Electromagnetic Induction
AP Physics C: Electricity and Magnetism Topic 13 9:11 English narration · English + 中文 subtitles burned in
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Almost every watt of electricity you have ever used began the same way, with a magnet and a coil of wire moving past each other.
你用过的几乎每一度电，都始于同一件事：一块磁体和一圈导线相对运动。
No fuel, no burning: only motion, and a changing magnetic field.
没有燃料，也没有燃烧，只有运动，以及不断变化的磁场。
Turn the coil, and a voltage appears. Stop turning, and it vanishes.
转动线圈，电压就出现；停止转动，它就消失。
So why should a moving magnet push electric charge?
那么，移动的磁体为什么能推动电荷呢？
This is Unit Thirteen: electromagnetic induction.
这是第十三单元：电磁感应。
One law does almost all of the work.
这里几乎所有的内容都出自一条定律。
From it come generators, contactless brakes, inductors, and circuits that ring.
从这一个想法出发，就有了发电机、无接触的制动、电感器，以及会振荡的电路。
Let's begin.
让我们开始吧。
Magnetic flux measures how much magnetic field passes through a surface.
磁通量衡量有多少磁场穿过一个面。
For a uniform field through a flat loop it is the field, times the area, times the cosine of the angle between them.
对于穿过平面线圈的均匀磁场，它等于磁场乘以面积，再乘以两者夹角的余弦。
The area vector does the real work: it points straight out of the surface, and the angle is measured from that vector, not from the surface.
真正起作用的是面积矢量：它垂直指向面的外侧， 而夹角是相对这个矢量来量的，不是相对这个面。
If the field is not uniform, cut the surface into tiny pieces and add up the field through each. That sum is the surface integral.
如果磁场不均匀，就把面切成许多小块，把穿过每一小块的磁场加起来， 这个总和就是面积分。
Watch what a change of flux does.
现在看磁通量的变化会带来什么。
Push the magnet into the coil and the meter jumps: a current flows, with no battery anywhere.
把磁体推入线圈，电表指针立刻偏转： 电路里没有电池，却有电流。
Hold the magnet still inside and the current dies, even though the field through the coil is strong.
把磁体停在线圈里不动，电流就消失了， 尽管此时穿过线圈的磁场很强。
Pull it out and the meter swings the other way.
把磁体抽出来，指针又向另一边偏转。
It is never the flux that matters. It is the change.
真正起作用的从来不是磁通量本身，而是它的变化。
That is Faraday's law. The induced emf — the electromotive force — equals minus the rate of change of the magnetic flux.
这就是法拉第定律：感应电动势等于磁通量变化率的相反数。
Look at what can change.
看看哪些量可以变化。
The field can grow or shrink. So can the area inside the loop. Or the loop can turn, so the angle changes.
磁场可以变强或变弱；线圈围出的面积可以变大或变小； 线圈也可以转动，使夹角改变。
Three routes, one law.
三条途径，一条定律。
Decide which one is changing, then differentiate.
先判断是哪一个在变，然后求导。
And in a coil of many turns, every turn sees the same change, so multiply by the number of turns.
对于多匝线圈，每一匝都经历同样的变化，所以要乘以匝数。
Now the minus sign.
现在来看那个负号。
It is Lenz's law, not decoration.
它不是装饰，它就是楞次定律。
The induced current always flows so that it fights the change that made it — the change in flux, not the flux itself.
感应电流的方向，总是反抗产生它的那个变化——反抗的是磁通量的变化，不是磁通量本身。
Push a north pole at a loop, and the loop pushes back; pull it away, and it pulls back in.
把磁体的北极推向线圈，线圈就把它推回去；把磁体拉走，线圈就把它拉住。
For a direction, point your right thumb along the magnetic field the loop must create. Your fingers then curl the way the current flows.
要判断方向，就让右手拇指指向线圈必须产生的磁场，四指弯曲的方向就是电流方向。
And why negative?
为什么必须是负号？
A current that helped the change would grow forever, making energy from nothing.
如果感应电流助长这个变化，它就会不断增大，凭空造出能量。
That is conservation of energy in disguise.
楞次定律其实就是能量守恒。
This is the case worth memorising: motional emf, a rod sliding across a magnetic field.
这是值得背下来的一个特例：一根导体棒在磁场中滑动。
The circuit's area grows, so the flux grows, so there is an emf — the field, times the rod's length, times the speed.
电路的面积不断变大，磁通量随之变大，于是产生电动势—— 磁场乘以棒长，再乘以速度。
Take a rod twenty centimetres long, sliding at three metres per second across a field of zero point five tesla.
