How careful do we have to be to eliminateaberrations? Is it possible to make an absolutely perfect optical system?Suppose we had built an optical system that is supposed to bring light exactlyto a point. Now, arguing from the point of view of least time, can we find acondition on how perfect the system has to be? The system will have some kindof an entrance opening for the light. If we take the farthest ray from the axisthat can come to the focus (if the system is perfect, of course), the times forall rays are exactly equal. But nothing is perfect, so the question is, howwrong can the time be for this ray and not be worth correcting any further?That depends on how perfect we want to make the image. But suppose we want to makethe image as perfect as it possibly can be made. Then, of course, ourimpression is that we have to arrange that every ray takes as nearly the sametime as possible. But it turns out that this is not true, that beyond a certainpoint we are trying to do something that is too fine, because the theory ofgeometrical optics does not work!
要消除偏差，我们该多么仔细呢？要建造一个绝对完美的光学系统，可能吗？假设我们已经建造了一个光学系统，且认为它，可以把光准确地带到一个点。现在，从最短时间的观点出发，来论证：对于‘一个系统可以有多么完美’这种问题，我们能找到一个条件，来判断，系统是否达到吗？{？}这个系统将有某种入口，对光开放。如果我们沿着轴，选取能来到焦点的最远的光线（当然，如果系统是完美的话），那么，对于所有的光线，时间就是准确相等的。但是，没有任何事情，是完美的，所以，问题就是，对于这个光线，时间的错误，可以有多大，且不值得进一步去修改了？这依赖于：我们希望图像，有多完美。但是，假设我们是想让图像，尽可能地完美。那么，当然，我们的印象就是，我们必须让每一条光线的路径，时间上尽可能一样。但结果则是，这并不是真实的，因为，我们想把事情做好，是有一定范围的，这个要求，已经超过了这个范围，因为，几何光学的理论，{在这里}并不起作用。

Remember that the principle of least timeis not an accurate formulation, unlike the principle of conservation of energyor the principle of conservation of momentum. The principle of least time isonly an approximation, and it is interesting to know how much error canbe allowed and still not make any apparent difference. The answer is that if wehave arranged that between the maximal ray—the worst ray, the ray that is farthestout—and the central ray, the difference in time is less than about the periodthat corresponds to one oscillation of the light, then there is no useimproving it any further. Light is an oscillatory thing with a definitefrequency that is related to the wavelength, and if we have arranged that thetime difference for different rays is less than about a period, there is no usegoing any further.
要记住，最短时间原理，并不像能量守恒原理、或动量守恒原理那样，是一个准确的公式。它只是一种近似，知道可以允许多少错误、且不会带来明显差别，这将非常有趣。答案就是，如果最大光线，就是最差的光线，或是最远的光线；那么，如果我们的安排，能让最大光线，与中间光线，在时间上的差别，比光的一个震荡周期，要短，那么，进一步的改进，就没有用。光是一个震荡的事物，具有确定的频率，且频率与波长相关，如果我们的安排，能让不同光线之间的时间差，大约小于一个周期，那么，进一步的工作，就没有用。

27–7Resolving power 27-7 分辨率
Another interesting question—a veryimportant technical question with all optical instruments—is how much resolvingpower they have. If we build a microscope, we want to see the objects thatwe are looking at. That means, for instance, that if we are looking at abacterium with a spot on each end, we want to see that there are twodots when we magnify them. 对于所有的光学设备，还有另外一个有趣的问题，这也是一个非常重要的技术问题，即它们的分辨率，是多少。如果我们建一个显微镜，那么，我们就是想看到，我们正在看的事物。这就意味着，比如，如果我们正在看一个细菌，其两端，各有一个点，那么，当我们把它们放大时，我们想看到两个圆点。 One might think that all we have to do is to get enoughmagnification—we can always add another lens, and we can always magnify againand again, and with the cleverness of designers, all the spherical aberrationsand chromatic aberrations can be cancelled out, and there is no reason why wecannot keep on magnifying the image. 有人可能会想，我们所要做的，就只是把它放的足够大—利用设计者的聪明才智，我们总是可以增加另外的透镜，我们总是可以一次又一次地放大，所有的球差和色差，都可被消去；为什么我们不能把图像不断地放大呢？好像没有理由吧。So the limitations of a microscope are not that it is impossible tobuild a lens that magnifies more than 2000 diameters. We can build asystem of lenses that magnifies 10,000 diameters, but we stillcould not see two points that are too close together because of the limitationsof geometrical optics, because of the fact that least time is not precise.
于是，显微镜的限制，并不是说：建一个透镜，能把直径，放大2000多倍，是不可能的。我们可以建一个透镜系统，它可以把直径，放大10，000倍，但是，我们仍然看不到，那两个离得很近的点，这是因为，几何光学的限制，及最短时间并不精确这一事实.

