Electric Fields
A-Level Physics Topic 18 11:02 English narration · English + 中文 subtitles burned in
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Deep inside a storm cloud, charge is building.
在雷暴云的深处，电荷正在积聚。
Negative charge gathers at the base, positive charge on the ground below.
负电荷聚集在云底，正电荷聚集在下方的地面。
Between them, in the empty air, an invisible tension grows — an electric field, stronger and stronger.
在它们之间空荡荡的空气里，一种看不见的张力在增长——一个电场，越来越强。
Air normally does not conduct electricity.
空气通常并不导电。
But push the field hard enough, and it tears the air's atoms apart, carving a path for a hundred million volts to leap to the ground.
但把电场推得足够强，它就把空气的原子撕裂， 为一亿伏特开辟出一条通向地面的路径。
That blinding flash is an electric field, unleashed.
那道刺眼的闪光，就是被释放出来的电场。
Electric charge fills the space around it with a field of force.
电荷用一个力场充满它周围的空间。
Today: electric fields and field lines, Coulomb's law, uniform fields, and electric potential.
今天：电场与场线、库仑定律、匀强电场，以及电势。
Let's begin.
让我们开始吧。
An electric field is a region where a charge feels a force.
电场是一个区域，在其中电荷会感受到力。
We measure its strength as the force on each unit of positive charge — the electric field strength, E, defined with a small positive test charge that does not disturb the field it is measuring.
我们用每单位正电荷所受的力来量度它的强度—— 这就是电场强度，E，它是用一个不会扰动被测电场的小正试探电荷来定义的。
It is a vector, so it has a direction as well as a size.
它是矢量，所以既有大小，也有方向。
So the force on any charge is simply the charge, times the field.
所以任何电荷所受的力，就等于电荷乘以电场。
We draw the field with field lines, and by convention they point the way a positive charge would move: away from positive, and toward negative.
我们用场线来画电场， 按照约定，场线指向正电荷会移动的方向：从正电荷发出，指向负电荷。
Four patterns you must be able to draw.
有四种图样是你必须会画的。
Between parallel plates the lines are parallel and equally spaced — a uniform field.
平行板之间，场线是平行且等间距的——这是匀强场。
A dipole, two opposite charges, has lines curving out of the positive and into the negative.
偶极子，也就是一对异号电荷，场线从正电荷弯出来、弯进负电荷。
A positive point charge has radial lines pointing straight out; a negative one has them pointing straight in.
正点电荷的场线沿半径笔直向外；负点电荷的则笔直向内。
And a charged sphere above an earthed plate has lines leaving the sphere and meeting the plate at right angles.
而一个带电球放在接地板上方时，场线从球上出发，垂直地落到板面上。
Three rules govern all of them: lines start on positive and end on negative or go to infinity; lines never cross, because the force at a point has only one direction; and closer lines mean a stronger field.
支配这一切的有三条规则：场线始于正电荷、终于负电荷，或者伸向无穷远； 场线永不相交，因为一点处的力只有一个方向；场线越密，场就越强。
Let us use it.
我们来用一下。
Two parallel plates five millimetres apart have a potential difference of two hundred volts between them.
两块相距五毫米的平行板之间有两百伏特的电势差。
Find the field strength, and the force on an electron in the gap.
求场强，以及间隙中一个电子所受的力。
For the field, E equals V over d — but convert the millimetres first, because five millimetres is five times ten to the minus three metres.
求场强用 E 等于 V 除以 d—— 但要先把毫米换算掉，因为五毫米就是五乘以十的负三次方米。
Two hundred divided by that gives four times ten to the fourth volts per metre.
两百除以它，得到四乘以十的四次方伏特每米。
Now the force on an electron: F equals q E, so one point six times ten to the minus nineteen coulombs times four times ten to the fourth, which is six point four times ten to the minus fifteen newtons.