取一根二十厘米长的棒，以每秒三米的速度在零点五特斯拉的磁场中滑动。
The induced voltage is zero point five, times zero point two, times three: zero point three volts.
感应电动势就是零点五乘零点二再乘三，等于零点三伏。
Now the other way round. If the flux through one loop falls from twenty milliwebers to eight, in thirty milliseconds, the average voltage is zero point four volts.
反过来看也一样：如果穿过单匝线圈的磁通量在三十毫秒内从二十毫韦伯降到八毫韦伯， 平均电动势就是零点四伏。
An induced current is a real current in a real field, so the field pushes on it.
感应电流是真实的电流，而且正处在真实的磁场里，所以磁场会对它施力。
Only the parts of the loop inside the field feel that push, which can drag the loop along, or spin it round.
只有位于磁场中的那几段导线才受到这个力，它可以把线圈拖着走，也可以让它转起来。
By Lenz's law, the push always opposes the motion that causes it — and the right-hand rule gives you the direction of the induced current before you argue about the sign.
根据楞次定律，这个力总是阻碍引起感应的运动； 而用右手定则，你可以先定出感应电流的方向，再去讨论正负号。
Slide a loop into a field, and it is held back; slide it out, and it is dragged in.
把线圈推进磁场，它被往回拽；把线圈拉出磁场，它被往里拖。
In a solid sheet of metal, the same thing happens in swirling loops called eddy currents, a brake with no contact and nothing to wear out.
在整块金属板中，同样的事发生在打旋的电流回路里，这叫涡电流—— 一种无接触、也不会磨损的制动。
Once you have the force, it is ordinary mechanics: Newton's second law.
求出这个力之后，剩下的就是普通力学：牛顿第二定律。
Let the rod go, and that same drag slows it down.
松开这根棒，同样的阻力会让它慢下来。
Now the full chain.
现在来走一遍完整链条。
A rod slides on two rails, twenty centimetres apart, joined by a zero point six ohm resistor. The field is zero point five tesla, and the rod is pushed at a steady three metres per second.
一根棒在相距二十厘米的两条导轨上滑动， 导轨由零点六欧的电阻连接，磁场为零点五特斯拉， 棒被以每秒三米的恒定速度推动。
Pause here and try it.
先暂停，自己试一试。
First the voltage: field times length times speed, zero point three volts.
首先是电动势：磁场乘棒长再乘速度，等于零点三伏。
Then the current: voltage over resistance, zero point five amps.
然后是电流：电动势除以电阻，等于零点五安。
Then the force on the rod: field times current times length, zero point zero five newtons, pointing backwards.
再是棒受到的力：磁场乘电流再乘棒长，等于零点零五牛，方向向后。
So the person pushing supplies zero point one five watts, and the resistor turns all of it into heat.
所以推棒的人提供零点一五瓦，而电阻恰好把这些变成热。
Mechanical work in, electrical energy out: a generator.
机械功进去，电能出来——这就是发电机。
In a loop that is standing perfectly still, what is actually pushing the charges around?
在一个完全静止不动的线圈里，到底是什么在推着电荷绕圈跑呢？
A changing magnetic flux creates a real electric field, one that loops back on itself instead of ending on charges.
变化的磁通量会产生真实的电场，这个电场自己首尾相接， 而不是从电荷出发、到电荷终止。
That is the third of Maxwell's equations.
这就是麦克斯韦方程组的第三个方程。
Put all four together, and they predict electromagnetic waves — fields that keep renewing each other, travelling through empty space at one fixed speed: the speed of light.
把四个方程放在一起，它们预言了电磁波——电场和磁场不断互相激发， 并以一个固定速度在真空中传播，那就是光速。
You are not asked to derive it.
考试不会要求你推导它。
A circuit can also induce a voltage in itself.
电路也能对自己产生感应电动势。
Change the current in a coil, and its own flux changes, so it induces a voltage that opposes the change.
改变线圈中的电流，它自己的磁场就变化， 磁通量也变化，于是感应出一个反抗这一变化的电动势。
This is inductance: resisting a change in its own current.
这就是电感：导体反抗自身电流变化的性质。
That voltage is minus the inductance, times the rate of change of the current.
自感电动势等于负的电感乘以电流的变化率。
An inductor is usually a solenoid, and its inductance comes purely from its geometry: turns squared, times area, divided by length, times the permeability of the core.
为此制作的元件叫电感器，通常是螺线管； 它的电感完全由几何形状决定：匝数的平方乘以横截面积，除以长度， 再乘以铁芯的磁导率。
A straight wire counts as none.
