We have also, in effect, solved anelectrical problem: if we have an electrically charged plate, with an amount σof charge per unit area, thenthe electric field at a point outside the sheet is equal to σ/2ε0, and is in the outward direction if the sheet is positivelycharged, and inward if the sheet is negatively charged. To prove this, wemerely note that −G, for gravity, plays the same role as 1/4πε0 for electricity.
实际上，我们还解决了一个电的问题：如果我们有一个充电平板，每单位面积电量为σ，那么，在这个薄板外的某点的电场，就等于σ/2ε0，如果板子带正电，场的方向就向外，负则向内。要证明这点，我们只需注意，1/4πε0在电场中所扮演的角色，与−G在重力场中的一样。

Now suppose that we have two plates, with apositive charge +σ onone and a negative charge −σon another at a distance D from the first. What is the field?Outside the two plates it is zero. Why? Because one attracts and the other repels,the force being independent of distance, so that the two balance out! Also,thefield between the two plates is clearly twice as great as that from one plate,namelyE = σ/ε0, and is directed from the positive plate to the negative one.
现在，假设我们有两个板， 一个带电+σ，另一个带电−σ，与前一板的距离为D 。场是什么？在两个板子之外，场是零。为什么？因为一个吸引，一个排斥，而力独立于距离，所以两个平衡了！另外，一个板子的场强是E = σ/ε0，而板子之间的场，很明显是其两倍，方向是从正板到负板。

Now we come to a most interesting andimportant problem, whose solution we have been assuming all the time, namely,that the force produced by the earth at a point on the surface or outside it isthe same as if all the mass of the earth were located at its center. Thevalidity of this assumption is not obvious, because when we are close, some ofthe mass is very close to us, and some is farther away, and so on. When we addthe effects all together, it seems a miracle that the net force is exactly the sameas we would get if we put all the mass in the middle!
现在，我们面临的问题，最有趣也最重要，其解决方案，我们一直是假定了的，即由地球，在地球表面、或地球外，所产生的力，与地球的质量好像全都集中在其中心时，所产生的力，是一样的。这个假设的有效性，并不明显，因为，当我们离得近时，有些质量距我们很近，有些则很远，如此等等。当我们把所有的影响都加在一起，似乎有一个奇迹，即净力，与我们把所有质量都放在中心时{的净力}，几乎完全一样。

Fig. 13-6. A thin spherical shell of massor charge.图3-6 一个薄的球形的质量壳或电壳。
We now demonstrate the correctness of thismiracle. In order to do so,however, we shall consider a thin uniform hollow shellinstead of the whole earth. Let the total mass of the shell be m, and let uscalculate the potential energy of a particle of mass m’ a distance R away fromthe center of the sphere (Fig. 13-6) and show that the potential energy is thesame as it would be if the mass m were a point at the center. (The potentialenergy is easier to work with than is the field because we do not have to worryabout angles, we merely add the potential energies of all the pieces of mass.)If we call x the distance of a certain plane section from the center, then allthe mass that is in a slice dx is at the same distance r from P, and thepotential energy due to this ring is −Gm’ dm/r. How much mass is in thesmall slice dx? An amount
(13.17.1)
where μ = m/4πa2 is the surface density of mass onthe spherical shell. (It is a general rule that the area of a zone of a sphereis proportional to its axial width.)Therefore the potential energy due to dm is
(13.17.2)
But we see that
r2 = y2 + (R − x)2 = y2 +x2 + R2 − 2Rx
= a2 + R2 − 2Rx.
Thus
2r dr = −2Rdx
Or
dx/r= −dr/R.
Therefore,
dW = Gm’2πaμ dr / R ,
and so
(13.18)
Thus, for a thin spherical shell, the potentialenergy of a mass m’, external to the shell, is the same as though the mass ofthe shell were concentrated at its center. The earth can be imagined as a seriesof spherical shells, each one of which contributes an energy which depends onlyon its mass and the distance from its center to the particle; adding them alltogether we get the total mass, and therefore the earth acts as though all thematerial were at the center!
我们现在演示这个奇迹的正确性。然而，为了这么做，我们将考虑一个薄的、均匀的空壳，而不是整个地球。设壳的总质量为m，另有质量为m’的粒子，距球心的距离为R（图13-6），让我们计算其势能，并指出，这个势能，与质量m全集中在球心时的势能，是一样的。（势能比场好处理，因为我们不用担心角度，只需把所有小的质量块的势能，加在一起就行。）如果我们称x，为某个平面到中心的距离，那么，在切片dx上的所有质量，到P的距离，都为r，可归于这个环的势能就是−Gm’ dm/r。小的切片dx上，有多少质量呢？一个量： (13.17.1)
这里，是球壳表面的质量密度。（一个球形地带的面积，正比于它的轴线的宽度，这是一条普遍规则。）因此，归于dm的势能就是：
(13.17.2)
但我们可以看到
r2 = y2 + (R − x)2 = y2 +x2 + R2 − 2Rx
= a2 + R2 − 2Rx.
这样
2r dr = −2Rdx
或者
dx/r= −dr/R.
因此
dW = Gm’2πaμ dr / R ,
于是
(13.18)
这样，对于一个薄的球形壳，一个处于壳外的质量m’，其势能，与壳的质量全集中在中心时的势能，是一样的。地球可以被想象为一系列球形壳，每一个都贡献一份能量，此能量，只依赖于该壳的质量，及其中心与粒子的距离；把这些质量全加起来，我们得到总质量，因此，地球的表现，就与其所有材料都在中心，是一样的。

