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121, 124–127ElectricityStretch

Electromagnetic Induction

Move a magnet into a coil and a current flows, even though no battery is connected. A changing\textit{changing} magnetic field through a circuit induces an e.m.f. — electromagnetic induction\textbf{electromagnetic induction} — and it is how almost all of the world’s electricity is generated.

40 min Topic 7 · Electric and Magnetic Fields

What you'll be able to do

  • Define magnetic flux and flux linkage and use Φ=BA\Phi = BA, NΦN\Phi
  • State and apply Faraday’s law, ε=−Δ(NΦ)/Δt\varepsilon = -\Delta(N\Phi)/\Delta t
  • State Lenz’s law and explain it using conservation of energy
  • Explain the operation of generators and transformers
1

Flux and flux linkage

Magnetic flux\textbf{Magnetic flux} Φ\Phi through an area is the flux density times the area perpendicular to the field: Φ=BAcos⁡θ\Phi = BA\cos\theta, measured in webers, where θ\theta is the angle between the field and the normal to the area.

For a coil of NN turns, each turn links the same flux, so the flux linkage\textbf{flux linkage} is NΦN\Phi, measured in weber-turns.

Flux linkage changes if the field strength changes, if the area changes, or if the coil rotates relative to the field. Any of these induces an e.m.f.

Φ=BAcos⁡θ\Phi = BA\cos\theta
Magnetic flux, in Wb.
NΦ=BANcos⁡θN\Phi = BAN\cos\theta
Flux linkage.
Example 1A 200-turn coil of area 4.0×10−3 m24.0\times10^{-3}\ \mathrm{m^{2}} is perpendicular to a 0.15 T field. Find the flux linkage.
1NΦ=200×0.15×4.0×10−3N\Phi = 200 \times 0.15 \times 4.0\times10^{-3}.
Answer0.12 Wb0.12\ \mathrm{Wb}-turns

Tip — When the field is perpendicular to the plane of the coil, θ=0\theta = 0 and flux is maximum. Check whether an angle is given to the plane or to the normal.

2

Faraday’s law

Faraday’s law states that the magnitude of the induced e.m.f. equals the rate of change of flux linkage: ε=Δ(NΦ)/Δt\varepsilon = \Delta(N\Phi)/\Delta t.

Faster change means a larger e.m.f. — moving a magnet quickly into a coil gives a bigger reading than moving it slowly, even though the total flux change is the same.

On a graph of flux linkage against time, the induced e.m.f. is the gradient. A constant flux linkage, however large, induces nothing.

ε=−Δ(NΦ)Δt\varepsilon = -\dfrac{\Delta(N\Phi)}{\Delta t}
Faraday’s law, with Lenz’s sign.
Example 1The field through the coil above falls from 0.15 T to zero in 0.050 s. Find the average induced e.m.f.
1Change in flux linkage: 0.12−0=0.120.12 - 0 = 0.12 Wb-turns.
2ε=0.120.050\varepsilon = \dfrac{0.12}{0.050}.
Answer2.4 V2.4\ \mathrm{V}
3

Lenz’s law

Lenz’s law\textbf{Lenz’s law}: the direction of the induced e.m.f. is such that any current it produces opposes the change in flux that caused it. That is the minus sign in Faraday’s law.

Push a north pole into a coil and the induced current makes the near end of the coil a north pole, repelling the magnet. Pull it out and the near end becomes south, attracting it back.

This is conservation of energy. You must do work against the opposing force to move the magnet, and that work becomes the electrical energy of the induced current. If the induced current attracted the incoming magnet instead, the magnet would accelerate, producing more current, and energy would be created from nothing.

Example 1Explain why a magnet dropped through a copper tube falls more slowly than one dropped outside it.
1As the magnet falls, the flux through each section of the tube changes.
2Eddy currents are induced in the copper.
3By Lenz’s law they create fields opposing the magnet’s motion, slowing it.
4The magnet’s lost gravitational energy is dissipated as heat in the tube.
AnswerInduced eddy currents oppose the change in flux, producing a braking force.

Tip — In a Lenz’s-law explanation, name what is changing, say the induced current opposes it, and link it to energy conservation.

4

Generators and transformers

A generator\textbf{generator} rotates a coil in a magnetic field. The flux linkage varies sinusoidally, so the induced e.m.f. is also sinusoidal — alternating current. Its peak value is BANωBAN\omega, so faster rotation gives a larger and more frequent e.m.f.

A transformer\textbf{transformer} has two coils on a shared iron core. An alternating current in the primary creates a changing flux in the core, which induces an alternating e.m.f. in the secondary. The ratio of voltages equals the ratio of turns.

Transformers only work with alternating current, because a steady current produces a constant flux and induces nothing. For an ideal transformer, power in equals power out, so stepping voltage up steps current down.

ε0=BANω\varepsilon_{0} = BAN\omega
Peak e.m.f. of a generator.
VsVp=NsNp\dfrac{V_{s}}{V_{p}} = \dfrac{N_{s}}{N_{p}}
Transformer turns ratio.
VpIp=VsIsV_{p}I_{p} = V_{s}I_{s}
Ideal transformer.
Example 1A transformer steps 230 V down to 11.5 V. The primary has 2000 turns. Find the secondary turns, and the primary current when the secondary supplies 2.0 A.
1Ns=2000×11.5230=100N_{s} = 2000 \times \dfrac{11.5}{230} = 100 turns.
2Ip=VsIsVp=11.5×2.0230I_{p} = \dfrac{V_{s}I_{s}}{V_{p}} = \dfrac{11.5 \times 2.0}{230}.
Answer100 turns; primary current 0.10 A.

Equation recap

Φ=BAcos⁡θ\Phi = BA\cos\theta
Flux.
ε=−Δ(NΦ)/Δt\varepsilon = -\Delta(N\Phi)/\Delta t
Faraday and Lenz.
ε=Blv\varepsilon = Blv
Moving straight conductor.
Vs/Vp=Ns/NpV_{s}/V_{p} = N_{s}/N_{p}
Transformer.

Common mistakes to avoid

Saying a strong steady magnetic field induces an e.m.f.
Only a changing flux linkage induces an e.m.f.
Forgetting the number of turns NN.
Flux linkage is NΦN\Phi; each turn contributes.
Saying the induced current helps the change.
By Lenz’s law it opposes the change.
Claiming transformers work on d.c.
A steady current gives constant flux and induces nothing.

Key takeaways

  • Flux Φ=BAcos⁡θ\Phi = BA\cos\theta; flux linkage NΦN\Phi.
  • Faraday: induced e.m.f. equals rate of change of flux linkage.
  • Lenz: the induced current opposes the change — a result of energy conservation.
  • Generators produce sinusoidal a.c. with peak e.m.f. BANωBAN\omega.
  • Transformers need a.c.; Vs/Vp=Ns/NpV_{s}/V_{p} = N_{s}/N_{p}.

Test yourself

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