Electromagnetism · Electrical Engineering

Faraday's Law and Lenz's Law: Electromagnetic Induction Explained with Interactive Calculators

Every power station, transformer, electric guitar pickup and wireless charger works because a changing magnetic flux induces a voltage. This guide explains magnetic flux, Faraday's law, the direction rule of Lenz's law, motional EMF, the AC generator, transformers, inductance and eddy currents, with worked examples and an animated generator you can control.

Faraday + LenzAnimated AC generatorMotional EMF calculatorWorked examples

By Bimal Ghimire · Published October 30, 2026 · 16 min read

Key facts

Faraday's law: the induced EMF in a coil of N turns is ε = −N dΦ/dt, where Φ = ∫B·dA is the magnetic flux in webers (1 Wb = 1 T·m²).
Lenz's law: the induced current opposes the change in flux that causes it; this is the minus sign, and it follows from energy conservation.
Motional EMF: a rod of length L moving at speed v across a field B develops ε = BLv.
AC generator: a coil rotating at ω gives ε = NBAω sin ωt, with peak NBAω and RMS NBAω/√2.
Ideal transformer: V₂/V₁ = N₂/N₁; it needs changing flux, so it works with AC, not steady DC.
Michael Faraday discovered induction in 1831; Heinrich Lenz stated the direction rule in 1834.

Foundation

Magnetic Flux

$$\Phi_B=\int_S\mathbf B\cdot d\mathbf A\qquad\text{uniform field, flat loop:}\qquad\Phi_B=BA\cos\theta$$

θ is the angle between B and the normal to the loop. Flux is maximum when the field passes straight through the loop (θ = 0) and zero when the field lies in the plane of the loop (θ = 90°).

Flux measures how much magnetic field passes through a surface. It can change in three ways, and each gives a family of devices: change B (transformers, induction cooktops), change the area A (a sliding rod on rails), or change the angle θ (rotating generators).

The Law

Faraday's Law of Electromagnetic Induction

$$\mathcal E=-N\frac{d\Phi_B}{dt}\qquad\text{average over }\Delta t:\qquad\bar{\mathcal E}=-N\frac{\Delta\Phi_B}{\Delta t}$$

In field form (Maxwell–Faraday equation): $\nabla\times\mathbf E=-\partial\mathbf B/\partial t$. A changing magnetic field creates a circulating, non-conservative electric field even where there is no wire.

Example: a 200-turn coil of area 0.01 m² sits in a 0.5 T field perpendicular to it. The field is switched off uniformly in 0.1 s. Average EMF: $\bar{\mathcal E}=200\times(0.5\times0.01)/0.1=$ 10 V. The same flux change in 1 ms would give 1000 V, which is why opening an inductive circuit produces sparks.

Direction

Lenz's Law: Which Way Does the Current Flow?

The induced current always flows so that its own magnetic field opposes the change in flux. A practical recipe:

Find the direction of B through the loop and whether the flux is increasing or decreasing.
Decide the induced field: opposite to B if the flux is increasing, the same direction as B if it is decreasing.
Use the right-hand rule (thumb along the induced field, fingers curl with the current) to get the current direction.

Why it must be so: push a magnet's north pole toward a coil. The induced current makes the near face of the coil a north pole, which repels the magnet, so you must do work, and that work becomes electrical energy and heat. If the coil attracted the magnet instead, the magnet would accelerate by itself and generate energy from nothing, violating conservation of energy. Magnet-in-a-copper-tube demonstrations and magnetic braking in trains rely on the same principle.

Moving Conductors

Motional EMF: ε = BLv

A conducting rod of length L slides at speed v along rails in a uniform field B perpendicular to the circuit. The free charges in the rod feel a magnetic force $q\mathbf v\times\mathbf B$, which drives them along the rod. Equivalently, the circuit area grows at rate Lv, so $d\Phi/dt=BLv$.

$$\mathcal E=BLv,\qquad I=\frac{BLv}{R},\qquad F_{drag}=BIL=\frac{B^2L^2v}{R},\qquad P=F_{drag}\,v=I^2R$$

Example: B = 0.8 T, L = 0.5 m, v = 4 m/s, R = 2 Ω: ε = 1.6 V, I = 0.8 A, retarding force 0.32 N, and mechanical power Fv = 1.28 W, exactly equal to the electrical power I²R = 1.28 W. Energy is conserved, as Lenz's law guarantees.

Power Generation

The AC Generator: ε = NBAω sin ωt

A coil of N turns and area A rotating at constant angular speed ω in a uniform field B has $\theta=\omega t$, so

$$\Phi=NBA\cos\omega t,\qquad\mathcal E=-\frac{d\Phi}{dt}=NBA\omega\sin\omega t,\qquad\mathcal E_{peak}=NBA\omega,\qquad\mathcal E_{rms}=\frac{NBA\omega}{\sqrt2}$$

The EMF is largest when the flux is zero (the coil plane is parallel to B) and zero when the flux is largest: EMF and flux are 90° out of phase. Example: N = 100, A = 0.02 m², B = 0.4 T, 3000 rpm (ω = 314.16 rad/s, f = 50 Hz): peak EMF = 251.3 V, RMS = 177.7 V. Run it in the generator below.

