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Electromagnetic induction: Class 12 Physics
Magnetic flux, Faraday's and Lenz's laws, motional EMF, eddy currents, self and mutual inductance, energy stored in an inductor, the AC generator, the DC motor and back EMF, and the transformer.
Key points
Magnetic flux Φ through a surface of area A in a uniform field B is Φ = BA cosθ, where θ is the angle between B and the normal to the surface. Its unit is the weber (Wb), where 1 Wb = 1 T m². Flux is greatest when the surface faces the field and zero when the surface is parallel to the field.
Electromagnetic induction is the production of an EMF in a circuit when the magnetic flux linked with it changes. Faraday found that moving a magnet into a coil, or changing the current in a nearby coil, makes the galvanometer deflect. There is no deflection when nothing changes.
Faraday's laws:
- An EMF is induced whenever the flux linked with a circuit changes.
- The size of the induced EMF equals the rate of change of flux linkage: E = −N ΔΦ/Δt for a coil of N turns.
Lenz's law gives the direction: the induced current flows in the direction that opposes the change which produces it. This is the meaning of the minus sign. Lenz's law follows from conservation of energy. If the induced current helped the change, energy would be created from nothing.
Motional EMF: a straight conductor of length L moving with speed v at right angles to a field B has an EMF E = BLv across its ends. The direction of the induced current is given by Fleming's right hand rule: first finger for field, thumb for motion, second finger for induced current.
Eddy currents are circulating currents induced in a solid piece of metal when the flux through it changes. They heat the metal and waste energy, so transformer cores are laminated, which means built of thin insulated sheets. Eddy currents are also used usefully, for example in induction furnaces and in electromagnetic braking.
Self induction: when the current in a coil changes, the flux through the same coil changes and an EMF is induced in it. The self inductance L of a coil is defined by E = −L ΔI/Δt. Its unit is the henry (H). One henry is the inductance of a coil in which a current changing at 1 A/s induces an EMF of 1 V. For a solenoid L = μ₀N²A/l.
Mutual induction: a changing current in one coil induces an EMF in a neighbouring coil. The mutual inductance M is defined by E₂ = −M ΔI₁/Δt, in henry. It depends on the number of turns, the size of the coils, their distance apart and the core material.
An inductor stores energy in its magnetic field. The energy is U = ½LI², in joules. This is like the energy ½CV² stored in a capacitor, which is stored in an electric field.
AC generator and DC motor:
- A generator turns a coil in a magnetic field. The flux changes continuously, so an alternating EMF E = E₀ sin ωt is produced, with E₀ = NBAω. Slip rings carry the current out.
- A DC motor uses a coil in a field with a split ring commutator. The current gives a torque that turns the coil, and the commutator reverses the current every half turn.
- As the motor coil turns it cuts field lines and an EMF is induced which opposes the supply voltage. This is the back EMF. Current I = (V − back EMF)/R, so the current is large at start and falls as the motor speeds up.
A transformer changes an alternating voltage by mutual induction. A primary coil and a secondary coil are wound on a laminated soft iron core. For an ideal transformer Vs/Vp = Ns/Np = Ip/Is. A step up transformer has Ns greater than Np and raises the voltage while lowering the current. A step down transformer does the opposite. A real transformer loses energy as heat in the windings, eddy currents, hysteresis and flux leakage. A transformer does not work on DC.
Study tip: Write the direction of the induced current with a clear reason from Lenz's law, since marks are given for the reasoning. For the generator and transformer, draw a neat labelled diagram. In numericals, find the change in flux or current first, then divide by the time taken.
Definitions and flashcards
Say the meaning first, then open the card to check.
Magnetic flux
The product BA cosθ, measured in weber.
Induced EMF
The EMF produced in a circuit by a changing magnetic flux.
Lenz's law
The induced current opposes the change that causes it.
Eddy current
A circulating current induced in a bulk conductor by changing flux.
Self inductance
The property of a coil by which it opposes a change of current in itself.
Henry
The unit of inductance, 1 V s/A.
Mutual inductance
The EMF induced in one coil per unit rate of change of current in another.
Back EMF
The EMF induced in a running motor that opposes the supply.
