There are our analogies, and the equationcorresponding to the oscillating circuit becomes the following, by directsubstitution of L for m , q for x , etc:
这里，就是我们的类比，有个方程，相当于震荡电路的，通过用L替换m , q 替换x，等等，就可变成下面的形式：

Now everything we learned about (25.14)can be transformed to apply to (25.15).Every consequence is the same; so much the same that there is a brilliantthing we can do.
现在，我们关于（25.14）所学到的所有的东西，都可被变换，而应用到（25.15）。所有的后果，都是一样的；一样的东西，是这么多，以至于，有一件辉煌的事情，有待我们去做。

Suppose we have a mechanical system whichis quite complicated, not just one mass on a spring, but several masses on severalsprings, all hooked together. What do we do? Solve it? Perhaps; but look, wecan make an electrical circuit which will have the same equations as thething we are trying to analyze! For instance, if we wanted to analyze a mass ona spring, why can we not build an electrical circuit in which we use aninductance proportional to the mass, a resistance proportional to thecorresponding mγ , 1/C proportional to k , all in the same ratio? 假设我们有一个力学系统，它非常复杂，不是有一个质量在一个弹簧上，而是有若干质量，在若干弹簧上，且全钩在一起。我们该如何解它呢？或许，我们可以建一个电路，它与我们要分析的事情，拥有同样的方程！例如，如果我们想分析弹簧上的一个质量，为什么我们就不能建一个电路，在其中，我们用电感，正比于质量，用电阻，正比于相应的mγ，用1/C ，正比于k，都是同样的比率。Then, of course, this electrical circuit will be the exact analog ofour mechanical one, in the sense that whatever q does, in response to V (V also is made to correspond to the forces that are acting), so the xwould do in response to the force! So if we have a complicated thingwith a whole lot of interconnecting elements, we can interconnect a whole lotof resistances, inductances, and capacitances, to imitate the mechanicallycomplicated system. 因此，当然，这个电路，将会是我们力学的系统是一个精确的类比，其意义就是，无论q，作为对V的反应（V被做的，相当于作用于力学系统中的力），而做了什么，那么x作为对力的反应，也要做类似的事情！所以，如果我们有一个复杂的力学性的系统，由很多相互连接的元素构成，那么，我们就可以把很多电阻、电感、和电容，连接起来，以模仿它。What is the advantage to that? One problem is just as hard (or as easy)as the other, because they are exactly equivalent. The advantage is not that itis any easier to solve the mathematical equations after we discover thatwe have an electrical circuit (although that is the method used by electricalengineers!), but instead, the real reason for looking at the analog is that itis easier to make the electrical circuit, and to change somethingin the system.
这样做，有什么好处呢？两个问题的难易，伯仲之间，因为它们是等价的。所谓的好处，并不在于，在我们发现了电路之后，解数学方程，变得更容易了（虽然这是电子工程师所用的方法！），而是在于，建一个电路、及改变此系统中的某些东西，更容易一些。