But notice what happens if our point is onthe inside of the shell. Making the same calculation, but with P on the inside,we still get the difference of the two r’s, but now in the form a−R−(a+R) = −2R,or minus twice the distance from the center. In other words, W comes out to be W= −Gm’m/a, which is independent of R and independent of position, i.e., thesame energy no matter where we are inside. Therefore no force; no work is donewhen we move about inside. If the potential energy is the same no matter wherean object is placed inside the sphere, there can be no force on it. So there isno force inside, there is only a force outside, and the force outside is thesame as though the mass were all at the center.
但是，如果我们的点，在壳内，要注意会发生什么。做同样的计算，但P在里面，我们将得到2个r的不同，但现在，在形式a−R−(a+R) = −2R中，或者是负的到中心距离的两倍。换句话说，W就是W = −Gm’m/a，独立于R和位置，亦即，无论我们在里面的什么地方，能量是一样的。因此，当我们在壳里面运动时，没有力；没有做功。如果无论对象被放在球内的什么地方，势能都是零，那么，就没有力作用于其上。所以，里面没有力，只在外面有力，且外面的力，与质量都集中在中心时的力，是一样的。

1 Chapter14. Workand Potential Energy(conclusion)第14章 功与势能（结论）
14–1Work 功
In the preceding chapter we have presented agreat many new ideas and results that play a central role in physics. These ideasare so important that it seems worthwhile to devote a whole chapter to a closerexamination of them. In the present chapter we shall not repeat the “proofs” orthe specific tricks by which the results were obtained, but shall concentrateinstead upon a discussion of the ideas themselves.
在上一章，我们提供了大量新的想法和结果，它们在物理学中，扮演着最重要的角色。这些想法，如此重要，以至于似乎值得用一章，来对它们进行更深的检查。上一章中，通过一些“证明”和技巧，我们得到了那些结果，本章中，我们将不重复这些证明等，而将集中在想法本身的讨论上。

