Faraday's Law of Induction Explained

Fig 1 — A magnet pushed into a coil increases the flux through it; the induced current flows so that its own field opposes the increase (Lenz's law)

Introduction to Electromagnetic Induction

The first three parts of this series dealt with fields that do not change in time. Electric fields came from charges (Part 2), and magnetic field lines were found to always close on themselves (Part 3). The electric and magnetic fields appeared to be independent of each other. Electromagnetic induction is the first link between them: a magnetic field that changes with time produces an electric field.

Induction is everywhere in electrical engineering. It generates almost all of the world's electricity, it is the reason transformers can step voltages up and down, it lets a phone charge on a pad and a payment card talk to a terminal, and it is one of the main mechanisms by which one circuit interferes with another. And because a changing magnetic field creates an electric field, induction is half of the mechanism that allows electromagnetic waves to exist.

Understanding Electromagnetic Induction

Discovery of Electromagnetic Induction

After Ørsted showed in 1820 that currents produce magnetic fields, many experimenters looked for the reverse effect: could a magnetic field produce a current? A steady field in a coil produced nothing. In August 1831 Michael Faraday noticed the crucial detail: a galvanometer connected to one coil wound on an iron ring gave a brief kick when the current in a second coil on the same ring was switched on or off, and nothing while the current was steady. He then showed that moving a magnet into or out of a coil, or moving the coil near a magnet, had the same effect. Joseph Henry in the United States made the same discovery independently at about the same time.

Faraday's conclusion was that it is not the magnetic field that matters, but its change. A current is induced only while the magnetic flux through the circuit is changing.

Relationship Between Magnetic Fields and Induced Electric Fields

What actually pushes the charges in the coil? In a stationary loop, it cannot be the magnetic force q𝐯×𝐁, because the charges are not moving to begin with. The answer is that a time-varying magnetic field creates an induced electric field in the space around it — whether or not a wire is present. A wire merely provides a path along which the charges can respond to that field and form a current.

This induced electric field is different from the electrostatic field of charges. Electrostatic field lines start and end on charges and the field is conservative: the work done carrying a charge around any closed path is zero. The induced electric field forms closed loops around the changing magnetic flux, and is not conservative — carrying a charge once around the loop gains energy. That energy per unit charge is the electromotive force (EMF).

Changing Magnetic Flux and Induced EMF

Since Φ_B = BA cos θ, the flux through a circuit can be changed in three ways, all of which induce an EMF:

Changing the field B — moving a magnet, or varying the current in a nearby coil (transformers, inductive coupling).
Changing the area A — a sliding conductor on rails, or a deforming loop (motional EMF).
Changing the orientation θ — rotating a coil in a field (generators, alternators).

Faraday's Law of Electromagnetic Induction

Statement of Faraday's Law

📌 Faraday's law

The electromotive force induced in a closed circuit is equal to the negative of the rate of change of the magnetic flux linked with the circuit.

Mathematical Expression of Faraday's Law

Faraday's law for a coil of N turns

ℰ = - N \frac{d Φ_{B}}{d t} = - \frac{d λ}{d t}

ℰ in volts, Φ_B in webers, t in seconds; λ = NΦ_B is the flux linkage. The EMF depends only on how fast the flux changes — not on how large the flux is. A huge steady flux induces nothing; a small flux changing rapidly (as at radio frequencies) can induce a large voltage.

For sinusoidal fields the derivative becomes a multiplication by jω. If the flux through a loop is Φ(t) = Φ₀ sin ωt, the induced EMF is −NωΦ₀ cos ωt, with peak value NωΦ₀ = 2πf NΦ₀. The induced voltage is proportional to frequency — the reason loop antennas and inductive coupling become more effective as frequency rises, and why a few millimetres of PCB loop matters far more at 1 GHz than at 1 kHz.

Understanding the Negative Sign in Faraday's Law

The minus sign tells us the direction of the induced EMF: it acts to drive a current whose own magnetic flux opposes the change in the original flux. If the flux through the loop is increasing, the induced current creates flux in the opposite direction; if the flux is decreasing, the induced current creates flux in the same direction to prop it up.

The sign is not an arbitrary convention — it is required by conservation of energy. If the induced current reinforced the change instead of opposing it, pushing a magnet into a coil would induce a current that pulled the magnet in faster, which would increase the current further, without limit: free energy. The minus sign ensures that you must do work against the induced current, and that work is exactly the electrical energy delivered to the circuit.

