Faraday's and Lenz's Laws of Electromagnetic Induction

Faraday's law and Lenz's law together formalise the relationship between a changing magnetic flux and the EMF it induces in a conductor. AQA specification 3.7.5 requires you to state both laws precisely, to derive consequences for moving conductors and rotating coils, and to apply Lenz's law to determine the direction of induced currents in physical setups. This lesson treats both laws as the central topic, building on the introductory induction lesson and preparing the ground for transformers and the AC generator.

Spec mapping: AQA 7408 A-Level Physics, Section 3.7.5 (Electromagnetic induction — Faraday's and Lenz's laws). This lesson develops the formal statements of both laws, the magnitude relation |ε| = N|ΔΦ/Δt|, the sign relation ε = −N dΦ/dt, the direction-finding procedure for induced currents (right-hand grip rule combined with Lenz), and worked examples for a rotating coil, a rod moving along rails, and the primary of a transformer. Refer to the official AQA 7408 specification document for the authoritative wording.

Synoptic links: (i) Energy conservation (3.4) — Lenz's law is a re-statement of the first law of thermodynamics applied to electromagnetism: induced currents must oppose their cause because otherwise a perpetual-motion machine would result. (ii) Mechanics — circular motion (3.6) — for a rotating-coil generator, the angular velocity ω of the coil maps directly to the angular frequency of the sinusoidal EMF. (iii) Transformers and AC transmission (3.7.5) — Faraday's law is the operating principle of every transformer, and Lenz's law explains why a current drawn from the secondary produces a "load" felt at the primary.

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Magnetic Flux Linkage — Revision

Before stating the laws formally, recall the key quantities:

• Magnetic flux through one turn of area A with field B at angle θ to the normal:

  Φ = BA cos θ (unit: weber, Wb)

• Flux linkage for N turns:

  N Φ = N B A cos θ (unit: weber-turn, often written Wb)

The flux linkage is what Faraday's law operates on, not the bare single-turn flux. Forgetting the factor of N is one of the most common errors in induction calculations.

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Faraday's Law — Formal Statement

Faraday's law of electromagnetic induction: The magnitude of the induced EMF across a closed circuit is equal to the rate of change of the magnetic flux linkage through the circuit.

Mathematically:

ε = −d(NΦ)/dt (sign-aware form)

|ε| = N |ΔΦ/Δt| (magnitude form, for finite changes)

The negative sign captures Lenz's law (see below) — it specifies the direction of the induced EMF relative to the direction of the flux change. The magnitude form is sufficient for numerical calculations of EMF size.

Practical Interpretation

Faraday's law tells us that an EMF is induced whenever the flux linkage changes. Three distinct mechanisms can change the flux linkage:

1. The magnetic field B can change — for instance, by switching on or off a nearby electromagnet, or by an AC current in a neighbouring coil.
2. The area A can change — for instance, a sliding rod that changes the area of a circuit it is part of.
3. The orientation angle θ can change — for instance, a rotating coil in a fixed magnetic field, where θ = ωt varies linearly with time.

Each mechanism gives rise to a different family of worked-example problems, and exam questions frequently combine them.

Derivation of ε = BLv from Faraday's Law

Consider a conducting rod of length L sliding along two parallel rails at speed v, completing a circuit through a stationary resistor. The rails and rod sit in a uniform magnetic field B perpendicular to the plane of the rails. In time Δt, the rod sweeps out a strip of area ΔA = L v Δt, increasing the flux through the circuit by ΔΦ = B ΔA = BLv Δt.

By Faraday's law:

|ε| = |ΔΦ/Δt| = BLv

This is the motional EMF of a moving conductor in a magnetic field — the same result derived in the introductory induction lesson, now seen as a special case of the general law.

Worked Example 1 — Coil in a Changing Field

Question: A flat coil of 250 turns and cross-sectional area 1.5 × 10⁻³ m² lies with its plane perpendicular to a magnetic field that increases uniformly from 0.10 T to 0.40 T in 0.20 s. Calculate the magnitude of the induced EMF.

