12 min readConcept sheetSection B long answer (5 marks), often with a choice: state Lenz's law and show that it follows from conservation of energy; very short answer (2 marks) on eddy currents; Section A MCQ (1 mark); also a NEET favourite

Faraday's and Lenz's laws, induced emf in a moving rod and eddy currents

A changing magnetic flux induces an emf, ε = −N dφ/dt. By Lenz's law the induced current opposes the change, which is energy conservation at work.

Magnetic fluxφ = BA cos θ, where θ is the angle between B and the normal to the area; SI unit weber, 1 Wb = 1 T m²

Faraday's lawε = −N dφ/dt for a coil of N turns; 1 V = 1 Wb s⁻¹

Lenz's lawThe induced current opposes the change that produces it; it follows from conservation of energy

emf in a moving rodε = Blv, for a rod of length l moving at speed v at right angles to B

Magnetic flux and Faraday's laws

Faraday found that a magnet can make a current, but only when something changes. Push a magnet into a coil joined to a galvanometer, and the pointer moves. Hold the magnet still, and the pointer returns to zero. Pull it out, and the pointer moves the other way. Switching a current on or off in a nearby coil does the same. Making an emf in this way is called electromagnetic induction. It is described by the magnetic flux φ, the amount of field passing through an area. For a flat area A in a uniform field, φ = BA cos θ. Here θ is the angle between B and the normal to the area. Its SI unit is the weber (Wb); 1 Wb = 1 T m². Faraday's first law: whenever the flux linked with a circuit changes, an emf is induced in it. Second law: the size of the emf equals the rate of change of flux. For a coil of N turns, ε = −N dφ/dt. If the circuit is closed, the emf drives an induced current.

Lenz's law and conservation of energy

**Lenz's law:** the induced current always flows in a direction that **opposes the change** that produces it. The minus sign in ε = −N dφ/dt stands for this.

1
Push the N pole of a magnet towards a closed coil. The flux through the coil increases.
2
To oppose this, the near face of the coil becomes a north pole and repels the magnet. Seen from the magnet, the induced current flows anticlockwise.
3
Pull the magnet away. The flux falls, so the near face becomes a south pole and attracts the magnet. The current is now clockwise.
4
Either way, the coil opposes the motion. So you must do work to move the magnet.
5
This mechanical work appears as the electrical energy of the induced current, and finally as heat in the coil.
6★
If the current helped the motion instead, the magnet would speed up by itself and energy would come from nothing. That is impossible. So, Lenz's law is a consequence of the law of conservation of energy.

★ key step

Derivation: emf induced in a moving rod

A rod PQ of length l slides on two parallel metal rails joined at one end. A uniform field B acts at right angles to the plane of the rails.

1
The rod and the rails form a closed loop. Let x be the distance of the rod from the closed end. The area of the loop is lx.
2
Flux through the loop: φ = Blx.
3
The rod moves with speed v, so x changes at the rate v.
4★
Size of the induced emf: ε = dφ/dt = Bl × (rate of change of x) = Blv.
5
Another view: each free charge moving with the rod feels a force qvB along the rod. The work done on each unit of charge over the length l is vBl, the same emf.
6
Its direction follows Fleming's right-hand rule: thumb along the motion, first finger along the field, middle finger along the induced current.

★ key step

Eddy currents

Induced currents also flow inside a solid piece of metal, such as a sheet or a block, when the flux through it changes. They flow in closed loops like whirlpools (eddies), so they are called eddy currents. Foucault first studied them, so they are also called Foucault currents. By Lenz's law, they flow so as to oppose the change in flux. Solid metal has a very low resistance, so eddy currents can be large and heat the metal. In motors and transformers this wastes energy. So their iron cores are built from thin sheets (laminations), insulated from each other, which break up the current paths. Eddy currents are also useful. Induction furnaces use them to melt metals and make alloys. Electromagnetic brakes in some electric trains use them to slow the wheels. Induction cooktops heat the steel base of a pan in the same way.

Worked example: a changing field and a moving rod

Problem

(a) A square coil of side 10 cm has 200 turns and a resistance of 8.0 Ω. A uniform field at right angles to its plane falls steadily from 0.50 T to 0.10 T in 0.20 s. Find the emf and the current. (b) A rod 40 cm long slides at 5.0 m s⁻¹ on rails in a field of 0.25 T at right angles to the rails. The circuit's resistance is 0.50 Ω. Find the emf, the current and the force needed to keep the rod moving.

Working

(a) Area A = 0.10 m × 0.10 m = 1.0 × 10⁻² m². The field is along the normal, so cos θ = 1.
Change in flux through each turn: Δφ = ΔB × A = (0.50 − 0.10) × 1.0 × 10⁻² Wb = 4.0 × 10⁻³ Wb.
ε = N Δφ/Δt = 200 × 4.0 × 10⁻³ Wb ÷ 0.20 s = 4.0 V.
I = ε/R = 4.0 V ÷ 8.0 Ω = 0.50 A.
(b) ε = Blv = 0.25 T × 0.40 m × 5.0 m s⁻¹ = 0.50 V.
I = ε/R = 0.50 V ÷ 0.50 Ω = 1.0 A.
The current in the rod feels a backward force F = BIl = 0.25 × 1.0 × 0.40 N = 0.10 N (Lenz's law). An equal push keeps v constant.
Energy check: power put in = Fv = 0.10 × 5.0 = 0.50 W; electrical power = εI = 0.50 × 1.0 = 0.50 W. They are equal.

Answer

(a) 4.0 V and 0.50 A; (b) 0.50 V, 1.0 A and a push of 0.10 N

How to draw: the magnet and coil experiment, and the rod on rails

Magnet and coil: draw a coil as a helix of 5 or 6 loops. Join its two ends to a circle with G inside, the galvanometer.
Draw a bar magnet beside the coil with N facing it. Draw an arrow towards the coil labelled 'motion', and show the pointer of G turned to one side.
On the face of the coil nearest the magnet, write N and draw an anticlockwise arrow, to show Lenz's law.
Rod on rails: draw two long parallel horizontal lines for the rails. Join them at the left end through a resistor R.
Draw the rod PQ as a vertical line across the rails and label its length l. Fill the region with crosses (×) for B into the page, and label B.
Draw an arrow on the rod showing its velocity v. Show the induced current round the loop with arrows, found by Fleming's right-hand rule.

In real life

The induction cooktop in many Indian kitchens heats the pan with eddy currents, and bicycle dynamos and power-station generators use Faraday's law. Metal detectors at railway stations and airports also work by induction.

Exam angle

5 marks: “State Lenz's law. Explain, with an example, why it is a consequence of the law of conservation of energy.” Statement; N pole approaching a coil: the near face becomes N and repels it; work must be done; the work becomes electrical energy and then heat; if the current helped the motion, energy would come from nothing, which is impossible; so Lenz's law follows from energy conservation.

2 marks: “What are eddy currents? Give one harmful effect and one use.” Induced closed loops of current in a solid metal placed in a changing flux. Harm: heating and energy loss in transformer cores (reduced by laminating). Use: induction furnace or electromagnetic brakes.

2 marks: “In a coil of 100 turns, the flux through each turn changes from 6 mWb to 2 mWb in 0.10 s. Find the average induced emf.” ε = N Δφ/Δt = 100 × 4 × 10⁻³/0.10 = 4.0 V.

Remember it

Change makes current, and the current fights the change. ε = −N dφ/dt; the minus sign is Lenz's law.