In learning any subject of a technicalnature where mathematics plays a role, one is confronted with the task ofunderstanding and storing away in the memory a huge body of facts and ideas,held together by certain relationships which can be “proved” or “shown” to existbetween them. It is easy to confuse the proof itself with the relationship whichit establishes. Clearly, the important thing to learn and to remember is the relationship,not the proof. In any particular circumstance we can either say “it can be shownthat” such and such is true, or we can show it. In almost all cases, theparticular proof that is used is concocted, first of all, in such form that itcan be written quickly and easily on the chalkboard or on paper, and so that itwill be as smooth-looking as possible. Consequently, the proof may look deceptivelysimple, when in fact, the author might have worked for hours trying differentways of calculating the same thing until he has found the neatest way, so as tobe able to show that it can be shown in the shortest amount of time! The thingto be remembered, when seeing a proof, is not the proof itself, but rather thatit can be shown that such and such is true. Of course, if the proofinvolves some mathematical procedures or “tricks” that one has not seen before,attention should be given not to the trick exactly, but to the mathematicalidea involved.
任何主题，如果它有技术性质，且数学在其中，扮演着重要角色，那么，在学习这种主题时，人们面对的任务，就是理解和记忆大量的事实和想法，这些事实和想法，被某些关系联系在一起，这些关系，可以“被证明”或“被指出”：是存在于这些事实和想法之间的。把证明本身，与其所建立的关系，弄混很容易。{有一点}很清楚，需要学习和记忆的事情，是那些关系，而不是证明。在任何具体的情形中，我们要么可以说，“可以指出”什么什么是真的，要么，我们可以指出它{？}。几乎在所有的案例中，被用到的具体的证明，都是编造出来的，首先，以这样的形式，它可以被快速地和容易地写在黑板或纸上，这样，它看上去就尽可能地顺溜。所以，证明看上去，可能会很简单，甚至有欺骗倾向，而事实上，作者可能花了很多时间，试了各种不同的方法，来计算同一个事情，直到找到了最清晰的方法，以便能够指出，该事情，可以在最短的时间内被指出来。当你看到一个证明，需要记住的事情，并不是证明本身，而不如说，这个证明告诉了我们：什么什么是真的。当然，如果证明牵扯到某些数学的过程、或你以前未曾见过的“技巧”，那么，需要注意的，恰恰不是这些技巧，而是所牵扯到的那些数学想法。

It is certain that in all thedemonstrations that are made in a course such as this, not one has been rememberedfrom the time when the author studied freshman physics. Quite the contrary: hemerely remembers that such and such is true, and to explain how it can be shownhe invents a demonstration at the moment it is needed. Anyone who has really learneda subject should be able to follow a similar procedure, but it is no useremembering the proofs. That is why, in this chapter, we shall avoid the proofsof the various statements made previously, and merely summarize the results.
在一个类似本讲座的课程中，会有很多演证；从作者开始研究一年级物理学起，没有一个演证，会被记得，这一点，是肯定的。完全相反的是，他只记住了：什么什么是真的，为了解释，这个真的，如何才能被指出，他在需要时候，发明了演证。任何人，如果他真的学了某个学科的话，应该能够走类似的过程，但是，记住证明，是没有用处的。这就是为什么，在本章中，我们将避免，前面已经做过的各种陈述的证明，而只是总结结果。

The first idea that has to be digested is workdone by a force. The physical word “work” is not the word in the ordinarysense of “Workers of the world unite!,” but is a different idea. Physical workis expressed as ∫F⋅ds , called “the line integral of F dot ds ,” which means that if the force, for instance, is in one directionand the object on which the force is working is displaced in a certaindirection, then only the component of force in the direction of the displacementdoes any work. If, for instance, the force were constant and the displacementwere a finite distance Δs , then the work done in moving the object through that distance isonly the component of force along Δs times Δs . The rule is “force times distance,” but we really mean only thecomponent of force in the direction of the displacement times Δs or, equivalently, the component of displacement in the direction offorce times F . It is evident that no work whatsoever is done by a force which is atright angles to the displacement.
第一个要消化的概念，就是力所做的功。物理学中的词汇功，并不是普通意义上的：“世界联合工会的工人”中的意思，而是不同的概念。物理学中的功，被表示为∫F⋅ds，被称为“F点积 ds的线性积分”，它意味，比如，如果力在一个方向，而这个力所作用的对象，在某方向发生了位移，那么，只有在位移方向上的力的分量，才做了功。例如，如果力是常数，而位移是一个有限的距离 Δs，那么，在移动对象通过此距离时所做的功，就只是沿着 Δs的力的分量乘以 Δs。规则就是“力乘以距离”，但是，我们的意思只是，位移方向上的力的分量，乘以 Δs，或者，可同等地说，力的方向上的位移分量，乘以F。很明显，如果力垂直于位移，那么，力就没有做功。