Suppose we have designed an automobile, andwant to know how much it is going to shake when it goes over a certain kind ofbumpy road. We build an electrical circuit with inductances to represent theinertia of the wheels, spring constants as capacitances to represent thesprings of the wheels, and resistors to represent the shock absorbers, and soon for the other parts of the automobile. Then we need a bumpy road. All right,we apply a voltage from a generator, which represents such and such akind of bump, and then look at how the left wheel jiggles by measuring thecharge on some capacitor. Having measured it (it is easy to do), we find thatit is bumping too much. Do we need more shock absorber, or less shock absorber?With a complicated thing like an automobile, do we actually change the shockabsorber, and solve it all over again? No!, we simply turn a dial; dial numberten is shock absorber number three, so we put in more shock absorber. The bumpsare worse—all right, we try less. The bumps are still worse; we change the stiffnessof the spring (dial 17 ), and we adjust all these things electrically, with merely theturn of a knob.
假设我们设计了一个汽车，我们想知道，当它走在某种颠簸不平的路上时，它会怎么摇晃？我们建了一个电路，用电感，代表轮子的惯性，用作为电容的弹簧常数，代表轮子的弹簧，用电阻，代表减震器，如此等等，对于汽车的其他部分，都如此。然后，我们需要一条颠簸的路。好，我们用发电机的电压，它代表着种种颠簸。然后，通过测量某些电容中电荷的变化，来查看左边轮胎，是如何左右摇摆的。在做过测量之后（这很容易做），我们发现，它颠簸的非常厉害，我们需要更多、还是更少的减震器呢？对于像汽车这种复杂的事情，我们是否需要实际改变减震器，把它全做一遍呢？不！我们只需旋转一下仪表盘；仪表盘的数字十，就是减震器的数字三，于是，我们放入更多的减震器。颠簸更厉害了，好，我们再放少一点试试。颠簸还是很厉害；那么，我们改变弹簧的硬度（仪表盘数字17）试试，所有这些，都是以电子的方式来调节，我们只是拧一拧球形把手而已！

This is called an analog computer.It is a device which imitates the problem that we want to solve by makinganother problem, which has the same equation, but in another circumstance ofnature, and which is easier to build, to measure, to adjust, and to destroy!
这被称为模拟计算机。对于我们想要解决的问题，该设备，通过制造出另一问题，来模拟它，两个问题，具有同样的方程，但此另一问题，在自然的另一种情形中，它更容易建立、测量、调整、和摧毁！

25–5Series and parallel impedances 25-5 串联和并联阻抗
Finally, there is an important item whichis not quite in the nature of review. This has to do with an electrical circuitin which there is more than one circuit element. For example, when we have aninductor, a resistor, and a capacitor connected as in Fig. 24–2, we note that all the charge went through every oneof the three, so that the current in such a singly connected thing is the sameat all points along the wire. Since the current is the same in each one, thevoltage across R is IR , the voltage across L is L(dI/dt) , and so on. So, the total voltage drop is the sum of these, and thisleads to Eq. (25.15).Using complex numbers, we found that we could solve the equation for thesteady-state motion in response to a sinusoidal force. We thus found that V^=Z^I^. Now Z^ is called the impedance of this particular circuit. It tells usthat if we apply a sinusoidal voltage, V^ , we get a current I^ .
最后，还有一项，很重要。它并不全在回顾中。当电路中的元件，多于一个时，就会有这个问题。例如，但我们有一个电感、一个电阻、和一个电容，如图24-2那样连接，我们注意到，所有的电荷，要依次经过每个元件，这样，在一个如此单独连接的事情中，沿线各点，电流都一样。由于每个元件上的电流一样，所以，R上的电压，就是 IR，L上的电压，就是 L(dI/dt)，如此等等。于是，总的电压下降，就是这些电压的和，这就会导致方程（25.15）。作为对正弦力的反应，会产生稳定状态的运动，关于此运动的方程，我们发现，我们可以利用复数来解。我们这样就发现了V^=Z^I^ 。现在，Z^ 被称为这个具体电路的阻抗。它告诉我们，如果我们加一个正弦波电压V^，我们就会得到电流I^。

