Conductors and Capacitors
AP Physics C: Electricity and Magnetism Topic 10 8:28 English narration · English + 中文 subtitles burned in
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A plane is struck by lightning about once a year, and the people inside feel almost nothing.
飞机大约每年会被闪电击中一次，而机舱里的人几乎毫无感觉。
A car in a thunderstorm is one of the safest places to sit.
雷雨天坐在汽车里， 也是最安全的地方之一。
Why?
为什么？
The metal around you is a conductor.
包围你的金属是导体。
In a tiny fraction of a second the charge spreads over its outer surface, and inside the shell the electric field falls to zero.
在极短的时间内， 电荷会分布到它的外表面，而在外壳内部，电场降为零。
This is Unit Ten: conductors and capacitors.
这是第十单元：导体与电容器。
First, the four rules every conductor in equilibrium obeys.
首先是处于平衡状态的导体必须遵守的四条规则。
Then we take two conductors, leave a gap between them, and build the device that stores charge and energy.
然后我们取两个导体，中间留出一个间隙，做成能够储存电荷和能量的器件。
Put extra charge on a conductor and the charges push apart until they stop moving.
给导体加上多余的电荷，它们会互相排斥，直到不再移动。
That state is electrostatic equilibrium, and it gives four rules.
这种状态叫做静电平衡， 由此得到四条规则。
One: the field inside the metal is zero — or the free electrons would still be moving.
第一：金属内部的电场为零——否则自由电子仍会继续移动。
Two: every bit of excess charge sits on the outer surface.
第二：所有多余的电荷都停留在外表面上。
Three: just outside, the field is perpendicular to the surface, with a size of the surface charge density over the permittivity of free space.
第三：紧靠表面外侧，电场垂直于表面， 大小等于面电荷密度除以真空介电常数。
Four: the whole conductor is one equipotential.
第四：整个导体是一个等势体。
Three consequences follow.
由此得到三个推论。
Put a conductor into an outside field and it polarizes: charge shifts on the surface until the inside is field-free again.
把导体放进外加电场中，它会被极化：电荷在表面重新分布， 直到内部再次没有电场。
That surface charge is not spread evenly — it crowds at points and sharp edges, so the field just outside is strongest there.
表面电荷并不是均匀分布的——它在尖端和棱角处最密集， 所以那里紧靠外侧的电场最强。
And a closed conducting shell keeps outside fields out completely — electrostatic shielding, the Faraday cage.
而一个封闭的导体壳能把外部电场完全挡在外面， 这就是法拉第笼。
Now let two charged conductors touch.
现在让两个带电导体接触。
Charge flows — but what stops it? Not equal charges.
电荷会流动——但是什么让流动停止呢？
It stops when both surfaces reach the same potential.
不是电荷相等。 当两个表面达到相同电势时，流动才停止。
A small sphere carries six microcoulombs. It touches a distant sphere of twice the radius, then they separate.
一个小球带六微库仑的电荷， 它与一个半径为两倍的远处金属球接触，然后分开。
A sphere's potential is its charge over its radius, times the Coulomb constant, so the big one must end with twice the charge.
球的电势等于电荷除以半径 再乘以库仑常量，所以大球最终带的电荷是小球的两倍。
Six microcoulombs split as two and four.
六微库仑分成二和四。
Ground is an idealized reference at zero potential; it can absorb or supply any amount of charge.
“地”是一个理想化的零电势参考点，它可以吸收或提供任意多的电荷。
Bring a positive rod near a neutral conductor, and its electrons are pulled to the near side.
把一根带正电的棒靠近一个中性导体，导体内的电子会被吸引到靠近的一侧。
Connect the conductor to ground, and more electrons flow up into it.
把导体接地，就会有更多电子从大地流上来。
Cut the wire first, then take the rod away, and a permanent negative charge is left behind.
先剪断导线，再把棒移走， 导体上就永久留下了负电荷。
It was induced — nothing ever touched it.
这是感应出来的电荷——从头到尾没有任何接触。
Two conductors with a gap between them make a capacitor.
两个导体中间隔着一个间隙，就构成了电容器。
Connect a battery and charge builds up: plus on one plate, an equal minus on the other.