Now if the vector displacement Δsis resolved into components, in other words, if the actualdisplacement is Δs and we want to consider it effectively as a component of displacement Δxin the x -direction, Δy in the y -direction, and Δz in the z -direction, then the work done in carrying an object from one place toanother can be calculated in three parts, by calculating the work done along x, along y , and along z . The work done in going along x involves only that component of force, namely Fx, and so on, so the work is FxΔx+FyΔy+FzΔz. When the force is not constant, and we have a complicated curved motion,then we must resolve the path into a lot of little Δs ’s, add the work done in carrying the object along each Δs, and take the limit as Δs goes to zero. This is the meaning of the “line integral.”
现在，如果矢量位移Δs 被分解成分量，换句话说，如果实际的位移是 Δs ，而我们想把它有效地考虑为：x方向的位移分量Δx ，y方向的位移分量Δy ，z方向的位移分量Δz ，那么，把一个对象，从一个地方移动到另一个地方所做的功，就可分成三部分来计算，即通过计算沿着x方向、y方向、和z方向所做的功。沿着x方向所做的功，只包含那个方向的力的分量，即 Fx , 如此等等，所以，功就是 FxΔx+FyΔy+FzΔz。当力不是常数，而我们的运动曲线又很复杂时，那么，我们就必须把路径分解成很多小的 Δs，把‘携对象沿着 Δs 所做的功’加起来，取Δs 趋向于零的极限。这就是“线性积分”的意思。

Everything we have just said is containedin the formula W=∫F⋅ds . It is all very well to say that it is a marvelous formula, but it isanother thing to understand what it means, or what some of the consequencesare.
我们刚才所说的一切，都包含在公式W=∫F⋅ds中。说它是一个了不起的公式，很好，完全没有问题，但是，理解其意义、或一些后果，则是另一回事，

The word “work” in physics has a meaning sodifferent from that of the word as it is used in ordinary circumstances that itmust be observed carefully that there are some peculiar circumstances in whichit appears not to be the same. For example, according to the physicaldefinition of work, if one holds a hundred-pound weight off the ground for awhile, he is doing no work. Nevertheless, everyone knows that he begins to sweat,shake, and breathe harder, as if he were running up a flight of stairs. Yetrunning upstairs is considered as doing work (in running downstairs,one gets work out of the world, according to physics), but in simply holding anobject in a fixed position, no work is done. Clearly, the physical definitionof work differs from the physiological definition, for reasons we shall brieflyexplore.
“功”这个词，在物理学中的意义，与在普通情况中的意义，完全不同，所以应该仔细观察，因为在有些具体情况下，意义似乎不一样。例如，依据物理学对功的定义，如果有人把100磅重量，（从地面上抱起来），抱一会儿，他并没有做功。尽管如此，每个人都知道，他要开始出汗，发抖，和气喘吁吁了，就好像他跑了一段楼梯一样。当然，向上跑楼梯，被认为是做了功（向下跑楼梯，依据物理学，是从世界上拿走功），但是，拿着一个东西，静止不动，没有做功。很清楚，功的物理定义，与生理定义不同，其原因，我们将做简单探索。

