Faraday's Law Calculator - Induced EMF and Magnetic Flux
Enter the parameters of your coil and changing magnetic field to find the induced electromotive force (EMF). Choose which variable to solve for: induced EMF, magnetic flux change, time interval, number of turns, field change rate, or coil area. The calculator applies Faraday's law with Lenz's law, includes the angle between the coil normal and the field, and shows every step of the working.
Formula
Worked example
A 50-turn coil with area 0.005 m^2 sits perpendicular to a field that changes by 4 T in 0.1 s. Flux change per loop: dPhi = 4 * 0.005 * cos(0) = 0.02 Wb. Rate: 0.02 / 0.1 = 0.2 Wb/s. EMF = 50 * 0.2 = 10 V.
What is Faraday's law?
Faraday's law of electromagnetic induction states that a changing magnetic flux through a closed loop induces an electromotive force (EMF) in that loop. The magnitude of the EMF is proportional to the rate at which the flux changes, and the number of turns in the coil multiplies the effect. Mathematically, EMF = -N * dPhi/dt, where N is the number of turns, dPhi is the change in flux, and dt is the time over which it changes. The negative sign, which comes from Lenz's law, indicates that the induced EMF drives a current that opposes the change in flux.
Magnetic flux and how it relates to field, area, and angle
Magnetic flux (Phi) measures how much of a magnetic field passes through a surface. For a uniform field B passing through a flat coil of area A, the flux through one loop is Phi = B * A * cos(theta), where theta is the angle between the magnetic field direction and the normal (perpendicular) to the coil surface. When theta = 0 degrees the field passes straight through the coil and the flux is at a maximum; when theta = 90 degrees the field is parallel to the coil plane and no flux passes through (zero induction). The expanded form of Faraday's law substitutes this expression directly: EMF = -N * A * cos(theta) * (dB/dt). This is useful when the area and angle are fixed and it is the field strength that is changing, which is the common situation in generators and transformers.
Six ways to use this calculator
This calculator solves for any one of six quantities by rearranging Faraday's law algebraically. Select 'Induced EMF' to find the voltage given a coil and flux change. Select 'Flux change' to find what flux shift a given EMF requires. Select 'Time interval' to find how quickly the flux must change to produce a target EMF. Select 'Number of turns' to find how many coil loops are needed. Select 'Magnetic field change' to find the required field shift given coil geometry. Select 'Coil area' to find the loop area needed for a target EMF. Entering the optional coil resistance adds the induced current (EMF / R) and dissipated power (EMF squared / R) to the results.
Applications of Faraday's law
Faraday's law underpins nearly every device that converts between mechanical motion and electrical energy. Electric generators spin a coil in a magnetic field, continuously changing the flux through the coil and producing alternating voltage. Transformers change a varying flux produced by a primary coil through an iron core into an EMF in a secondary coil, with the turns ratio controlling the voltage step-up or step-down. Inductive sensors, such as those used in metal detectors and non-contact position sensors, detect objects by sensing changes in flux. Wireless charging pads induce current in a receiver coil by rapidly switching the field in the transmitter. In all of these, the same relationship governs the result: more turns, faster flux change, or larger area each increase the induced voltage.
Faraday's law variables reference
| Symbol | Quantity | SI unit | Typical range |
|---|---|---|---|
| EMF (epsilon) | Induced electromotive force | Volt (V) | mV to kV |
| N | Number of turns | dimensionless | 1 to 10,000+ |
| dPhi | Magnetic flux change | Weber (Wb) | uWb to mWb |
| dt | Time interval | Second (s) | us to s |
| B | Magnetic field strength | Tesla (T) | uT to tens of T |
| A | Coil cross-sectional area | Square metre (m^2) | cm^2 to m^2 |
| theta | Angle - coil normal to field | Degree | 0 to 90 deg |
| Phi = B*A*cos(theta) | Magnetic flux through one loop | Weber (Wb) | uWb to Wb |
Standard symbols, SI units, and typical value ranges encountered in undergraduate physics and electrical engineering.
Frequently asked questions
Why does Faraday's law have a negative sign?
The negative sign comes from Lenz's law, which is a consequence of the conservation of energy. It tells you that the induced EMF drives a current whose magnetic field opposes the change in flux that caused it. If a flux increase induced a current that further increased the flux, the system would accelerate without limit, violating energy conservation. The negative sign is essential for the correct direction of the induced current, but when you only need the magnitude of the EMF (which is what most calculators display), you can drop it.
What is the difference between EMF and voltage?
EMF (electromotive force) is the energy per unit charge supplied by a source such as an inducing magnetic flux or a chemical reaction inside a battery. Voltage (potential difference) is the energy per unit charge between two points in a circuit. The induced EMF drives a current through any resistance present in the loop; the terminal voltage across that resistance equals EMF minus the voltage drop across the coil's own resistance. For an ideal (zero-resistance) coil they are numerically equal.
Does the shape of the coil matter?
For a uniform magnetic field, only the cross-sectional area of the coil matters, not its shape. A circular coil and a square coil with the same area in the same uniform field produce the same flux. Shape does matter in non-uniform fields, where you need to integrate B over the surface, but for the standard Faraday's law formula used here a flat area in a uniform field is assumed.
What happens at a 90-degree angle?
When the coil plane is parallel to the magnetic field (theta = 90 degrees between the field and the coil normal), the flux through the coil is zero regardless of the field strength. There is no flux to change, so no EMF is induced. This is why the calculator returns an indeterminate result for the 'solve for field change' and 'solve for area' modes when theta = 90 degrees - you would need infinite field change or area to produce any EMF.
How does increasing the number of turns change the induced EMF?
The number of turns N acts as a direct multiplier. Doubling the turns doubles the induced EMF for the same rate of flux change, because each loop contributes its own induced voltage and the loops are connected in series. This is exactly how a transformer steps up voltage: the secondary coil has more turns than the primary, so the same rate of flux change induces a higher voltage across it.
Can I use this calculator for AC generators?
Yes, with the understanding that this calculator uses finite differences (dPhi/dt) rather than the instantaneous derivative. For a coil rotating at angular frequency omega in a field B, the instantaneous EMF is EMF = N * B * A * omega * sin(omega * t), which peaks at N * B * A * omega. You can evaluate the peak value by entering the peak rate of flux change. For sinusoidal signals, the RMS EMF is the peak value divided by the square root of 2.