Magnetic Fields and Electromagnetism
AP Physics C: Electricity and Magnetism Topic 12 12:45 English narration · English + 中文 subtitles burned in
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Look north on a clear winter night, far enough north, and the sky glows green.
在晴朗的冬夜向北看，只要纬度够高，你会看到天空泛起绿光。
Those lights are switched on by a magnetic field.
点亮这些光的，是磁场。
The Sun throws charged particles at the Earth, day after day. Earth's own field catches them, bends them, and steers them down toward the poles, where they slam into the air and make it shine.
太阳日复一日地把带电粒子抛向地球， 地球自己的磁场抓住它们、让它们偏转，并把它们导向两极—— 在那里它们撞进大气，使天空发亮。
Notice what the field did. It never sped those particles up. It only turned them.
注意磁场做了什么： 它从未让这些粒子加速，它只是让它们转向。
Learn how a magnetic field turns a moving charge, and this whole unit opens up.
学会磁场如何让运动电荷转向，整个单元就会豁然开朗。
This is Unit Twelve: magnetic fields and electromagnetism.
这是第十二单元：磁场与电磁学。
Where magnetism comes from, the force it puts on a moving charge, and the two tools that give you the field of any current — the Biot-Savart law, and Ampère's law.
磁性从哪里来，磁场对运动电荷施加什么力， 以及求任意电流磁场的两件工具——毕奥-萨伐尔定律和安培定律。
Let's begin.
让我们开始吧。
A magnetic field is a vector field.
磁场是一个矢量场。
At every point it has a size and a direction, and it decides the force on any moving charge you put there.
它在每一点都有大小和方向， 并决定你放在那里的任何运动电荷所受的力。
We draw it with field lines.
我们用磁感线来画它。
Outside a bar magnet they leave the north pole and enter the south pole; inside the magnet the line keeps going, from south back to north, so every line closes on itself.
在条形磁铁外部，磁感线从北极出发、进入南极； 在磁铁内部，磁感线继续向前，从南极回到北极——所以每一条线都自我闭合。
Where the lines crowd together, the field is stronger.
磁感线越密集的地方，磁场越强。
A compass is a tiny magnet that is free to turn, so it swings until it lines up with the field around it.
指南针就是一块可以自由转动的小磁铁， 它会一直转，直到与周围的磁场方向一致。
That is why a compass finds north: the Earth itself behaves like a giant bar magnet.
这就是指南针能指北的原因： 地球本身就像一块巨大的条形磁铁。
Those closed loops are a law.
这些闭合的回路本身就是一条定律。
Gauss's law for magnetism — the second of Maxwell's four equations — says that the magnetic flux through any closed surface is exactly zero.
磁场的高斯定律—— 麦克斯韦四个方程中的第二个——指出：穿过任意闭合曲面的磁通量恰好为零。
As many field lines leave the surface as enter it.
有多少条磁感线穿出，就有多少条穿入。
Compare that with an electric field, where a single charge sitting inside the surface gives you a flux that is not zero.
把它和电场比较： 电场中只要有一个电荷位于闭合面内，通量就不为零。
The difference is that there is no magnetic charge.
区别在于，世界上没有磁荷。
No isolated north pole has ever been found.
从来没有人找到过孤立的磁北极。
Cut a bar magnet in half, and you do not get a north piece and a south piece; you get two smaller magnets, each with both poles.
把一块条形磁铁切成两半， 你不会得到一块北极和一块南极，而是得到两块更小的磁铁，各自都有两个极。
Every magnetic source is a dipole, and every dipole is really charge going round in a circle.
每一个磁源都是磁偶极子，而每一个磁偶极子，本质上都是做圆周运动的电荷。
So where does a permanent magnet get its dipoles?
那么，永久磁铁的磁偶极子从哪里来？
From electrons moving inside it.
来自其内部电子的运动。
How a material answers an outside field depends on what those tiny dipoles do.
一种材料如何回应外加磁场，取决于这些微小偶极子的行为。
In a ferromagnetic material — iron, nickel, cobalt — the field lines up whole domains at once, and the alignment stays after the field is taken away. That is a permanent magnet.
在铁磁性材料中——铁、镍、钴——外场会让整块整块的磁畴一起排列整齐， 而且撤去磁场后这种排列仍然保留，这就是永久磁铁。
In a paramagnetic material, like aluminium or titanium, the dipoles line up only weakly, and they relax as soon as the field is gone.
在顺磁性材料中，比如铝或钛，偶极子只会微弱地排列，磁场一撤就恢复混乱。
And every material is also diamagnetic: its electron structure sets up a weak alignment against the field.
