Electric Potential
AP Physics C: Electricity and Magnetism Topic 9 8:02 English narration · English + 中文 subtitles burned in
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Look at these numbers. A small battery says one point five volts. A car battery says twelve.
看这些数字：一节小电池写着一点五伏，汽车电池是十二伏，高压输电线是四十万伏， 一道闪电大约一亿伏。
A high-voltage transmission line carries four hundred thousand. A lightning bolt, about a hundred million. But what does a voltage actually mean?
可是电压到底是什么意思？
It is not a force, and it is not an energy.
它不是力，也不是能量。
One volt means one joule of energy for every coulomb of charge.
一伏表示每一库仑电荷携带一焦耳的能量。
<slow>Voltage is energy per unit charge.</slow>
电压就是单位电荷的能量。
This is Unit Nine: electric potential.
这是第九单元：电势。
We start with the energy stored by a pair of charges, turn it into potential, read it off a map, and finish by launching a proton.
我们先从一对电荷储存的能量出发，把它变成电势， 再学会从等势图上读出它，最后用它把一个质子加速出去。
Notice one thing early: potential is a scalar, so there are no components to resolve.
请先记住一点：电势是标量，所以不需要分解分量。
Let's begin.
让我们开始吧。
Two charges near each other store energy, like a stretched spring.
两个电荷靠近时会储存能量，就像被拉伸的弹簧。
To define it, we bring them in from infinitely far apart and count the work you must do.
为了给它下定义， 我们把它们从无穷远处拿过来，并统计外力必须做的功。
That work is the pair's electric potential energy: Coulomb's constant times the two charges, over their separation.
这个功就是这对电荷的电势能： 库仑常量乘以两个电荷，再除以它们之间的距离。
Now the sign.
现在看符号。
Two like charges give a positive energy — you pushed them together, and letting go returns it as kinetic energy.
两个同号电荷的电势能为正——你把它们推到一起；一旦松手，它又变回动能。
Two opposite charges give a negative energy: the pair is bound.
两个异号电荷的电势能为负：它们是被束缚的。
And the electric force is conservative, so the work depends only on the end points.
而且电场力是保守力，所以这个功只取决于起点和终点。
Let's use it.
我们来用一用。
Three identical charges, each plus two point zero microcoulombs, sit at the corners of a triangle of side thirty centimetres.
三个相同的电荷，每个都是正的二点零微库仑， 位于边长三十厘米的三角形的三个顶点上。
How much energy is stored?
储存了多少能量？
Start with one pair: Coulomb's constant, times the charge squared, divided by the distance — zero point one two joules.
先看一对：库仑常量乘以电荷的平方，再除以距离——得到零点一二焦耳。
There are three pairs, all identical, so the total is zero point three six joules.
一共有三对，而且每一对都相同，所以总能量是零点三六焦耳。
In general, add the energy of every separate pair of charges.
一般来说，要把每一对电荷的电势能都加起来。
That is the work needed to build the triangle from infinity — and notice the energy belongs to the SYSTEM of charges, not to any one of them.
这就是把这个三角形从无穷远处组装起来所需要的功。
Now the step that makes everything easier.
接下来这一步会让一切都变简单。
Divide the potential energy by the charge you carry, and you get electric potential: the potential energy per unit charge at a point in space.
把电势能除以你携带的电荷量，就得到电势： 空间某点处单位电荷的电势能。
Its unit is one joule per coulomb, which we call one volt.
它的单位是焦耳每库仑，我们称之为伏特。
Around a single point charge it is Coulomb's constant times the charge, divided by the distance.
在一个点电荷周围，电势等于库仑常量乘以电荷量，再除以距离。
It falls off as one over r.
它随距离按一比 r 下降。
And potential is a scalar: a sign, but no direction — which is why it is easier than the field.
而且电势是标量：它有正负，却没有方向——这正是它比电场好用得多的原因。
Several charges?
有好几个电荷怎么办？
Add their potentials, with their signs. No components, no angles.
把它们的电势按符号相加，没有分量，也没有角度。
Take a point ten centimetres from a plus three nanocoulomb charge, and thirty centimetres from a minus three nanocoulomb charge.
取一个点，它离一个正三纳库仑的电荷十厘米，离一个负三纳库仑的电荷三十厘米。
The positive charge contributes plus two hundred and seventy volts. The negative charge contributes minus ninety volts.
正电荷贡献正的二百七十伏，负电荷贡献负的九十伏。
Add them: one hundred and eighty volts.
相加得到一百八十伏。
For a continuous charge distribution, cut the object into tiny pieces, treat each piece as a point charge, and integrate — superposition again, and because potential is a scalar, this integral is much friendlier than the one for field.
如果电荷是连续分布的，就把物体切成许多小块，把每一小块当作点电荷，然后积分。 因为电势是标量，这个积分比电场的积分友好得多。
Take a thin ring of charge and a point on its axis.
取一个细圆环，以及它轴线上的一点。
Every piece of the ring is the same distance from that point.
圆环上的每一小块到该点的距离都相同。
That distance is constant, so it comes out of the integral, and all that is left is the total charge.
这个距离是常数，所以可以提到积分号外面，剩下的就只是总电荷量了。
Saying that out loud is the marked step.
把这句话说出来，正是评分点。
The course asks for four shapes only: a wire, a ring, an arc, and a line.
课程只要求四种形状： 长直导线、轴线上的圆环、圆心处的圆弧，以及有限长的直线。
Before we read potential off a map, look at the field once more.
在学会从图上读出电势之前，先再看一次电场。
These arrows show which way a positive charge would be pushed at each point: away from the positive charge, toward the negative one.
