Electromagnetic Induction: Faraday’s and Lenz’s Laws

Electromagnetic induction is the production of an electromotive force (emf) in a conductor by a changing magnetic flux; Faraday’s law states ε = −N ΔΦ ÷ Δt, where Φ = BA cos θ is the magnetic flux, and Lenz’s law says that the induced current flows in the direction that opposes the change that produced it. In brief: an emf is induced when a magnet moves relative to a coil, when the field changes, when the area of the loop changes or when the coil rotates. For a straight conductor of length L moving at speed v perpendicular to a field B, ε = BLv. The minus sign in Faraday’s law expresses Lenz’s law and conservation of energy.

Generators, transformers, induction cooktops, wireless chargers and microphones all work by induction. This guide defines magnetic flux, states Faraday’s and Lenz’s laws, explains motional emf, compares the factors affecting induced emf in a table, works through numerical examples and corrects common mistakes.

What is magnetic flux?

The magnetic flux through a surface of area A in a uniform field B is Φ = BA cos θ, where θ is the angle between the field and the normal to the surface. It measures how many field lines pass through the surface. The SI unit is the weber (Wb): 1 Wb = 1 T·m². The flux is largest when the surface is perpendicular to the field (θ = 0°), and zero when the surface is parallel to the field (θ = 90°). For a coil of N turns, the flux linkage is NΦ.

What does Faraday’s law say?

Michael Faraday found that an emf is induced in a circuit when the magnetic flux through it changes. The induced emf equals the rate of change of the flux linkage: ε = −N ΔΦ ÷ Δt. A faster change gives a larger emf, and more turns give a larger emf. If the circuit is closed, a current I = ε ÷ R flows. The flux can change in several ways: by moving a magnet toward or away from the coil, by changing the current in a nearby coil (and so the field), by changing the area of the loop, or by rotating the loop so that the angle θ changes. The emf in a straight rod of length L moving with speed v at right angles to the field is ε = BLv (motional emf), because the rod sweeps out area at a rate Lv.

What does Lenz’s law say?

Heinrich Lenz stated in 1834 that the direction of the induced current is such that its magnetic field opposes the change in flux that produced it. If a north pole of a magnet approaches a coil, the induced current creates a north pole on the nearest side of the coil, repelling the magnet. If the magnet is pulled away, the coil makes a south pole, attracting it back. This opposition is a consequence of the conservation of energy: if the induced current assisted the change, energy would be created from nothing. The work done in moving the magnet against the opposing force is converted into electrical energy. Eddy currents, induced in solid pieces of metal, also oppose the change, and are used for braking in trains and rollercoasters, and for induction cooktops, where they heat the pan.

Ways to change the flux compared

Worked example: induced emf

1. Changing field. A coil has N = 200 turns and an area of 0.010 m², with its plane perpendicular to the field. The field increases from 0 to 0.50 T in 0.10 s. ΔΦ = 0.50 × 0.010 = 5.0 × 10⁻³ Wb. ε = N ΔΦ ÷ Δt = 200 × 5.0 × 10⁻³ ÷ 0.10 = 10 V.
2. Current. If the resistance of the coil circuit is 5.0 Ω, I = 10 ÷ 5.0 = 2.0 A. If the field is increasing into the page, the current flows so as to produce a field out of the page (counterclockwise as seen from the front).
3. Moving rod. A rod 0.50 m long moves at 4.0 m/s perpendicular to a field of 0.20 T: ε = BLv = 0.20 × 0.50 × 4.0 = 0.40 V.
4. Rotating coil. The flux through a coil of area 0.020 m² in a field of 0.50 T changes from the maximum (θ = 0°) to zero (θ = 90°) in one quarter of a turn, taking 5.0 ms. Average emf for a single turn: 0.50 × 0.020 ÷ 0.0050 = 2.0 V.
5. Flux. A square loop of side 0.10 m is tilted so that the normal makes 60° with a field of 0.30 T. Φ = BA cos 60° = 0.30 × 0.010 × 0.5 = 1.5 × 10⁻³ Wb.

How was induction discovered?

Michael Faraday discovered electromagnetic induction in 1831 at the Royal Institution in London, by switching a current in a coil wound on an iron ring and observing a momentary current in a second coil, and by moving a magnet in and out of a coil. Joseph Henry in the US made similar discoveries independently, around 1830. Faraday’s disc, the first electric generator, produced a continuous current in 1831. James Clerk Maxwell expressed the law in a mathematical form in the 1860s, as one of his four equations.

Which mistakes are common?

Mistake 1: Saying that a strong field alone induces an emf

A steady field through a fixed coil induces nothing. The flux must change.

Mistake 2: Ignoring the angle in the flux

Use the component of the field perpendicular to the surface: Φ = BA cos θ with θ from the normal.

Mistake 3: Getting the direction of the induced current the wrong way

Lenz’s law: the induced field opposes the change of flux, not the flux itself.

Mistake 4: Forgetting the number of turns

The emf is proportional to N for a coil.