Electric Force, Field, and Potential
AP Physics 2 Topic 10 8:04 English narration · English + 中文 subtitles burned in
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A thundercloud is a charge factory.
雷雨云是一座电荷工厂。
Ice crystals crash inside it, and electrons collect at the bottom of the cloud. The ground below turns positive.
云里冰晶不断碰撞，电子聚集到云的底部， 下面的地面则带上正电。
Nothing moves — air is an insulator.
此刻还没有电流——空气是绝缘体。
But charge keeps building, and the invisible field between cloud and ground grows.
但电荷不断积累，云与地之间那个看不见的电场越来越强。
At three million volts per metre, the air gives up.
当电场达到每米约三百万伏时，空气终于顶不住了。
This is Unit Ten: electric force, field, and potential.
这是第十单元：电场力、电场与电势。
One law for the force between charges, then two shortcuts — the field, a vector, and the potential, a scalar.
先是一条描述电荷之间作用力的定律， 再是两个强大的工具——电场，它是矢量；电势，它是标量。
Keep those two apart, and this unit is easy.
把这两者分清楚，这个单元就不难了。
Charge is a fundamental property of matter, and it comes in two kinds.
电荷是物质的一种基本性质，而且只有两种。
Like charges repel; opposite charges attract.
同种电荷互相排斥，异种电荷互相吸引。
Charge is quantized: every charge is a whole number of elementary charges — a proton plus one, an electron minus one, a neutron zero.
电荷是量子化的：任何电荷都是元电荷的整数倍——质子带正一，电子带负一，中子为零。
Charge is conserved: you never create it, you only move electrons.
电荷还守恒：你从不创造电荷，只是移动电子。
And a tiny object is modelled as a point charge: an object small enough that its size does not matter here.
而很小的物体可以简化成一个点电荷：小到可以忽略自身大小。
One consequence to keep: although the electric force is vastly stronger than gravity, gravity dominates at large scales, because large objects are usually electrically neutral.
有一个结论要记住：虽然电力远远强于引力， 但在大尺度上占主导的是引力， 因为大的物体通常在电性上是中性的。
Coulomb's law says the force between two point charges is Coulomb's constant, times both charges, divided by the distance squared.
库仑定律说：两个点电荷之间的力，等于库仑常量乘以两个电荷量，再除以距离的平方。
It is inverse square: double the separation, and the force drops four times.
它是平方反比的：距离加倍，力就减小到四分之一。
The direction lies along the line joining them: apart for like signs, together for opposite.
力的方向沿着两电荷的连线：同号相斥，异号相吸。
Gravity has the same shape but only pulls, and is far weaker; it still rules the universe, because large objects are neutral.
引力具有同样的形式，但只会吸引，而且弱得多； 它之所以仍然主宰宇宙，是因为大尺度的物体是电中性的。
Let's use it.
我们来用一用。
A plus three microcoulomb charge sits ten centimetres from a minus two microcoulomb charge.
一个正三微库的电荷，与一个负二微库的电荷相距十厘米。
Use only the sizes: three microcoulombs times two microcoulombs, times Coulomb's constant, divided by zero point one squared. That is five point four newtons.
先只用电荷量的大小：三微库乘以二微库，再乘以库仑常量，除以零点一的平方， 得到五点四牛顿。
Now the direction, from the signs: opposite charges, so the force is attractive.
再由正负号判断方向：两个电荷异号，所以这个力是吸引力。
This is the process of charging.
这就是起电的过程。
An object gets charged by moving electrons, in three ways.
物体靠转移电子来带电，共有三种方式。
Friction: rubbing drags electrons from one surface to another.
摩擦起电：摩擦把电子从一个表面拉到另一个表面。
Conduction: touching shares charge.
接触起电：接触让电荷分配到新的物体上。
Induction: a nearby charge pushes electrons to one side of a neutral object; ground it, and it stays charged.
感应起电：附近的电荷把中性物体里的电子推到一侧， 再把它接地，它就带上了电。
In a conductor, electrons move freely. In an insulator they are stuck, but can still shift inside each atom.
在导体中电子可以自由移动； 在绝缘体中电子被束缚住，但仍能在每个原子内部略微偏移。
That shift is polarization; permittivity measures how easily it happens.
这种偏移叫做极化；介电常数衡量极化发生的难易程度。
Rather than the force between every pair of charges, ask a better question: what does one charge do to the space around it?
与其去追踪每一对电荷之间的力，不如问一个更好的问题： 一个电荷对它周围的空间做了什么？
Each arrow shows the push a small positive test charge would feel — away from positive, toward negative.
每一支箭头表示一个很小的正试探电荷在该处会受到的推力—— 背离正电荷，指向负电荷。
Together they map the electric field.
它们合在一起，就画出了电场。
The electric field is the force per unit charge: the force on a small positive test charge, divided by that charge — small enough not to disturb the field.
电场是单位电荷所受的力：把一个很小的正试探电荷受到的力，除以它的电荷量—— 这个试探电荷要小到不会扰动电场。
Turn that around, and you get the tool you will use most: the force on any charge is that charge times the field.
把定义反过来，就得到你最常用的工具： 任意电荷受到的力，等于这个电荷量乘以电场强度。
For a point charge, the field is Coulomb's constant times the charge over the distance squared.
对点电荷，电场等于库仑常量乘以电荷量，再除以距离的平方。
Where field lines crowd together, the field is stronger, and fields from several charges add as vectors.
电场线越密集，电场就越强；多个电荷的电场要按矢量相加。
Conductors are special: their electrons move until the field inside the metal is zero.
导体很特别：里面的电子会一直移动，直到金属内部的电场为零。
So on a charged conductor at rest, the extra charge sits on the outside surface, and at that surface the field is perpendicular to the metal.
