D.4Induction HL only
Almost all the world's electricity comes from generators that work by electromagnetic induction: moving a wire through a magnetic field, or changing the field through a coil, produces an emf. This topic explains magnetic flux, Faraday's law and Lenz's law, and how a rotating coil generates alternating current. It builds on D.2 and D.3. The whole of D.4 is HL only.
Knowledge and science
Nature of science
Experiments. In 1831 Michael Faraday wound two coils on an iron ring. A meter connected to one coil flicked only at the moment the current in the other was switched on or off, never while it was steady. Careful attention to that brief flick led him to the law of induction.
Patterns and trends. In 1820 Hans Christian Ørsted had shown that a current produces a magnetic field. Faraday looked for the reverse: could magnetism produce a current? It could, but only when something was changing.
Models. Faraday had little formal mathematics. He reasoned with pictures of field lines being "cut" by wires. Maxwell later turned those pictures into precise equations.
Global impact. Induction made large-scale electricity generation possible, powering factories, cities and the modern world.
ToK: questions to think about
- Does terminology help or hinder? Flux, flux density and flux linkage sound alike but mean different things. What effect can unclear terminology have on communicating science to the public?
- Pictures or equations? Faraday made major discoveries by thinking in images, without mathematics. Are visual reasoning and mathematical reasoning equally valid ways of knowing?
- Is Lenz's law a law? It can be derived from the conservation of energy. When one principle follows from another, which is more fundamental?
- Who benefits from discoveries? Faraday's work transformed industry, but its benefits were shared unequally around the world. Do scientists have responsibilities for how their discoveries are used?
1. Magnetic flux and flux linkage HL
The magnetic flux $\Phi$ through an area measures how much magnetic field passes through it, like counting the field lines that cross it:
$B$ is the field strength, $A$ the area and $\theta$ the angle between the field and the normal (the line at right angles) to the area. Flux is measured in webers: $1\ \text{Wb} = 1\ \text{T m}^2$. It is greatest when the field passes straight through the area ($\theta = 0$) and zero when the field is parallel to the surface ($\theta = 90°$).
A coil of $N$ turns has flux linkage $N\Phi$: each turn has the same flux through it.
Worked example: flux through a tilted loop HL
A loop of area 0.050 $\text{m}^2$ is in a uniform field of 0.30 T. The normal to the loop is at 60° to the field. Find the flux, and the flux linkage of a 40-turn coil of the same area in the same position.
$\Phi = BA\cos\theta = 0.30 \times 0.050 \times \cos 60^\circ = 7.5 \times 10^{-3}$ Wb.
Flux linkage $= N\Phi = 40 \times 7.5 \times 10^{-3} = 0.30$ Wb.
2. Faraday's law HL
An emf is induced in a circuit whenever the magnetic flux linking it changes. A steady flux, however large, induces nothing.
Faraday's law: the induced emf equals the rate of change of flux linkage. The minus sign is Lenz's law (section 4): the induced emf opposes the change. If the circuit is complete, the emf drives an induced current.
Since $\Phi = BA\cos\theta$, the flux can change in three ways:
- the field strength changes (a magnet moved towards a coil, or the current in a nearby coil switched on or off);
- the area in the field changes (a coil moved into or out of a field, or a rod sliding along rails);
- the angle changes (a coil rotating in a field, as in a generator).
The faster the change, the bigger the emf. Pushing a magnet into a coil quickly gives a bigger reading on a meter than pushing it in slowly, although the total change in flux is the same.
Worked example: a changing field HL
A 200-turn coil of area $4.0 \times 10^{-3}\ \text{m}^2$ is at right angles to a magnetic field. The field falls steadily from 0.50 T to 0.10 T in 0.20 s. Find the size of the induced emf.
Change in flux through each turn: $\Delta\Phi = \Delta B \times A = 0.40 \times 4.0 \times 10^{-3} = 1.6 \times 10^{-3}$ Wb.
$\varepsilon = N\dfrac{\Delta\Phi}{\Delta t} = 200 \times \dfrac{1.6 \times 10^{-3}}{0.20} = 1.6$ V.
3. A conductor moving through a field HL
When a straight conductor of length $L$ moves at speed $v$ at right angles to a uniform field $B$, the free charges inside it move with it, so each feels a magnetic force $qvB$ along the conductor (D.3). Charge builds up at the ends until the electric field $E = \frac{\varepsilon}{L}$ it creates balances the magnetic force: $qE = qvB$. So:
for a straight conductor moving perpendicular to a uniform field. The same result follows from Faraday's law: in time $\Delta t$ the rod sweeps out an area $Lv\Delta t$, so the flux through the circuit changes at a rate $BLv$.
Worked example: a sliding rod HL
In the arrangement above, $B = 0.50$ T, the rails are 0.20 m apart, the rod moves at 3.0 $\text{m s}^{-1}$ and $R = 2.0\ \Omega$ (the rod and rails have negligible resistance). Find the emf, the current, the force needed to keep the rod moving at constant speed, and the power supplied.
$\varepsilon = BvL = 0.50 \times 3.0 \times 0.20 = 0.30$ V, so $I = \dfrac{0.30}{2.0} = 0.15$ A.
The current in the rod feels a force $F = BIL = 0.50 \times 0.15 \times 0.20 = 0.015$ N, which opposes the motion (Lenz's law). To keep the speed constant, you must push with 0.015 N.
Power supplied: $Fv = 0.015 \times 3.0 = 0.045$ W. Power dissipated in the resistor: $I^2R = 0.15^2 \times 2.0 = 0.045$ W. Your work becomes thermal energy in the resistor: energy is conserved.
