Faraday’s law: a changing magnetic flux through a coil induces an EMF equal to ε = −N ΔΦ/Δt, where N is the number of turns and ΔΦ/Δt is the rate of change of flux. With the flux change written as ΔΦ = A·ΔB, this free calculator solves for the induced EMF, the turns, the coil area, the field change or the time — and shows every step.
Worked default: N = 200, A = 0.010 m², ΔB = 0.40 T, Δt = 0.20 s gives ε = 4.00 V. Related formulas: flux Φ = B·A·cos θ, motional EMF ε = B·L·v, and the peak EMF of a rotating coil ε₀ = N·A·B·ω with ω = 2πf.
Faraday’s law says the EMF induced in a coil is the number of turns times the rate at which the magnetic flux through it changes: ε = −N ΔΦ/Δt. The magnetic flux through one turn is Φ = B·A·cos θ; when the field is perpendicular to the coil (θ = 0) and only its strength changes, the flux change is simply ΔΦ = A·ΔB. Substituting gives the form this calculator uses, ε = N·A·ΔB/Δt.
There are three steps. First, choose which quantity you want — EMF, turns, area, field change or time — in the Solve for menu; the other four become your inputs. Second, enter the values you know and pick their units (tesla or millitesla for the field, seconds or milliseconds for the time, and so on); the calculator converts everything to SI. Third, read the answer with the worked steps and the direction note. The minus sign in Faraday’s law is Lenz’s law: the induced current always opposes the change that created it, so the sign gives the direction while the number gives the size.
The equation rearranges four ways. To find how many turns give a target EMF, use N = ε·Δt/(A·ΔB). To find the coil area, A = ε·Δt/(N·ΔB). To find the field change, ΔB = ε·Δt/(N·A). To find how quickly the flux must change, Δt = N·A·ΔB/ε. Because the EMF depends on 1/Δt, a faster change always induces a larger EMF — and dividing by Δt = 0 is undefined, so a positive time is required.
A 200-turn coil of area 0.010 m² sits in a magnetic field that rises from 0.10 T to 0.50 T (a change of ΔB = 0.40 T) over 0.20 s. The flux change through one turn is ΔΦ = A·ΔB = 0.010 × 0.40 = 0.0040 Wb, so the induced EMF is ε = N·ΔΦ/Δt = 200 × 0.0040 / 0.20 = 4.0 V. Reading it the other way, to induce 4.0 V you would need the flux to change over Δt = N·A·ΔB/ε = 200 × 0.010 × 0.40 / 4.0 = 0.20 s.
Electromagnetic induction is how almost all of the world’s electricity is generated: rotating coils in the magnetic fields of power-station generators, wind turbines and bicycle dynamos all rely on ε = −N ΔΦ/Δt. The same law runs transformers, induction cooktops, electric guitar pickups, metal detectors and the ignition coil in a petrol engine — anywhere a changing magnetic field needs to be turned into a voltage.
Induced EMF is measured in volts (V), the same unit as any other voltage. Faraday’s law, ε = −N ΔΦ/Δt, multiplies the number of turns by the rate of change of magnetic flux (in webers per second, Wb/s), and one Wb/s equals one volt.
The minus sign is Lenz’s law: the induced EMF (and the current it drives) always acts to oppose the change in flux that produced it. It sets the direction of the EMF, not its size — so for the magnitude you can drop the sign, which is what this calculator reports.
Using the flux itself instead of the rate of change of flux. A steady magnetic flux, however large, induces no EMF at all — only a changing flux does. Faraday’s law depends on ΔΦ/Δt, so both how much the flux changes and how quickly matter.
For a fixed flux change, the induced EMF ε = N ΔΦ/Δt grows without bound as Δt gets smaller — a very sudden change induces a very large EMF (the principle behind ignition coils and inductive spikes). Because dividing by Δt = 0 is undefined, the calculator guards that case and asks for a positive time.
Choose “time interval” in the Solve for menu and enter the EMF you want along with N, the area and the field change. The calculator rearranges Faraday’s law to Δt = N·A·ΔB/ε and returns the interval over which the flux must change to produce that EMF.