Magnetic Fields
A-Level Physics Topic 20 13:21 English narration · English + 中文 subtitles burned in
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High above the poles, the sky comes alive — sheets of green light, rippling in the dark.
在两极的高空，天空变得鲜活起来——一片片绿色的光，在黑暗中荡漾。
This is the aurora.
这就是极光。
Its cause?
它的成因是什么？
Our planet is a giant magnet.
我们的星球是一块巨大的磁铁。
Streams of charged particles race toward us from the Sun, and Earth's magnetic field grabs them, bending their paths, funnelling them down toward the poles.
一束束带电粒子从太阳向我们飞奔而来， 地球的磁场抓住它们，弯折它们的路径，把它们汇聚到两极。
There, they crash into the air and make it glow.
在那里，它们撞进空气，使空气发光。
A magnetic field, steering charges through space — that is what this whole topic is about.
磁场，在空间中操纵着电荷的运动——这正是这一整章要讲的东西。
Magnetic fields push on moving charges and on currents — and moving magnets push back, making electricity.
磁场对运动的电荷和电流施力——而运动的磁铁也会反过来施力，产生电。
Today: field lines, the force on a current and on a charge, and electromagnetic induction.
今天：场线、 电流和电荷所受的力，以及电磁感应。
Let's begin.
让我们开始吧。
First, the shape of a magnetic field.
首先，是磁场的形状。
We draw it with field lines.
我们用场线来画它。
Outside a bar magnet — permanent magnets get their field from tiny atomic currents — the lines curve from the north pole around to the south, crowding together where the field is strong.
在条形磁铁的外面，场线从北极弯曲绕到南极， 在磁场强的地方挤在一起。
A straight wire carrying a current wraps the field in circles around itself — point your right thumb along the current, and your fingers curl the way the field points.
一根通电的直导线，把磁场以圆圈的形式缠绕在自己周围—— 让右手拇指指向电流方向，手指弯曲的方向就是磁场的方向。
And a coil of wire, a solenoid, makes a field almost uniform inside, just like a bar magnet.
而一圈圈的导线，也就是螺线管， 在内部产生几乎匀强的磁场，就像一根条形磁铁。
Four patterns you must be able to draw.
有四种图样是你必须会画的。
A bar magnet: curved lines from N to S outside, closest together near the poles.
条形磁铁：外部是从 N 到 S 的弯曲线，越靠近磁极越密集。
A long straight wire: concentric circles around it, and the direction comes from the right-hand grip rule — thumb along the current, fingers curl the way the field points.
长直导线：绕着它的一圈圈同心圆，方向由右手螺旋定则给出—— 拇指指向电流，四指弯曲的方向就是磁场的方向。
A flat circular coil: the field through the centre is at right angles to the coil, and the coil behaves like a small bar magnet.
平面圆形线圈： 穿过圆心的磁场与线圈平面垂直，而这个线圈的行为就像一块小条形磁铁。
And a long solenoid: nearly uniform along the axis inside, like a stretched bar magnet, falling off fast outside.
还有长螺线管：内部沿轴线几乎是匀强的，像一块被拉长的条形磁铁，外部则迅速衰减。
One more fact worth a mark: an iron core inside a solenoid greatly increases the field, because the iron's own atomic magnets line up and add to it — which is why electromagnets and transformers have iron cores.
还有一个值一分的事实：在螺线管里放一个铁芯会大大增强磁场， 因为铁自身的原子磁体会排列整齐并叠加上去—— 这正是电磁铁和变压器都带铁芯的原因。
A wire of length nought point two metres carries a current of three amps at right angles to a field of flux density nought point five tesla.
一根长零点二米的导线通有三安培的电流，与磁通密度为零点五特斯拉的磁场垂直。
Find the force.
求它受到的力。
Because it is at right angles, sine theta is one, so F equals B I L.
因为是垂直的，sin theta 等于一，所以 F 等于 B I L。
Nought point five times three times nought point two is nought point three newtons.
