Electromagnetic induction
Introduced| English | Meaning |
|---|---|
| magnetic flux/mæɡˈnetɪk flʌks/ | A measure of the magnetic field passing through a surface, accounting for field strength, area and orientation. |
| flux linkage/flʌks ˈlɪŋkɪdʒ/ | The sum of magnetic fluxes through the turns of a coil; for identical turns it is the flux per turn multiplied by the number of turns. |
| Faraday's law/ˈfærədeɪz lɔː/ | The induced electromotive force equals the negative rate of change of magnetic flux linkage. |
| Lenz's law/ˈlentsɪz lɔː/ | An induced current or electromotive force acts in a direction that opposes the change in magnetic flux that produces it. |
| weber/ˈveɪbə/ | weber |
| conservation of energy/ˌkɒnsəˈveɪʃn ɒv ˈenədʒi/ | The principle that energy cannot be created or destroyed, only transferred or transformed. |
Hold the magnet still and the current stops
- Push a magnet into a coil and a galvanometer needle swings. Hold the magnet inside the coil, perfectly still, and the needle returns to zero, even though the coil now sits in the strongest field it has felt.
- Pull the magnet out and the needle swings the other way.
- So it is not the field that induces an e.m.f. It is the change in the field, and the two laws of this lesson say how big and which way.
- This lesson is magnetic flux 磁通量, flux linkage, Faraday's law 法拉第定律 and Lenz's law 楞次定律.
Magnetic flux and flux linkage
- Magnetic flux is the product of the magnetic flux density and the area perpendicular to the field:
- Say perpendicular, or "normal to the field". If the area's normal is at angle $\theta$ to $B$, use $\Phi = BA\cos\theta$. The unit is the weber 韦伯, $\text{Wb} = \text{T m}^2$.
- For a coil of $N$ turns, the flux linkage 磁链 is $N\Phi = NBA$, measured in weber-turns.
- The two-mark definitions want exactly those words: flux is flux density times perpendicular area; flux linkage is flux times number of turns.
A needle that moves only while the magnet does
The magnetic flux through an area $A$ at right angles to a field $B$ is:
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$\Phi = BA$ (weber). For a coil of $N$ turns the flux linkage is $N\Phi$.
A $0.50\ \text{m}^2$ coil sits at right angles to a $0.20\ \text{T}$ field. What is the flux?
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$\Phi = BA = 0.20 \times 0.50 = 0.10\ \text{Wb}$.
Which words are needed in the definitions of flux and flux linkage? Select all that apply.
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"Perpendicular" is marked, because BA is only true for the component of the area at right angles to B. Flux is defined whether or not it is changing.
Faraday's law
- Faraday's law: the induced e.m.f. equals the rate of change of flux linkage:
- "Rate of change of flux linkage", not "of flux", is the phrase that earns the mark, since the number of turns matters.
- Three things can change the flux: a changing $B$, a changing area, or a changing orientation. The third one is the a.c. generator.
Electromagnetic induction
Move the magnet through the coil — a current is induced only while the field is changing. Faster gives more current; flip the magnet to reverse it.
The induced e.m.f. equals the rate of change of flux ____.
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$|\varepsilon| = N\dfrac{d\Phi}{dt}$ — the rate of change of the flux linkage $N\Phi$.
Worked example: a collapsing field
- A coil of $200$ turns and area $0.010\ \text{m}^2$ lies with its plane at right angles to a $0.50\ \text{T}$ field. The field falls steadily to zero in $0.20\ \text{s}$. Find the average induced e.m.f.
- Flux linkage at the start: $N\Phi = NBA = 200 \times 0.50 \times 0.010 = 1.0\ \text{Wb}$. At the end it is zero.
- Faraday's law: $|\varepsilon| = \dfrac{\Delta(N\Phi)}{\Delta t} = \dfrac{1.0}{0.20} = 5.0\ \text{V}$.
