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Faraday's law, $\mathcal{E} = -N\,\Delta\Phi/\Delta t$, Lenz's law and the direction of induced currents, eddy currents, and induction at work in pickups, cooktops and safety outlets.
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By the end of this lesson you will be able to find the size and direction of induced emfs and currents from changing flux.
From the last lesson you know magnetic flux, $\Phi = BA\cos\theta$, and that it changes when the field, the area or the angle changes. You know that an emf drives current around a circuit. This lesson connects the two: a changing flux is itself a source of emf.
| Term | What it means |
|---|---|
| Electromagnetic induction | The production of an emf by a changing magnetic flux. |
| Faraday's law | $\mathcal{E} = -N\,\Delta\Phi/\Delta t$: the emf equals the rate of change of flux linkage. |
| Lenz's law | The induced current's field opposes the change in flux that caused it. |
| Induced current | The current an induced emf drives, $I = \mathcal{E}/R$. |
| Eddy currents | Swirling induced currents in solid conductors. |
| Generator | A device that turns motion into electricity by induction. |
In 1831 Michael Faraday in London and Joseph Henry in Albany, New York, each discovered that a magnet moving near a coil makes a current flow in it, while a stationary magnet does nothing. What matters is change. Faraday's law says the induced emf equals the rate of change of flux linkage:
$$\mathcal{E} = -N\frac{\Delta\Phi}{\Delta t}.$$
The flux can change because the field changes, the area changes, or the loop turns. The minus sign is Lenz's law: the induced current flows in the direction whose own magnetic field opposes the change. Push a magnet in and the coil pushes back; pull it out and the coil pulls.
Another way: picture
Picture a coil as stubborn about the flux through it. Whatever flux it has, it tries to keep. Push more field in, and it drives a current to cancel the increase; take field away, and it drives a current to replace it. The faster you try to change its flux, the harder it resists, and that resistance is the induced emf.
Another way: steps
Faraday wound two coils on an iron ring. When he connected a battery to one, a compass near the other twitched, then settled back; when he disconnected it, the compass twitched the other way. A steady current did nothing; only switching it on or off did. The changing flux in the ring induced a current in the second coil.
He then showed that moving a magnet in and out of a coil, or moving a coil near a magnet, did the same. Joseph Henry, a teacher at the Albany Academy, found the effect independently at about the same time. The unit of inductance, the henry, is named for him.
Since $\Phi = BA\cos\theta$, flux changes when any of the three factors changes. Change the field strength, as by moving a magnet or switching an electromagnet. Change the area, as by sliding a rod along rails or squeezing a flexible loop. Change the angle, as by spinning a coil in a field.
All three obey the same law and produce emfs in the same way. Generators turn coils; transformers change fields; the next lesson's sliding rod changes area. A large steady field through a still coil does nothing, however strong.
Lenz's law is energy conservation in disguise. Suppose the induced current helped the change instead of opposing it. Pushing a magnet toward a coil would induce a current that pulled the magnet in faster, inducing more current, pulling harder: energy from nothing.
Instead the induced current always opposes, so you must do work to push the magnet in, and that work becomes the electrical energy in the circuit. Generators are hard to turn when they deliver current for exactly this reason; the energy comes from whatever turns them.
To find an induced current's direction, ask three questions. Which way does the field through the loop point? Is the flux increasing or decreasing? Which way must the induced current flow so its own field opposes that change? Then use the right-hand grip rule to turn the needed field into a current direction.
For a north pole pushed toward a coil, the flux through the coil toward the far end increases, so the induced current makes a field pointing back toward the magnet: the near face of the coil becomes a north pole, repelling the approaching magnet.
Checking an answer. No change in flux must mean no emf. A faster change must give a larger emf. The induced current must oppose the change, never help it.
Faraday's law is one of Maxwell's four equations, tested for nearly two centuries. A changing magnetic field creates a circulating electric field, which pushes the charges in the wire around the loop. The emf is the work that field does per coulomb in one trip around.
Using the average $\Delta\Phi/\Delta t$ gives the average emf over the interval. If the flux changes steadily, the emf is constant and the average is exact; if not, the emf varies, and calculus gives it moment by moment.
Induction happens in any conductor, not just coils. When a magnet moves near a sheet of metal, swirling currents called eddy currents are induced in it, and by Lenz's law they oppose the motion. Drop a strong magnet down a copper pipe and it falls slowly, braked by the currents it induces.
Roller coasters such as those at Cedar Point in Ohio use magnetic brakes: fins on the cars pass between strong magnets, and eddy currents slow them smoothly with no contact and no wear. Trains, exercise bikes and some truck brakes use the same principle.
An induction cooktop has a coil under its glass surface carrying alternating current at about $25$ kHz. The rapidly changing flux induces eddy currents in the bottom of an iron or steel pan, and the pan's resistance turns them into heat. The glass itself stays relatively cool.