Fig. 27–9.The resolving power of an opticalsystem. 图27-9 一个光学系统的分辨率。
To discover the rule that determines howfar apart two points have to be so that at the image they appear as separatepoints can be stated in a very beautiful way associated with the time it takesfor different rays. 图像中的两个点，要离开一定的距离，这样，才能显得是分开的，要规定这个距离，需要一条规则，而要发现此规则，可以用一种非常漂亮的方式来陈述，而此方式，则与不同光线所花时间，有联系。Suppose that we disregard the aberrations now, and imagine that fora particular point P (Fig. 27–9) allthe rays from object to image T take exactly the same time. (It isnot true, because it is not a perfect system, but that is another problem.) Nowtake another nearby point, P′ , and ask whether its image will bedistinct from T . In other words, whether we can make out the differencebetween them. 现在，假设我们不管偏差，而去想象，对于一个具体的点P（图27-9），从对象到图像T的所有光线，所花时间，基本一样（这不是真的，因为它不是一个完美的系统，但这是另外一个问题。）现在，在附近，另取一点P′，问其图像，是否可区别于T？换句话说，我们是否可以把这个差别，做出来。Ofcourse, according to geometrical optics, there should be two point images, butwhat we see may be rather smeared and we may not be able to make out that thereare two points. 当然，依据几何光学，应该是有两个点的图像，但是，我们看到的，则更可能是一团模糊，且我们无法搞清楚，这里有两个点。The condition that the second point is focused in a distinctlydifferent place from the first one is that the two times for the extreme rays P′STand P′RT on each side of the big opening of the lenses to gofrom one end to the other, must not be equal from the two possibleobject points to a given image point. 第二个点聚焦的位置，要与第一个点的，明显不同，这要求一个条件，那就是：两条极端光线P′ST 和 P′RT，经过透镜的两端，从轴的这端，走向那端，这两条光线的时间，应该不相当于：两个可能的对象点，到一个被给予的图像点。{？} Why? Because, if thetimes were equal, of course both would focus at the same point. So thetimes are not going to be equal. But by how much do they have to differ so thatwe can say that both do not come to a common focus, so that we candistinguish the two image points? 为什么？因为，如果时间是相等的，那么当然，两条光线，就应该聚焦在同一个点。所以，时间将不会相等。但是，它们应该差多少，才能让我们就可以说，两条光线不会来到同一个焦点呢，这样，我们就可以区分这两个图像的点？The general rule for the resolution of any optical instrument isthis: two different point sources can be resolved only if one source is focusedat such a point that the times for the maximal rays from the other source toreach that point, as compared with its own true image point, differ by morethan one period. It is necessary that the difference in time between the topray and the bottom ray to the wrong focus shall exceed a certain amount,namely, approximately the period of oscillation of the light:
任何光学设备，都有分辨率，其规则就是：对于两个不同的点源，要想分辨它们，只有这样：设一个源的光线，聚焦在这样一个点上，使得从另外一个源，到该点的最大光线所花时间，与它的真正的图像点相比，要差一个周期以上。到达错误焦点的，有两条光线，一条走顶部，一条走底部，这两条光线的时间差，有必要超过一定的量，亦即，大约就是光的震荡周期：
t2−t1>1/ν, (27.17)
where ν is the frequency of thelight (number of oscillations per second; also speed divided by wavelength). 这里ν是光的频率（每秒振荡的数目；也是速度除以波长。）Ifthe distance of separation of the two points is called D , and if theopening half-angle of the lens is called θ , then one can demonstratethat (27.17)is exactly equivalent to the statement that D must exceed λ/2nsinθ, where n is the index of refraction at P and λ isthe wavelength. The smallest things that we can see are therefore approximatelythe wavelength of light. A corresponding formula exists for telescopes, whichtells us the smallest difference in angle between two stars that can just bedistinguished.1
1. The angle is about λ/D , where D is the lensdiameter. Can you see why?
如果两个点被分开的距离，被称为D，且如果透镜张开的角的一半，被称为θ，那么，我们就可以演证，（27.17），与‘D应该超过λ/2nsinθ’这一陈述，完全等价，这里，n是P点的折射指数，λ是波长。因此，我们所能看到的最小的事物，大约就是光的波长。对于望远镜，存在相应的公式，它告诉我们，两个恒星，要被区分开，所需要的角度上的最小差别。（脚注1）{？pdf中，是λ/nsinθ}
脚注1：这个角度，大约就是λ/D，这里D是透镜的直径。你能看出为什么吗？