而所有材料还都具有抗磁性：其电子结构会产生一种与外场方向相反的微弱排列。
We measure the response with the permeability.
我们用磁导率来衡量这种回应。
Empty space has one fixed value. Matter does not — its permeability changes with temperature, with direction, and with how strong the field is.
真空只有一个固定值； 物质则不然——它的磁导率会随温度、方向以及外场强弱而变化。
Now the idea this whole unit is built on: magnetism comes from moving charge.
现在来看这个单元赖以建立的核心思想：磁性来自运动的电荷。
A single charge, simply moving, makes a magnetic field around itself.
一个电荷只要在运动，就会在自己周围产生磁场。
At any point, that field is perpendicular to two things at once — the velocity of the charge, and the line from the charge to that point.
在任意一点上，这个磁场同时垂直于两样东西——电荷的速度， 以及从电荷指向该点的连线。
Right hand again: fingers along the velocity, curl them toward the point, and your thumb gives the field there.
还是用右手：四指沿速度方向，向该点弯曲， 拇指就给出那里的磁场方向。
The size follows the same geometry. It is strongest out to the side, where the velocity and that line are at right angles, and it falls to zero straight ahead of the charge, and straight behind it.
大小也遵循同样的几何关系： 在侧向最强，也就是速度与那条连线互相垂直的地方； 而在电荷的正前方和正后方，磁场为零。
A moving charge also feels a force when it sits inside someone else's field.
运动电荷处在别人的磁场里时，也会受到力。
The magnetic force is the charge, times the cross product of the velocity and the field.
磁力等于电荷量乘以速度与磁场的叉积。
Its size is the charge, times the speed, times the field, times the sine of the angle between them.
它的大小等于电荷量、速率、磁感应强度， 再乘以两者夹角的正弦。
So a charge moving straight along the field feels no force at all.
所以，沿着磁场方向运动的电荷根本不受力。
Point your right fingers along the velocity, curl them toward the field, and your thumb is the force on a positive charge; flip it for a negative one.
右手四指沿速度方向，向磁场方向弯曲，拇指就是正电荷所受力的方向； 负电荷则相反。
And here is the property the exam loves: the force is always at right angles to the velocity, so it never does any work.
下面是考试最爱考的性质：这个力始终与速度垂直， 因此它永远不做功。
It cannot change the speed. It can only change the direction.
它无法改变速率，只能改变方向。
Watch what that does.
看看这会带来什么。
The charge enters a uniform field, feels a sideways push, and turns — and the push turns with it, always aiming at the same centre.
电荷进入匀强磁场，受到一个侧向的推力而转弯—— 而这个推力也跟着一起转，始终指向同一个圆心。
That is circular motion, with the magnetic force playing the part of the centripetal force.
这就是圆周运动， 磁力扮演了向心力的角色。
Set the magnetic force equal to the mass times the speed squared over the radius, and the radius comes out as the momentum divided by the charge times the field.
令磁力等于质量乘以速率平方再除以半径， 就得到半径等于动量除以电荷量与磁感应强度之积。
The time for one lap is even nicer: it does not depend on the speed at all.
转一圈所需的时间更妙：它完全不依赖于速率。
A faster particle simply rides a bigger circle, and comes round in exactly the same time.
更快的粒子只是走更大的圆，绕一圈用的时间一模一样。
Let's put numbers on it.
我们来代入数字。
A proton enters a field of zero point five tesla at two times ten to the fifth metres per second, at right angles to the field.
一个质子以每秒二乘十的五次方米的速率， 垂直进入零点五特斯拉的磁场。
Find the force on it, and the radius of its path.
求它受到的力，以及运动轨迹的半径。
The force is the charge times the speed times the field: one point six times ten to the minus fourteen newtons.
力等于电荷量乘以速率再乘以磁感应强度：一点六乘十的负十四次方牛顿。
For the radius, set that force equal to the centripetal force. The radius is the mass times the speed, over the charge times the field — which comes to four point two millimetres.
求半径时，令这个力等于向心力，于是半径等于质量乘速率， 除以电荷量乘磁感应强度——结果是四点二毫米。
Small, but easy to measure.
很小，但很容易测量。
This is exactly how a mass spectrometer weighs an ion.
质谱仪正是用这种方法来称量离子的。
Put an electric field and a magnetic field in the same region, and the two forces act completely independently — you simply add them as vectors.
把电场和磁场放在同一区域，两个力互不干扰、各自独立作用—— 你只需把它们按矢量相加。
Cross them over, and you have built a velocity selector.
让它们互相正交，就造出了一台速度选择器。
The electric force pushes one way, the magnetic force pushes the other, and only particles at one exact speed feel no net force and fly straight through.