Electromagnetic Induction Calculators

Turns N

Field B (T)

Coil area A (m²)

Speed (rpm)

Load resistance (Ω)

The animation is slowed down so you can follow it; the chart uses the true frequency. Load power assumes a purely resistive load and neglects the coil's own resistance and inductance.

Field B (T)

Rod length L (m)

Speed v (m/s)

Circuit resistance R (Ω)

Turns N

Area A (m²)

Initial B (T)

Final B (T)

Initial angle θ₁ (°)

Final angle θ₂ (°)

Time Δt (s)

Primary voltage V₁ (V)

Primary turns N₁

Secondary turns N₂

Secondary current I₂ (A)

Solenoid turns N

Radius r (m)

Length ℓ (m)

Current I (A)

Applications

Transformers, Self-Inductance and Eddy Currents

Transformers

Both coils share the same changing core flux, so each turn has the same EMF: $V_2/V_1=N_2/N_1$. An ideal (lossless) transformer conserves power, so $I_2/I_1=N_1/N_2$. Example: 230 V on 1000 turns with 50 secondary turns gives 11.5 V. Real transformers lose a few percent to winding resistance, core hysteresis and eddy currents.

Self-inductance

$$\mathcal E=-L\frac{dI}{dt},\qquad L_{solenoid}\approx\frac{\mu_0N^2A}{\ell},\qquad U=\tfrac12LI^2$$

For a long solenoid (ℓ ≫ r). Example: N = 1000, r = 2 cm, ℓ = 30 cm gives L ≈ 5.26 mH and stores 10.5 mJ at 2 A.

Eddy currents

Changing flux also induces circulating currents inside solid conductors. They cause heating (useful in induction cooktops and furnaces, wasteful in transformer cores, which are therefore laminated) and braking forces (eddy-current brakes in trains and roller coasters, the slow fall of a magnet in a copper tube).

Going Deeper

When the "Flux Rule" Needs Care

The compact rule ε = −dΦ/dt combines two different physical effects: the electric field induced by a time-varying B (Maxwell–Faraday equation) and the magnetic force on charges in a moving conductor (motional EMF). For thin wire circuits both give the same answer, which is why the rule is so useful. Feynman points out cases where it must be applied with care, such as the Faraday disk (homopolar generator) or circuits whose conducting path changes as they move. In those cases, go back to the fundamental statement: the EMF is the integral of the force per unit charge, $\mathcal E=\oint(\mathbf E+\mathbf v\times\mathbf B)\cdot d\boldsymbol\ell$.

Keep Learning

Related Tools and Guides

FAQ

Frequently Asked Questions

What does Faraday's law of induction state?

The electromotive force (EMF) induced in a closed loop equals the negative rate of change of the magnetic flux through it: ε = −dΦ/dt, or ε = −N dΦ/dt for a coil of N turns. Flux can change because the field changes, the area changes, or the orientation of the loop changes.

What is Lenz's law?

Lenz's law says the induced current flows in the direction that opposes the change in magnetic flux that produced it. It is the reason for the minus sign in Faraday's law and is a consequence of conservation of energy: if the induced current reinforced the change, it would create energy from nothing.

What is the unit of magnetic flux?

The weber (Wb). One weber is one tesla times one square metre (1 Wb = 1 T·m²). A flux changing at 1 Wb/s through one turn induces 1 volt, so 1 V = 1 Wb/s.

How do you calculate the EMF of an AC generator?

For a coil of N turns and area A rotating at angular speed ω in a uniform field B, the flux is Φ = NBA cos ωt and the EMF is ε = NBAω sin ωt. The peak EMF is NBAω and the RMS value is NBAω/√2. For 100 turns, 0.02 m², 0.4 T and 3000 rpm the peak is about 251 V.

What is motional EMF?

When a conductor of length L moves with velocity v perpendicular to a magnetic field B, the magnetic force on its charges produces an EMF ε = BLv. If the circuit is closed with resistance R, the current I = BLv/R experiences a retarding force BIL, and the mechanical power Fv equals the electrical power I²R.

Why do transformers only work with AC?

A transformer relies on Faraday's law: a changing current in the primary coil produces a changing flux in the core, which induces an EMF in the secondary. A steady DC current produces a constant flux, so no EMF is induced after the initial switch-on transient. For an ideal transformer, V₂/V₁ = N₂/N₁.

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

Yes. In differential form it is the Maxwell–Faraday equation, ∇×E = −∂B/∂t: a time-varying magnetic field produces a curling electric field. The circuit form ε = −dΦ/dt also includes motional EMF from the magnetic force on moving charges, which comes from the Lorentz force law rather than from Maxwell's equations alone.

Sources

References

Faraday, M. (1832). Experimental researches in electricity. Philosophical Transactions of the Royal Society of London, 122, 125–162.

Griffiths, D. J. (2017). Introduction to Electrodynamics (4th ed.), Chapter 7: Electrodynamics. Cambridge University Press.

Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol. II, Chapter 17: The Laws of Induction. feynmanlectures.caltech.edu

Ling, S. J., Sanny, J., & Moebs, W. (2016). University Physics Volume 2, Chapter 13: Electromagnetic Induction, and Chapter 14: Inductance. OpenStax. openstax.org

Cite this article: Ghimire, B. (2026). Faraday's Law & Lenz's Law: Electromagnetic Induction Explained. Engineers Views. https://engineersviews.com/blog/faradays-law-lenzs-law-electromagnetic-induction/