Aluminum and copper pans do not work well on most induction cooktops because they conduct too well, so little heat is made, and glass or ceramic pans do not work at all. Induction cooking is efficient and quick, and several American cities now encourage it over gas stoves.
A dynamic microphone is a loudspeaker in reverse: sound waves push a diaphragm attached to a coil that moves in a magnet's field, and the changing flux induces a small voltage that copies the sound. Shure, based near Chicago, has made such microphones since the 1930s.
The magnetic stripe on older credit cards stores data as tiny magnetized regions. Swiping the card past a coil changes the flux through it and induces pulses that the reader decodes. Swipe too slowly and the pulses are too weak to read, a direct consequence of Faraday's law.
The outlets near American sinks and bathtubs have test and reset buttons: they are ground-fault circuit interrupters. Inside, both the outgoing and returning wires pass through a small iron ring with a sensing coil. Normally their currents are equal and opposite, so their fields cancel and the flux in the ring is zero.
If some current leaks through a person to ground, the two currents no longer balance, the ring carries a changing flux, and a current is induced in the sensing coil. A circuit detects it and cuts the power within a fortieth of a second. The National Electrical Code has required them near water since the 1970s.
Changing magnetic fields induce currents wherever there are conductors, including the human body. Transcranial magnetic stimulation, approved to treat depression, places a coil against the scalp and pulses it, inducing small currents in brain tissue that make nerve cells fire.
On a planetary scale, solar storms shake Earth's magnetic field, and the changing flux induces currents in long power lines and pipelines. In March 1989 such induced currents knocked out the Hydro-Québec grid, leaving six million people without power, and utilities now watch space-weather forecasts from NOAA.
Many intersections in the United States detect waiting cars with a rectangle of wire cut into the pavement, visible as a thin sealed groove in the asphalt. A small current alternating tens of thousands of times a second flows in the loop, making a changing field above it. When a car stops over the loop, its steel body and the eddy currents induced in it change the loop's flux and shift the circuit's natural frequency.
A detector in the roadside cabinet senses that shift and tells the signal controller a vehicle is waiting. Motorcycles and bicycles, with less metal, sometimes fail to trigger the loop, which is why some intersections paint a symbol showing riders exactly where to stop so that as much metal as possible sits over the wire.
Leo Fender and Gibson's engineers built the pickups that defined American rock and roll. Each pickup is a magnet wrapped in about $5000$ turns of hair-thin copper wire. The magnet magnetizes the steel string above it, and as the string vibrates, it changes the flux through the coil.
If the A string, vibrating at $110$ Hz, changes the flux through each turn by $10^{-8}$ Wb every quarter vibration, Faraday's law gives an average emf of about $22$ mV, which an amplifier boosts to drive a speaker. Higher strings vibrate faster and induce more, and nylon strings, which are not magnetic, induce nothing, which is why classical guitars need microphones instead.
On March 13, 1989, a burst of charged particles from the Sun struck Earth's magnetic field, making it fluctuate rapidly. Across Canada and the northern United States, the changing flux through the enormous loops formed by long transmission lines and the ground induced currents of tens of amperes.
These currents saturated transformer cores in Quebec's grid, and in ninety seconds the system collapsed, leaving six million people without power for nine hours. A transformer at a New Jersey nuclear plant was also damaged. Since then, NOAA's Space Weather Prediction Center issues geomagnetic storm warnings, and utilities adjust their grids when a storm is coming.
It is natural to think a strong magnet placed inside a coil makes a current flow. A steady flux, however large, induces nothing. Only while the flux is changing, as the magnet moves in or out, does an emf appear, and it is larger the faster the change.
A related error is to think the induced current always opposes the field. It opposes the change in the field. When the flux is decreasing, the induced current makes a field in the same direction as the original, trying to keep it from falling.
A magnet falling through a $400$-turn coil changes the flux through each turn by $2.0 \times 10^{-4}$ Wb in $0.050$ s. Find the rate of change.
$\dfrac{\Delta\Phi}{\Delta t} = \dfrac{2.0 \times 10^{-4}}{0.050} = 4.0 \times 10^{-3}\ \text{Wb/s}$
Per turn.
Multiply by the turns.
$\mathcal{E} = 400 \times 4.0 \times 10^{-3} = 1.6\ \text{V}$
Faraday's law.
The coil and meter have $20$ Ω. Find the current.
$I = \dfrac{1.6}{20} = 0.080\ \text{A}$
Ohm's law.
Describe the direction as it enters.
$\text{the coil repels the approaching magnet}$
Opposing the increase.
Describe the direction as it leaves.
$\text{the coil attracts the departing magnet}$
Opposing the decrease; the current reverses.
A patient's arm, a loop of area $0.010$ m², sits in a field that changes by $0.030$ T in $1.0$ ms. Find the flux change.