这些箭头表示正电荷在每一点会被推向哪个方向：远离正电荷，指向负电荷。
That is one map of the space around these charges.
这是描述电荷周围空间的一张图。
The same physics has a second map, made of numbers instead of arrows.
而同样的物理还有第二张图， 它由数字构成，而不是箭头。
Here is that second map.
这就是第二张图。
An equipotential line, or isoline, joins the points that share one potential.
等势线——也叫等值线——把电势相同的点连起来。
Around a single charge they are circles.
在单个电荷周围，它们是一圈圈的圆。
Four rules read any map.
四条规则就能读懂任何一张图。
First, isolines join points of equal potential.
第一，等势线连接电势相等的点。
Second, they cross field lines at right angles.
第二，它们与电场线处处垂直。
Third, the field points from high potential to low potential — downhill.
第三，电场由高电势指向低电势，永远沿着下坡方向。
Fourth, along an isoline the field has no component, so moving a charge along one costs no work.
第四，沿等势线方向电场没有分量，所以沿它移动电荷不做功。
Closely spaced isolines mean a strong field.
等势线挨得越近，电场就越强。
Because the two maps always meet at right angles, you can draw either one from the other.
正因为两张图处处垂直，你可以由其中一张画出另一张。
Field and potential hold the same information; two operations move between them.
电场和电势携带同样的信息，有两种运算把它们联系起来。
One way: the field is the negative gradient of the potential — in any direction, the negative rate of change of V in that direction, which is the negative slope of the graph.
一个方向：某方向上的电场等于电势沿该方向变化率的负值，也就是图像斜率的负值。
The other way: the change in potential is minus the integral of the field along the path.
另一个方向：电势的变化等于电场沿路径积分的负值。
Look at the graph: where the potential drops steeply, the field is strong; where it is flat, the field is zero.
看这张图： 电势下降得越陡，电场就越强；电势平坦的地方，电场为零。
Two quick ones.
两个小例子。
Along the x-axis the potential is three x squared minus two x volts.
沿 x 轴，电势为三倍 x 的平方减去两倍 x 伏。
Find the field at fifty centimetres.
求五十厘米处的电场。
Pause here and try it.
先暂停，自己试一试。
Differentiate the potential and flip the sign: the field is two minus six x volts per metre.
把电势求导并取负号：电场等于二减去六倍 x 伏每米。
At x equals zero point five that is minus one point zero volts per metre, pointing the negative way.
代入 x 等于零点五，得到负一点零伏每米——它是负的，所以指向 x 的负方向。
Now the map version: isolines every ten volts, two centimetres apart.
再看读图的版本：等势线每隔十伏、相距两厘米。
Ten over zero point zero two is five hundred volts per metre, from the higher line to the lower one.
十除以零点零二等于五百伏每米，方向由数值高的线指向数值低的线。
Now conservation of electric energy. When a charge moves between two points at different potentials, the potential energy of the system — charge and field together — changes by the charge times the potential difference.
当电荷在电势不同的两点之间移动时， 电荷与电场组成的系统，其电势能的变化等于电荷量乘以电势差。
If only the electric force acts, the kinetic energy changes by the opposite amount.
如果只有电场力做功，动能的变化就恰好与它大小相等、符号相反。
Watch the two signs. A positive charge speeds up moving to lower potential.
注意这两个符号：正电荷向低电势移动时加速；负电荷向高电势移动时加速。
A negative charge speeds up moving to higher potential. Both are the same trade.
两者其实是同一笔交易：电势能变成动能。
A proton starts at rest and is accelerated through a potential drop of five hundred volts.
一个质子从静止开始，经过五百伏的电势降落被加速。
How fast does it end up?
它最后有多快？
Pause here and try it.
先暂停，自己试一试。
The kinetic energy gained is the charge times the potential difference: one point six times ten to the minus nineteen, times five hundred — eight times ten to the minus seventeen joules.
获得的动能等于电荷量乘以电势差： 一点六乘以十的负十九次方，再乘以五百，等于八乘以十的负十七次方焦耳。
Set that equal to one half m v squared, and the speed is about three point one times ten to the fifth metres per second.
令它等于二分之一乘以质量乘以速度的平方，解得速度约为三点一乘以十的五次方米每秒。
Where does this show up?
这些知识会在哪里出现？
Inside a battery, chemistry pushes positive and negative charge apart and holds the terminals at a fixed potential difference.
在电池内部，化学反应把正负电荷分开， 并让两个电极之间保持固定的电势差。
On the grid, power travels at hundreds of thousands of volts, because at a high potential difference the same energy rides on less charge.
在电网上，电能以几十万伏输送， 因为电势差越高，同样的能量所需要的电荷就越少。
And in an accelerator, every volt of drop hands the particle more kinetic energy.
在加速器里，每一伏的电势降落都会给粒子增加动能。
All those opening numbers were energy per unit charge.
我们一开始看到的那些数字，全都是单位电荷的能量。
Four habits that save marks.
四个能保住分数的习惯。
First, potential is a scalar — add contributions with their signs, and never resolve components.
第一，电势是标量——把各项按符号相加，绝不要分解分量。
Second, state where you put your zero of potential — usually at infinity.
第二，写明你把电势零点取在哪里；对点电荷来说通常取在无穷远。
Third, no work is done along an equipotential, and an equipotential is perpendicular to the field.
第三，沿等势面移动电荷不做功，而等势面与电场垂直。
Fourth, when the field is not uniform, go through energy, not kinematics.
第四，当电场不均匀时，走能量的路线——电荷量乘以电势差——而不是运动学。
Get those four, and this unit is yours.
做到这四点，这个单元就是你的了。