Now suppose we have a more complicatedcircuit which has two pieces, which by themselves have certain impedances, Z^1and Z^2 and we put them in series (Fig. 25–6a) andapply a voltage. What happens? It is now a little more complicated, but if I^is the current through Z^1 , the voltage difference across Z^1 , is V^1=I^Z^1 ; similarly, the voltage across Z^2 is V^2=I^Z^2 . The same current goes through both. Therefore the total voltageis the sum of the voltages across the two sections and is equal to V^=V^1+V^2=(Z^1+Z^2)I^. This means that the voltage on the complete circuit can be written V^=I^Z^s, where the Z^ s of the combined system in series is the sum of the two Z^ ’s of the separate pieces:
现在，假设我们有一个更复杂的电路，它包含两部分，每部分都有一定的阻抗Z^1和Z^2，我们把它们串联起来（图25-6a），并加上一个电压。会发生什么呢？现在，情况更复杂了，但是，如果I^是经过Z^1的电流，那么，Z^1上的电压差，就是 V^1=I^Z^1；类似地，Z^2上的电压差，就是 V^2=I^Z^2。经过两者的电流，是一样的。因此，总的电压，就是通过两部分的电压之和，它等于V^=V^1+V^2=(Z^1+Z^2)I^。这就意味着，整个电路上的电压，可被写作V^=I^Z^s，这里，组合的串联系统中的Z^ s，就是两个分开部分的 Z^之和：
(25.16)

This is not the only way things may be connected. We may also connectthem in another way, called a parallel connection (Fig. 25–6b).Now we see that a given voltage across the terminals, if the connecting wiresare perfect conductors, is effectively applied to both of the impedances, andwill cause currents in each independently. Therefore the current through Z^1is equal to I^1=V^/Z^1. The current in Z^2 is I^2=V^/Z^2 . It is the same voltage. Now the total current which is suppliedto the terminals is the sum of the currents in the two sections: I^=V^/Z^1+V^/Z^2. This can be written as
这并不是事物连接的唯一方法。我们还可以用另外一种方法，来连接它们，称为并连（图25-6b）。现在，我们看到，在终端上，有给定电压，如果连接线是完美的导体，并被有效地应用在两个阻抗上，那么，它将在两个阻抗上，分别独立地引起电流。因此，通过Z^1的电流就等于 I^1=V^/Z^1。Z^2 的电流是 I^2=V^/Z^2。它们的电压是一样的。现在，被提供给终端的总的电流，就是两部分电流之和：I^=V^/Z^1+V^/Z^2。这可被写作：

Thus 这样
(25.17)

More complicated circuits can sometimes be simplified by taking piecesof them, working out the succession of impedances of the pieces, and combiningthe circuit together step by step, using the above rules. If we have any kind ofcircuit with many impedances connected in all kinds of ways, and if we includethe voltages in the form of little generators having no impedance (when we passcharge through it, the generator adds a voltage V ), then the following principles apply: (1) At any junction, thesum of the currents into a junction is zero. That is, all the current which comesin must come back out. (2) If we carry a charge around any loop, and backto where it started, the net work done is zero. These rules are called Kirchhoff’slaws for electrical circuits. Their systematic application to complicatedcircuits often simplifies the analysis of such circuits. We mention them herein conjunction with Eqs. (25.16)and (25.17),in case you have already come across such circuits that you need to analyze inlaboratory work. They will be discussed again in more detail next year.
更复杂的电路，有时，可被简化为由上面的两部分组成，即得出各部分的阻抗的相继{？}，使用上面的规则，一步一步地组成电路。如果我们的电路，是由各种阻抗、以各种方式连接起来的，如果我们包含电压的方式，是没有阻抗的小的发电机（当我们让电荷通过它时，发电机加了一个电压V），那么，下面的原理，就可应用：（1）在任何节点，流入、流出的电流之和，为零。也就说呢，所有流进的电流，都应该流出去。（2）如果，在任何循环中，我们载着一个电荷走，那么，当它返回其起点时，所做净功为零。这些规则，被称为电路的基尔霍夫规律。对于复杂电路，系统性地应用它们，通常会简化对这种电路的分析。我们在这里提到它们，因为，它们与方程（25.16）和（25.17），有连带关系，假定你在实验室工作中，已经碰到过这种电路，并需要分析它们。明年，我们将更仔细地讨论它们。

