo is a real Safeguard CLI session: it installs the CLI, opens the interactive terminal and asks it to fix a lodash CVE in a repository. It reads the finding and advisory, asks before each command, edits package.json to lodash 4.17.21, runs npm install and npm test with approval, prints a summary and pushes nothing. The same flow applies to a charger test dashboard or a smart charging web app your team maintains: scan the repository, then let the assistant propose the dependency upgrade under your approval.

Click through it at your own pace, or open it full screen.

The second demo sets up the Safeguard runner on your own machine, approves it with its device code and then scans from the terminal. That suits teams who keep coil simulation scripts, instrument control code or charger test tools on lab machines and want the scan to run where that code lives.

Click through it at your own pace, or open it full screen.

The third demo connects Claude to the Safeguard MCP server with OAuth and lets you choose which tools Claude can use. For an engineer who already works with Claude while writing test or analysis code, it brings dependency findings into the same conversation, limited to the tools you allow.

Click through it at your own pace, or open it full screen.

To try it on your own repositories, download Safeguard and follow the CLI documentation.

Quick check

Sources

• OpenStax, University Physics Volume 2, 13.1 Faraday's Law
• OpenStax, University Physics Volume 2, 13.2 Lenz's Law
• OpenStax, University Physics Volume 2, 13.3 Motional Emf
• OpenStax, University Physics Volume 2, 13.4 Induced Electric Fields
• OpenStax, University Physics Volume 2, 13.5 Eddy Currents
• OpenStax, University Physics Volume 2, 13.6 Electric Generators and Back Emf
• OpenStax, University Physics Volume 2, 14.1 Mutual Inductance
• OpenStax, University Physics Volume 2, 15.6 Transformers
• OpenStax, University Physics Volume 2, 16.1 Maxwell's Equations and Electromagnetic Waves
• HyperPhysics, Georgia State University: Faraday's Law
• Royal Institution: Michael Faraday's ring-coil apparatus
• Royal Institution: Michael Faraday at the Royal Institution
• Michael Faraday, Experimental Researches in Electricity, Volume 1 (Project Gutenberg text)
• Wireless Power Consortium: Qi wireless charging
• Wireless Power Consortium: Qi Specifications and version history
• Wireless Power Consortium: How Qi Works
• Wireless Power Consortium: Principle of Inductive Power
• Wireless Power Consortium: Coupling Factor
• NFC Forum: NFC Technology
• NIST: CODATA value of the vacuum magnetic permeability
• Linux kernel documentation: Linux power supply class
• Android Developers: dumpsys
• Albert Einstein, On the Electrodynamics of Moving Bodies (1905), English translation hosted by Fourmilab

Last reviewed: 24 October 2026.