It is a fact that when one holds a weighthe has to do “physiological” work. Why should he sweat? Why should he need to consumefood to hold the weight up? Why is the machinery inside him operating at fullthrottle, just to hold the weight up? Actually, the weight could be held up withno effort by just placing it on a table; then the table, quietly and calmly,without any supply of energy, is able to maintain the same weight at the sameheight! The physiological situation is something like the following. There aretwo kinds of muscles in the human body and in other animals: one kind, called striatedor skeletal muscle, is the type of muscle we have in our arms, forexample, which is under voluntary control; the other kind, called smoothmuscle, is like the muscle in the intestines or, in the clam, the greater adductormuscle that closes the shell. The smooth muscles work very slowly, but they canhold a “set”; that is to say, if the clam tries to close its shell in a certainposition, it will hold that position, even if there is a very great forcetrying to change it. It will hold a position under load for hours and hours withoutgetting tired because it is very much like a table holding up a weight, it “sets”into a certain position, and the molecules just lock there temporarily with nowork being done, no effort being generated by the clam. The fact that we haveto generate effort to hold up a weight is simply due to the design of striatedmuscle. What happens is that when a nerve impulse reaches a muscle fiber, thefiber gives a little twitch and then relaxes, so that when we hold something up,enormous volleys of nerve impulses are coming in to the muscle, large numbersof twitches are maintaining the weight, while the other fibers relax. We can seethis, of course: when we hold a heavy weight and get tired, we begin to shake.The reason is that the volleys are coming irregularly, and the muscle is tiredand not reacting fast enough. Why such an inefficient scheme? We do not knowexactly why, but evolution has not been able to develop fast smoothmuscle. Smooth muscle would be much more effective for holding up weights becauseyou could just stand there and it would lock in; there would be no work involvedand no energy would be required. However, it has the disadvantage that it isvery slow-operating.
当一个人提起一个重量时，他必须要做生理上的功，这是一个事实。为什么他会出汗？为什么他需要消耗食物，以提起这个重量？为什么它里面的那个机制，要开足马力，只是为了拿起这个重量？实际上，这个重量，可以被拿起来，放在桌子上，没有任何其他效果；然后，这个桌子，可以默默无闻地，把这个重量，保持在这个高度，无需任何能量支持！生理的情况，大致如下。在人类和其他动物的身体中，有两类肌肉，一类被称为条纹肌或骨骼肌，这类肌肉，可随意控制，例如，我们的膀子上的肌肉；另一类，被称为平滑肌，如肠中的肌肉，或者，蚌的肌肉，最大的闭壳肌，用来关闭壳的。平滑肌工作很慢，但是，它们可以举起一个“集”{？}，也就是说，如果蚌尝试在一个位置，关掉它的壳，它会保持那个位置，即便那些非常大的力量，想改变之{也不行}。这种肌肉，在负载下，能保持这个位置，达数小时而不会变得疲倦，因为，这很像桌子，载着重量，它会设置到一定的位置，分子只是暂时地锁在那里，没做任何功，蚌并没有做任何努力。我们必须努力，才能举起一个重量，这一事实，可简单地要归于条纹肌的设计安排。实际发生的是这样，当神经脉冲达到肌纤维的时候，纤维有一个小的抽动，然后放松，这样当我们提起某物时，巨大的神经脉冲齐射，就会来到肌肉，大量的抽动，就会维持这个重量，而其他的纤维则会放松。当然，我们可以看到这一点：当我们拿起一个非常重的东西时，就会变得疲劳，然后，就会颤抖。原因就是，齐射不规则地到来，而肌肉变得疲劳，反应也不够快。为什么是这样一种低效的方案？我们并不准确地知道为什么，但是，进化还没有发展出快的平滑肌。对于拿起重量来说，平滑肌可以更有效，因为，你只需站在那里，他就会锁住{？}，不会牵扯到功，对能量也没有要求。然而，它也有的不利的一点，即它是慢操作型的。

Returning now to physics, we may ask whywe want to calculate the work done. The answer is that it is interesting anduseful to do so, since the work done on a particle by the resultant of all theforces acting on it is exactly equal to the change in kinetic energy of thatparticle. That is, if an object is being pushed, it picks up speed, and
Δ(v2)=2F⋅Δs /m.
现在，返回物理学。我们可以问，为什么我们想计算所做的功。答案是，这样做，既有用又有趣，由于一个粒子，可以受很多力，这些力的合力所做的功，准确地等于其动能的变化。也就是说，如果一个对象被推了，它会得到速度，且
Δ(v2)=2F⋅Δs /m.