Electromagnetic induction · 电磁感应
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| magnetic flux/mæɡˈnetɪk flʌks/ | 磁通量 | cí tōng liàng |
| flux linkage/flʌks ˈlɪŋkɪdʒ/ | 磁链 | cí liàn |
| Faraday's law/ˈfærədeɪz lɔː/ | 法拉第定律 | fǎ lā dì dìng lǜ |
| Lenz's law/ˈlentsɪz lɔː/ | 楞次定律 | léng cì dìng lǜ |
| weber/ˈveɪbə/ | 韦伯 | wéi bó |
| conservation of energy/ˌkɒnsəˈveɪʃn ɒv ˈenədʒi/ | 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.
把磁铁拿住不动,电流就停了
- 把磁铁推进线圈,检流计指针就摆动。把磁铁放在线圈里面、完全不动,指针就回到零——尽管线圈此刻正处在它感受过的最强的磁场里。
- 把磁铁抽出来,指针朝相反方向摆动。
- 所以感生电动势的不是磁场。是磁场的变化,而这一课的两条定律说的是它有多大、朝哪个方向。
- 这一课讲磁通量(magnetic flux)、磁链、法拉第定律(Faraday's law)和楞次定律(Lenz's law)。
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
磁通量与磁链
- 磁通量是磁通密度与垂直于磁场的面积的乘积:
- 要说"垂直",或"与场法向一致"。若面积的法线与 $B$ 成 $\theta$ 角,用 $\Phi = BA\cos\theta$。单位是韦伯(weber),$\text{Wb} = \text{T m}^2$。
- 对 $N$ 匝的线圈,磁链(flux linkage)是 $N\Phi = NBA$,单位是韦伯匝。
- 两分的定义要的正是这些词:磁通量是磁通密度乘垂直面积;磁链是磁通量乘匝数。
只在磁铁动的时候才动的指针
The magnetic flux through an area $A$ at right angles to a field $B$ is: · 与场 $B$ 成直角的面积 $A$ 上的磁通量是:
$\Phi = BA$ (weber). For a coil of $N$ turns the flux linkage is $N\Phi$. · $\Phi = BA$(韦伯)。对于 $N$ 匝的线圈,磁链是 $N\Phi$。
A $0.50\ \text{m}^2$ coil sits at right angles to a $0.20\ \text{T}$ field. What is the flux? · 一个 $0.50\ \text{m}^2$ 的线圈与 $0.20\ \text{T}$ 的场成直角。磁通量是多少?
$\Phi = BA = 0.20 \times 0.50 = 0.10\ \text{Wb}$. · $\Phi = BA = 0.20 \times 0.50 = 0.10\ \text{Wb}$。
Which words are needed in the definitions of flux and flux linkage? Select all · 所有 that apply. · 磁通量和磁链的定义需要哪些词?选出所有适用的。
"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. · "垂直"是要评分的,因为 BA 只对与 B 垂直的那部分面积成立。磁通量的定义与它是否在变化无关。
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.
法拉第定律
- 法拉第定律:感生电动势等于磁链的变化率:
- "磁链的变化率"而不是"磁通量的变化率",才是得分的措辞,因为匝数是要紧的。
- 有三样东西能改变磁通量:变化的 $B$、变化的面积,或变化的取向。第三种就是交流发电机。
Electromagnetic induction · 电磁感应
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. · 将磁铁穿过线圈——只有当磁场发生变化时才会产生感应电流。速度越快，电流越大；翻转磁铁可反向电流。
The induced e.m.f. equals the rate of change of flux ____. · 感应电动势等于磁 ____ 的变化率。
$|\varepsilon| = N\dfrac{d\Phi}{dt}$ — the rate of change of the flux linkage $N\Phi$. · $|\varepsilon| = N\dfrac{d\Phi}{dt}$——磁链 $N\Phi$ 的变化率。
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.
例题:塌缩的磁场
- 一个 $200$ 匝、面积 $0.010\ \text{m}^2$ 的线圈,其平面垂直于 $0.50\ \text{T}$ 的磁场。磁场在 $0.20\ \text{s}$ 内匀速降到零。求平均感生电动势。
- 起始磁链:$N\Phi = NBA = 200 \times 0.50 \times 0.010 = 1.0\ \text{Wb}$。终了为零。
- 法拉第定律:$|\varepsilon| = \dfrac{\Delta(N\Phi)}{\Delta t} = \dfrac{1.0}{0.20} = 5.0\ \text{V}$。
- 注意线圈平面垂直于磁场,所以它的法线沿着磁场,$\Phi = BA$ 不带余弦。给出与平面而不是与法线夹角的题,考的正是这一点。
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? · 一个 200 匝、面积 0.010 m^2 的线圈垂直于 0.50 T 的磁场,该磁场在 0.20 s 内降到零。平均感生电动势是多少伏?
