Physics
Electromagnetic induction calculator
Flux, Faraday's law, motional emf, self-inductance and coil energy — all with the formula reversible: clear whichever quantity the problem withholds.
It is not the field that induces: it is the change
Faraday's law says one thing, but says it well: the induced electromotive force equals the rate at which the linked flux changes. How strong the field is does not matter; how fast it varies does. A coil sitting in the field of a very powerful but motionless magnet produces nothing; the same coil moved a few centimetres in a tenth of a second generates a measurable voltage.
The flux can change in three ways, and every exercise picks one: the field changes, the area crossed changes, or the orientation changes. The third case is the alternator, where the loop rotates and the flux varies sinusoidally — that is how mains alternating current is produced, and the frequency is simply the rate of rotation.
The minus sign in the law is Lenz's law: the induced current always opposes the cause that produced it. This is not a quirk, it is conservation of energy. If the induced current reinforced the change, the system would feed itself; by opposing it, it demands work from whoever moves the magnet, and that work is what becomes electrical energy.
Common mistakes
- Looking for an emf when the flux is constant: if nothing changes, the induced emf is zero however large the field.
- Measuring the angle between the field and the plane of the loop instead of between the field and the normal. With the loop facing the field the angle in the formula is zero, not ninety.
- Forgetting the number of turns: a coil's emf is N times that of a single loop, and in a solenoid the inductance goes as N² outright.
Frequently asked questions
What is the formula for the Faraday-Lenz law?
ε = −N·ΔΦ/Δt. The magnitude gives the size of the induced emf; the minus sign says the induced current opposes the change of flux that produced it.
What does Lenz's law say?
That the induced current flows in whichever direction fights the cause that produced it. Push a magnet's north pole towards a loop and the loop behaves as a north pole and pushes back. It is conservation of energy written for induction.
How do you work out the inductance of a solenoid?
L = μ₀·N²·A/ℓ, where N is the total number of turns, A the cross-section and ℓ the length. It grows with the square of the turns, so doubling the winding quadruples the inductance.
What is the difference between flux and magnetic field?
The field B is defined at every point and measured in tesla; the flux Φ is the field times the area it crosses, measured in webers. The flux therefore also depends on the size and orientation of the loop.
How this calculation works
Magnetic flux: Φ = B·A·cos θ, with θ measured between B and the normal to the surface; measured in webers (1 Wb = 1 T·m²). Faraday-Lenz law: ε = −N·ΔΦ/Δt. Motional emf of a bar on rails: ε = B·L·v. Self-induction: ε = −L·ΔI/Δt. Inductance of a solenoid: L = μ₀·N²·A/ℓ, in henries. Energy stored in a coil: U = ½·L·I². Energy density of a magnetic field: u = B²/(2μ₀). Permeability of free space: μ₀ = 4π × 10⁻⁷ T·m/A.