直导线按零电感处理。
An inductor carrying current stores energy in its magnetic field, one half the inductance, times the current squared.
通有电流的电感器把能量储存在磁场中：二分之一乘电感再乘电流的平方。
Close the switch on a resistor and an inductor in series with a battery.
把电阻和电感器与电池串联，然后合上开关。
Kirchhoff's loop rule gives the battery voltage as the resistor voltage, plus the inductor voltage, which holds the rate of change of the current.
基尔霍夫回路定则给出：电池电压等于电阻上的电压加上电感上的电压， 而电感那一项里含有电流的变化率。
That is a differential equation, and its solution is a rising exponential. The current climbs toward the battery voltage over the resistance, at a pace set by the time constant, inductance over resistance.
这个微分方程的解是一条上升的指数曲线： 电流向着「电池电压除以电阻」攀升，快慢由时间常数决定，也就是电感除以电阻。
After one time constant, it is about sixty-three percent of the way. A falling current drops to thirty-seven percent of where it started.
经过一个时间常数，电流大约走完全程的百分之六十三； 正在衰减的电流则降到原来的百分之三十七。
The instant the switch closes, the current cannot jump, so the inductor blocks it. Long afterwards, nothing changes, and the inductor is just a wire, the opposite of a capacitor.
开关刚合上的瞬间，电流不能突变，电感把它挡住； 很久以后一切都不再变化，电感就只是一根导线——和电容器恰好相反。
Try the numbers.
来算一算。
A twelve volt battery, a six ohm resistor and a three henry inductor, in series.
一个十二伏的电池、一个六欧的电阻和一个三亨的电感器串联。
The final current is twelve divided by six: two amps.
最终电流是十二除以六，等于二安。
The time constant is three divided by six: zero point five seconds.
时间常数是三除以六，等于零点五秒。
So after half a second, the current has climbed sixty-three percent of the way to two amps, about one point three amps.
所以经过半秒，电流上升到终值的百分之六十三左右，约为一点三安。
At the first instant, the inductor holds all twelve volts, and the current is zero; long afterwards, it holds nothing.
最初的瞬间，电感承担全部十二伏，电流为零；很久以后它上面的电压为零。
Now swap the resistor for a capacitor.
现在把电阻换成电容器。
Nothing dissipates energy, so nothing ever settles: the circuit oscillates, and it rings on forever.
不再有任何东西消耗能量，所以电路永不停歇：它开始振荡。
Energy pours out of the capacitor's electric field, into the inductor's magnetic field, and back again.
能量从电容器的电场倾泻进电感器的磁场，然后又流回来。
Watch the two bars trade while the total stays the same.
看这两根能量条来回交换，而总量始终不变。
Write the loop rule, and you get an equation you already know. The second derivative of the charge equals minus the charge, over the inductance times the capacitance. That is simple harmonic motion, with charge playing the part of displacement.
写出这个电路的回路方程，你会得到一个已经熟悉的方程： 电荷的二阶导数等于负的电荷除以「电感乘电容」。
The angular frequency is one, over the square root of the inductance times the capacitance. Try it with a two henry inductor, and an eight microfarad capacitor.
这就是简谐运动，其中电荷扮演位移的角色， 所以角频率等于一除以「电感乘电容」的平方根。
The angular frequency is two hundred and fifty radians per second, and the period is about twenty-five milliseconds.
试一试：两亨的电感器和八微法的电容器，角频率是二百五十弧度每秒， 周期约二十五毫秒。
And if the capacitor began at twelve volts, energy conservation alone gives the largest current. It is about twenty-four milliamps.
如果电容器起初被充到十二伏， 仅凭能量守恒就能求出最大电流，约二十四毫安。
Four habits that save marks.
四个能保住分数的习惯。
First, always start from the flux: write it down, then differentiate.
第一，永远从磁通量出发：先写出它是什么，再求导。
Second, say which of the three things is changing — the field, the area, or the angle.
第二，说清楚三样东西中是哪一样在变化——磁场、面积，还是夹角。
Third, walk the chain in order. Flux, then voltage, then current, then force, then Newton's second law. Write each link separately.
第三，按顺序走完整条链条：磁通量、电动势、电流、力，最后是牛顿第二定律， 每一环都单独写出来。
Fourth, know the two limits of an inductor: at the first instant it blocks the current, long afterwards it is just a wire.
第四，记住电感器的两个极限：最初的瞬间它挡住电流，很久以后它只是一根导线。
Get those, and this unit is yours.
做到这些，这个单元就是你的了。