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Faraday's Law EMF Calculator

Physics

The induction scenario

One set of entries feeds every panel, so the EMF, the flux table, the circuit block and the charts all describe the same coil. Switching mode keeps what you have already typed.

EMF = −N · A · cos θ · (B₂ − B₁) / Δt

The coil sits still while the field through it is ramped — the classic solenoid or electromagnet problem. Only the field changes, so the area and the angle come out of the flux as constants.

200 turns of 120 cm² wound coil while the pole field is driven from 0.05 T to 0.35 T over a quarter of a second — the standard laboratory induction demonstration.
0 to 10; also used by the copy and download buttons.

The coil and the change

A whole number of at least 1. The EMF scales straight with it.
Negative values mean a field pointing the other way.
ΔB is taken as B₂ − B₁, so a rising field gives a negative EMF.
Enter a dimension and the enclosed area of one turn is worked out for you.
The area enclosed by one turn, not by the whole winding.
Measured from the coil's normal — 0° threads the most flux, 90° threads none.
Must be greater than zero — an instantaneous change would give an infinite EMF.
Leave blank to skip the current, power and charge block.
cos θ falls to zero as the coil turns edge-on to the field, and so does the flux.

Induced EMF (magnitude)

2.8800 V

Lenz-signed: -2.8800 Vflux rising

EMF = −N · A · cos θ · (B₂ − B₁) / Δt

Direction of the induced current

Flux is rising, so the induced current runs clockwise viewed with B coming out of the page towards you.

A clockwise current makes its own field into the page, opposing the increase. The signed EMF is negative for exactly this reason.

Magnetic flux

Initial flux Φ₁ (per turn)600.0000 µWb
Final flux Φ₂ (per turn)4.2000 mWb
Flux change ΔΦ = Φ₂ − Φ₁3.6000 mWb
Rate of change dΦ/dt14.4000 mWb/s
Flux linkage NΦ₂840.0000 mWb-turns
Linkage change NΔΦ720.0000 mWb-turns
Interval Δt250.0000 ms

The circuit

Induced current I = |EMF| / R240.0000 mA
Power dissipated EMF² / R691.2000 mW
Charge transferred N·ΔΦ / R60.0000 mC

The EMF in other units

EMF in µV2880000.0000
EMF in mV2880.0000
EMF in V2.8800
EMF in kV0.0029

The flux change in other units

ΔΦ in Wb0.0036
ΔΦ in mWb3.6000
ΔΦ in µWb3600.0000
ΔΦ in Mx360000.0000

The coil in the field

Bnθ = 0.0°200 turns· θ is measured from the coil's normal n, not from its planeinduced current: clockwise, seen with B coming out of the page towards you

The field-line spacing tracks the strength you entered, and the loop is drawn with its long axis at right angles to its normal — square across the field at θ = 0°, flat along it at θ = 90°.

Flux and EMF over the interval

0.00 s125.00 ms250.00 mstimeflux ΦEMF

An average EMF assumes a steady rate of change, so the flux is a straight ramp and the EMF is the flat line whose height is its slope.

Formula, substitution and the flux values in between.
StepWorking
Flux through one turnΦ = B · A · cos θ
Initial flux Φ₁0.05 T × 0.012 m² × cos 0° = 0.0006 Wb
Final flux Φ₂0.35 T × 0.012 m² × cos 0° = 0.0042 Wb
Flux changeΔΦ = Φ₂ − Φ₁ = 0.0036 Wb
Rate of changedΦ/dt = 0.0036 Wb ÷ 0.25 s = 0.0144 Wb/s
Faraday's lawEMF = −N · dΦ/dt = −200 × 0.0144 Wb/s = -2.88 V
Magnitude|EMF| = 2.88 V

About This Tool

Faraday's Law Calculator – Induced EMF from Changing Magnetic Flux

Faraday's law of induction is the reason electricity is generated at all. Every power station alternator, transformer, induction hob, guitar pickup, metal detector and contactless charger is one equation at work: EMF = −N · ΔΦ / Δt. This induced EMF calculator evaluates it in every form a student or engineer actually meets — a field being ramped, a loop changing area, a coil turned between two angles, a coil spinning continuously, a rod sliding on rails, and a flux rate you already know — and then rearranges it to find whichever quantity is missing.

What magnetic flux really is

Magnetic flux through one turn is Φ = B · A · cos θ, measured in webers. It counts how much field actually threads the loop, which is why the angle appears: a coil square across the field catches everything, and a coil edge-on to it catches nothing at all. Multiply by the turns and you get the flux linkage Λ = N · Φ, the quantity Faraday's law really differentiates. A 200-turn coil of 0.012 m² in a 0.35 T field links 0.84 Wb-turns, and if that field arrived from 0.05 T over a quarter of a second, the average induced EMF is 2.88 V.

θ is measured from the normal, not the plane

The angle in cos θ is between the field and the line sticking out at right angles to the coil face. At θ = 0° the flux is at its maximum; at θ = 90°it is exactly zero. Read it off the coil's plane by mistake and every answer comes out with sine and cosine swapped.

