Drag the bar magnet through the coil — or enable auto-oscillation.
A changing magnetic flux Φ induces an EMF
ε = −N dΦ/dt (Faraday's law).
The induced current always opposes the change (Lenz's law).
Flux Φ0.00 mWb
EMF ε0.00 V
dΦ/dt0.00 Wb/s
EMF ε waveform (scrolling oscilloscope)
Magnetic flux Φ (bar graph)
Controls
Stats
Flux Φ0.000 mWb
EMF ε0.000 V
dΦ/dt0.000 Wb/s
Current dir.—
Position0.0 cm
Keyboard
P Pause / Play R Reset
Faraday's Law
ε = −N dΦ/dt — the induced electromotive force in a coil of
N turns equals the negative rate of change of magnetic flux Φ through the coil.
The minus sign is Lenz's law: the induced current flows to oppose the change that caused it.
Magnetic flux is Φ = B · A · cos θ. Here the magnet moves along the
coil axis (θ = 0) and the field follows a dipole-like falloff:
Φ(x) = B₀ A / (1 + (x/L)²)^(3/2).
Lenz's Law in Action
When the N-pole approaches, flux increases; the coil drives current to create a field
opposing the increase (repelling the approaching magnet). When the magnet is withdrawn,
flux decreases; the current reverses to sustain the flux (attracting the retreating magnet).
Energy is conserved: every joule appearing as EMF costs mechanical work.
Applications
Electric generators, transformers, wireless chargers (Qi), induction cooktops,
metal detectors, and MRI gradient coils all exploit Faraday induction.
The oscilloscope waveform you see here is exactly the signal shape produced by a
hand-cranked AC generator.
About this simulation
This interactive model demonstrates electromagnetic induction by letting you slide a bar magnet along the axis of a solenoid coil. As the magnet moves, the magnetic flux linking the coil changes and an electromotive force appears, computed in real time from Faraday's law, ε = −N dΦ/dt. A scrolling oscilloscope plots the induced EMF while current-direction arrows reveal Lenz's law — the induced current always opposes the change in flux that produced it.
🔬 What it shows
The flux through the coil is modelled with a dipole-like axial profile, Φ(x) = B₀·A / (1 + (x/L)²)^(3/2). Each frame the EMF is found numerically as ε = −N·ΔΦ/Δt, so the waveform, flux bar and current arrows all stay consistent with the magnet's motion and the chosen parameters.
🎮 How to use
Drag the bar magnet through the coil with the mouse or touch, or press the Auto-Oscillate button to drive it sinusoidally. Three sliders set the magnet strength B₀ (0.2–3.0 T), the number of coil turns N (5–200) and the auto frequency (0.1–3.0 Hz). Pause (P) and Reset (R) control the run.
💡 Did you know?
Faraday discovered induction in 1831, but the minus sign — Lenz's law — was stated independently by Heinrich Lenz in 1834. It is a direct expression of energy conservation: the opposing force means you must do mechanical work to generate electrical energy.
Frequently asked questions
What is Faraday's law of induction?
Faraday's law states that a changing magnetic flux through a circuit induces an electromotive force in it. For a coil of N turns the induced EMF equals minus the number of turns times the rate of change of flux, ε = −N dΦ/dt. The faster the flux changes, the larger the induced voltage.
Why does the EMF only appear while the magnet is moving?
Because induction depends on the rate of change of flux, not on flux itself. When the magnet is stationary the flux Φ is constant, dΦ/dt is zero, and so is the EMF. Only motion — or a changing field — produces a non-zero ε, which is why the oscilloscope trace flattens to the zero line whenever the magnet stops.
What do the green and red current arrows mean?
They show the direction of the induced current set by Lenz's law. Green arrows (counter-clockwise) appear when the flux is rising and the coil pushes back against the approaching magnet; red arrows (clockwise) appear when the flux falls and the current reverses to sustain it. The current always opposes the change that created it.
How do the sliders change the result?