Transformer
A device that changes an AC voltage using mutual induction.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Magnetic flux | Φ = BA cosθ | θ is the angle between B and the normal. |
| Faraday's law | E = −N ΔΦ / Δt | For N turns. |
| Motional EMF | E = BLv | Conductor moving perpendicular to the field. |
| Self induced EMF | E = −L ΔI / Δt | L is self inductance. |
| Mutual induced EMF | E₂ = −M ΔI₁ / Δt | M is mutual inductance. |
| Energy in inductor | U = ½LI² | Energy stored in the magnetic field. |
| Peak EMF of generator | E₀ = NBAω | ω is angular speed of the coil. |
| Transformer | Vs / Vp = Ns / Np | Ideal transformer. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
State Faraday's law and Lenz's law. (3 marks)
Faraday's law: the induced EMF in a circuit equals the rate of change of magnetic flux linkage, E = −NΔΦ/Δt. Lenz's law: the direction of the induced current is such that it opposes the change in flux that produced it.
How does Lenz's law agree with conservation of energy? (2 marks)
When a magnet is pushed toward a coil, the induced current makes a field that repels it. Work must be done against this force and that work appears as electrical energy. If the force helped the motion, energy would be gained without any work, which is impossible.
Why is the core of a transformer laminated? (2 marks)
Thin insulated sheets make the paths for eddy currents narrow and raise their resistance, so the eddy currents and the heat they cause are greatly reduced. This saves energy.
Define the henry. (2 marks)
One henry is the self inductance of a coil in which an EMF of 1 volt is induced when the current changes at the rate of 1 ampere per second.
What is back EMF in a motor and why is it useful? (3 marks)
The turning coil of a motor cuts field lines, so an EMF is induced in it which opposes the supply voltage. This is the back EMF. It limits the current, which is I = (V − E)/R. At start the back EMF is zero and the current is large. As the motor speeds up the back EMF rises and the current falls. It acts as a natural control of the current.
Explain the working of an AC generator. (5 marks)
A generator has a coil of N turns and area A that rotates at angular speed ω in a uniform field B. The flux through the coil is Φ = BA cos ωt. By Faraday's law the induced EMF is E = −N dΦ/dt = NBAω sin ωt, so it is alternating with peak value E₀ = NBAω. The ends of the coil are joined to two slip rings, and carbon brushes press on the rings to take the current to the external circuit. The EMF is zero when the coil plane is perpendicular to the field and maximum when it is parallel to the field. Greater N, B, A or speed gives a larger EMF.
Numericals with worked steps
A coil of 200 turns has a flux through it that changes from 0.05 Wb to 0.01 Wb in 0.2 s. Find the average induced EMF.
- ΔΦ = 0.01 − 0.05 = −0.04 Wb.
- E = −NΔΦ/Δt = −200 × (−0.04) / 0.2.
- 200 × 0.04 = 8. 8 / 0.2 = 40.
Answer: E = 40 V (magnitude)
A rod of length 0.60 m moves at 5.0 m/s at right angles to a field of 0.30 T. Find the EMF across its ends.
- E = BLv.
- E = 0.30 × 0.60 × 5.0.
- 0.30 × 0.60 = 0.18. 0.18 × 5.0 = 0.90.
Answer: E = 0.90 V
A transformer has 100 turns in the primary and 2500 turns in the secondary. The primary is connected to 220 V. Find the secondary voltage, and the primary current if the secondary current is 0.4 A (ideal case).
- Vs = Vp × Ns/Np = 220 × 2500/100 = 220 × 25 = 5500 V.
- Ip/Is = Ns/Np, so Ip = 0.4 × 25 = 10 A.
- Check power: 220 × 10 = 2200 W and 5500 × 0.4 = 2200 W.
Answer: Vs = 5500 V, Ip = 10 A
An inductor of 0.5 H carries 4 A. Find the energy stored.
- U = ½LI².
- I² = 16.
- U = 0.5 × 0.5 × 16 = 4.
Answer: U = 4 J
Common mistakes
- Saying that a magnet at rest in a coil induces an EMF. Flux must be changing.
- Ignoring the number of turns N in Faraday's law.
- Treating the minus sign in Lenz's law as a numerical negative, instead of as showing opposition.
- Using a transformer with DC. It needs a changing current to work.
- Forgetting that a step up transformer lowers the current by the same factor, so power is not increased.