Lenz's Law

Statement of Lenz's Law

📌 Lenz's law (Heinrich Lenz, 1834)

The direction of an induced current is always such that it opposes the change in magnetic flux that produced it.

Direction of Induced Current

Lenz's law gives a quick procedure for finding the current direction without careful sign bookkeeping:

Find the direction of the original magnetic field through the loop.
Decide whether the flux through the loop is increasing or decreasing.
The induced field inside the loop points opposite to the original field if the flux is increasing, and along it if the flux is decreasing.
Use the right-hand rule (thumb along the induced field, fingers curl along the current) to find the current direction.

In Fig 1, the north pole of a magnet moves toward the coil, so the rightward flux through the coil grows. The induced current makes the face of the coil nearest the magnet a north pole — creating a leftward induced field — and the two north poles repel. You feel this as a resisting force when pushing the magnet in. Pull the magnet away and the coil's near face becomes a south pole, attracting the magnet and resisting its withdrawal.

Relationship Between Faraday's Law and Lenz's Law

Faraday's law gives the magnitude of the induced EMF; Lenz's law is the physical meaning of its minus sign. Together they describe a kind of electromagnetic inertia: any circuit resists changes in the flux linking it. The same principle explains self-inductance (a coil resists changes in its own current, V = L dI/dt), eddy-current braking, induction heating and the damping of moving-coil meters.

Integral Form of Faraday's Law

The EMF around a closed path C is the line integral of the induced electric field around that path. Writing the flux as a surface integral over any surface S bounded by C gives the integral form — the third of Maxwell's equations:

Faraday's law — integral form

\oint_{C} 𝐄 \cdot d 𝐥 = - \frac{d}{d t} \int_{S} 𝐁 \cdot d 𝐀

The direction of d𝐥 around C and the direction of d𝐀 are related by the right-hand rule: curl the fingers along C and the thumb gives the normal. The path C does not have to be a wire — the law holds for any imaginary loop in space, which is what makes it a field equation.

Compare this with electrostatics, where ∮𝐄·d𝐥 = 0 (the work done round a closed path is zero, and voltage between two points is uniquely defined). Once magnetic fields change, the circulation of 𝐄 is no longer zero, and the "voltage" between two points depends on the path taken. This has direct practical consequences: the voltage read by an oscilloscope probe depends on the loop formed by the probe tip and its ground lead, because that loop picks up changing flux.

Differential Form of Faraday's Law

Stokes' theorem converts the circulation of a field around a loop into the flux of its curl through the enclosed surface: ∮𝐄·d𝐥 = ∫(∇×𝐄)·d𝐀. For a stationary surface the time derivative can be moved inside the integral, and since the result holds for every surface, the integrands must be equal:

Faraday's law — differential form

\nabla \times 𝐄 = - \frac{\partial 𝐁}{\partial t} phasor form: \nabla \times 𝐄 = - j ω 𝐁

The curl measures how much a field circulates around a point. Wherever 𝐁 is changing, 𝐄 swirls around it. In the frequency domain (sinusoidal steady state, time dependence e^{jωt}), ∂/∂t becomes jω — the form used throughout RF engineering.

Applications of Faraday's Law

Electrical Generators

A coil of N turns and area A rotating at angular speed ω in a uniform field B links flux Φ = BA cos ωt. Faraday's law gives a sinusoidal EMF:

AC generator EMF

ℰ (t) = N B A ω \sin ω t ℰ_{peak} = N B A ω

This is why mains electricity is sinusoidal. The mechanical work done turning the shaft against the Lenz's-law torque is converted into electrical energy.

Transformers

An alternating current in the primary winding produces an alternating flux in the core, which links the secondary. With the same flux through every turn, Faraday's law gives the same voltage per turn on both windings:

Ideal transformer

\frac{V_{2}}{V_{1}} = \frac{N_{2}}{N_{1}} \frac{I_{2}}{I_{1}} = \frac{N_{1}}{N_{2}} Z_{in} = {(\frac{N_{1}}{N_{2}})}^{2} Z_{L}

The impedance transformation by the square of the turns ratio is used constantly at RF: a 1:2 turns-ratio transformer converts 50 Ω to 200 Ω, and a 1:1 transmission-line transformer forms a balun for feeding a balanced dipole from unbalanced coax.

Inductive Coupling

When two coils are near each other, a changing current in one induces a voltage in the other. The strength of this coupling is the mutual inductance M, defined by V₂ = −M dI₁/dt, and the fraction of flux shared is the coupling coefficient k = M/√(L₁L₂), between 0 and 1. Inductive coupling is used deliberately in RF transformers, coupled-resonator filters and NFC/RFID (13.56 MHz), and appears unintentionally as crosstalk between adjacent circuits.