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. · 磁链从 NBA = 1.0 Wb 降到零,所以电动势是 1.0/0.20 = 5.0 V。线圈平面垂直于 B,所以它的法线沿 B,不需要余弦。
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.
楞次定律
- 楞次定律:感生电动势的方向总是阻碍产生它的那个变化。与法拉第定律合起来:
- 理由是能量守恒(conservation of energy)。如果感生的效应反过来加强那个变化,电流就会无限增长,能量就凭空而来了。
- 实际上,这就是你必须做功才能把磁铁推进线圈的原因:感生电流阻碍你,而那份功正是电能的来源。
Lenz's law says the induced e.m.f.: · 楞次定律说,感应电动势:
It opposes the change — the minus sign in $\varepsilon = -\dfrac{d(N\Phi)}{dt}$. · 它反抗变化——即 $\varepsilon = -\dfrac{d(N\Phi)}{dt}$ 中的负号。
Lenz's law follows from conservation of energy. · 楞次定律源于能量守恒。
If the induced effect reinforced the change, energy would be created from nothing — so it must oppose it. · 如果感应效应增强变化,能量就会凭空产生——所以它必须反抗变化。
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.
从磁通图上读电动势
- 感生电动势是磁链对时间图的斜率的负值。画电动势图就是用眼睛对那张图求导。
- 磁通匀速上升给出恒定的电动势。磁通恒定给出零电动势,不管磁通有多大。下降更快给出更大且反号的电动势。
- 中间那种情形正是开头那个谜题的全部要点:静止在强而稳定的磁场中的线圈,磁链很大,而电动势完全为零。
Select all · 所有 the changes that increase the induced e.m.f. · 选出所有能增大感应电动势的改变。
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. · 更大的 $N$、$B$ 或变化率都会增大电动势。静止的磁铁没有磁通变化——没有电动势。
A coil held still in a strong, steady magnetic field has a large induced e.m.f. · 静止在强而稳定的磁场中的线圈会有很大的感生电动势。
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. · 磁链很大但恒定,而电动势是它的变化率,为零。只有变化的磁通才能感生出东西。
Match each flux-linkage graph to the e.m.f. it produces. · 把每种磁链图与它产生的电动势配对。
The e.m.f. is minus the gradient, so sketching it is differentiating the graph by eye. · 电动势是斜率的负值,所以画它就是用眼睛对图求导。
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.
例题:切割磁通
- 一架翼展 $60\ \text{m}$ 的飞机以 $250\ \text{m/s}$ 水平飞行,当地地磁场的竖直分量是 $45\ \mu\text{T}$。求两翼尖之间的电动势。
- 在时间 $\Delta t$ 内机翼扫过面积 $Lv\Delta t$,切割磁通 $BLv\Delta t$,所以磁通的变化率是:
- 只有垂直于扫过面积的 $B$ 分量算数,这就是题目给出竖直分量的原因。
- 两翼尖之间有电压却没有电流,因为没有完整的电路。
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? · 一架翼展 60 m 的飞机以 250 m/s 飞行,当地地磁场竖直分量为 45 微特斯拉。两翼尖之间的电动势是多少伏?
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. · 机翼每秒扫过面积 Lv,所以电动势 = BLv = 45e-6 x 60 x 250 = 0.68 V。只有垂直于扫过面积的 B 分量算数,而没有电路就没有电流。
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.
容易丢掉的分
- 磁通量是磁通密度乘垂直面积,而法拉第定律是磁链的变化率。这两个词都要评分。
- 恒定的磁通不管多大,感生的电动势都是零。只有变化才能感生。
- 楞次定律是能量守恒的推论。被问为什么时要说出来。
- 留意题目给的角是与线圈平面的还是与它法线的;两者差 90 度,余弦要变成正弦。
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 $\varepsilon = BLv$
你掌握了
- 磁通量是磁通密度乘以垂直于磁场的面积,$\Phi = BA$,单位韦伯;磁链是 $N\Phi$
- 法拉第定律:感生电动势是磁链的变化率,所以恒定的磁通什么也感生不出
- 楞次定律:电动势阻碍产生它的变化,这由能量守恒得出
- 电动势是磁链图斜率的负值,而导体扫过磁场给出 $\varepsilon = BLv$