零点五乘以三再乘以零点二，等于零点三牛顿。
Now the general case, which is where marks are lost: the full formula carries the sine theta, with theta the angle between the WIRE and the FIELD.
现在看一般情形，丢分往往就在这里： 完整的公式带有 sin theta，其中 theta 是导线与磁场之间的夹角。
So the force is largest at right angles, and exactly zero when the wire lies ALONG the field — a current parallel to the field feels nothing at all.
所以垂直时力最大，而当导线沿着磁场方向时力恰好为零—— 与磁场平行的电流完全不受力。
That same equation does something more: it DEFINES the magnetic flux density.
同一个方程还做了另一件事：它定义了磁通密度。
Rearranged, B equals F over I L for a wire at right angles — so B is the force per unit current per unit length on a wire perpendicular to the field.
移项之后， 对垂直放置的导线，B 等于 F 除以 I L——所以 B 是垂直于磁场的导线 每单位电流、每单位长度所受的力。
That is the definition an exam wants, not "how strong the magnet is".
这才是考试想要的定义，而不是"磁铁有多强"。
The unit is the tesla, and it follows straight from that: newtons per amp per metre.
单位是特斯拉，它也直接由此而来：牛顿每安培每米。
For direction, use your LEFT hand.
判断方向要用左手。
First finger, Field.
食指，磁场。
Second finger, Current.
中指，电流。
Thumb, the force or thrust.
拇指，力，也就是推力。
Hold all three at right angles to each other.
三者互相垂直。
The commonest error is reaching for the right hand — that one is for induction, not for force.
最常见的错误是伸出右手——右手是用在电磁感应上的，不是用在受力上的。
Put a current-carrying wire in a magnetic field, and it feels a force.
把一根通电的导线放进磁场里，它就会受到一个力。
The force is the flux density, times the current, times the length in the field — biggest when the wire sits at right angles to the field, and zero when it lies along it.
这个力等于磁通密度，乘以电流， 再乘以在磁场中的长度——当导线与磁场垂直时最大，当它沿着磁场时为零。
That flux density, measured in tesla, is really the strength of the field.
那个磁通密度，以特斯拉为单位，其实就是磁场的强度。
For the direction, use Fleming's left hand: first finger for field, second finger for current, and the thumb points the way of the force.
至于方向，用弗莱明左手定则： 食指代表磁场，中指代表电流，拇指指向力的方向。
A single moving charge feels the same kind of force — the field, times the charge, times its speed. Same left hand, but the second finger is the motion of a positive charge; reverse it for a negative one.
单个运动的电荷会感受到同样的力——磁场，乘以电荷，再乘以它的速度。
Now, back to the aurora.
现在，回到极光。
In a uniform field, this force always pulls at right angles to the motion, so it never speeds the charge up or slows it down — it only turns it. The charge loops around in a circle.
在匀强磁场里，这个力总是垂直于运动方向来拉它，所以它从不让电荷加速或减速—— 它只让电荷转向。
Balance the magnetic force against the centripetal force, and the radius is the momentum divided by the field and the charge.
电荷就绕成一个圆。 让磁力与向心力相平衡， 半径就等于动量除以磁场和电荷。
Faster particles simply carve wider circles.
更快的粒子，只不过绕出更大的圆。
A charge moving at right angles to a uniform field feels a force at right angles to both its velocity and the field.
一个垂直于匀强磁场运动的电荷，受到的力同时垂直于它的速度和磁场。
Follow the logic, because the whole result comes from it.
跟着这条逻辑走，因为整个结论都是从它来的。
A force always perpendicular to the motion does NO work.
始终垂直于运动方向的力不做功。
No work means the kinetic energy is constant, so the speed never changes — only the direction does.
不做功意味着动能不变，所以速率永远不变——变的只是方向。
A constant-magnitude force always turning the velocity is exactly circular motion.