Raising the magnet strength B₀ increases the peak flux and therefore the EMF amplitude. Adding coil turns N multiplies the EMF directly, since ε scales with N. In Auto-Oscillate mode the frequency slider sets how quickly the magnet sweeps back and forth, which raises dΦ/dt and the EMF for faster motion.
Is the physics in this simulation accurate?
The relationships are faithful: ε = −N dΦ/dt, Φ = B·A·cosθ, and Lenz's opposing current are all correctly applied. The flux uses a simplified one-dimensional dipole profile along the coil axis rather than a full three-dimensional field integral, so it captures the shape and trends well but the absolute voltage values are illustrative rather than laboratory-exact.
About Faraday's Law of Induction
Faraday's law of electromagnetic induction, discovered by Michael Faraday in 1831, states that a changing magnetic flux through a conducting loop induces an electromotive force (EMF) in that loop. Mathematically, EMF = −dΦ/dt, where Φ = ∫B·dA is the magnetic flux. The negative sign, expressed by Lenz's law, means the induced current flows in a direction that opposes the change in flux, conserving energy and preventing perpetual motion.
The induced EMF can arise from a changing magnetic field strength, a loop moving through a non-uniform field, a loop rotating in a field, or a changing loop area. The magnitude of EMF is proportional to the rate of change of flux and to the number of turns in the coil (EMF = −N dΦ/dt for an N-turn solenoid). This principle is the operating basis of electrical generators, transformers, induction motors, and wireless charging systems.
This simulator lets you move a magnet toward and away from a coil, vary field strength and loop area, and observe the induced current direction and magnitude in real time. You can verify Lenz's law by noting that the induced current always acts to repel an approaching magnet and attract a retreating one, and explore how the rate of change—not the absolute flux—determines the EMF.
Frequently Asked Questions
What is magnetic flux and how is it measured?
Magnetic flux Φ is the total magnetic field passing through a surface, measured in webers (Wb) or volt-seconds. It equals the integral of the magnetic field component perpendicular to the surface: Φ = B·A·cos(θ), where B is field strength in teslas, A is area in square meters, and θ is the angle between the field and the surface normal. A field perfectly perpendicular to the loop gives maximum flux; a parallel field gives zero flux.
Why does the induced current oppose the change in flux (Lenz's law)?
Lenz's law is a consequence of energy conservation. If the induced current reinforced the flux change rather than opposing it, you could extract unlimited energy from a simple coil without doing work—violating the first law of thermodynamics. The opposing current creates a force on any moving magnet that resists its motion, requiring you to do mechanical work to maintain the motion. This work is converted to electrical energy in the circuit.
How does Faraday's law explain how a generator works?
An electrical generator rotates a coil in a magnetic field. As the coil turns, the angle between the field and the coil's normal changes continuously, causing the flux to vary sinusoidally: Φ = BA·cos(ωt). Differentiating gives EMF = BAω·sin(ωt)—a sinusoidal alternating voltage. The mechanical rotation energy is converted to electrical energy via the induced EMF. Larger generators use stronger magnets, more coil turns, and higher rotation speeds to maximize EMF.
What is mutual induction and how does it relate to transformers?
Mutual induction occurs when a changing current in one coil induces an EMF in a nearby second coil through their shared magnetic flux. A transformer uses this principle: an AC current in the primary coil creates a changing flux in an iron core, which links to the secondary coil and induces an EMF proportional to the turns ratio. A step-up transformer has more secondary than primary turns, increasing voltage while reducing current (conserving power).
What is eddy current and is it useful?
Eddy currents are closed-loop currents induced in bulk conducting materials (not just coils) by a changing magnetic field. By Lenz's law, they oppose the flux change, creating a braking force on moving conductors in magnetic fields. While often undesirable in transformer cores (causing heating), eddy currents are exploited in induction cooktops (heating pots directly), magnetic brakes (contactless braking in trains and rollercoasters), and metal detectors (detecting subsurface conductors by their eddy-current signatures).