所以静止的带电导体，多余的电荷全部分布在外表面， 而且在表面处电场垂直于金属表面。
Outside a uniformly charged sphere, the field is the same as a point charge at its centre.
在均匀带电球体的外部，电场与把全部电荷集中在球心的点电荷完全相同。
An insulator is different: its charge stays spread inside, and the field there is not zero.
绝缘体则不同：电荷会散布在材料内部，内部电场并不为零。
Your turn.
轮到你了。
Find the field twenty centimetres from a plus five microcoulomb charge, then the force on a plus two nanocoulomb charge placed there.
求距离一个正五微库电荷二十厘米处的电场， 再求把一个正二纳库的电荷放在那里所受的力。
Pause here and try it.
先暂停，自己试一试。
For the field: Coulomb's constant times five microcoulombs, divided by zero point two squared — one point one million newtons per coulomb, pointing away.
电场：库仑常量乘以五微库，除以零点二的平方， 得到每库仑一百一十万牛顿，方向背离该电荷。
For the force: multiply by two nanocoulombs, giving two point two millinewtons.
力：把电场乘以二纳库，得到二点二毫牛。
Fields give force. Now energy.
电场给出力，现在来看能量。
Two charges store electric potential energy, and it has a precise meaning: it is the work an outside force must do to bring them together from infinitely far apart.
两个电荷储存着电势能，它有一个精确的含义： 把它们从无穷远处移到现在的位置，外力必须做的功。
The formula looks like Coulomb's law, but with distance to the first power, and the signs kept.
公式看起来像库仑定律，但距离是一次方，而且保留电荷的正负号。
Like charges give positive energy: they will fly apart.
同号电荷的电势能为正：一放手它们就会飞开。
Opposite charges are bound, with negative energy.
异号电荷被束缚在一起，电势能为负。
For more charges, add up every pair.
电荷更多时，把每一对都加起来。
Divide that energy by the charge, and you get electric potential: the potential energy per unit charge, measured in volts.
把这个能量除以电荷量，就得到电势：单位电荷的电势能，单位是伏特。
Near a point charge, it is Coulomb's constant times the charge over the distance.
在点电荷附近，电势等于库仑常量乘以电荷量，再除以距离。
Here is the payoff: potential is a scalar.
好处在这里：电势是标量。
It has no direction, so potentials from several charges simply add, keeping their signs.
它没有方向，所以多个电荷产生的电势可以直接相加， 只要带上正负号。
A positive charge, left alone, moves from high potential to low.
一个正电荷若不受约束，总是从高电势移向低电势。
Field and potential are two views of one thing.
电场与电势是同一件事的两种看法。
The average field between two points is the potential difference divided by the distance, so a field can be measured in volts per metre.
两点之间的平均电场， 等于电势差除以两点之间的距离，所以电场也可以用伏特每米来量度。
Join the points of equal potential and you draw equipotential lines, like contours on a map.
把电势相同的点连起来，就画出等势线，就像地图上的等高线。
They cross field lines at right angles, and moving a charge along an equipotential costs no work, because the potential itself has not changed.
等势线与电场线垂直，而沿着等势线移动电荷完全不做功， 因为电势本身没有改变。
Two conducting plates facing each other, with a gap between.
两块面对面的导体极板，中间隔着一段空隙。
Connect a battery, and charge piles up — positive on one plate, negative on the other.
接上电池后电荷开始堆积—— 一块极板带正电，另一块带负电。
The voltage climbs fast, then levels off as the plates fill.
电压先快速上升，随着极板充满而逐渐趋于平稳。
This device is a capacitor.
这个器件就是电容器。
Capacitance is the charge stored for each volt across the plates, in farads.
电容是极板上每一伏特电压所储存的电荷量，单位是法拉。
Here is the exam point: capacitance does not depend on the charge you put on it. It is fixed by the geometry alone — the plate area, the gap, and the dielectric between.
考试要点：电容与你放上去的电荷量无关，它只由几何结构决定—— 极板面积、极板间距，以及中间的电介质。
Between the plates the field is uniform, so a charge in there moves like a projectile in gravity.
两板之间的电场是匀强的，所以其中的电荷运动就像重力场中的抛体。
The stored energy is one half of the charge times the voltage.
储存的能量等于二分之一乘以电荷量再乘以电压。
Last idea: energy conservation.
最后一个概念：能量守恒。
Move a charge through a potential difference, and its electric potential energy changes by charge times potential difference.
让一个电荷经过一段电势差， 它的电势能变化量等于电荷量乘以这段电势差。
Released from rest, that energy becomes kinetic energy: charge times voltage equals one half times mass times speed squared.
若从静止释放， 这份能量就变成动能：电荷量乘以电压，等于二分之一乘以质量再乘以速度的平方。
An electron starts at rest and is accelerated through one hundred volts. The speed comes out at five point nine million metres per second — about two percent of the speed of light.
一个电子从静止出发，经过一百伏特被加速， 速度约为每秒五百九十万米——大约是光速的百分之二。
Four marks students throw away.
每年都有四类分白白丢掉。
First, field is a vector, potential is a scalar — add fields with direction, add potentials as signed numbers.
第一，电场是矢量，电势是标量—— 电场要按方向相加，电势只需带符号直接相加。
Second, put magnitudes into Coulomb's law and take the direction from the signs.
第二，代入库仑定律时用电荷量的大小，方向由正负号判断。
Third, force and field are inverse square, but energy and potential go as one over the distance.
第三，力和电场是平方反比，但能量和电势与距离的一次方成反比。
Fourth, for a charge accelerated through a voltage, go straight to energy conservation.
第四，遇到电荷被电压加速，直接用能量守恒。