Faraday's law of electromagnetic induction states that the EMF induced in a closed loop equals the negative rate of change of magnetic flux through it: ε = −dΦB/dt. When a conducting coil rotates at angular frequency ω in a uniform magnetic field B, the flux varies sinusoidally as ΦB = NBA cos(ωt), producing a sinusoidal EMF ε = NBAω sin(ωt)—the principle of the AC generator. This animation makes the link between geometry, flux change, and output voltage vivid and immediate.
Rotate the coil at different speeds, change the number of turns and the field strength, and watch the real-time EMF plot evolve. The animation clearly shows how the EMF peaks when the coil plane is parallel to B (maximum rate of flux change) and is zero when the coil is perpendicular to B (flux is at its maximum but momentarily not changing).
Why is the induced EMF maximum when the coil is parallel to the field?
The flux is Φ = NBA cosθ, where θ is the angle between the coil normal and B. The EMF is ε = −dΦ/dt = NBAω sinθ. At θ = 90° (coil plane parallel to B), sinθ = 1 and the flux is changing most rapidly, giving maximum EMF. At θ = 0° (coil perpendicular to B), sinθ = 0: the flux is at a peak but momentarily stationary, so EMF = 0.
How does a rotating coil in a magnetic field generate AC?
As the coil rotates at constant angular velocity ω, the angle θ = ωt changes linearly with time. The flux varies as cos(ωt) and the induced EMF as sin(ωt)—a pure sinusoid at frequency f = ω/(2π). This is exactly the waveform delivered by AC generators (alternators) in power stations, which rotate at 50 rev/s (3000 rpm) in the UK to produce 50 Hz mains electricity.
What is magnetic flux and how is it measured?
Magnetic flux ΦB through a surface is defined as ΦB = ∫∫ B · dA, the integral of the normal component of B over the surface. For a uniform field and flat coil of area A: ΦB = BA cosθ. The SI unit is the weber (Wb = V·s = T·m²). One weber per second of flux change induces one volt, so a rapidly changing flux of 1 Wb in 1 ms gives 1000 V.
Faraday's law relates a changing magnetic field to an induced electric field: ∇ × E = −∂B/∂t. Ampère's law (with Maxwell's displacement current term) relates a changing electric field and electric current to a magnetic field: ∇ × B = μ₀J + μ₀ε₀∂E/∂t. Together these two equations, along with Gauss's laws for E and B, form Maxwell's equations describing all classical electromagnetism.
A generator converts mechanical rotation into electrical energy via Faraday's law: the rotating coil's changing flux induces an EMF. A motor does the reverse: an externally supplied current in the coil (in the presence of a magnetic field) experiences a Lorentz force torque, causing rotation (converting electrical to mechanical energy). The same machine can act as both: regenerative braking in electric vehicles uses the motor as a generator to recover kinetic energy.
The negative sign encapsulates Lenz's law: the induced EMF drives a current whose magnetic field opposes the change in flux. If a magnet approaches a coil, the induced current creates a field repelling the magnet, requiring work to be done against this force. Without this opposition, energy would be created from nothing. The negative sign is therefore a direct mathematical statement of energy conservation in electromagnetic induction.
As the coil rotates, its leads would otherwise twist and break. Slip rings are smooth conductive rings attached to the rotating shaft, in contact with fixed brushes that carry current to the external circuit. Unlike the commutator in DC generators (which reverses connections each half-turn to rectify the output), slip rings allow the full sinusoidal AC to pass through. Brush wear is a maintenance requirement in large alternators.
A dynamic microphone uses a small coil attached to a diaphragm, suspended in a permanent magnet's field. Sound waves move the diaphragm, moving the coil and changing the flux through it. By Faraday's law, this induces a tiny EMF that mirrors the sound waveform—converting acoustic pressure variations directly into a voltage. The output is typically 1–10 mV and needs preamplification.
Flux linkage λ = NΦB is the product of the number of turns and the flux through each turn. For a coil with N turns: ε = −dλ/dt = −N dΦB/dt. The unit is the weber-turn (Wb). Flux linkage is a useful concept in analysing inductors and transformers: the self-inductance L is defined by λ = LI, so ε = −L dI/dt, linking flux change to current change and impedance.
In MRI, a radiofrequency (RF) pulse tips nuclear magnetic moments away from the main field B0. As they precess and relax back, their changing magnetisation induces a tiny EMF in receiver coils surrounding the patient—directly via Faraday's law. Modern receive arrays use up to 128 coils simultaneously, each detecting signals from different body regions to speed up imaging. Signal levels are femtovolts, requiring cryogenically cooled preamplifiers in some high-field systems.
In August 1831 Faraday wound two coils on an iron ring. When he connected the primary coil to a battery, the sudden change in current (and flux) induced a brief current pulse in the secondary—demonstrating mutual induction. He also showed that physically moving a bar magnet in and out of a coil generated a sustained current as long as motion continued. These two experiments established the principle of electromagnetic induction.
This model recreates Michael Faraday's 1831 discovery: a bar magnet travels along the axis of a coil, and the changing magnetic flux it produces induces a measurable electromotive force. The governing relationship, EMF = −N dΦ/dt, ties the number of coil turns and the rate of flux change directly to the voltage produced. Flip the magnet's poles or reverse its direction of travel and the induced current reverses too, a direct demonstration of Lenz's law: induced effects always oppose the change that causes them. Whilst real magnetic fields are three-dimensional, the field here is modelled as an axial dipole so flux, EMF and current direction can be read off instantly.
A magnet moves along the coil's axis, either oscillating automatically or dragged by hand, whilst the coil's flux rises and falls. A scope trace beneath the coil plots either the induced EMF or the flux itself in real time, alongside live readouts of EMF, flux, current direction and velocity.
Set the coil's turns and radius and the magnet's strength with the sliders, then flip polarity with the N↓S / S↓N buttons. Switch between Auto motion (adjustable frequency and amplitude) and manual Drag, and choose whether the scope displays EMF or Flux.
Faraday's original 1831 apparatus used two coils wound on a single iron ring: switching the current in one coil induced a brief pulse in the other with no physical contact between them, a discovery that underpins every transformer and generator built since.
The equation EMF = −N dΦ/dt depends on the rate of change of flux with time, not the flux itself. Since flux depends on position, its time derivative equals the flux gradient with position multiplied by velocity. A magnet held stationary produces zero EMF even if it sits exactly where flux is greatest, because there is no change happening at that instant.
Swapping which pole faces the coil flips the sign of the flux through it, so the rate of change of flux also flips sign. Because EMF is proportional to that rate of change, the induced current switches direction too, from clockwise to anticlockwise or vice versa, exactly as Lenz's law predicts.
Increasing the turns count N increases the induced EMF proportionally, since each turn links the same changing flux and their individual electromotive forces add together. Doubling the turns roughly doubles the EMF for identical magnet motion, which is why real generators and transformers use many hundreds or thousands of turns.
In Auto mode the magnet's velocity follows a cosine function that reaches zero exactly when its position reaches a turning point. Since EMF depends directly on velocity, it drops to zero at that instant even though the magnet may be close to the coil, where flux itself is near its peak.
They set the magnet's sinusoidal motion: amplitude controls how far it travels from the coil's centre, and frequency controls how quickly it oscillates back and forth. Both increase the magnet's peak velocity, which in turn increases the peak induced EMF, since EMF scales with speed rather than distance travelled.