Chapter26. Optics:The Principle of Least Time
26–1Light
This is the first of a number of chapterson the subject of electromagnetic radiation. Light, with which we see,is only one small part of a vast spectrum of the same kind of thing, thevarious parts of this spectrum being distinguished by different values of acertain quantity which varies. This variable quantity could be called the “wavelength.”As it varies in the visible spectrum, the light apparently changes color fromred to violet. If we explore the spectrum systematically, from long wavelengthstoward shorter ones, we would begin with what are usually called radiowaves.Radiowaves are technically available in a wide range of wavelengths, some evenlonger than those used in regular broadcasts; regular broadcasts havewavelengths corresponding to about 500 meters. Then there are theso-called “short waves,” i.e., radar waves, millimeter waves, and so on. Thereare no actual boundaries between one range of wavelengths and another, becausenature did not present us with sharp edges. The number associated with a givenname for the waves are only approximate and, of course, so are the names wegive to the different ranges.
下面若干章，都是关于电磁辐射的，本章是第一个。我们看东西，凭借的，就是可见光，光这一类事物，有一个巨大的光谱，可见光，只是其中的一小部分，这个光谱的不同部分，可以通过某个变化的量的不同的值，而被区分开来。这个变化的量，就被称“波长”。当波长，在可见光光谱变化时，光是明显地，从赤色，变到紫色。如果我们从长波，到短波，来系统性地探索光谱，我们开始的地方：通常被称为无线电波。无线电波的波长范围，非常宽，有的波长，甚至比通常广播所用的，都要长；通常广播的波长，相当于500米。因此，就有了所谓的“短波”，即雷达波，毫米波，等等。一个波长范围，与另一个波长范围之间，并没有实际的边界，因为，自然并没有给我们提供严格的界限。对于波长，与某个名字相关联的数字，只是近似的，当然，我们给不同的范围所起的名字，同样也是近似的。

Then, a long way down through themillimeter waves, we come to what we call the infrared, and thence tothe visible spectrum. Then going in the other direction, we get into a regionwhich is called the ultraviolet. Where the ultraviolet stops, the x-raysbegin, but we cannot define precisely where this is; it is roughly at 10−8 m, or 10−2 μ m. These are “soft” x-rays; thenthere are ordinary x-rays and very hard x-rays; then γ-rays, and so on, for smallerand smaller values of this dimension called the wavelength.
然后，是漫长的毫米波之路，通过它之后，我们来到了红外区域，由此，再来到可见光光谱。然后，又到了另外一个方向，此区域，被称为紫外区。在紫外区停止的地方，就是x射线开始的地方，但是，我们不能精确地定义其位置；大约是在10−8 m, 或 10−2 μ m。这些是“软的”x射线，然后，就是通常的x射线，和非常硬的x射线；然后，是γ射线，如此等等，它们都被称为波长，其值越来越小。

Within this vast range of wavelengths,there are three or more regions of approximation which are especiallyinteresting. In one of these, a condition exists in which the wavelengthsinvolved are very small compared with the dimensions of the equipment availablefor their study; furthermore, the photon energies, using the quantum theory,are small compared with the energy sensitivity of the equipment. Under theseconditions we can make a rough first approximation by a method called geometricaloptics. 这个波长范围，非常巨大，在其中，有三个以上的近似区域，特别有趣。在其中之一，存在着这种情况：所牵扯到的波长，与可用来研究它们的仪器的尺寸相比，非常小。另外，利用量子理论，可以得到光子能量，此能量，与此仪器对能量的敏感度相比，较小。在这些条件之下，我们可以通过几何光学，做一个初步的近似。 If, on the other hand, the wavelengths are comparable to thedimensions of the equipment, which is difficult to arrange with visible lightbut easier with radiowaves, and if the photon energies are still negligiblysmall, then a very useful approximation can be made by studying the behavior ofthe waves, still disregarding the quantum mechanics. This method is based onthe classical theory of electromagnetic radiation, which will bediscussed in a later chapter. 另一方面，如果波长与仪器的尺寸，是可比的，当然，这一点对可见光来说，比较难以做到，但是对于无线电波来说，比较容易，以及如果光子能量，还是小到可以忽略，那么，一个很有用的近似，就可以通过研究波的表现，而做到，当然，仍不管量子力学。这种方法，基于经典的电磁辐射理论，后面某章，将讨论它。Next, if we go to very short wavelengths, where we can disregard thewave character but the photons have a very large energy compared withthe sensitivity of our equipment, things get simple again. This is the simple photonpicture, which we will describe only very roughly. The complete picture, whichunifies the whole thing into one model, will not be available to us for a longtime.
下面，看波长非常短的区域，在这里，我们可以忽略波的特性，但是，光子的能量，与我们仪器的敏感度相比，非常大，事情又变的简单了。这就是简单的光子的图像，我们也只是非常粗略地描述一下。其完整图像，将会把所有的事情，都统一到一个模式中，但是，要达到这点，对我们来说，还需很长时间。