电场力往一边推，磁场力往另一边推，只有速率恰好等于某个值的粒子受合力为零， 能够笔直穿过。
That speed is the electric field divided by the magnetic field.
这个速率就是电场强度除以磁感应强度。
The same physics runs inside a solid conductor: a field across a current pushes the moving carriers to one face, charge piles up there, and a voltage appears across the conductor.
同样的物理也发生在固体导体内部：与电流方向交叉的磁场把运动的载流子推向一侧， 电荷在那一面堆积，导体两侧就出现电压。
That is the Hall effect, and the sign of that voltage tells you whether the carriers are positive or negative.
这就是霍尔效应； 而这个电压的正负，能告诉你载流子带正电还是负电。
A current is nothing but moving charges, so a current makes a magnetic field too.
电流不过就是运动的电荷，所以电流同样会产生磁场。
Look carefully at the shape. The field does not point away from the wire, and it does not point along it.
仔细看这个形状： 磁场既不指向远离导线的方向，也不沿着导线方向。
It wraps around the wire, in concentric circles centred on it, and at any point it is tangent to the circle through that point.
它环绕着导线，形成以导线为中心的一圈圈同心圆。
Grip the wire with your right hand, thumb along the current, and your fingers curl the way the field goes.
用右手握住导线，拇指指向电流方向，四指弯曲的方向就是磁场的方向。
To get the size, we add up the field from every short piece of the wire. That is the Biot-Savart law.
要求出大小，就要把导线上每一小段产生的磁场加起来，这就是毕奥-萨伐尔定律。
Each little element contributes a field along the cross product of that element with the direction to the point, and it falls off as one over the distance squared. Adding those contributions is superposition, exactly as with electric fields.
每一小段贡献的磁场，方向沿着这一小段与指向该点方向的叉积， 大小按距离的平方反比衰减。
The exam only asks you to do this integral in kind cases, and the kindest is the centre of a circular loop.
考试只会让你在很友好的情形下做这个积分， 其中最友好的就是圆形线圈的圆心。
Every element sits the same distance from the centre, and every element is perpendicular to the line to the centre, so every sine is one.
每一小段到圆心的距离都相同， 而且每一小段都与指向圆心的连线垂直，所以每个正弦值都是一。
The integral collapses to the length of the wire, and the field is the permeability of free space, times the current, over twice the radius.
积分于是收缩成导线的长度，磁场等于真空磁导率乘以电流，再除以两倍半径。
Say those two symmetry reasons out loud in the exam — that is where the marks are.
在考试里把这两条对称性理由写出来——分数就在那里。
And an arc that is a quarter of a circle gives you a quarter of the answer.
而一段四分之一圆的圆弧，就给出四分之一的结果。
Just as a field pushes on a moving charge, a field pushes back on a current.
正如磁场会推动运动电荷，磁场也会反过来推动电流。
Every element of wire feels the current, times that element, crossed with the field; and for a straight wire in a uniform field it is simply current, times length, crossed with field.
导线上的每一小段所受的力，等于电流乘以这一小段与磁场的叉积； 对于匀强磁场中的直导线，就简化为电流乘以长度再与磁场作叉积。
Now put two long wires side by side, ten centimetres apart, each carrying five amps in the same direction.
现在把两根长直导线并排放置，相距十厘米，各自沿同一方向通有五安培的电流。
Wire one makes a field at wire two of one times ten to the minus five tesla, pointing into the page.
导线一在导线二处产生的磁场是一乘十的负五次方特斯拉，方向指向纸内。
So every metre of wire two feels five times ten to the minus five newtons, and it is pulled toward wire one.
于是导线二每米受到五乘十的负五次方牛顿的力，被吸向导线一。
Currents in the same direction attract; currents in opposite directions repel.
同向电流相互吸引，反向电流相互排斥。
When the geometry is symmetric, there is a much faster tool.
当几何形状具有对称性时，还有一件快得多的工具。
Ampère's law says that if you walk once around a closed loop, adding up the part of the field that lies along your path, the total equals the permeability of free space, times the current your loop encloses.
安培定律说： 如果你沿一条闭合路径绕行一周，把沿路径方向的磁场分量累加起来， 总和就等于真空磁导率乘以这条回路所包围的电流。
Choose that Amperian loop to match the symmetry, so the field has one constant size everywhere on it and points along it — then the field comes straight out of the integral.
选取与对称性相匹配的回路，使磁场在回路上处处大小相同、方向沿着回路—— 这样磁场就能直接从积分号里提出来。
Take a long straight wire carrying five amps, and ask for the field ten centimetres away.
取一根通有五安培电流的长直导线， 求距它十厘米处的磁场。
The field lines are circles, so use a circle.