Chapter28. Electromagnetic Radiation第28章 电磁辐射
28–1Electromagnetism 28-1 电磁学
The most dramatic moments in thedevelopment of physics are those in which great syntheses take place, wherephenomena which previously had appeared to be different are suddenly discoveredto be but different aspects of the same thing. The history of physics is thehistory of such syntheses, and the basis of the success of physical science ismainly that we are able to synthesize.
在物理学的发展中，最有戏剧性的瞬间，就是伟大综合发生之时，彼时，以前看上去不同的现象，突然间被发现，只是同一个事物的不同表现。物理学的历史，就是这种综合的历史，物理科学成功的基础，主要就是：我们能够去综合。

Perhaps the most dramatic moment in thedevelopment of physics during the 19th century occurred to J. C.Maxwell one day in the 1860s, when he combined the laws of electricity andmagnetism with the laws of the behavior of light. As a result, the propertiesof light were partly unravelled—that old and subtle stuff that is so importantand mysterious that it was felt necessary to arrange a special creation for itwhen writing Genesis. Maxwell could say, when he was finished with hisdiscovery, “Let there be electricity and magnetism, and there is light!”
或许，在19世纪物理学的发展中，最有戏剧性的时刻，就是在1860年代的某一天，发生在麦克斯韦身上的，即当他的把电和磁的规律，与光的表现的规律，结合在一起之时。作为结果，光的属性，被部分地揭示了—那些老的和不易察觉的事情，是如此重要和神秘，以至于对于它们，在写《创世纪》时，都有必要安排一个特殊的创造{章节}。当麦克斯韦完成了其发现时，他可以说：“让电来，让磁来，然后，光也来了”。{模仿圣经}

For this culminating moment there was along preparation in the gradual discovery and unfolding of the laws ofelectricity and magnetism. This story we shall reserve for detailed study nextyear. However, the story is, briefly, as follows. The gradually discoveredproperties of electricity and magnetism, of electric forces of attraction andrepulsion, and of magnetic forces, showed that although these forces wererather complex, they all fell off inversely as the square of the distance. Weknow, for example, that the simple Coulomb law for stationary charges is thatthe electric force field varies inversely as the square of the distance. 在达到这一顶点时刻之前，电和磁的规律，是逐步被发现和展开的，这个准备期很长。这个故事的细节，我们保留到明年再讲。然而，简明扼要地说，这个故事，就是这样。电和磁的一些属性、电力的吸引和排斥的属性、即磁力的属性，是逐渐被发现的，它们指出了，虽然这些力，相当复杂，但是，它们全与距离的平方，成反比。例如，我们知道，对于净电荷的简单的库伦规律，就是电场的变化，与距离的平方成反比。 As a consequence, for sufficiently great distances there is verylittle influence of one system of charges on another. Maxwell noted that theequations or the laws that had been discovered up to this time were mutuallyinconsistent when he tried to put them all together, and in order for the wholesystem to be consistent, he had to add another term to his equations. With thisnew term there came an amazing prediction, which was that a part of theelectric and magnetic fields would fall off much more slowly with the distancethan the inverse square, namely, inversely as the first power of the distance!And so he realized that electric currents in one place can affect other chargesfar away, and he predicted the basic effects with which we are familiartoday—radio transmission, radar, and so on.
作为一种后果，当距离足够大时，一个电荷系统，对另一个的影响，就会很小。对于那时所发现的方程或规律，麦克斯韦发现，当他尝试把它们放在一起时，它们相互并不一致，为了让整个系统，能够一致，他就必须给他的方程组，另增一项。这个新项，带来的预测，令人惊奇，那就是，电场与磁场的一部分，随着距离的变化，与平方反比相比，下降地很慢，即只是与距离的一次方成反比。于是，他意识到，在一个地方的电流，可以影响很远地方的电荷，他预测了一个基本的作用，这我们今天都很熟悉—无线电发射、雷达等。

