Faraday's Law Calculator — induced EMF from changing magnetic flux

Faraday's law of induction states that the electromotive force induced in a circuit equals the negative rate of change of the magnetic flux through it. Written for a coil of N turns it is EMF = −N × dΦ/dt. It is the principle behind every electrical generator, every transformer, every induction motor and every wireless charger, and it is the reason a bicycle dynamo produces nothing at all until the wheel starts turning.

This calculator computes induced EMF from three different starting points, because in practice you rarely measure flux directly — you measure a field, an area and a time, or a rotation speed. Arb Digital publishes it alongside the site's other electromagnetism tools, and it covers Faraday's law of induction specifically, not the separate law of electrolysis that shares his name.

What This Faraday's Law Calculator Does

The first mode is the everyday laboratory case: a coil of fixed area sits in a field that changes from one value to another over some interval. The tool computes the flux change as area multiplied by the field change, divides by the time, and multiplies by the turns.

The second mode takes the flux change in webers directly, for when you already have it or are working from a flux-linkage figure. The third handles a coil rotating steadily in a uniform field, which is the generator case: here the flux varies sinusoidally and the tool returns the peak EMF, N B A ω.

Adding a circuit resistance turns the EMF into a current and a power. This matters because an induced EMF and an induced current are different things, and the distinction is where a lot of confusion lives. The EMF exists whenever the flux changes, whether or not the circuit is closed. The current only flows if there is a path, and it is the current — not the EMF — that produces the opposing force you feel when you move a magnet near a closed loop.

How to Use It

1. Pick the mode matching your measurement. Field-and-area is the usual one. Use the rotating mode only for a coil turning at constant angular speed in a steady field.
2. Enter the turns as a count. N multiplies the result directly, so it is the cheapest way to increase induced voltage and the first thing to check if your answer is off by a round factor.
3. Use teslas for the field and square metres for the area. Their product is webers. If your field is in gauss, divide by 10,000 first; if your area is in square centimetres, divide by 10,000 as well.
4. Keep the time interval realistic. EMF is inversely proportional to it, so the same flux change over a millisecond instead of a second produces a thousand times the voltage. This is not a rounding detail — it is how ignition coils work.
5. Add a resistance only if the circuit is closed. Leaving it at zero reports the open-circuit EMF with no current, which is the honest answer for an unconnected coil.

The Formula: EMF = −N dΦ/dt

Magnetic flux through a flat coil is Φ = B A cos θ, where θ is the angle between the field and the normal to the coil. The flux is measured in webers, one weber being one tesla square metre, a derived SI unit set out in the BIPM SI Brochure. Faraday's law says the induced EMF is the negative of the rate at which that flux changes, multiplied by the number of turns. Georgia State University's HyperPhysics page on Faraday's Law puts it as the induced EMF in a coil being equal to the negative of the rate of change of magnetic flux times the number of turns.

There are exactly three ways to change that flux, and they correspond to the three terms in the product B A cos θ. Change the field strength, change the area the circuit encloses, or change the orientation. A transformer uses the first, a sliding-rail circuit the second, a generator the third.

Work the defaults. A 200-turn coil of area 0.01 m² sits in a field that rises from 0 to 0.5 T in 0.1 s. The flux change is 0.01 × 0.5 = 0.005 Wb. The rate of change is 0.005 ÷ 0.1 = 0.05 Wb/s. The EMF is 200 × 0.05 = 10 V. Into a 5 Ω circuit that drives 2 A and dissipates 20 W. The tesla itself is defined through the vacuum magnetic permeability, whose 2022 CODATA value of 1.25663706127 × 10⁻⁶ N A⁻² is published by NIST.

The Minus Sign Is Lenz's Law, and It Is Not Decoration

The negative sign in Faraday's law is a separate physical statement usually credited to Lenz: the induced current flows in whatever direction opposes the change that produced it. If the flux through a loop is increasing, the induced current creates a field opposing the increase. If it is decreasing, the induced current tries to maintain it.

This is energy conservation wearing a disguise. Suppose the sign were positive. An induced current that reinforced the change causing it would strengthen the flux, which would induce more current, which would strengthen the flux further — free energy from a runaway loop. The minus sign is what forbids that, and it is why pushing a magnet into a coil takes work: the induced current opposes you, and the mechanical effort you spend is exactly the electrical energy that appears.

This calculator reports the magnitude of the EMF and uses the sign to indicate direction relative to the flux change, because in most practical problems what you want is the size of the voltage. Just remember the opposition is real and shows up as drag on a generator shaft. A generator with nothing connected spins freely; connect a load and it becomes noticeably harder to turn, and the extra torque required is precisely the electrical power being delivered.

Why Time Matters More Than Field Strength

The most useful thing this law tells you is that induced voltage depends on the rate of flux change, not on the amount. A large flux change spread over a long time induces very little; a small flux change collapsed suddenly induces a great deal.

This is the operating principle of the automotive ignition coil. The magnetic field in the coil is not extraordinary and the flux is modest, but when the current is interrupted the field collapses in tens of microseconds. Divide even a small flux change by a microsecond-scale interval and multiply by tens of thousands of turns and you get tens of kilovolts from a twelve-volt supply.

It is also why switching off an inductive load is hazardous to the switch. Opening a relay coil circuit forces the current to zero almost instantly, and the coil responds with a voltage spike far above the supply, which is what arcs across the contacts. The flyback diode fitted across every relay coil exists to give that current somewhere to go so the collapse is gradual. The inductor energy calculator quantifies the energy stored in that field, which is what has to be dissipated somewhere.

This Is Induction, Not Electrolysis

Michael Faraday has two entirely separate laws named after him and they get confused constantly in search results. The law on this page is Faraday's law of induction, part of Maxwell's equations, relating induced EMF to changing magnetic flux, with results in volts.

Faraday's laws of electrolysis are a different subject altogether: they relate the mass of a substance deposited at an electrode to the electric charge passed through the cell, using the Faraday constant of about 96,485 coulombs per mole. That is electrochemistry, its inputs are current, time and molar mass, and its output is a mass in grams. If you are electroplating something or calculating deposition rates, this is not the calculator you want.

The shared name reflects nothing more than one man's productivity. If your problem involves a magnet, a coil, or a changing field, you are in the right place. If it involves an electrolyte and an electrode, you are not.

Common Mistakes to Avoid

• Expecting an EMF from a steady field — only the rate of change matters. A stationary coil in a constant field induces nothing, however strong the field.
• Forgetting the number of turns — N multiplies the answer directly, and omitting it on a thousand-turn coil is a thousandfold error.
• Mixing gauss with teslas or square centimetres with square metres — both mistakes are factors of ten thousand, and both are easy to make silently.
• Confusing EMF with current — the EMF exists whenever flux changes. Current needs a closed circuit, and it is the current that produces the opposing drag.
• Reaching for this page for electrolysis — Faraday's electrolysis laws are unrelated and use the Faraday constant to find deposited mass from charge.

Related Free Tools From Arb Digital

To convert field units before entering them, use the magnetic field converter, and for the field a current-carrying conductor produces in the first place, the magnetic field of a wire calculator. Coil properties are covered by the inductance converter and the energy in a coil's field by the inductor energy calculator. Once you have an EMF, the Ohm's law calculator and the electrical power calculator take it through to current and heat. For the electrostatic side of the subject, see the electric field calculator and the electric potential calculator. The full free online tools hub lists everything Arb Digital publishes.

Frequently Asked Questions

This tool is provided for educational and study use. It assumes a uniform field over the whole coil area and a linear change over the interval entered, and it does not model coil inductance, eddy currents, core saturation or the loading effect of the circuit on the source.