磁感线是圆，所以就取一个圆。
The field times the circumference equals the permeability times the current, and the answer is one times ten to the minus five tesla — the same number as before, in a single line of work.
磁场乘以周长等于磁导率乘以电流，答案是一乘十的负五次方特斯拉—— 和刚才同样的数字，却只用了一行运算。
It also handles a slab or a cylinder carrying a current density — current spread over an area rather than down a thin wire.
它同样能处理载有电流密度的平板或圆柱——电流分布在一个面上， 而不是集中在一根细导线里。
The other classic is the solenoid: a long coil of wire.
另一个经典例子是螺线管：一根绕成长长线圈的导线。
We assume the field inside is uniform and runs along the axis, and that the field outside is small enough to ignore.
我们假设内部磁场是匀强的、沿着轴线方向，而外部磁场小到可以忽略。
Now draw a rectangular loop, with one long side inside the coil and the opposite side outside it.
现在画一个矩形回路，一条长边在线圈内部，对面那条长边在外部。
The outside side sits where the field is about zero, so it contributes nothing. The two short sides cross the field at right angles, so they contribute nothing either.
外部那条边所在处磁场约为零，所以没有贡献； 两条短边与磁场垂直，所以也没有贡献。
Only the inside side is left.
只剩下内部那条边。
It gives the field times its length, and the current the loop encloses is the turns per metre, times that same length, times the current.
它给出磁场乘以边长；而回路包围的电流等于每米匝数，乘以同样的边长，再乘以电流。
The lengths cancel, and the field inside is the permeability of free space, times the turns per metre, times the current.
长度相互抵消，于是内部磁场等于真空磁导率乘以每米匝数再乘以电流。
Notice what is missing from that answer: the radius.
注意答案里少了什么：半径。
Anywhere inside a long solenoid, the field is the same.
在长螺线管内部，任何位置的磁场都一样。
One more, because examiners love it.
再来一个，因为出题人很爱考它。
A solid cylinder of radius R carries a current spread evenly over its whole cross-section.
一根半径为 R 的实心圆柱导体， 电流均匀地分布在整个横截面上。
Find the field inside it, and outside it.
求圆柱内部和外部的磁场。
Pause here and try it.
先暂停，自己试一试。
Outside, your loop encloses the whole current, so you get the straight-wire answer, falling off as one over the distance.
在外部，你的回路包围了全部电流， 所以得到的就是长直导线的结果，按距离的一次方反比衰减。
Inside, the loop encloses only the fraction of the current that fits inside its area — the radius squared, over the big radius squared.
在内部，回路只包围了落在它面积之内的那一部分电流—— 即半径的平方比上圆柱半径的平方。
Put that in, and the field grows in a straight line, from zero on the axis up to a maximum at the surface.
代入之后，磁场沿直线增长， 从轴线处的零一直升到表面处的最大值。
Sketch that graph: straight up, then one over the distance on the way down.
把这张图画出来： 先是一条上升的直线，然后按距离的反比下降。
Step back and see where you are.
退一步，看看你已经走到哪里。
Two of Maxwell's four equations now belong to you.
麦克斯韦四个方程中，已经有两个属于你了。
The second one says that magnetic field lines close on themselves — there are no monopoles.
第二个方程说：磁感线自我闭合——不存在磁单极子。
The fourth is Ampère's law, with one extra idea from Maxwell: a changing electric field also creates a magnetic field, just as a current does.
第四个是安培定律，外加麦克斯韦补充的一个想法： 变化的电场同样会产生磁场，就像电流那样。
The exam will not ask you to compute with that extra term. It may very well ask you to say it in words.
考试不会要求你用这一项做计算，但很可能要求你用文字把它说清楚。
Four habits that save marks.
四个能保住分数的习惯。
First, the magnetic force does no work, because it is always perpendicular to the velocity — so it can never appear in an energy equation.
第一，磁力不做功，因为它始终与速度垂直—— 所以它绝不能出现在能量方程里。
Second, when you use Ampère's law, name your loop and say why the field is constant along it; the setup is where the marks live.
第二，用安培定律时， 要写出你选的回路，并说明为什么磁场沿回路处处相同；分数就在这一步。
Third, keep the two roles apart: the force a field puts on a current, and the field that current creates.
第三，把两件事分清楚：磁场对电流施加的力，和电流本身产生的磁场。
And fourth, check every cross product with your right hand, and check the sign of the charge before you write a direction down.
第四，每一个叉积都用右手检查一遍，写方向之前先确认电荷的正负。
Get those four, and Unit Twelve is yours.
做到这四点，第十二单元就是你的了。