Wireless Power Transfer

Wireless chargers such as Qi phone pads (operating roughly in the 87–205 kHz range) are loosely coupled transformers with an air gap. Because k is small (typically 0.2–0.7 for a pad), both coils are tuned to resonance with capacitors so that a large circulating current builds up and enough flux reaches the receiver. Resonant magnetic coupling extends this to larger gaps for electric-vehicle charging and medical implants.

Electromagnetic Interference

Any conducting loop is an unintended receiving antenna for magnetic fields. The voltage induced in a loop of area A by a sinusoidal field B at frequency f is 2πf·B·A. A PCB trace and its return path, a cable shield grounded at both ends, or a probe and its ground lead all form such loops. Faraday's law gives three remedies, all used in good EMC practice:

Minimise loop area — route return currents directly under signal traces on a solid ground plane, and twist wire pairs so that adjacent twists induce opposing EMFs.
Increase distance from the source, since near-field magnetic fields fall off rapidly with distance.
Shield — eddy currents induced in a conductive enclosure create fields that oppose the incident field (Lenz's law), cancelling it inside.

Numerical Examples

EMF From a Changing Magnetic Field

Problem: A 200-turn coil of area 0.01 m² is placed perpendicular to a magnetic field that rises uniformly from 0 to 0.5 T in 0.1 s. Find the induced EMF. If the coil resistance is 4 Ω and it is short-circuited, what current flows?

ℰ = N ΔΦ / Δt = N A ΔB / Δt

| ℰ | = 200 \times \frac{0.01 \times 0.5}{0.1} = 200 \times 0.05 = 10 V I = \frac{10}{4} = 2.5 A

Answer: ℰ = 10 V for the 0.1 s the field is changing; I = 2.5 A

Once the field stops changing at 0.5 T, the EMF drops to zero even though the flux is at its largest. By Lenz's law the current circulates so that its field opposes the rising applied field.

AC Generator Output Voltage

Problem: A generator coil has 100 turns of area 0.05 m² and rotates at 3000 rpm in a uniform field of 0.4 T. Find the frequency, the peak EMF and the rms EMF.

f = 3000/60 = 50 Hz; ω = 2πf = 314.16 rad/s

ℰ_{peak} = N B A ω = 100 \times 0.4 \times 0.05 \times 314.16 = 628.3 V ℰ_{rms} = \frac{628.3}{\sqrt{2}} = 444.3 V

Answer: f = 50 Hz | ℰ_peak = 628.3 V | ℰ_rms = 444.3 V

Doubling the speed doubles both the frequency and the voltage, because the EMF is proportional to ω. Note the factor N·B·A·2π/√2 = 4.44 f N Φ_peak — exactly the transformer equation from Part 3.

Motional EMF in a Sliding Bar

Problem: A metal bar of length 0.5 m slides at 10 m/s along two parallel rails joined by a 2 Ω resistor. A uniform field of 0.2 T points into the page, perpendicular to the rails. Find the EMF, the current, and the force needed to keep the bar moving at constant speed.

The enclosed area grows at rate Lv, so dΦ/dt = BLv

ℰ = B L v = 0.2 \times 0.5 \times 10 = 1.0 V I = \frac{1.0}{2} = 0.5 A F = B I L = 0.2 \times 0.5 \times 0.5 = 0.05 N

Answer: ℰ = 1.0 V | I = 0.5 A (counter-clockwise) | F = 0.05 N

The flux into the page is increasing, so the induced current creates a field out of the page inside the loop: counter-clockwise. Energy check: mechanical power F·v = 0.05 × 10 = 0.5 W equals electrical power I²R = 0.5² × 2 = 0.5 W. Lenz's law guarantees the books balance.

Voltage Induced in an NFC Loop at 13.56 MHz

Problem: A single-turn circular loop of radius 1 cm sits in a 13.56 MHz magnetic field with peak flux density 1 μT normal to the loop. Find the peak induced EMF. What is the EMF for a 5-turn loop of the same size?

Sinusoidal flux: ℰ_peak = N ω B A, with A = π(0.01)² = 3.142 × 10⁻⁴ m²

ℰ_{peak} = 2 π \times 13.56 \times 10^{6} \times 10^{- 6} \times 3.142 \times 10^{- 4} = 85.2 \times 3.142 \times 10^{- 4} = 26.8 mV

Answer: 26.8 mV peak (1 turn); 134 mV peak (5 turns)

A field of 1 μT corresponds to H = B/μ₀ ≈ 0.8 A/m — a weak NFC field at the edge of the reading range. This is why NFC tag antennas use several turns and are tuned to resonance with a capacitor: at resonance the voltage across the tag chip is multiplied by the loaded Q of the coil, lifting tens of millivolts to the volt level needed to power the chip.