- Note that the plane of the coil is at right angles to the field, so its normal is along the field and $\Phi = BA$ with no cosine. A question that gives the angle to the plane rather than the normal is testing exactly this.
A coil of 200 turns and area 0.010 m^2 is at right angles to a 0.50 T field which falls to zero in 0.20 s. What is the average induced e.m.f., in volts?
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The flux linkage falls from NBA = 1.0 Wb to zero, so the e.m.f. is 1.0/0.20 = 5.0 V. The coil's plane is perpendicular to B, so its normal is along B and no cosine is needed.
Lenz's law
- Lenz's law: the induced e.m.f. acts in the direction that opposes the change producing it. Combined with Faraday:
- The reason is conservation of energy 能量守恒. If the induced effect reinforced the change, the current would grow without limit and energy would come from nothing.
- Practically, it is why you must do work to push a magnet into a coil: the induced current opposes you, and that work is where the electrical energy comes from.
Lenz's law says the induced e.m.f.:
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It opposes the change — the minus sign in $\varepsilon = -\dfrac{d(N\Phi)}{dt}$.
Lenz's law follows from conservation of energy.
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If the induced effect reinforced the change, energy would be created from nothing — so it must oppose it.
Reading an e.m.f. off a flux graph
- The induced e.m.f. is minus the gradient of the flux-linkage against time graph. Sketching the e.m.f. is differentiating the graph by eye.
- A steadily rising flux gives a constant e.m.f. A constant flux gives zero e.m.f., however large the flux is. A faster fall gives a larger e.m.f. of the opposite sign.
- That middle case is the whole point of the opening puzzle: a coil sitting still in a strong steady field has a large flux linkage and no e.m.f. at all.
Select all the changes that increase the induced e.m.f.
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A bigger $N$, $B$, or rate of change all raise the e.m.f. A stationary magnet gives no change of flux — no e.m.f.
A coil held still in a strong, steady magnetic field has a large induced e.m.f.
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The flux linkage is large but constant, and the e.m.f. is its rate of change, which is zero. Only a changing flux induces anything.
Match each flux-linkage graph to the e.m.f. it produces.
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The e.m.f. is minus the gradient, so sketching it is differentiating the graph by eye.
Worked example: flux cutting
- An aircraft of wingspan $60\ \text{m}$ flies horizontally at $250\ \text{m/s}$ where the vertical component of the Earth's field is $45\ \mu\text{T}$. Find the e.m.f. between the wingtips.
- In time $\Delta t$ the wing sweeps out an area $Lv\Delta t$, cutting flux $BLv\Delta t$, so the rate of change of flux is:
- Only the component of $B$ perpendicular to the swept area counts, which is why the question gives the vertical component.
- There is a p.d. between the wingtips but no current, because there is no complete circuit.
An aircraft of wingspan 60 m flies at 250 m/s where the vertical component of the Earth's field is 45 microtesla. What is the e.m.f. between the wingtips, in volts?
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The wing sweeps area Lv per second, so emf = BLv = 45e-6 x 60 x 250 = 0.68 V. Only the component of B perpendicular to the swept area counts, and there is no current without a circuit.
Marks that slip away
- Flux is flux density times the perpendicular area, and Faraday's law is the rate of change of flux linkage. Both words are marked.
- A constant flux, however large, induces zero e.m.f. Only change induces.
- Lenz's law is a consequence of conservation of energy. Say so when asked why.
- Watch whether an angle is given to the plane of the coil or to its normal; they differ by 90 degrees and the cosine flips to a sine.
You've got it
- magnetic flux is flux density times the area perpendicular to the field, $\Phi = BA$ in webers; flux linkage is $N\Phi$
- Faraday's law: the induced e.m.f. is the rate of change of flux linkage, so a constant flux induces nothing
- Lenz's law: the e.m.f. opposes the change that produces it, which follows from conservation of energy
- the e.m.f. is minus the gradient of a flux-linkage graph, and a conductor sweeping through a field gives $\varepsilon = BLv$