Worked example: an aircraft wing HL
An aircraft with a wingspan of 35 m flies horizontally at 240 $\text{m s}^{-1}$ where the vertical component of the Earth's magnetic field is 45 µT. Find the emf between its wingtips.
$\varepsilon = BvL = 45 \times 10^{-6} \times 240 \times 35 = 0.38$ V. It can't drive a current, because any wire connected across the wings would have the same emf induced in it.
4. Lenz's law and energy HL
- Push the north pole of a magnet towards a coil: the induced current makes the near end of the coil a north pole, which repels the approaching magnet.
- Pull the magnet away: the near end becomes a south pole, which attracts it back.
Why it must be so: if the induced current helped the change instead, the magnet would be pulled in faster, inducing a bigger current, pulling harder still: energy would appear from nothing. Lenz's law is a consequence of the conservation of energy. The electrical energy produced always comes from the work done against the opposing force, as in the sliding-rod example.
Lenz's law is behind electromagnetic braking: a magnet falls slowly down a copper pipe, and trains and roller coasters can be slowed by currents induced in metal plates, with no contact and no wear.
5. Coils moving into and out of a field HL
A rectangular coil moving at constant speed into a uniform field region, through it and out the other side:
- Entering: only the leading edge is in the field, so $\varepsilon = NBvL$, where $L$ is the length of that edge.
- Fully inside (a uniform field): the flux isn't changing, so $\varepsilon = 0$. Both sides cut field lines, and their emfs cancel around the coil.
- Leaving: $\varepsilon = NBvL$ again, in the opposite direction.
Worked example: a coil crossing a field HL
A square 50-turn coil of side 10 cm moves at 0.20 $\text{m s}^{-1}$ into a 0.40 T field, perpendicular to its plane. Find the emf while it is entering, and how long the emf lasts.
$\varepsilon = NBvL = 50 \times 0.40 \times 0.20 \times 0.10 = 0.40$ V.
The coil is fully inside after it has moved its own width: $t = \dfrac{0.10}{0.20} = 0.50$ s.
6. The ac generator HL
A coil rotating at a steady rate in a uniform magnetic field has the angle $\theta$ in $\Phi = BA\cos\theta$ changing continuously: $\theta = \omega t$, where $\omega = 2\pi f$. The flux linkage varies sinusoidally, so the induced emf does too: this is an alternating emf.
- When the plane of the coil is at right angles to the field, the flux is maximum but momentarily not changing: $\varepsilon = 0$.
- When the plane of the coil is parallel to the field, the flux is zero but changing fastest: $\varepsilon$ is maximum.
Changing the rotation frequency: spinning the coil twice as fast makes the flux change twice as fast, so the peak emf doubles, and the period halves (the frequency of the ac doubles). The peak emf can also be increased by using more turns, a larger coil area or a stronger field.
Worked example: speeding up a generator HL
A generator coil produces an alternating emf of peak value 63 V at a frequency of 50 Hz. The coil is now turned at 25 Hz. State the new peak emf and the new period of the ac.
Halving the frequency halves the rate of change of flux, so the peak emf halves to 31.5 V. The period doubles from $\frac{1}{50} = 0.020$ s to $\frac{1}{25} = 0.040$ s.
In a power station, a turbine driven by steam, water or wind turns the generator. To keep the mains frequency fixed (50 or 60 Hz), generators are run at a constant speed.
7. Self-induction HL
A coil carrying a changing current produces a changing magnetic field, and that changing field links the coil itself. So a changing current in a coil induces an emf in the same coil. This is self-induction. By Lenz's law, the self-induced emf (sometimes called a back emf) opposes the change in current: it slows the rise of current when a circuit is switched on, and tries to keep the current flowing when it is switched off. Breaking the current in a large coil suddenly can induce an emf big enough to cause a spark across the switch. (Only a qualitative understanding is needed.)
8. Common mistakes HL
- Thinking a large flux gives a large emf. It's the rate of change of flux linkage that matters. A huge steady flux induces nothing.
- Measuring θ from the plane of the coil in $\Phi = BA\cos\theta$. θ is between the field and the normal to the coil.
- Forgetting N. Faraday's law uses the flux linkage $N\Phi$.
- Saying a coil fully inside a uniform field has an induced emf because it is moving. The flux is constant, so $\varepsilon = 0$.
- Getting the Lenz direction wrong. The induced current opposes the change in flux, not the flux itself: if the flux is decreasing, the induced field acts to keep it up.
- Saying doubling the generator's frequency only changes the frequency of the ac. The peak emf doubles too.
9. Check your understanding HL
A magnet is held still inside a coil. Is an emf induced? What if the coil is moved along with the magnet?
No, in both cases. There is no relative motion, so the flux linkage doesn't change.
A coil is turned from facing the field ($\theta = 0$) to edge-on ($\theta = 90°$) in 0.10 s. Its flux linkage when facing the field is 0.050 Wb. Find the average emf.
The flux linkage falls from 0.050 Wb to 0, so $\varepsilon = \frac{0.050}{0.10} = 0.50$ V.
Why does a magnet fall more slowly through a copper tube than through a plastic one?
As the magnet falls, the changing flux induces currents in the copper (a conductor). By Lenz's law these currents create fields that oppose the magnet's motion. Plastic is an insulator, so no currents flow.
At what positions of a rotating generator coil is the induced emf zero, and why?
When the plane of the coil is at right angles to the field. The flux through it is at a maximum (or minimum) there, so for an instant it isn't changing.
A metal rod falls vertically while held horizontal in a horizontal magnetic field, with no circuit connected. What happens?
An emf $BvL$ is induced between its ends, increasing as it speeds up. No current flows (no complete circuit), so there is no opposing force and it falls freely.