一个大小恒定、始终使速度转向的力，正好就是圆周运动。
So set the magnetic force equal to the centripetal force: B Q v equals m v squared over r, and one v cancels to give r equals m v over B Q.
于是令磁场力等于向心力：B Q v 等于 m v 平方除以 r，约掉一个 v， 就得到 r 等于 m v 除以 B Q。
A proton moves at two times ten to the sixth metres per second at right angles to a nought point five tesla field.
一个质子以每秒二乘以十的六次方米的速度垂直射入零点五特斯拉的磁场。
Find the radius of its circular path.
求它圆周路径的半径。
Use r equals m v over B Q.
用 r 等于 m v 除以 B Q。
Substituting the proton mass and charge gives about nought point nought four three metres.
代入质子的质量和电荷，得到大约零点零四三米。
Two things to read off that formula.
从这个公式可以读出两件事。
First, the radius depends on the momentum m v — which is exactly why a bubble chamber measures momentum from the curvature of a track.
第一，半径取决于动量 m v—— 这正是气泡室能从径迹的弯曲程度测出动量的原因。
Second, the period is two pi m over B Q, and the period does NOT depend on the speed at all.
第二，周期是二 pi m 除以 B Q， 而周期完全不依赖于速率。
A faster particle travels a bigger circle but takes the same time per turn.
跑得更快的粒子画出更大的圆，但转一圈用的时间是一样的。
And if the velocity also has a component along the field, that component is untouched, so the path becomes a helix.
而如果速度还有一个沿磁场方向的分量，那个分量不受影响，于是路径就变成一条螺旋线。
Now the Hall effect. Take a slab of conductor carrying a current, and put it in a field at right angles to that current.
取一块通有电流的导体薄片，把它放进与电流垂直的磁场中。
The moving charges feel a magnetic force B Q v-drift, so they pile up on one face.
运动的电荷受到磁场力 B Q v 漂移，于是在一个面上堆积起来。
That build-up creates an electric field across the slab which pushes back, and it grows until the two balance. At that steady state, e E equals B e v-drift, so E equals B v-drift.
这种堆积在薄片两面之间产生一个电场，把后来的电荷往回推， 电场不断增强，直到两者平衡。
Combine that with V-Hall equals E times the width, and with the current expression, and you get V-Hall equals B I over n t q.
在这个稳态下，e E 等于 B e v 漂移， 所以 E 等于 B 乘以 v 漂移。
A Hall probe uses exactly this to MEASURE a field: pass a known current through a thin semiconductor slab and read the Hall voltage — largest when the slab is at right angles to the field, which is how you find the field's direction as well as its size.
再结合霍尔电压等于 E 乘以宽度，以及电流的表达式， 就得到霍尔电压等于 B I 除以 n t q。 霍尔探头正是用这个来测量磁场： 让已知的电流通过一片薄薄的半导体，读出霍尔电压—— 当薄片垂直于磁场时读数最大，所以你既能测出磁场的大小，也能找到它的方向。
A velocity selector uses crossed electric and magnetic fields to let through just one speed.
速度选择器用正交的电场和磁场，只让某一个速率的粒子通过。
Set them so the electric force q E and the magnetic force q v B oppose each other.
把它们布置成电场力 q E 与磁场力 q v B 互相抵抗。
The electric force does not depend on speed; the magnetic one does.
电场力与速率无关，而磁场力与速率有关。
So there is exactly one speed at which they cancel: q E equals q v B, giving v equals E over B.
所以恰好只有一个速率能让两者相消： q E 等于 q v B，得到 v 等于 E 除以 B。
Particles at that speed feel no net force and go straight through the slit.
以这个速率运动的粒子不受净力， 径直穿过狭缝。
Anything faster has too much magnetic force and is deflected one way; anything slower is deflected the other.
比它快的，磁场力过大，被偏向一边；比它慢的，被偏向另一边。
Notice the charge cancels entirely, so it selects by speed regardless of the particle.