Magnetic Pickup in a PCB Signal Loop

Problem: A trace and its return path enclose a loop area of 2 cm² on a PCB near a switching circuit that produces a 50 MHz magnetic field of 0.1 μT peak. Find the induced noise voltage. How much does re-routing the return path directly under the trace, reducing the area to 0.2 cm², help?

ℰ_peak = ω B A

ℰ_{1} = 2 π \times 50 \times 10^{6} \times 0.1 \times 10^{- 6} \times 2 \times 10^{- 4} = 6.28 mV ℰ_{2} = 0.628 mV

Answer: 6.28 mV before, 0.63 mV after — a 20 dB improvement

Induced voltage is directly proportional to loop area, so a 10× reduction in area gives 20 log₁₀(10) = 20 dB less pickup — with no shielding and no extra components. The same loop radiates 20 dB less too, since a small loop's radiated field is proportional to its area. This is the single most effective EMC layout rule.

Summary

Concept	Expression	Key point
Faraday's law	ℰ = −N dΦ_B/dt	EMF from changing flux
Integral form	∮ 𝐄·d𝐥 = −d/dt ∫ 𝐁·d𝐀	Induced 𝐄 circulates; not conservative
Differential form	∇ × 𝐄 = −∂𝐁/∂t	Phasor: ∇ × 𝐄 = −jω𝐁
Lenz's law	(minus sign)	Induced current opposes the change
Generator	ℰ_peak = NBAω	Rotation gives sinusoidal EMF
Motional EMF	ℰ = BLv	Changing area
Transformer	V₂/V₁ = N₂/N₁; Z_in = (N₁/N₂)²Z_L	Shared flux, impedance transformation
Loop pickup	ℰ = 2πf·N·B·A	Grows with frequency and area

A changing magnetic flux induces an EMF; the size depends on the rate of change, not on the amount of flux.
The minus sign (Lenz's law) makes induced currents oppose the change that caused them, which is required by conservation of energy.
In field form, a time-varying magnetic field creates a circulating electric field: ∇ × 𝐄 = −∂𝐁/∂t. This is Maxwell's third equation.
Generators, transformers, inductive coupling, wireless charging, NFC and much EMI are all applications of the same law.

Faraday's law tells us that changing magnetic fields create electric fields. In Part 5: Ampere's Circuital Law we look at the reverse link — how currents create magnetic fields — and in Part 6 discover that a changing electric field does so too.

Put It Into Practice

Use the NEXT ANTENNAS calculators to apply the ideas from this article to real designs.

Antenna Design Patch Antenna Calculator Width, length, effective permittivity and inset feed of a rectangular microstrip patch for your frequency and substrate. Open calculator → Transmission Lines Microstrip Calculator Trace width for a target impedance, effective permittivity and guided wavelength on any PCB substrate. Open calculator → Impedance Matching Smith Chart Visualiser Plot impedances, reflection coefficient and VSWR, and build matching networks step by step. Open calculator → All Tools All RF Calculators Browse every NEXT ANTENNAS calculator for RF, microwave and antenna design in one place. Open calculator →

Faraday's Law of Electromagnetic Induction

Introduction to Electromagnetic Induction

Understanding Electromagnetic Induction

Discovery of Electromagnetic Induction

Relationship Between Magnetic Fields and Induced Electric Fields

Changing Magnetic Flux and Induced EMF

Faraday's Law of Electromagnetic Induction

Statement of Faraday's Law

Mathematical Expression of Faraday's Law

Understanding the Negative Sign in Faraday's Law

Lenz's Law

Statement of Lenz's Law

Direction of Induced Current

Relationship Between Faraday's Law and Lenz's Law

Integral Form of Faraday's Law

Differential Form of Faraday's Law

Applications of Faraday's Law

Electrical Generators

Transformers

Inductive Coupling

Wireless Power Transfer

Electromagnetic Interference

Numerical Examples

EMF From a Changing Magnetic Field

AC Generator Output Voltage

Motional EMF in a Sliding Bar

Voltage Induced in an NFC Loop at 13.56 MHz

Magnetic Pickup in a PCB Signal Loop

Summary

Put It Into Practice

Keep Learning