Electric Charges, Fields, and Gauss's Law
AP Physics C: Electricity and Magnetism Topic 8 9:52 English narration · English + 中文 subtitles burned in
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A lightning bolt hits a car. Inside, the people feel nothing at all. Why?
闪电击中一辆汽车，车里的人却毫无感觉。
The metal shell around them is a conductor.
为什么？ 包住他们的金属外壳是导体。
Charge races over its outside surface, and inside the field is exactly zero.
电荷迅速铺满它的外表面，而车内的电场恰好为零。
By the end of this lesson you will prove that in one line.
学完这节课，你将能用一行式子把它证明出来。
This is Unit Eight: electric charges, fields, and Gauss's law.
这是第八单元：电荷、电场与高斯定律。
We start with the force between two charges, and finish with a law that gives the field of a whole charged object in one line.
我们从两个电荷之间的力开始， 最后用一条定律，一行就写出整个带电体的电场。
Charge comes in two kinds, positive and negative.
电荷分两种：正电荷和负电荷。
Like charges push apart; opposite charges pull together.
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.
电荷是量子化的：任何电荷都是元电荷的整数倍。
And charge is conserved — that is the conservation of charge: never created, never destroyed, only moved.
电荷还守恒——这就是电荷守恒：不能创造，不能消灭，只能搬走。
Charge is a fundamental property of matter, like mass.
电荷是物质的基本属性，就像质量一样。
It has a size and a sign, and no direction.
它有大小和正负，没有方向。
A point charge is a model, used when the object's size does not matter.
点电荷是一种模型，当物体的大小无关紧要时就用它。
Coulomb's law gives the force between two point charges.
库仑定律给出两个点电荷之间的力。
It grows with each charge and falls with the square of the distance: move twice as far away, and the force drops to one quarter.
电荷越大力越大，距离的平方越大力越小： 距离变为两倍，力就只剩四分之一。
That makes it an inverse-square force, exactly like gravity.
所以它是平方反比的力，和引力一样。
It acts along the line joining them — apart for like charges, together for opposite ones.
力沿着两电荷的连线—— 同种电荷相斥，异种电荷相吸。
With several charges, add the forces as vectors; that rule has a name, superposition.
有多个电荷时，把这些力按矢量相加； 这条规则有个名字，叫叠加。
And the electric force is enormously stronger than gravity; gravity only rules the stars because large objects are neutral.
而且电力比引力强得多；引力之所以主宰星辰，只是因为大天体几乎完全中性。
Let's use it.
我们来用一用。
A one microcoulomb charge sits at a right-angle corner.
一个一微库仑的电荷位于直角的顶点。
A positive three microcoulomb charge is zero point three metres away on one side. A negative three microcoulomb charge is the same distance away on the other.
一边零点三米处有一个正三微库仑的电荷，另一边同样距离处有一个负三微库仑的电荷。
Each gives a force of the same size: zero point three newtons.
每一个产生的力大小相同：零点三牛顿。
One pushes, one pulls, at right angles.
一个推，一个拉，而且互相垂直。
So combine them with Pythagoras: about zero point four two newtons.
于是用勾股定理合成：约零点四二牛顿。
Never add the sizes — components first.
千万不要直接把大小相加——要先分解成分量。
How does an object become charged?
物体怎样带上电？
Three ways.
有三种方式。
Charging by friction: rubbing moves electrons between the surfaces.
摩擦起电：摩擦把电子从一个表面转移到另一个表面。
Charging by conduction: touching a charged object shares charge of the same sign.
传导起电：接触带电体，同号电荷被分享。
Charging by induction: bring a charge close, ground the far side, and remove the ground wire first — the object keeps the opposite sign, with no contact at all.
感应起电：把电荷靠近，把远端接地，然后先移走接地线—— 物体就保留了相反的电性，而你从未碰过它。
An insulator cannot pass charge, but it can develop polarization: its molecules turn, so one face goes slightly positive and the other slightly negative.
绝缘体不能传导电荷，但它可以发生极化：分子转向， 使一面略带正电、另一面略带负电。
That is why a charged comb lifts paper.
这就是带电的梳子能吸起纸屑的原因。
An electroscope shows net charge by its leaves repelling — and on any isolated conductor the excess charge sits entirely on the outer surface.
验电器靠金属箔片张开来显示净电荷—— 而在任何孤立导体上，多余的电荷全部分布在外表面。
How easily a material polarizes is called its permittivity.
材料极化的难易程度，就叫做它的介电常数。
A charge changes the space around it. We call that the electric field.
电荷改变它周围的空间，我们用电场来描述这种改变。
The field at a point is the force per unit charge — what a small positive test charge would feel there, divided by its charge.
