Spin a bicycle wheel with a dynamo light fitted to the fork and the lamp glows. Pedal harder and it glows brighter. There is no battery anywhere in the system. The light is powered entirely by the rider's legs, converted into electricity by a small magnet sweeping past a coil of copper wire. The same trick, scaled up by a factor of a billion, lights entire cities.
This article explains the physical law behind that conversion — Faraday's law of induction — shows how to compute the induced voltage, and works through a numerical example you can check by hand. It also clears up the sign confusion that trips up almost everyone the first time.
Why this calculation matters
Faraday's law is the operating principle of nearly every machine that generates or transforms electrical power. Generators in power stations, the alternator under a car hood, transformers on utility poles, induction motors, inductive charging pads, metal detectors, and the read heads in older hard drives all rely on the same idea: a changing magnetic flux through a loop of wire drives a voltage around that loop.
If you design or analyze any of these devices, you need to predict that voltage. Size the coil wrong, or misjudge how fast the flux changes, and a generator delivers too little voltage or a transformer overheats. The law also sets hard limits: it tells you why a transformer cannot work on direct current, why a faster-spinning generator produces a higher voltage, and why a coil with more turns is more sensitive. None of that intuition is reliable without the equation behind it.
The core formula
Faraday's law states that the electromotive force, or EMF, induced in a coil equals the negative rate of change of magnetic flux linkage:
EMF = -N * (dPhi/dt)
Here N is the number of turns in the coil, Phi is the magnetic flux through a single turn, and dPhi/dt is how quickly that flux is changing with time. EMF is measured in volts.
The magnetic flux itself is the amount of magnetic field passing through the loop area:
Phi = B * A * cos(theta)
B is the magnetic flux density in tesla, A is the loop area in square metres, and theta is the angle between the field and the normal to the loop. Flux is measured in webers, where one weber equals one tesla times one square metre.
Put the two together and you see there are three independent ways to change the flux and therefore induce a voltage: change the field strength B, change the loop area A, or rotate the loop to change the angle theta. A rotating generator uses the third route; a transformer uses the first.
The minus sign is Lenz's law. It says the induced current flows in whatever direction opposes the change that created it. If the flux through the coil is increasing, the induced current sets up a field that pushes back against the increase. This is not a mathematical decoration — it is conservation of energy. If the induced current reinforced the change instead of opposing it, you would get free energy. For finding the magnitude of the voltage you can drop the sign and use absolute values; keep it only when the direction of current matters.
A worked example
Take a flat coil with N = 200 turns. A magnetic field threads through it, and the flux through one turn falls steadily from 0.05 Wb to 0 over a time interval of 0.1 s. Find the magnitude of the induced EMF.
Step 1 — find the rate of change of flux. The flux drops by 0.05 Wb in 0.1 s, and it does so at a constant rate, so the magnitude of dPhi/dt is simply the change divided by the time:
|dPhi/dt| = 0.05 / 0.1 = 0.5 Wb/s
Step 2 — multiply by the number of turns. Each turn sees the same changing flux, and the turns are in series, so their contributions add:
|EMF| = N * |dPhi/dt| = 200 * 0.5
|EMF| = 100 V
So a coil of 200 turns, with the flux through it collapsing in a tenth of a second, produces 100 volts. That is a substantial voltage from a fairly modest setup, and it points straight at the two design levers: a faster flux change or more turns both raise the EMF. Halve the collapse time to 0.05 s and the voltage doubles to 200 V. Double the turns to 400 and it doubles again. This is exactly the principle every generator and transformer exploits — wind more turns, spin faster, or use a stronger field.
Common mistakes
Expecting a steady field to induce a voltage. A magnet held motionless next to a coil produces nothing, no matter how strong it is. Only a changing flux drives an EMF. This is why transformers need alternating current — direct current gives a constant flux and zero induced voltage on the secondary.
Dropping the number of turns. The flux per turn might be small, but the EMF scales directly with N. Forgetting the factor of N underestimates the voltage of a multi-turn coil by orders of magnitude. The product N times Phi is sometimes called the flux linkage for exactly this reason.
Confusing flux and flux density. B is the field strength in tesla; Phi is the flux in webers, equal to B times area. Faraday's law uses the rate of change of flux, not of field strength alone. Mixing them up scrambles the units.
Reading the minus sign as a real negative voltage. The sign in Faraday's law encodes direction, not a numerical value to subtract. For a magnitude calculation, work with absolute values and use Lenz's law separately to decide which way the current flows.
Assuming the flux changes linearly. The simple formula change-over-time gives the average EMF. If the flux varies sinusoidally, as in a real rotating generator, the instantaneous EMF is the derivative of that sinusoid and the peak value is higher than the average.
Try the interactive NovaSolver calculator
Working one case by hand is useful, but seeing how turns, field strength, and speed trade off in real time builds far better intuition. The Electromagnetic Induction (Faraday's Law) Simulator on NovaSolver animates a magnet moving through a coil: you set the speed mode, the number of turns, the coil area, and the field strength, and it returns the induced EMF, the magnetic flux, and the induced current live, with a chart of flux and EMF against time.
Related calculators
- Electromagnetic Induction & Faraday's Law Simulator — a complementary view of induction that emphasises how flux linkage and Lenz's law shape the induced current.
- Coulomb Law Simulator — for the electrostatic force between point charges, the starting point of the whole electromagnetic story.
- Solenoid Magnetic Field Calculator — to find the field B inside a coil before you feed it into a flux calculation.
You can browse the full set in the electromagnetism tools hub.
Closing note
Faraday's law is compact, but it carries an enormous amount of engineering. The single equation EMF equals minus N times the rate of change of flux explains how a bicycle dynamo, a power-station generator, and a phone charger all do the same job. Three ideas are worth keeping: only a changing flux induces a voltage, the voltage scales with both the number of turns and the speed of the change, and the minus sign is conservation of energy in disguise. Get those right and most induction problems become a matter of identifying what the flux is doing and how fast.
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