Electromagnetic Induction

How the model works

Magnetic flux through one loop is the field passing through it: Φ = ∫ B·dA. Faraday's law says a changing flux induces an EMF in a coil of N turns:

ε = −N · dΦ/dt

The minus sign is Lenz's law: the induced current opposes the change. As the magnet's north pole approaches, flux increases and the induced current makes the near face of the coil a north pole that pushes back. As the magnet leaves, the sign reverses.

The model treats the magnet as a magnetic dipole on the coil's axis. The flux through a loop of radius a at axial distance z is Φ = μ₀ m a² / (2(a² + z²)^{3/2}). The EMF follows from the chain rule dΦ/dt = (dΦ/dz)·v.

Idealised: point dipole, thin coil, no current-induced forces on the magnet, and an open-circuit voltmeter. Numbers are illustrative, of school-lab magnitude.

Guided experiments

1. Faster means bigger. Pass the magnet at 1 m/s, then 2 m/s. Prediction: peak voltage doubles and each pulse lasts half as long. The area under each pulse (total flux change) stays the same. Pass at 2 m/s
2. Two pulses, opposite signs. Watch one full pass. Prediction: one pulse while entering and an opposite-sign pulse while leaving. At the instant the magnet is centred, the flux is at its maximum but ε = 0. One slow pass
3. Hold it still. Stop the magnet inside the coil. Prediction: zero voltage, despite the large flux. Only change induces an EMF. Magnet at rest inside

Challenge

You double the number of turns and halve the speed. What happens to the peak EMF and to the total flux change during the pass?

References

• OpenStax, University Physics Volume 2, §13.1 “Faraday's Law” and §13.2 “Lenz's Law”.