This interactive tutorial visualizes Faraday's Law of electromagnetic induction: moving a magnet near a coil induces a voltage (EMF) and current. The simulation bridges the physical action (moving the magnet) and the abstract result (voltage and current flow) by emphasizing the rate of change of magnetic flux.

 

Mathematical foundation

1. Magnetic flux (Φ)

The magnetic flux through a surface is the surface integral of the magnetic field B over the area A of the loop. For a uniform field perpendicular to the loop, Φ = B · A. In general, Φ = ∫∫ B · dA. For a magnet moving along the coil axis, we approximate the field using a dipole and integrate B_z over the coil cross-section.

2. Faraday's Law

Faraday's Law states that the induced electromotive force (EMF) in a loop equals the negative rate of change of magnetic flux through the loop:

ε = − dΦ/dt

For a coil with N turns, ε = − N (dΦ/dt). So the faster the flux changes (e.g. the faster you move the magnet), the larger the induced voltage. The minus sign is from Lenz's Law.

3. Lenz's Law

Lenz's Law says the induced current opposes the change in flux. When you push the North pole into the coil, the coil generates a magnetic field that opposes that motion (as if it were repelling the magnet). So the induced EMF has a sign such that the resulting current creates a field in the opposite direction. Moving the magnet in gives one polarity; moving it out gives the opposite. The simulation shows this with a center-zero galvanometer: direction of motion determines the sign of the deflection.

4. Dipole field and flux calculation

For a bar magnet moving along the z-axis, we use a simplified dipole approximation: the z-component of the field at a point is B_z ∝ strength / r³, where r is the distance from the magnet. We sample B_z at points on the coil face and integrate to get Φ, then compute ε = −ΔΦ/Δt from frame to frame.

 

Worked idea (direction)

North pole in: Flux through the coil increases. By Lenz's Law, the induced current creates a field that opposes this increase, so it points out of the coil on the magnet side. The galvanometer deflects one way (e.g. negative). North pole out: Flux decreases. The induced field tries to maintain it, so it points in. The galvanometer deflects the other way (positive). The “Aha!” moment is that direction of motion determines polarity of the current.

 

Usage

Follow these steps to explore Faraday's Law:

1. Drag the magnet: Click and drag the bar magnet (red = North, blue = South) along the coil axis. Moving it into the coil increases flux and induces one polarity; moving it out induces the opposite. The voltmeter needle and EMF readout show the sign and magnitude.
2. Coil loops (N): Use the slider to change the number of turns (1–10). More turns give a larger effective flux change and EMF for the same magnet motion. The coil and its wires to the bulb update immediately, even when the simulation is stopped or before you press Run.
3. Magnet strength: Adjust the slider to change the field strength (0.1–5). Stronger magnet gives larger flux and EMF when moving.
4. Preset: Use the dropdown to load Default, Strong magnet, Many loops, or Weak magnet. Then drag the magnet or use Step/Run to see the effect.
5. Step Fwd / Step Bwd: When the animation is stopped, these move the magnet a small step toward or away from the coil and update flux/EMF. Step stops the Run animation if it is running.
6. Run / Stop: Toggle continuous animation. Run advances the magnet slowly; Stop freezes it so you can drag or step.
7. 3D view: Use Iso, Front, Top, Side and Zoom+ / Zoom− to change the camera. You can also drag to rotate and scroll to zoom.
8. Flux vs time and EMF vs time: The two graphs show recent history. When you move the magnet, flux changes and EMF spikes in the direction consistent with ε = −dΦ/dt.
9. Lenz's Law: Green arrows inside the coil (when EMF is significant) indicate the direction of the induced field opposing the change in flux.

Tips: Push the North pole in quickly to see a negative EMF spike; pull it out to see a positive spike. The center-zero meter makes the direction of motion vs polarity very clear. Use presets to compare strong vs weak magnet or few vs many loops.

Parameters

• Coil loops (N): Number of turns (1–10). Default 3. Larger N multiplies the effective flux change. Changing this slider updates the 3D coil and wires to the bulb at any time (run or stop).
• Magnet strength: Scaling factor for the dipole field (0.1–5). Default 1.0. Affects flux magnitude.
• Magnet position (z): Set by dragging or Step Fwd/Bwd. Limited to the range [−6, 6] along the coil axis. The coil center is at z = 0.
• Flux (Φ): Computed by sampling the dipole B_z over the coil face and summing over turns. Units are arbitrary but proportional to real flux.
• EMF (ε): Induced EMF = −ΔΦ/Δt, estimated from frame-to-frame flux change. Positive when flux is decreasing (e.g. magnet pulled out).
• Voltmeter: Center-zero display. Needle color: blue for positive EMF, red for negative. Rotation scale maps EMF to ±60°.

The “Double Spike” in EMF vs Time

The EMF graph’s double spike is the visual signature of differentiation. Since Faraday’s Law is ε = − dΦ/dt, the EMF is the slope (rate of change) of the flux curve.

Each peak in the Flux Φ vs Time graph is when the magnet passes through the center of the coil. That produces two EMF spikes:

• First spike (approach): As the magnet approaches the coil, flux increases rapidly. That steep positive slope in Φ creates a large negative EMF spike (downward peak).
• Zero crossing (center): When the magnet is centered in the coil, flux is at its maximum. The slope of a curve at its peak is zero, so the EMF graph crosses the horizontal axis at that moment.
• Second spike (departure): As the magnet leaves, flux decreases rapidly. That steep negative slope creates a large positive EMF spike (upward peak).

The EMF crossing zero lines up with the apex of the Flux peak. The sharpness of the EMF spikes reflects how fast the magnet is moving or how concentrated the field is.

Why Flux Peaks Stay Constant While EMF Spikes Grow

When you increase the Cycles/sec (frequency) slider, the peak magnetic flux Φ stays roughly constant, while the EMF spikes get much larger. That matches the physics.

Why flux peaks stay the same: Flux depends on geometry and position: same magnet strength, same coil, and the same peak position (magnet through center) each time. So the maximum Φ is unchanged.

Why EMF spikes grow: EMF = −ΔΦ/Δt. When you move the magnet faster, the change in flux (ΔΦ) from zero to max is the same, but the time interval (Δt) shrinks. A smaller Δt in the denominator gives a larger EMF. So faster motion → higher voltage spikes even though the flux amplitude is unchanged.

On the graphs: at slow cycles the blue flux peaks are wide and the green EMF spikes are small; at fast cycles the flux peaks are narrow (same height) and the EMF spikes shoot up to a higher magnitude.