Solution:

ΔB = 0.40 − 0.10 = 0.30 T ΔΦ (per turn) = A × ΔB = 1.5 × 10⁻³ × 0.30 = 4.5 × 10⁻⁴ Wb ΔΦ/Δt = 4.5 × 10⁻⁴ / 0.20 = 2.25 × 10⁻³ Wb s⁻¹ |ε| = N × |ΔΦ/Δt| = 250 × 2.25 × 10⁻³ = 0.5625 V ≈ 0.56 V

Worked Example 2 — Moving Rod on Rails

Question: A conducting rod of length 0.40 m slides along two parallel rails at 1.5 m s⁻¹, perpendicular to a magnetic field of 0.60 T. The rails and rod complete a circuit through a 2.0 Ω resistor. Calculate the induced EMF and the current that flows.

Solution:

|ε| = BLv = 0.60 × 0.40 × 1.5 = 0.36 V

I = ε/R = 0.36 / 2.0 = 0.18 A

Worked Example 3 — Rotating Coil

Question: A coil of 100 turns and area 4.0 × 10⁻³ m² rotates at 40 rev s⁻¹ in a uniform field of 0.25 T about an axis perpendicular to the field. Calculate the peak EMF.

Solution:

ω = 2π f = 2π × 40 = 80π = 251.3 rad s⁻¹

The flux linkage is NΦ = NBA cos(ωt), so the EMF is:

ε(t) = −d(NΦ)/dt = NBAω sin(ωt)

Peak EMF: ε₀ = NBAω = 100 × 0.25 × 4.0 × 10⁻³ × 251.3 = 25.1 V

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Lenz's Law — Formal Statement

Lenz's law: The direction of the induced current is such that its magnetic effect opposes the change in flux that is producing it.

Lenz's law is the statement of energy conservation applied to electromagnetic induction. If the induced current enhanced the flux change rather than opposing it, the system would amplify itself indefinitely — producing electrical energy without a corresponding input of mechanical or chemical energy. This violates the first law of thermodynamics. Lenz's law is therefore not an independent postulate but a consequence of energy conservation.

Procedure for Applying Lenz's Law

To determine the direction of the induced current:

1. Identify the direction of the original magnetic flux through the circuit and its trend (increasing or decreasing).
2. The induced current creates a flux that opposes the change — if the original flux is increasing, the induced flux opposes (points the opposite way); if the original flux is decreasing, the induced flux supports (points the same way).
3. Use the right-hand grip rule to determine the current direction that produces the required induced flux.

Worked Example 4 — Magnet Pushed into a Solenoid

Setup: A bar magnet with its north pole facing a solenoid is pushed towards the solenoid (entering with the north pole first).

Application of Lenz's law:

• The flux from the bar magnet through the solenoid is in the direction pointing away from the magnet's north pole — into the solenoid from the magnet end.
• As the magnet approaches, this flux increases in magnitude.
• The induced current must therefore create a flux opposing the increase — i.e., flowing in the direction away from the magnet through the solenoid, so out the magnet-facing end of the solenoid.
• Using the right-hand grip rule, this corresponds to a current flowing in a specific direction in the solenoid windings — and the magnet-facing end of the solenoid becomes a north pole (like poles repel, opposing the incoming magnet).

You can feel this opposition directly: pushing a magnet into a solenoid requires work, and the work done equals the electrical energy dissipated in the circuit's resistance.

Worked Example 5 — Magnet Pulled Away from a Solenoid

Setup: The same magnet is now pulled away from the solenoid.

• The flux from the magnet through the solenoid is now decreasing.
• The induced current must create a flux supporting the original — i.e., in the same direction as the magnet's original flux through the solenoid.
• The current direction therefore reverses compared to the previous example, and the magnet-facing end of the solenoid becomes a south pole (attracting the receding magnet — opposing its withdrawal).