某点的电场就是单位电荷所受的力——一个小的正检验电荷在那里受到的力，除以它的电量。
It must be small enough not to disturb what it measures.
检验电荷必须小到不会扰乱它所测量的场。
Turn it around for the rule you will use most: the force on a charge equals the charge times the field.
反过来，就得到你最常用的规则： 电荷受到的力等于电量乘以电场。
In a uniform field that force is constant, so a charge fired sideways bends into a parabola.
在匀强电场中这个力是恒定的， 所以横向射入的电荷会弯成抛物线。
And fields simply add as vectors.
而多个电荷的场，直接按矢量相加。
Field lines make the field visible.
电场线让电场看得见。
They start on positive charge and end on negative charge.
它们起于正电荷，止于负电荷。
Where they crowd together the field is strong; where they spread out it is weak.
线越密，场越强；线越疏，场越弱。
They never cross, because the field has one direction at each point.
它们永不相交，因为每一点的电场只有一个方向。
Here two opposite charges form a dipole.
这里两个异号电荷组成一个电偶极子。
Conductors are special.
导体有一个特别之处。
Inside a conductor in equilibrium the field is exactly zero — otherwise the free electrons would keep moving.
处于静电平衡的导体内部，电场恰好为零—— 否则自由电子就会继续移动。
So all the extra charge is pushed to the outer surface, and there the field points straight out, at ninety degrees to the metal.
所以多余的电荷全部被推到外表面， 而在那个表面上，电场垂直向外，与金属成九十度。
Outside a charged sphere, the field is the same as a point charge at the centre.
在带电球体外部，电场与放在球心的一个点电荷完全相同。
Real objects are not points.
真实物体不是点。
For a continuous charge distribution we cut the object into tiny pieces, treat each as a point charge, and add their fields with an integral.
对于连续电荷分布，我们把物体切成很小的小块，把每一块当作点电荷， 再用积分把它们的场加起来。
Write each piece with the right density: the linear charge density, lambda, per unit length for a wire or a ring; the surface charge density, sigma, per unit area for a sheet; the volume charge density, rho, per unit volume for a solid.
每一小块要用合适的密度来写： 细线或圆环用线电荷密度 λ 乘长度，薄面用面电荷密度 σ 乘面积， 实体用体电荷密度 ρ 乘体积。
Then use symmetry before you integrate: if every piece has a mirror piece that cancels its sideways field, say so — that sentence earns a mark.
然后在积分之前先用对称性：如果每一小块都有一个镜像小块，把横向的场抵消掉， 就把这句话写出来——这句话是能得分的。
And AP asks for only a handful of shapes: an infinite wire or cylinder, a ring on its axis, an arc at its centre, a finite line.
而且 AP 只要求少数几种形状： 无限长直导线或圆柱、轴上的圆环、圆心处的圆弧、有限长直线。
A thin ring carries charge spread evenly around it.
一个细圆环上均匀带电。
Find the field on its axis, a distance from the centre.
求轴线上距圆心一定距离处的电场。
Pause here and try it.
先暂停，自己试一试。
Every piece of the ring is the same distance from that point, and each has a partner opposite it, so the sideways fields cancel and only the axial part survives.
圆环上每一小块到该点的距离都相同，而且每一块在对面都有一个搭档， 所以横向的场互相抵消，只剩下沿轴线的分量。
That gives the formula on screen.
于是得到屏幕上这个公式。
Now check both limits: at the centre the field is zero; very far away it becomes a simple point charge.
现在检验两个极限：在圆心处场为零；在很远处，它变成一个简单的点电荷。
Both checks pass.
两个检验都通过。
One more idea before the big law: electric flux.
在讲那条大定律之前，还有一个概念：电通量。
Flux measures how much field passes through a surface.
通量衡量有多少电场穿过一个面。
For a flat area in a uniform field it is the field, times the area, times the cosine of the angle to the direction perpendicular to that surface.
对匀强电场中的一块平面，它等于电场乘以面积， 再乘以电场与该面法线方向夹角的余弦。
Face on it is largest; edge on it is zero.
正对时最大，侧对时为零。
In general we write it as an integral.
一般情况下，我们把它写成对整个面的积分。
On a closed surface that direction points outward, so lines leaving count positive and lines entering count negative.
在闭合曲面上，法线方向朝外，所以穿出的线算正，穿入的线算负。
Now wrap a closed surface around a charge and count the lines crossing it.
现在用一个闭合曲面把电荷包起来，数一数穿过它的线。
Change the shape — stretch it, dent it, make it lumpy.
改变这个曲面的形状——拉长它、压凹它、弄得坑坑洼洼。
The same lines still cross.
