The Faraday’s law formula states that the EMF induced in a coil equals the number of turns multiplied by the rate at which magnetic flux through it changes: ε = -N(ΔΦ/Δt). EMF is measured in volts, flux in webers and time in seconds. The minus sign gives direction only, not size.
Wave a magnet through a coil of wire and a voltmeter twitches. Wave the same magnet through twice as fast and the needle jumps roughly twice as far — the magnet is no stronger, only quicker. That single observation is why the equation contains a rate of change rather than a field strength.
The trouble starts when you have to put numbers in. Plenty of students who can explain electromagnetic induction perfectly still drop marks by feeding the equation B when it wants ΔB, or by quietly forgetting the coil’s 350 turns. This page is about the arithmetic — what to substitute, how to rearrange, and how to spot a nonsense answer.
What Is the Faraday’s Law Formula?
The Faraday’s law formula is ε = -N(ΔΦ/Δt): the induced EMF equals the number of turns times the rate of change of magnetic flux through the coil. Everything else in this article is that one line, rearranged.
Notice what the equation does not contain. There is no B on its own and no Φ on its own — only a change in flux divided by the time it took. A coil parked in the strongest magnet you own produces exactly zero volts, because nothing is changing.
Almost every calculation therefore has two stages. First work out the magnetic flux, then work out how fast it changed.
Substituting the flux into Faraday’s law gives the working form you will actually use most often. When only the field strength changes and the coil faces the field square-on, cos θ = 1 and the area is fixed:
This is the version that solves the majority of exam questions. The other versions come from letting A change instead of B, or letting θ change instead of either.
Every Symbol in the Faraday’s Law Formula (and Its SI Unit)
Each symbol in the Faraday’s law formula has one SI unit, and mixing them up is the single biggest source of wrong answers. Learn the table below and half the marks look after themselves.
| Symbol | Quantity | SI unit | Watch out for |
|---|---|---|---|
| ε | Induced EMF | volt (V) | Answers in mV are common for small coils. |
| N | Number of turns | none (a pure count) | Easy to forget entirely. N multiplies your answer. |
| Φ | Magnetic flux through one turn | weber (Wb) | 1 Wb = 1 T·m2 = 1 V·s. |
| ΔΦ | Change in flux (final minus initial) | weber (Wb) | A decrease is negative. Keep the sign until the end. |
| Δt | Time over which the change happened | second (s) | Not the total experiment time. Convert ms to s. |
| B | Magnetic flux density | tesla (T) | Convert mT and gauss before substituting. |
| A | Area of one turn | square metre (m2) | 1 cm2 = 10-4 m2, not 10-2. |
| θ | Angle between the field and the coil’s normal | degrees or radians | Measured from the normal, not from the coil’s face. |
One term deserves its own name. The product NΦ is called the flux linkage, measured in webers (or weber-turns), and Faraday’s law can be read as “EMF equals the rate of change of flux linkage”.
That reading is worth adopting, because it stops N drifting out of your working. If you calculate the flux linkage before and after, the N is baked in from the start.
Every symbol in the Faraday’s law formula, with the SI unit it must be substituted in.
How to Calculate Induced EMF in Four Steps
To calculate induced EMF, find the flux before and after, subtract to get ΔΦ, divide by the time interval, then multiply by the number of turns. The four steps below work for every standard induced-EMF question.
- Find the flux at the start and at the end. Use Φ = B·A·cos θ for each moment. Convert cm2 to m2 and mT to T now, before anything else.
- Subtract to get ΔΦ. Always final minus initial. If the flux fell, ΔΦ is negative — that is information, not an error.
- Divide by Δt, then multiply by N. This gives ε = -N(ΔΦ/Δt). Only the time during which the flux was actually changing counts.
- Sanity-check the size and the sign. Quote the magnitude unless the question asks for direction.
That last step is the one most people skip, and it is the cheapest mark on the page. A rough feel for typical magnitudes catches a slipped power of ten instantly:
- A single loop and a hand-waved magnet: millivolts.
- A few hundred turns and a field switched off in a fraction of a second: a few volts to tens of volts.
- A mains generator coil: hundreds of volts.
- Anything above a few kilovolts from a classroom coil means a unit slipped somewhere.
Once you have a number, it is worth checking your working against our Faraday’s Law Calculator, which solves ε = N·A·(ΔB/Δt) and rearranges for the turns, area, field change or time interval so you can see exactly which substitution went astray.
Drag the turns slider and watch the peak EMF scale in exact proportion. Then drag the frequency slider — the field and the coil are unchanged, yet the voltage climbs, which is the rate-of-change idea made visible.