In both cases, the induced current direction is exactly what is needed to oppose the change. The phrase "Lenz opposes; nature conserves" is a useful mnemonic.

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Lenz's Law and the Motional EMF Direction

For a rod sliding along rails in a magnetic field, Lenz's law also determines the direction of the induced current. The induced current flowing in the moving rod itself experiences a magnetic force F = BIL — by Lenz's law, this force opposes the motion of the rod. The rod therefore experiences a braking effect, and the agent moving the rod must do work against this opposing force. The work done is the electrical energy dissipated in the circuit's resistance.

This is the operating principle of electromagnetic braking used in trains and lorries — a copper or aluminium disc spins between magnet poles; the induced eddy currents in the disc produce a Lenz-law braking force that slows the disc without any mechanical contact. Modern hybrid vehicles use this principle to recover braking energy back into the battery.

Worked Example 6 — Force on a Moving Rod

Question: The rod from Worked Example 2 (L = 0.40 m, v = 1.5 m s⁻¹, B = 0.60 T, R = 2.0 Ω) is moved by an external agent. Calculate the force the agent must apply.

Solution:

I = 0.18 A (calculated above)

Force on the rod due to the induced current in the field: F_mag = BIL = 0.60 × 0.18 × 0.40 = 0.0432 N

By Lenz's law this force opposes the motion. The agent must apply a force of 0.0432 N in the direction of motion to maintain constant velocity.

Check: power supplied by the agent = F v = 0.0432 × 1.5 = 0.0648 W Power dissipated in resistor = I²R = (0.18)² × 2.0 = 0.0648 W ✓

The two are equal — confirming energy conservation.

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The Transformer as an Application of Faraday's Law

A transformer consists of two coils wound on a shared iron core. An alternating current I_p in the primary coil (Np turns) creates an alternating magnetic flux Φ(t) in the iron core. This same flux links the secondary coil (Ns turns), and by Faraday's law induces an EMF:

ε_s = −Ns dΦ/dt

The primary coil itself also has an EMF induced in it by the same flux (the back-EMF):

ε_p = −Np dΦ/dt

Since both EMFs are produced by the same dΦ/dt, the ratio is fixed:

ε_s / ε_p = Ns / Np

For an ideal transformer with no losses, the primary back-EMF equals the applied AC voltage V_p, and the secondary EMF equals the secondary terminal voltage V_s, giving the standard transformer equation:

V_s / V_p = N_s / N_p

The full quantitative treatment of transformer operation, including efficiency and losses, is developed in the next lesson.

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Specimen Exam Question

Specimen question modelled on the AQA A-Level Physics 7408 Paper 2 format. Not from any published paper.

Q. (a) State Faraday's law of electromagnetic induction. (2 marks) (b) State Lenz's law, and explain in terms of energy conservation why it must be true. (3 marks) (c) A search coil of 5000 turns and cross-sectional area 1.2 × 10⁻⁴ m² is placed inside a long solenoid carrying an alternating current. The flux density inside the solenoid varies sinusoidally as B(t) = 0.080 sin(120π t) T. Calculate the peak EMF induced in the search coil. (4 marks) (d) A copper ring is held above the north pole of a strong vertical bar magnet so that the magnet's axis passes through the centre of the ring. The ring is released and falls under gravity. Describe and explain in detail, using both Faraday's and Lenz's laws, the magnitude and direction of the induced current in the ring as it falls, and the consequent effect on the ring's motion. (9 marks)

(Total: 18 marks)

AO Breakdown

• AO1 (Knowledge / definitions): 5 marks — parts (a), (b) and the explanation in (d).
• AO2 (Numerical application): 4 marks — part (c) calculation.
• AO3 (Analysis / extended explanation): 9 marks — the detailed application of both laws in (d).

Grade-band Model Answers (Part (d), 9-mark explanation)

Grade C response (~200 words):