再求电子受到的力：F 等于 q E， 也就是一点六乘以十的负十九次方库仑乘以四乘以十的四次方， 结果是六点四乘以十的负十五次方牛顿。
A charge in a uniform field feels a constant force, F equals q E, and therefore a constant acceleration, a equals q E over m.
匀强场中的电荷受到恒定的力，F 等于 q E，因而有恒定的加速度，a 等于 q E 除以 m。
That is the same situation as a mass in a uniform gravitational field, and the same two cases follow.
这和一个质量在匀强重力场中的处境完全相同，于是也有同样的两种情形。
Released at rest, it simply speeds up along the field if it is positive, or against it if negative, gaining kinetic energy.
从静止释放，如果它带正电就沿着场加速，带负电就逆着场加速，从而获得动能。
Entering at right angles to the field, it keeps its sideways speed while accelerating downfield — and that gives a parabolic path, exactly like a projectile under gravity.
如果垂直于场方向射入，它会保持横向速度，同时沿场方向加速—— 于是走出一条抛物线，和重力作用下的抛体运动一模一样。
This is how a cathode-ray tube used to steer its electron beam across the screen.
过去的阴极射线管正是这样把电子束在屏幕上偏转的。
Two charges push or pull on each other, and Coulomb's law gives the force.
两个电荷会相互推拉，而库仑定律给出这个力。
It is the two charges multiplied together, divided by the square of the distance, with a constant out front — one over four pi epsilon nought, where epsilon nought is the permittivity of free space.
它等于两个电荷相乘，除以距离的平方， 前面再乘一个常量——一除以四 pi epsilon 零，其中 epsilon 零是自由空间的介电常数。
Like charges repel; opposite charges are attractive.
同种电荷相斥；异种电荷相吸。
And just like gravity, it is an inverse-square law — double the distance, and the force drops to a quarter.
而且就像引力一样，它是平方反比定律—— 距离加倍，力就降到四分之一。
A worked example.
来做一道例题。
Find the electrostatic force between point charges of two point nought nanocoulombs and three point nought nanocoulombs placed four point nought centimetres apart.
求两个点电荷之间的静电力，一个是二点零纳库仑，另一个是三点零纳库仑， 相距四点零厘米。
Write down Coulomb's law, then convert the units first — this is where the marks go.
先写下库仑定律，然后先做单位换算——分数就丢在这一步。
A nanocoulomb is ten to the minus nine coulombs, and four centimetres is nought point nought four metres.
一纳库仑是十的负九次方库仑，四厘米是零点零四米。
Substituting, with one over four pi epsilon nought as nine times ten to the ninth, gives three point four times ten to the minus five newtons.
把这些代进去， 取一除以四 pi epsilon 零为九乘以十的九次方，得到三点四乘以十的负五次方牛顿。
And state the direction: both charges are positive, so the force is repulsive, along the line joining them.
还要说明方向：两个电荷都是正的，所以力是排斥的，沿着它们的连线。
One result saves you a lot of work.
有一个结论能替你省下很多功夫。
Outside a charged spherical conductor, the field is exactly the same as that of a point charge of the same total charge sitting at the centre — so you can use all the point-charge formulae, provided r is measured from the centre, not from the surface.
在一个带电球形导体的外部， 场与把同样多的总电荷集中在球心的点电荷所产生的场完全相同—— 所以你可以用全部的点电荷公式，前提是 r 要从球心量起，而不是从表面量起。
That last detail is a common slip.
最后这个细节是常见的失误。
And inside a hollow charged conductor the field is zero everywhere.
而在中空带电导体的内部，场处处为零。
Since no work is needed to move a charge around inside, the whole conductor is an equipotential — every point on and inside it sits at the same potential.
既然在里面移动电荷不需要做功，整个导体就是一个等势体—— 它表面上和内部的每一点都处于同一个电势。
Divide Coulomb's law by the test charge and what is left is the field of the source charge alone.
把库仑定律除以试探电荷，剩下的就是源电荷单独产生的电场。
A single point charge makes its own field, spreading out in every direction.