注意电荷完全约掉了，所以它是按速率来筛选的，与粒子是什么无关。
Two long parallel wires: each wire sits in the other's magnetic field, so each feels a force.
两根长的平行导线：每一根都处在另一根产生的磁场里，所以每一根都受到力。
Work it out with the right-hand grip rule for the field and Fleming's left hand for the force, and the answer is the opposite of what most people guess.
用右手螺旋定则判断磁场、用左手定则判断受力，把它推出来， 答案和大多数人猜的正好相反。
Parallel currents — flowing the same way — attract.
同向的平行电流——朝同一个方向流的——互相吸引。
Antiparallel currents, flowing opposite ways, repel.
反向的平行电流，朝相反方向流的，互相排斥。
It is the reverse of magnets and of charges, where like repels like.
这与磁体和电荷正好相反， 那里同类是互相排斥的。
And this is not a curiosity: this force is the basis of the SI definition of the ampere.
而这并不是个趣闻：这个力正是国际单位制中安培定义的基础。
Now flip it around.
现在反过来看。
Magnetic flux measures how much field passes through a loop — the flux density times the area.
磁通量量度有多少磁场穿过一个回路——磁通密度乘以面积。
Here is the magic: when that flux through a coil changes, a voltage appears from nowhere.
神奇的地方在这里： 当穿过线圈的磁通量发生变化时，一个电压就凭空出现了。
Push a magnet into a coil, and the needle jumps.
把磁铁推进线圈，指针就跳动。
Pull it out, and the needle swings the other way.
把它抽出来，指针就朝另一个方向摆。
Move it faster, and the kick is bigger.
动得更快，这一下就更大。
A changing magnetic field makes electricity — this is electromagnetic induction.
变化的磁场产生电—— 这就是电磁感应。
Two laws pin this down.
有两条定律把这件事定死。
Faraday's law says the induced electromotive force — the e.m.f. — equals how fast the flux linkage changes — more turns, a stronger field, a larger area, or a faster change, all give more.
法拉第定律说，感应电压等于磁链变化的快慢——更多的匝数、 更强的磁场、更大的面积，或更快的变化，都会让它更大。
Lenz's law fixes the direction: the induced current always opposes the very change that made it.
楞次定律确定方向： 感应电流总是反抗产生它的那个变化。
Push the magnet in, and the coil pushes back.
把磁铁推进去，线圈就往回推。
That is simply energy conservation — you must do work to generate the electricity.
这不过就是能量守恒—— 你必须做功，才能产生电。
Before Faraday, get the two quantities straight.
在讲法拉第之前，先把两个量分清楚。
The magnetic flux through a flat area at right angles to the field is simply Phi equals B A.
垂直于磁场的一块平面所穿过的磁通量， 就是 Phi 等于 B A。
If the normal to the area is at an angle theta to the field, use Phi equals B A cos theta — note it is COS here, unlike the sine in the force equation, because the angle is measured to the normal rather than to the surface.
如果这块面积的法线与磁场成 theta 角， 就用 Phi 等于 B A cos theta——注意这里是余弦，和受力公式里的正弦不同， 因为这个角是相对于法线量的，而不是相对于表面。
The unit is the weber, which is a tesla metre squared.
单位是韦伯，也就是特斯拉平方米。
And for a coil of N turns, the flux linkage is N Phi, equal to N B A.
而对于 N 匝的线圈，磁链是 N Phi，等于 N B A。
Flux linkage is the quantity Faraday's law actually uses, so dropping the N is a common way to lose the mark.
法拉第定律真正用到的量正是磁链，所以漏掉那个 N 是常见的丢分方式。
A coil of two hundred turns and area nought point zero one square metres sits with its plane at right angles to a nought point five tesla field.
一个二百匝、面积为零点零一平方米的线圈，其平面与零点五特斯拉的磁场垂直。
The field falls steadily to zero in nought point two seconds.