Five ways the flux can change

Faraday's law does not care which factor moves. Ramp the field and you get EMF = −N·A·cos θ·ΔB/Δt. Change the area — a loop pulled clear of a magnet gap — and it is EMF = −N·B·cos θ·ΔA/Δt. Turn the coil and the cos θ term does the work. Spin it continuously and the flux follows cos ωt while the EMF follows N·B·A·ω·sin ωt, giving a peak EMF of N·B·A·ω and an RMS value of EMF_peak / √2. Slide a rod of length L at speed v across the field and the swept-area form collapses to EMF = B·L·v.

What the minus sign is telling you

The minus sign is Lenz's law: the induced current flows in whatever direction opposes the change that created it. Rising flux drives the current one way, falling flux the other, and the tool states which — clockwise or counter-clockwise, seen with the field coming towards you. It is really a statement about energy: drop the sign and a loop would amplify its own flux for nothing, which is also why dragging a rod through a field takes real mechanical work equal to the electrical power the circuit dissipates.

From EMF to current, power and charge

Close the loop through a resistance and Ohm's law finishes the job: I = |EMF| / R, P = EMF² / R, and the total charge pushed round the circuit is q = N·ΔΦ / R. That last result has no time in it — halve the interval and the current doubles while it flows for half as long — which is the principle behind a ballistic galvanometer and behind every search coil used to measure a field from a single flick of the needle.

Units, scale and where the numbers land

Fields are accepted in tesla, millitesla, microtesla and gauss; areas in m², cm², mm², in² and ft²; intervals from microseconds to minutes; flux in webers down to microwebers and in maxwells for older texts. Everything is normalised to SI before the arithmetic and converted back only for display, so a 12 mT swing through 3.5 cm² is handled exactly like a 100 T pulse. Laboratory search coils typically produce microvolts to volts, a bicycle dynamo a few volts, and a mains alternator hundreds — the tool restates the answer in volts, millivolts, microvolts and kilovolts so the magnitude is never in doubt.

Frequently Asked Questions

Is the Faraday's Law EMF Calculator free?

Yes, Faraday's Law EMF Calculator is totally free :)

Can I use the Faraday's Law EMF Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Faraday's Law EMF Calculator?

Yes, any data related to Faraday's Law EMF Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does this Faraday's law calculator work?

Pick the induction scenario you have — a ramped field, a changing loop area, a coil turned between two angles, a coil spun continuously, a rod sliding on rails, or a flux rate you already know — and every entry is normalised to tesla, square metres and seconds before Φ = B·A·cos θ and EMF = −N·ΔΦ/Δt are evaluated. A 200-turn coil of 0.012 m² whose field goes from 0.05 T to 0.35 T in 0.25 s gives 2.88 V. That same unrounded figure then drives the induced current, the dissipated power and the charge transferred, so nothing downstream is rebuilt from a rounded intermediate.

Is θ measured from the coil's face or from its normal?

From the normal — the line sticking out at right angles to the coil face. A coil lying square across the field has θ = 0° and threads the most flux; at θ = 90° the field skims along the plane of the coil and threads none at all. This is the single most common mistake with this law, because the picture in your head is usually of the coil's plane rather than of its normal. The tool says so on the angle field and reports a flux of exactly zero at 90° rather than the 6 × 10⁻¹⁷ that a naive cosine would produce.

What does the minus sign in Faraday's law actually do?

It is Lenz's law: the induced current flows the way that opposes the change that caused it. The tool takes every change as final minus initial — ΔΦ = Φ₂ − Φ₁ — and carries the minus through, so a rising flux gives a negative EMF and a falling flux a positive one. Both numbers are shown: the signed value tells you the direction, and the magnitude is what an answer key quotes. Drop the sign and you would have a loop that amplifies its own flux for free, which is why the minus is really a statement about energy conservation.

Why does the average EMF differ from the peak EMF of a spinning coil?

Turning a coil from 0° to 90° and stopping gives an average EMF over that interval — the total flux change divided by the time it took. A coil that keeps spinning produces a sinusoid whose value passes through zero when the coil is face-on and peaks a quarter turn later, so its peak N·B·A·ω is higher than the average of the same swing. The generator mode reports peak, RMS and frequency separately for exactly this reason, and the waveform chart shows the EMF as the negative slope of the flux curve.

Why does the charge transferred not depend on how fast the change is?

Because q = N·ΔΦ/R has no Δt in it. Halving the interval doubles the EMF and therefore doubles the current, but the current flows for half as long, so the same total charge goes round the loop. This is the principle a ballistic galvanometer works on: a single flick of the needle measures the flux that changed, no matter how quickly the magnet was moved. The tool reports the charge whenever you supply a loop resistance and a finite flux change.

How accurate is this for a real coil or generator?

The arithmetic is exact for the numbers you enter, but the model is the textbook one: a uniform field over the whole coil, every turn enclosing the same area, no self-inductance, no eddy currents in the core and no winding resistance beyond the loop resistance you supply. A real search coil sits in a field that varies across its face, a real alternator has a back-EMF that fights the drive, and a real transformer core saturates. Treat the answer as the first-order figure a full magnetic model should be checked against, not as a measured value.