Faraday's Law of Induction — The Complete Physics Guide
In 1831, Michael Faraday made one of the most consequential discoveries in the history of technology: a changing magnetic field can generate an electric current, even with no battery or other voltage source in sight. This phenomenon — electromagnetic induction — is the physical principle behind virtually every electric generator on Earth, from the massive turbines at a hydroelectric dam to the tiny dynamo on a bicycle wheel. Without Faraday's discovery, the entire electrical grid as we know it simply wouldn't exist.
Faraday's insight was as elegant as it was powerful: it isn't a magnetic field itself that generates current, but a changing magnetic field (or equivalently, a coil moving through a steady field) — a subtlety that took considerable experimental ingenuity to uncover, since a stationary magnet near a stationary coil produces no current at all, no matter how strong the field.
What is Electromagnetic Induction?
Electromagnetic induction is the generation of an electromotive force (EMF) — essentially, a voltage capable of driving current — in a conductor due to a changing magnetic flux through it. Magnetic flux (Φ) measures the total amount of magnetic field passing through a given area: for a uniform field perpendicular to a flat area, Φ = BA, measured in webers (Wb). Whenever this flux changes — because the field strength changes, the area changes, or the orientation between field and area changes — an EMF is induced in any conductor bounding that area.
This can happen in three distinct ways: the magnetic field strength itself can change (as when a magnet is moved toward or away from a coil), the area of the circuit can change (as when a conducting loop is stretched or shrunk within a field), or the angle between the field and the coil can change (as when a coil rotates within a fixed field — precisely the mechanism used in every rotating generator).
The Formula Explained
N is the number of turns in the coil — more turns means proportionally more induced EMF, since each turn contributes its own share to the total. ΔΦ is the change in magnetic flux through a single turn. Δt is the time interval over which this change occurs — faster changes produce larger EMFs, which is why briskly plunging a magnet into a coil produces a much bigger spike in induced voltage than slowly sliding it in.
The complete form of Faraday's Law includes a minus sign, EMF = −N ΔΦ/Δt, known as Lenz's Law: the induced EMF always acts to oppose the change in flux that created it. This calculator reports the magnitude of the induced EMF; the direction (and therefore the sign) is determined separately by Lenz's Law.
How to Use This Calculator
Use "N, B₁, B₂, A & Δt" to find the induced EMF given the number of coil turns, the initial and final field strengths, the coil's cross-sectional area, and the time over which the change occurs. Use "EMF, B₁, B₂, A & Δt" to find how many turns a coil would need to produce a target EMF. Use "EMF, N, B₁, B₂ & A" to find how quickly the field must change to produce a given EMF.
Worked Example 1 — EMF from a Changing Field
Problem: A 100-turn coil of area 0.02 m² sits in a field that changes from 0.1 T to 0.5 T over 0.5 s. Find the induced EMF.
ΔΦ = |0.5 − 0.1| × 0.02 = 0.008 Wb
EMF = NΔΦ/Δt = (100)(0.008)/0.5 = 1.6 V
Worked Example 2 — Designing a Coil for a Target EMF
Problem: How many turns are needed to produce 12 V from the same changing field (0.1 T to 0.5 T, area 0.02 m²) over 0.4 s?
ΔΦ = 0.4 × 0.02 = 0.008 Wb
N = EMF × Δt / ΔΦ = (12)(0.4)/0.008 = 600 turns
Lenz's Law — The Direction of Induced Current
Lenz's Law states that induced current always flows in the direction that opposes the change producing it — if flux through a coil is increasing, the induced current creates its own magnetic field opposing that increase; if flux is decreasing, the induced current tries to maintain it. This isn't an arbitrary rule but a direct consequence of energy conservation: if induced current instead reinforced the change, it would create a runaway feedback loop generating energy from nothing, violating the most fundamental law in all of physics.
Lenz's Law has a very tangible physical consequence: it always takes work to induce a current, since the induced magnetic field always resists the motion causing it. This is precisely why turning a generator's crank (or a wind turbine's blades) gets harder as more electrical current is drawn from it — you're literally fighting against the induced field with every turn.
Real-World Applications
Electric generators: power stations of every kind — coal, nuclear, hydroelectric, wind — ultimately generate electricity by rotating a coil within a magnetic field (or vice versa), continuously changing the flux and inducing an alternating EMF.
Transformers: use a changing current in one coil to induce a changing flux, which induces an EMF in a second nearby coil, allowing voltage to be efficiently stepped up or down throughout the electrical grid.
Wireless charging and induction cooktops: both use rapidly alternating magnetic fields to induce currents in a nearby coil or in the base of a cooking pot, transferring energy without any direct electrical contact — the same fundamental physics scaled from a smartphone charging pad to an industrial induction furnace capable of melting metal.
Mutual Induction and Transformers
When two coils are placed near each other, a changing current in one coil produces a changing flux that induces an EMF in the second — a phenomenon called mutual induction, and the working principle of every transformer. A transformer's primary coil is connected to an AC source, whose constantly alternating current produces a constantly changing magnetic flux; this changing flux, guided through a shared iron core, induces an alternating EMF in the secondary coil. The ratio of induced voltage to input voltage equals the ratio of secondary to primary turns (Vs/Vp = Ns/Np), allowing voltage to be stepped up or down with remarkable efficiency and almost no moving parts.
This is precisely why electrical grids transmit power at extremely high voltages (hundreds of thousands of volts) over long distances, then step voltage back down to safe household levels near the point of use — high voltage transmission dramatically reduces resistive power losses in the wires, and transformers make converting between voltage levels essentially free of energy waste.
Eddy Currents
Faraday's Law applies not just to coils of wire but to any conductor experiencing a changing flux — including solid metal objects, where induction produces swirling loops of current called eddy currents. These currents generate their own opposing magnetic field (per Lenz's Law), producing a braking effect whenever a conductor moves through a magnetic field or experiences a rapidly changing one. This effect is a nuisance in some contexts (eddy currents in transformer cores waste energy as heat, which is why cores are built from thin, electrically insulated laminations to suppress them) but genuinely useful in others.
Eddy current braking is used in roller coasters, some train systems, and industrial machinery to provide smooth, contact-free braking without any mechanical wear — a strong magnet near a moving metal plate induces eddy currents that oppose the plate's motion, converting kinetic energy directly into heat with no friction or moving parts to maintain.