接上电池，电荷开始积累： 一个极板带正电，另一个带等量的负电。
The voltage climbs quickly at first, then levels off at the battery's voltage.
电压先快速上升，然后稳定在电池电压。
Nothing crosses the gap — the charge waits there, held by the field between the plates.
没有任何电荷穿过那个间隙——电荷只是停在那里，被极板之间的电场固定住。
How much charge can it hold?
它能储存多少电荷？
That is its capacitance: the charge on one plate, divided by the potential difference between the plates.
这由电容决定：一个极板上的电荷量，除以两极板之间的电势差。
One coulomb per volt is one farad — an enormous unit, so real parts are labelled in microfarads and picofarads.
每伏特一库仑就是一法拉——这个单位非常大，所以实际元件都标成微法和皮法。
And here is the point students miss: capacitance is fixed by geometry alone — the area, the gap, and the material between, never the charge you put on.
还有一点学生常常忽略：电容只由几何形状决定——极板面积、间隙和中间的材料， 而不取决于你放上去多少电荷。
For parallel plates the capacitance comes out in three steps, and the exam wants all three.
对于平行板电容器，电容可以分三步推出来，而考试要的正是这三步。
Step one: Gauss's law with a pillbox, plus superposition, gives the field between the plates.
第一步：用高斯定律加上一个药盒形高斯面，再用叠加原理，得到极板之间的电场。
Each plate makes half of it; the two halves add between the plates and cancel outside.
每个极板产生其中的一半，两个一半在两板之间相加，在外面相互抵消。
This field is the same everywhere except close to the edges, as long as the gap is much smaller than the plates.
只要间距远小于极板的尺寸，这个电场就处处相同，只有靠近边缘的地方例外。
Step two: multiply that uniform field by the gap to get the potential difference.
第二步：把这个匀强电场乘以间距，就得到电势差。
Step three: divide the charge by it.
第三步：用电荷除以它。
The charge cancels, leaving the permittivity of free space times the area, over the gap.
电荷约掉，剩下真空介电常数乘以极板面积再除以间距。
Now the numbers.
来算一算。
The plates have an area of zero point zero two zero square metres, and the gap is one millimetre.
极板面积是零点零二零平方米，间距是一毫米。
Pause and work out the capacitance.
先暂停，自己求出电容。
The formula gives about one hundred and eighty picofarads.
代入公式得到大约一百八十皮法。
Charge it to one hundred volts, and each plate holds capacitance times voltage: about eighteen nanocoulombs.
把它充到一百伏，每个极板上的电荷等于电容乘以 电压：大约十八纳库仑。
Tiny — which is why a real capacitor uses a thin gap and a large rolled-up area.
数值很小——这正是为什么真正的电容器要用很薄的间隙 和卷起来的大面积极板。
The same three steps handle the other two shapes this course expects.
同样这三步也适用于本课程要求的另外两种形状。
For two concentric spheres, the field in the gap is the point-charge field, and integrating it gives a capacitance built from the two radii.
对于两个同心球壳， 间隙中的电场就是点电荷的电场，积分后得到的电容由两个半径决定。
For a coaxial cable, the field falls off as one over the distance, so the integral brings in a logarithm.
对于同轴电缆，电场按距离的倒数减小，所以积分中会出现对数。
Same method every time: field, potential difference, divide.
方法始终相同：先求电场，再积分得到电势差，然后相除。
Because the field between the plates is uniform, a charge inside feels a constant force — and constant force means constant acceleration.
因为两板之间的电场是匀强的，里面的电荷受到恒定的力——恒力意味着匀加速。
So the path is a parabola: this is projectile motion, exactly like a ball thrown sideways in gravity.
所以它的轨迹是一条抛物线，就像水平抛出的小球在重力中的运动一样。
An electron enters halfway between the plates at twenty million metres per second.
一个电子以每秒两千万米的速度从两板正中间射入。
The field is one thousand newtons per coulomb, and the plates are four centimetres long.
电场是每库仑一千牛顿， 极板长四厘米。
Dividing the electric force by that tiny mass gives about one point eight times ten to the power fourteen metres per second squared.