磁场在零点二秒内均匀地降到零。
Find the average induced e.m.f.
求平均感应电动势。
Work with flux LINKAGE, not flux.
要用磁链来做，而不是磁通量。
The flux linkage starts at N B A: two hundred times nought point five times nought point zero one, which is one point nought weber.
磁链的初值是 N B A： 二百乘以零点五再乘以零点零一，等于一点零韦伯。
It ends at zero.
末值是零。
So the change is one point nought weber, and you divide by the time: one point nought over nought point two gives five point zero volts.
所以变化量是一点零韦伯，再除以时间：一点零除以零点二，得到五点零伏特。
Two demonstrations show both laws at once.
有两个演示能同时展示这两条定律。
Push a bar magnet into a coil connected to a galvanometer and the needle deflects.
把一块条形磁铁推进一个接着检流计的线圈，指针会偏转。
Pull it out and the deflection REVERSES — that is Lenz's law, the induced e.m.f. opposing the change.
把它拉出来，偏转方向会反过来——这就是楞次定律，感应电动势反抗着这个变化。
Move it faster and the deflection is LARGER — that is Faraday's law, the rate of change.
推得更快，偏转就更大——这就是法拉第定律，变化率。
Second: a copper disc swinging into a field is rapidly slowed.
第二个：一块铜圆盘摆进磁场时会迅速慢下来。
Eddy currents are induced in the metal, and by Lenz's law they oppose the motion that created them.
金属中被感应出涡流， 而根据楞次定律，它们反抗产生它们的那个运动。
Note the disc is not magnetic — copper is not attracted to a magnet at all — yet it is stopped, and this is exactly how eddy-current brakes work.
注意这个圆盘并不是磁性的—— 铜根本不会被磁铁吸引——可它还是被制止了，而电涡流刹车正是这样工作的。
Finally, a question that comes up every year: how do you make the induced e.m.f. larger?
最后是一道每年都会出现的题：怎样让感应电动势变大？
Read it straight off epsilon equals N d Phi by d t, with Phi equal to B A.
直接从 epsilon 等于 N 乘以 d Phi 比 d t 里读出来，其中 Phi 等于 B A。
There are four ways.
有四种办法。
More turns N.
匝数 N 更多。
A stronger field B.
磁场 B 更强。
A larger area A.
面积 A 更大。
Or a faster change — the same flux change in less time.
或者变化更快—— 同样的磁通变化在更短的时间内完成。
And one last point worth stating properly: Lenz's law is not an extra rule, it is conservation of energy.
还有最后一点值得说清楚： 楞次定律不是一条额外的规则，它就是能量守恒。
If the induced e.m.f. reinforced the change instead of opposing it, a small push would grow without limit and energy would come from nothing.
如果感应电动势不是反抗变化而是加强变化，那么轻轻一推就会无限增长， 能量就凭空产生了。
That is why the minus sign is in the equation.
这正是方程里那个负号的来由。
Three marks to secure.
三个要拿稳的分。
First, the force on a current is flux density times current times length; on a moving charge it is field times charge times speed — direction from Fleming's left hand.
第一，电流所受的力等于磁通密度乘电流乘长度；运动电荷所受的力等于磁场乘电荷乘速度—— 方向用弗莱明左手定则。
Second, a charge crossing a field moves in a circle.
第二，横穿磁场的电荷做圆周运动。
Third, for induction, the e.m.f. equals the rate of change of flux linkage, and it opposes the change.
第三，对于电磁感应， 电压等于磁链的变化率，而且它反抗那个变化。
Flux linkage changes three ways: a changing field, a changing area, or a changing orientation — a coil turning in a field, which is the a.c. generator.
磁链有三种变化方式：磁场变化、面积变化，或者取向变化—— 线圈在磁场中转动，那就是交流发电机。
Master these, and magnetic fields are yours.
掌握这些，磁场就是你的了。