It seems a miracle that someone talking inEurope can, with mere electrical influences, be heard thousands of miles awayin Los Angeles. How is it possible? It is because the fields do not vary as theinverse square, but only inversely as the first power of the distance. Finally,then, even light itself was recognized to be electric and magnetic influencesextending over vast distances, generated by an almost incredibly rapidoscillation of the electrons in the atoms. All these phenomena we summarize bythe word radiation or, more specifically, electromagnetic radiation,there being one or two other kinds of radiation also. Almost always, radiationmeans electromagnetic radiation.
只借助于电的影响，一个人在欧洲说话，就会在千里之外的洛杉矶，被听到，这似乎是一个奇迹。这是如何可能的呢？这是因为，场并不是与距离的平方成反比，而只是与距离的一次方成反比。因此，最终，甚至光本身，都被认为，是电和磁的影响，此影响，产生于原子中的电子的不可思议的高速震荡，然后，在广阔的距离上扩展。所有这些现象，我们总结为一个词：辐射，或者，更具体点，就是电磁辐射。只有一种或两种辐射。辐射，几乎总是意味着电磁辐射。

And thus is the universe knit together. Theatomic motions of a distant star still have sufficient influence at this greatdistance to set the electrons in our eye in motion, and so we know about thestars. If this law did not exist, we would all be literally in the dark aboutthe exterior world! And the electric surgings in a galaxy five billion lightyears away—which is the farthest object we have found so far—can stillinfluence in a significant and detectable way the currents in the great “dish”in front of a radio telescope. And so it is that we see the stars and thegalaxies.
宇宙就是这样，被编织在一起。一个遥远恒星上的原子的运动，在这么巨大的距离处，仍有充分的影响，可已让我们眼睛中的电子，进行运动，于是，我们就知道了这些恒星。如果这个规律不存在，那么，关于外部世界，我们理论上就是处于黑暗中。50亿光年，是目前我们能找到对象的最远距离；射电望远镜前，有巨大的“盘子”，盘中有电流；银河系中、50亿光年之外的电子浪涌，仍能以一种明显的、可探测到的方式，影响此电流。我们就是这样，看到恒星和银河系的。

This remarkable phenomenon is what we shalldiscuss in the present chapter. At the beginning of this course in physics weoutlined a broad picture of the world, but we are now better prepared tounderstand some aspects of it, and so we shall now go over some parts of itagain in greater detail. We begin by describing the position of physics at theend of the 19th century. All that was then known about the fundamentallaws can be summarized as follows.
这个引人注目的现象，就是在当前这一章，我们要讨论的。在这个物理课程的开始部分，我们勾画了世界的广阔图景，然而现在，对于理解世界的某些方面，我们已经准备的更充分了，所以现在，我们将更仔细地重温某些部分。我们从描述物理学在19世纪末的位置，开始。那时，关于基础规律所知道一切，可总结如下。

First, there were laws of forces: one forcewas the law of gravitation, which we have written down several times; the forceon an object of mass m , due to another of mass M , isgiven by
首先，有力的规律：一个力，就是万有引力的规律，这我们已经写了好几次了；可归于质量M，而作用于质量为 m的对象上的力，就是：
(28.1)
where er isa unit vector directed from m to M , and r is thedistance between them.
这里，er是从m到 M的单位向量，r是两者之间的距离。

