Electromagnetic induction
| English | Chinese | Pinyin |
|---|---|---|
| magnetic flux | 磁通量 | cí tōng liàng |
| flux linkage | 磁链 | cí liàn |
| Faraday's law | 法拉第定律 | fǎ lā dì dìng lǜ |
| Lenz's law | 楞次定律 | léng cì dìng lǜ |
| weber | 韦伯 | wéi bó |
| conservation of energy | 能量守恒 | néng liàng shǒu héng |
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:
$\Phi = BA$ (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?
$\Phi = BA = 0.20 \times 0.50 = 0.10\ \text{Wb}$.
Which words are needed in the definitions of flux and flux linkage? Select all that apply.
"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 ____.
$|\varepsilon| = N\dfrac{d\Phi}{dt}$ — 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?
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.:
It opposes the change — the minus sign in $\varepsilon = -\dfrac{d(N\Phi)}{dt}$.
Lenz's law follows from conservation of energy.
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.
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.
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.
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?
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 $\varepsilon = BLv$