单个点电荷产生自己的电场，向四面八方扩散。
Its strength is the charge, divided by the distance squared, times that same constant.
它的强度等于电荷除以距离的平方， 再乘以那个相同的常量。
For a positive charge, the field lines burst straight outward, like the spokes of a wheel.
对于正电荷，场线笔直地向外迸发，像车轮的辐条。
Close in, they crowd together and the field is strong; far away, they thin out and it weakens.
靠得近时，场线挤在一起，场就强；离得远时，场线稀疏，场就弱。
Between two parallel charged plates, something simpler happens: the field is uniform — the same strength and direction everywhere.
在两块平行带电板之间，会发生更简单的事：电场是匀强的——处处的强度和方向都相同。
Its strength is just the voltage across the plates, divided by the gap between them.
它的强度就等于两板之间的电压，除以它们之间的间隙。
Fire a charged particle into this field, and it feels a constant force, bending its path into a smooth curve, exactly like a ball thrown through gravity.
把一个带电粒子射入这个电场， 它就受到一个恒定的力，把它的路径弯成一条平滑的曲线，就像一个抛过重力场的球一样。
Electric potential at a point is the work done per unit positive charge in bringing a small positive test charge from infinity to that point.
某一点的电势，是把一个小正试探电荷从无穷远处带到该点时，每单位正电荷所做的功。
Find the electric potential four centimetres from a point charge of three point nought nanocoulombs.
求距离一个三点零纳库仑的点电荷四厘米处的电势。
The formula is V equals Q over four pi epsilon nought r — and notice it is r, not r squared, which is the difference between potential and field.
公式是 V 等于 Q 除以四 pi epsilon 零 r—— 注意分母是 r，不是 r 平方，这正是电势与场强的区别。
Nine times ten to the ninth, times three times ten to the minus nine, divided by nought point nought four metres, gives six point eight times ten squared volts.
九乘以十的九次方，乘以三乘以十的负九次方，再除以零点零四米， 得到六点八乘以十的二次方伏特。
One more thing that makes life easy: potential is a scalar.
还有一点让计算变得轻松：电势是标量。
With several charges you just add the potentials, each with its own sign, and there are no directions to resolve — quite unlike fields, where you need a vector sum.
有多个电荷时，你只要把各自的电势连同符号相加，没有方向需要分解—— 这和场很不一样，场必须做矢量求和。
Field and potential are not two separate ideas — one is the gradient of the other.
场强和电势并不是两个各自独立的概念——一个是另一个的梯度。
E equals minus d V by d x: the field strength is the negative potential gradient.
E 等于负的 d V 比 d x： 场强等于电势梯度的负值。
Look at the uniform field between plates.
来看平行板之间的匀强场。
The potential falls steadily from plus V down to zero across the gap, so the graph is a straight line of gradient minus V over d — and the field, being minus that gradient, is a constant V over d.
电势在间隙中从正 V 均匀地降到零， 所以图像是一条斜率为负 V 除以 d 的直线——而场强是它的负值， 于是就是常数 V 除以 d。
That is exactly the formula we used earlier, now derived rather than quoted.
这正是我们前面用过的那个公式，现在是推出来的，而不是背来的。
The minus sign carries real meaning: the field always points towards lower potential.
那个负号有实实在在的含义：场永远指向电势较低的方向。
Check it on a point charge too — differentiate Q over four pi epsilon nought r and negate it, and you get Q over four pi epsilon nought r squared, which is E.
也在点电荷上验算一下——把 Q 除以四 pi epsilon 零 r 求导再取负号， 你得到 Q 除以四 pi epsilon 零 r 平方，那正是 E。
A charge q sitting at a point of potential V has electric potential energy E P equals q V, and for two point charges that becomes Q q over four pi epsilon nought r.
一个电荷 q 位于电势为 V 的点上，就具有电势能 E P 等于 q V， 而对于两个点电荷，它变成 Q q 除以四 pi epsilon 零 r。
The sign is where the physics lives.