用电场力除以电子极小的质量，得到大约一点八乘以十的十四次方 米每二次方秒。
The electron is inside for two nanoseconds, and drifts about zero point three five millimetres towards the positive plate.
电子在板间只停留两纳秒，向正极板偏移大约零点三五毫米。
Charging a capacitor takes work.
给电容器充电是要做功的。
The first bit of charge crosses almost free, but every later bit is pushed against the charge already there, so the last bit costs the full voltage.
第一份电荷几乎不费力就搬过去了， 但后面每一份都必须克服已经存在的电荷，所以最后一份要付出完整的电压。
Plot charge against voltage and you get a straight line; the work done is the triangle underneath.
把电荷对电压作图，得到一条直线；所做的功就是它下面的三角形面积。
That is where the one half comes from.
这就是那个二分之一的来历。
The stored energy is half the charge times the voltage, and capacitance rewrites that two other ways.
储存的能量等于二分之一乘以电荷乘以电压， 再用电容的定义还可以改写成另外两种形式。
Use whichever form the question gives you.
题目给了哪一种，就用哪一种。
Capacitors combine the opposite way to resistors.
电容器的组合方式与电阻正好相反。
In parallel, both feel the same voltage, the charges add, and the capacitances add.
并联时，两者的电压相同，电荷相加， 电容也直接相加。
In series, the same charge sits on every plate, the voltages add, and the reciprocals add — so the total is always smaller than the smallest one.
串联时，每个极板上的电荷相同，电压相加， 而相加的是电容的倒数——所以总电容总是比其中最小的那个还要小。
Two microfarads and four microfarads give four thirds of a microfarad in series, and six in parallel.
二微法与四微法串联得到三分之四微法，并联则得到六微法。
Now slide an insulating material into the gap — a dielectric.
现在把一块绝缘材料塞进间隙——这就是电介质。
Its charges cannot travel, but every molecule stretches or turns a little.
它的电荷不能自由移动， 但每个分子都会稍稍拉伸或转动。
The material becomes polarized, and the lined-up molecules make their own field, pointing the opposite way, so the field inside is weaker.
材料被极化，排列起来的分子产生自己的电场， 方向与外场相反，于是内部的电场被削弱。
The dielectric constant is exactly that ratio: the original field divided by the reduced one.
介电常数正是这个比值： 原来的电场除以减弱后的电场。
Weaker field, less voltage for the same charge — so the capacitance rises by that factor.
电场弱了，同样的电荷对应的电压就小了—— 所以电容按同样的倍数增大。
This is where the exam sets its trap.
考试的陷阱就设在这里。
What happens next depends on one thing: which quantity is held fixed.
接下来会发生什么，只取决于一件事：哪个量被固定住。
If the battery stays connected, the voltage cannot change, so the charge and the energy both rise by the dielectric constant.
如果电池一直连着，电压不能变，于是电荷和能量都按介电常数倍增大。
If the battery is removed first, the charge is trapped, so the voltage, the field and the energy all fall by that factor.
如果先把电池断开，电荷被困住，于是电压、电场和能量都按同样的倍数减小。
Take a one hundred picofarad capacitor charged to twelve volts.
取一个一百皮法的电容器，充到十二伏。
Disconnect it, then fill the gap with a material whose dielectric constant is three.
先断开电源，再往间隙里填入介电常数 为三的材料。
The voltage drops to four volts, and the stored energy falls to one third.
电压降到四伏，储存的能量降到三分之一。
Three marks students throw away.
三个学生常常白丢的分。
First, in electrostatic equilibrium, say both halves: the field inside a conductor is zero, and all the excess charge is on the surface.
第一，讲静电平衡时两句话都要说：导体内部电场为零， 而且所有多余电荷都在表面上。
Second, pick a Gaussian surface that matches the symmetry, so the field is constant on it.
第二，选高斯面时要与对称性匹配， 使电场在面上处处相同。
Third, in any dielectric question, write down whether the charge or the voltage is fixed before you touch a formula.
第三，任何电介质题目，先写下是电荷固定还是电压固定， 再动公式。