Next, the laws of electricity andmagnetism, as known at the end of the 19th century, are these: theelectrical forces acting on a charge q can be described by twofields, called E and B , and thevelocity v of the charge q , by the equation
其次，是电和磁的规律，在19世纪末，我们知道了它们，即：作用于电荷q上的力，可以用两个场来描述，它们被称为E和B，且电荷q的矢速v，通过下面的方程来描述：
F=q(E+v×B). (28.2)
To complete this law, we have to say whatthe formulas for E and B are in a givencircumstance: if a number of charges are present, E and the Bare each the sum of contributions, one from each individual charge. So if wecan find the E and B produced by a singlecharge, we need only to add all the effects from all the charges in theuniverse to get the total E and B ! This isthe principle of superposition.
要完成这条规律，我们必须说，对于E和B的公式，是在一个被给予的情形中：如果若干电荷在场，E和B，就分别为若干贡献之和，每一个贡献，来自一个单独的电荷。所以，如果我们可以找到一个独立电荷所产生的E和B，那么，我们就只需要把宇宙中所有电荷的效果，加起来，就可以得到总的E和B！这就是叠加的原理。

What is the formula for the electric andmagnetic field produced by one individual charge? It turns out that this isvery complicated, and it takes a great deal of study and sophistication toappreciate it. But that is not the point. We write down the law now only toimpress the reader with the beauty of nature, so to speak, i.e., that it is possibleto summarize all the fundamental knowledge on one page, with notations that heis now familiar with. This law for the fields of an individual charge iscomplete and accurate, so far as we know (except for quantum mechanics) but itlooks rather complicated. 由一个单独的电荷所产生的电场和磁场公式，是什么呢？结果显示，这很复杂，需要大量的研究、和复杂的技巧，来鉴赏它。但这还不是问题的关键。我们现在把规律写出来，只是想让读者，对大自然的美，有所印象，这么说吧，用读者现在已经熟悉的表示法，把所有的基础知识，总结到一张纸上，是可能的。就我们所知（除量子力学外），关于一个独立电荷所产生的电场的规律，是完整的和准确的，但是，它看上去，相当复杂。We shall not study all the pieces now; we only write it down to givean impression, to show that it can be written, and so that we can see ahead oftime roughly what it looks like. As a matter of fact, the most usefulway to write the correct laws of electricity and magnetism is not the way weshall now write them, but involves what are called field equations,which we shall learn about next year. But the mathematical notations for theseare different and new, and so we write the law in an inconvenient form forcalculation, but in notations that we now know.
现在，我们不会研究所有的部分；我们写下它，只是为了给个印象，以表示，它是可以写出来的，这样，我们就可以提前粗略地看到，它长什么样子。事实上，对于电场和磁场的正确规律，我们书写它们的方式，并不是最有用的方式，那些最有用的方式，包含了我们被称为场方程的东西，这些方程，我们大概明年才学。但是，关于这些方程的数学表示法，是不同的和新的，所以，我们写这些规律，用的不是便于计算的形式，而是现在已知的表示法。

The electric field, E , isgiven by
电场E，由下式给出：
(28.3)

What do the various terms tell us? Take thefirst term, E=−qer′/4πϵ0r′2. That, of course, is Coulomb’s law, which we already know: q is thecharge that is producing the field; er′is the unit vector in the direction from the point P where Eis measured, r is the distance from P to q . But,Coulomb’s law is wrong. The discoveries of the 19th century showed thatinfluences cannot travel faster than a certain fundamental speed c, which we now call the speed of light. It is not correct that the first termis Coulomb’s law, not only because it is not possible to know where the chargeis now and at what distance it is now, but also because the onlything that can affect the field at a given place and time is the behavior ofthe charges in the past. How far in the past? The time delay, or retardedtime, so-called, is the time it takes, at speed c , to get fromthe charge to the field point P . The delay is r′/c.
方程中的各项，告诉了我们什么呢？先看第一项E=−qer′/4πϵ0r′2。这就是库仑规律，当然我们已经知道：q就是产生电场的电荷；我们是在点P，测量E，er′是从点P出发的单位矢量，r是从P到q的距离。但是，库伦规律是错误的。19世纪的发现，已经指出，影响的传播速度，不能比一个基础速度 c快，这我们现在称为光速。第一项是库仑规律，这是不正确的，不仅是因为，现在要知道电荷在什么地方、及它的距离是多少，是不可能的，而且因为，一个电场，是在给定的位置、和时间中，唯一能影响该场的，是电荷过去的表现。这个过去有多远呢？时间延迟，或所谓的迟滞时间，就是以光速 c、从电荷处到点P处所花的时间。这个延迟就是 r′/c。