物理意义就藏在符号里。
For like charges the energy is positive: a store of energy that would be released if you let them fly apart.
同号电荷的能量为正：这是一份储存起来的能量，一旦让它们飞开就会释放出来。
For opposite charges it is negative — a bound system, a well you must supply energy to climb out of.
异号电荷的能量为负——这是一个束缚系统，一口你必须提供能量才能爬出来的势阱。
In both cases the energy tends to zero as r tends to infinity, which is exactly what "zero at infinity" means as a choice of reference.
两种情形下，当 r 趋于无穷时能量都趋于零，而这正是选取"无穷远处为零"这个参考点的含义。
Here is a pattern worth memorising, because it recurs in every exam.
下面这个套路值得背下来，因为它每次考试都会出现。
An electron orbits a nucleus of charge plus Z e at radius r.
一个电子以半径 r 绕电荷为正 Z e 的原子核运动。
The Coulomb attraction provides the centripetal force — that sentence is the whole method.
库仑吸引力提供向心力——这一句话就是整个方法。
So equate the two: Z e squared over four pi epsilon nought r squared equals m v squared over r.
于是把两者等同起来：Z e 平方除以四 pi epsilon 零 r 平方，等于 m v 平方除以 r。
Cancel one r and solve for the speed: v is the square root of Z e squared over four pi epsilon nought m r.
约掉一个 r，解出速度：v 等于 Z e 平方除以四 pi epsilon 零 m r 的平方根。
Then the total energy is kinetic plus potential — a half m v squared, plus the potential energy which is minus Z e squared over four pi epsilon nought r.
再求总能量，等于动能加势能——二分之一 m v 平方，加上势能， 而势能是负的 Z e 平方除以四 pi epsilon 零 r。
Substituting gives minus Z e squared over eight pi epsilon nought r.
代入后得到 负的 Z e 平方除以八 pi epsilon 零 r。
It is negative, so it is bound: you would have to supply energy to pull the electron free.
它是负的，所以是束缚态： 你必须提供能量，才能把这个电子拉走。
Electric potential is the energy per unit positive charge to bring a charge from infinitely far away, to a point.
电势是把一个电荷从无穷远处带到某一点，每单位正电荷所做的功。
For a point charge it is the charge, over the distance, times the constant.
对于点电荷， 它等于电荷除以距离，再乘以那个常量。
Bring two charges together, and their combined electric potential energy is the product of the charges, over the distance.
把两个电荷放到一起，它们合起来的电势能， 就是电荷之积除以距离。
And the field is simply the rate at which potential changes with distance — the steeper the potential, the stronger the field.
而电场就等于电势随距离变化的快慢——电势越陡，电场越强。
You may have noticed something: these equations look exactly like gravity's.
你可能已经注意到一件事：这些方程看起来和引力的一模一样。
And they should — both are inverse-square fields, described by the very same mathematics.
它们本该如此—— 两者都是平方反比的场，由完全相同的数学描述。
But there is one crucial difference.
但有一个关键的区别。
Gravity only ever pulls.
引力只会拉。
Electric charge can do both: like charges push apart, and opposite charges pull together.
而电荷两样都能做：同种电荷相斥推开，异种电荷相吸拉近。
That single difference is why electricity is so much richer.
正是这一个区别， 让电现象丰富得多。
Three marks to secure.
三个要拿稳的分。
First, the electric field is force per unit positive charge, so field lines point away from positive.
第一，电场是每单位正电荷所受的力，所以场线从正电荷指出去。
Second, Coulomb's law is an inverse-square law, just like gravity.
第二，库仑定律是平方反比定律，和引力一样。
Third, for a uniform field between plates, the field strength is simply voltage over the gap.
第三，对于板间的匀强电场， 场强就等于电压除以间隙。
Master these, and electric fields are yours.
掌握这些，电场就是你的了。