穿过的仍然是同样多的线。
The total flux does not care about the shape or the size, only about the charge inside.
总通量与曲面的形状和大小无关，只与里面的电荷有关。
That is Gauss's law, the first of Maxwell's equations: the total flux out of any closed surface equals the charge enclosed, divided by the permittivity of free space.
这就是高斯定律，麦克斯韦方程组的第一条： 穿出任意闭合曲面的总通量，等于该曲面所包围的电荷除以真空介电常数。
It is true for every closed surface.
它对每一个闭合曲面都成立。
But it only finds the field when you choose the surface cleverly.
但只有聪明地选择曲面，它才能帮你求出电场。
The surface you draw has a name — a Gaussian surface — and it is not a real object; you invent it.
你画的这个面有个名字——高斯面——它不是真实的物体，是你自己想象出来的。
Pick one where the field has the same size everywhere and points straight through.
要选电场在上面处处大小相同、并且垂直穿过的那种面。
Then the integral becomes E times the area.
这时积分就变成 E 乘以面积。
The exam uses exactly three symmetries — spherical symmetry, cylindrical symmetry and planar symmetry — and a full-credit answer has four parts: name the surface, give the symmetry argument, count the enclosed charge, then solve.
考试只用三种对称——球对称、柱对称和平面对称—— 而满分的答案总是有四个部分：写出所选的面、给出对称性论证、 数清所包围的电荷，然后求解。
Take a solid ball with charge spread evenly through it.
取一个电荷均匀分布的实心球。
Outside the ball, draw a spherical Gaussian surface around it — spherical symmetry. By symmetry the field is the same size everywhere on it, so the flux is the field times that area.
在球外画一个包住它的球面： 由对称性，电场在这个面上处处大小相同， 所以通量就是电场乘以这个面积。
Set it equal to the enclosed charge over the permittivity, and the field is exactly that of a point charge at the centre. Now go inside.
让它等于所包围的电荷除以真空介电常数， 得到的电场与放在球心的一个点电荷完全一样。
A smaller sphere encloses only part of the charge, so the field rises in a straight line from zero at the centre to the surface value.
现在进到球内。 较小的球面只包住一部分电荷，所以电场从球心的零开始，沿直线上升到表面处的值。
That is the graph to know by heart.
这就是你应该背下来的那张图。
Two more results worth knowing by heart.
还有两个结果值得背下来。
For a very long charged wire, wrap a coaxial cylinder around it — cylindrical symmetry.
对很长的带电直导线，用一个同轴圆柱面把它包起来—— 这就是柱对称。
The flat ends contribute nothing, because the field only skims along them, so only the curved side counts — and the field falls as one over the distance.
两个平的端面没有贡献，因为电场只是擦着它们过去，所以只有侧面算数—— 电场按距离的一次方反比下降。
For a large flat sheet, use a pillbox straddling both faces — planar symmetry; there the field is uniform, with no fall-off at all.
对很大的带电平面，用一个穿过两面的药盒面—— 这就是平面对称； 那里的电场是均匀的，完全不随距离减小。
Same law, same four steps, three different patterns.
同一条定律、同样四个步骤，三种不同的规律。
Now the question we started with.
现在回到一开始的问题。
Lightning hits the car, and the charge spreads over the metal shell almost instantly.
闪电击中汽车，电荷几乎瞬间铺满金属外壳。
Draw a closed surface just inside the metal: the field on it is zero, so the flux through it is zero, so the charge it encloses must be zero.
在金属内侧画一个闭合曲面：它上面的电场为零，所以穿过它的通量为零， 所以它所包围的电荷必须为零。
All the charge is outside, and nothing pushes on the people.
全部电荷都在外面，对车里的人没有任何作用力。
That is a Faraday cage — it protects aeroplanes, laboratories and shielded cables.
这就是法拉第笼——它保护着飞机、实验室和屏蔽电缆。
Four habits that save marks.
四个能保住分数的习惯。
First, state the symmetry argument before you use Gauss's law — say why E is constant on your surface; that sentence is worth a mark by itself.
第一，用高斯定律之前先写出对称性的论证—— 单是这句话就值一分。 先说清楚为什么你的面上 E 是恒定的。
Second, add forces and fields as vectors, using components.
第二，力和场都要按矢量相加，要用分量。
Third, keep two ideas apart: the field a charge creates, and the force a field puts on a charge.
第三，把两个概念分开：电荷产生的场，和场作用在电荷上的力。
Fourth, check every answer at its limits — far away it must look like a point charge.
第四，在极限情形检验每一个答案——在很远处，它必须看起来像一个点电荷。
Get those four, and this unit is yours.
做到这四点，这个单元就是你的了。