Rearranging the Faraday’s Law Formula
The Faraday’s law formula rearranges into four useful forms, one for each quantity you might be asked to find. The physics never changes; only the subject of the equation does.
| To find | Rearranged formula | Typical question |
|---|---|---|
| EMF, ε | ε = N · ΔΦ/Δt | “Find the EMF induced in the coil.” |
| Turns, N | N = ε · Δt / ΔΦ | “How many turns are needed to reach 9.0 V?” |
| Time, Δt | Δt = N · ΔΦ / ε | “How quickly must the field collapse?” |
| Field change, ΔB | ΔB = ε · Δt / (N · A) | “What field change produced this reading?” |
| Area, A | A = ε · Δt / (N · ΔB) | “What coil area would you need?” |
The magnitude signs are dropped in the table on purpose. Rearranging is easier when you work with sizes, then restore direction at the end using the physical argument rather than the algebra.
A practical tip: rearrange before you substitute. Putting numbers in first and then trying to unpick them is how sign errors and stray factors of N creep in.
What the Minus Sign Does to Your Answer
The minus sign in Faraday’s law changes the direction of the induced EMF, never its size. If a question asks “find the induced EMF”, the expected answer is almost always the magnitude — 2.8 V, not -2.8 V.
So what is it doing there? It encodes Lenz’s law: the induced EMF pushes current the way that opposes whatever change created it. Drop the minus sign and the equation would let a coil amplify its own flux forever, which would be free energy.
In practice, treat the sign as a separate question. Compute the size from the numbers, then decide the direction from the physics of the situation.
One caution when you carry the result forward. If you feed a negative EMF into a current calculation, the minus survives — so state your positive direction first, or work with magnitudes and note the direction in words. Remember too that an EMF is an energy-per-charge quantity, closely related to potential difference but produced by a changing flux rather than a chemical cell.
Average EMF or Peak EMF? Choosing the Right Version
Use ΔΦ/Δt when you want the average EMF over an interval, and dΦ/dt when you want the EMF at one instant. Choosing the wrong one is a quiet source of lost marks, because both give “an EMF” and neither looks obviously wrong.
The distinction is really about gradients. Flux linkage plotted against time has a slope at every point, and the induced EMF is that slope.
A steeper flux-linkage graph means a larger induced EMF; a horizontal section means no EMF at all.
A rotating coil is the case where the difference really bites. Its flux varies sinusoidally, so the EMF does too, and the peak value is set by the angular frequency ω = 2πf:
The average over a full cycle is zero, because the EMF spends as long negative as positive. That is why rotating coils are quoted by peak or by RMS value, with the RMS equal to the peak divided by sqrt(2), or about 0.707 of the peak.
This sinusoidal output is precisely why generators produce alternating rather than direct current: the sign of the gradient flips twice per revolution, whatever you do.
Common Mistakes When Using the Faraday’s Law Formula
Most wrong answers in Faraday’s law questions come from five specific substitution errors, not from misunderstanding the physics. Each one is worth checking before you commit an answer.
1. Substituting B instead of ΔB
A field of 0.60 T that grew from 0.12 T gives ΔB = 0.48 T, not 0.60 T. The equation only ever cares about the change. If the question quotes a starting value, it is there to be subtracted.
2. Leaving the area in cm2
Squared units bite twice: 1 cm2 is 10-4 m2, not 10-2. A coil of 25 cm2 is 2.5 × 10-3 m2. Getting this wrong scales your answer by a factor of 100.
3. Losing the number of turns
Faraday’s law uses flux linkage, NΦ, not flux. With 350 turns, forgetting N makes your answer 350 times too small — and an 8.0 mV answer to a “few volts” question rarely triggers alarm bells on its own.
4. Measuring θ from the wrong line
The angle in Φ = B·A·cos θ is measured between the field and the normal to the coil, the line sticking straight out of its face. A coil lying flat in a vertical field has θ = 0 and maximum flux, not zero.
5. Using the wrong Δt
Δt is the time the flux took to change, not how long the experiment lasted. If a magnet sits still for 5 s and is then yanked out in 0.10 s, the interval you want is 0.10 s.
How the Faraday’s Law Formula Connects to Other Equations
Faraday’s law sits at the centre of a small family of equations that share its variables. Knowing which neighbour to reach for turns a hard question into an easy one.
Magnetic flux density. Before any flux calculation you need B in teslas, which is the quantity the magnetic field equations supply — for a solenoid, for instance, B depends on the current and the turns per metre.
Motional EMF. When a straight conductor of length L slides at speed v across a perpendicular field, the changing area gives ε = B·L·v. It is Faraday’s law with the area doing the changing, not a separate rule.
Ohm’s law. An EMF on its own drives nothing until the circuit is closed. Once it is, the induced current follows from Ohm’s law as I = ε/R, which is how nearly every multi-part induction question ends.
For a fuller treatment of the phenomenon itself — Lenz’s law, transformers and the rest — see the companion guide to electromagnetic induction. The formal derivation appears in MIT OpenCourseWare’s chapter on Faraday’s law (PDF), and the University of Tennessee’s Faraday’s law module works through the flux-linkage sign conventions in detail.