$\Delta\Phi = 0.010 \times 0.030 = 3.0 \times 10^{-4}\ \text{Wb}$
One loop of tissue.
Find the emf around the loop.
$\mathcal{E} = \dfrac{3.0 \times 10^{-4}}{1.0 \times 10^{-3}} = 0.30\ \text{V}$
A single turn.
Tissue around the loop has $500$ Ω. Find the current.
$I = \dfrac{0.30}{500} = 6.0 \times 10^{-4}\ \text{A}$
Well under a milliampere.
Compare with nerve thresholds.
$\text{faster switching could twitch nerves}$
Scanners are limited to avoid it.
Find the power dissipated.
$P = 0.30 \times 6.0 \times 10^{-4} = 1.8 \times 10^{-4}\ \text{W}$
Negligible heating.
Explain the tapping noise of MRI.
$\text{gradient coils jolt as their currents switch}$
Forces on currents in the strong field.
A $200$-turn coil of area $30$ cm² sits in an electromagnet's $0.80$ T field, which drops to zero in $0.020$ s. Find the flux per turn at the start.
$\Phi = 0.80 \times 30 \times 10^{-4} = 2.4 \times 10^{-3}\ \text{Wb}$
Facing the field.
Find the change.
$\Delta\Phi = -2.4 \times 10^{-3}\ \text{Wb}$
Falls to zero.
Find the emf.
$\mathcal{E} = 200 \times \dfrac{2.4 \times 10^{-3}}{0.020} = 24\ \text{V}$
Faraday's law.
Find the direction.
$\text{current makes a field in the original direction}$
Opposing the decrease.
Switch off ten times faster. Find the emf.
$\mathcal{E} = 240\ \text{V}$
Proportional to the rate.
Explain sparks at switches.
$\text{sudden switching induces large emfs}$
Why relays need protection diodes.
Write Faraday's law.
$\mathcal{E} = N\dfrac{\Delta\Phi}{\Delta t}$
Rate of change of linkage.
Substitute the values.
$\mathcal{E} = 50 \times \dfrac{0.010}{0.25}$
Turns, flux change and time.
Evaluate the emf.
A coil of $500$ turns, each of area $10$ cm², faces a magnetic field that increases by $0.2$ T in $0.05$ s. What average emf is induced in the coil, in V?
Complete the worked solution: a coil of $50$ turns, each of area $100$ cm², faces a field that changes steadily from $0.8$ T to $0.2$ T in $0.3$ s. The coil's circuit has resistance $2$ Ω. Find the change in flux per turn in mWb, the size of the emf in V, and the current in A.
Find the change in flux per turn.
$\Delta\Phi = A(B_2 - B_1) =$ f
In milliwebers, with its sign.
Find the size of the emf.
$\mathcal{E} = N\dfrac{|\Delta\Phi|}{\Delta t} =$ e
Faraday's law.
Find the current.
$I = \dfrac{\mathcal{E}}{R} =$ i
Ohm's law.
Find the current's direction.
$\text{its field opposes the change in flux}$
Lenz's law.
Match each situation to what induction does.
| the induced current repels the magnet | the induced current attracts the magnet | no emf is induced | a larger emf is induced | |
|---|---|---|---|---|
| a magnet pushed toward a coil | ||||
| a magnet pulled away from a coil | ||||
| a magnet held still inside a coil | ||||
| a magnet moved in faster |
A coil of $500$ turns, each of area $0.002$ m², faces a field increasing steadily at $3$ T/s. The coil's circuit has resistance $10$ Ω. Fill in the emf in V, the current in A, and the power dissipated in W.
| value | |
|---|---|
| emf (V) | |
| current (A) | |
| power (W) |
A coil of $100$ turns, each of area $50$ cm², faces a field rising steadily at $0.6$ T/s. Write the induced current, in A, as a function of the circuit's total resistance $R$ in ohms.
Answer:
A coil of $200$ turns, each of area $40$ cm², faces a $0.5$ T field and is turned edge-on in $0.05$ s. What average emf is induced, in V?
Answer: V
An electric guitar pickup has $5000$ turns of fine wire around a magnet. The vibrating steel A string, at $110$ Hz, changes the flux through each turn by $1.0 \times 10^{-8}$ Wb in a quarter of a vibration. What average emf does the pickup produce over that quarter, in mV?
Answer: mV
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A coil of $500$ turns, each of area $10$ cm², faces a field rising steadily at $0.3$ T/s. Write the induced current, in A, as a function of the circuit's total resistance $R$ in ohms.
Answer:
You can apply Faraday's and Lenz's laws. Explain to someone why a magnet held still inside a coil induces no current.
23. Your turn: the flux through a $50$-turn coil changes by $0.010$ Wb per turn in $0.25$ s. What emf is induced?, step 3
$\mathcal{E} = 2.0\ \text{V}$
Volts.