In this chapter our discussion is limitedto the geometrical optics region, in which we forget about the wavelength andthe photon character of the light, which will all be explained in due time. Wedo not even bother to say what the light is, but just find out how itbehaves on a large scale compared with the dimensions of interest. All thismust be said in order to emphasize the fact that what we are going to talkabout is only a very crude approximation; this is one of the chapters that weshall have to “unlearn” again. But we shall very quickly unlearn it, because weshall almost immediately go on to a more accurate method.
本章，我们的讨论，将局限于几何光学范围之内，在这里，我们将忘记光的波长、光子等特性，这些，将在合适的时候解释。我们甚至都不会去说，光是什么，而只是找出，光在一个大的尺度上，是如何表现的，这个尺度，指与我们感兴趣的事情的大小相比而言。说这些的目的，是为了强调以下事实，即我们将要谈论的，只是一个非常粗略的近似；有些章，是我们要“故意忘却”的，本章就是其中之一。但很快，我们就会地故意地忘却它，因为，我们几乎是马上，就要去学习一个更加准确的方法。

Although geometrical optics is just anapproximation, it is of very great importance technically and of great interesthistorically. We shall present this subject more historically than some of theothers in order to give some idea of the development of a physical theory orphysical idea.
虽然几何光学，只是一种近似，但是，从技术上看，它很重要，从历史上看，意义重大。与其他主题相比，我们在谈论这个主题时，会谈到更多的历史，为的是让大家，对物理理论的发展、或物理想法的发展，能有一个概念。

First, light is, of course, familiar toeverybody, and has been familiar since time immemorial. Now one problem is, bywhat process do we see light? There have been many theories, but itfinally settled down to one, which is that there is something which enters theeye—which bounces off objects into the eye. 首先，当然，光对每个人来说，都很熟悉，且自古以来，就很熟悉。现在的问题就是，我们看到光的过程，是什么？曾经有过很多理论，但最终，都落到一个，那就是，有某物，进入了眼睛—它从对象上弹开，进入了眼睛。We haveheard that idea so long that we accept it, and it is almost impossible for usto realize that very intelligent men have proposed contrary theories—thatsomething comes out of the eye and feels for the object, for example. 很久以来，我们听到的，就是这种想法，且我们也接受了它，以至于，我们几乎不可能意识到，非常聪明的人，都曾建议过完全相反的理论—例如，有东西从眼睛里出来，去感觉对象。Someother important observations are that, as light goes from one place to another,it goes in straight lines, if there is nothing in the way, and that therays do not seem to interfere with one another. That is, light is crisscrossingin all directions in the room, but the light that is passing across our line ofvision does not affect the light that comes to us from some object. 还有其它一些观察，也很重要，它们就是，当光从一地，走向另一地时，如果路上，没有任何东西，那么，它走的是直线，且光线之间，互不干涉。也就是说，在房间中，光在所有方向，交叉移动，但是，通过我们视线的光，并不影响，从物体来到我们眼睛中的光。This was once a most powerful argument against the corpuscular theory;it was used by Huygens. If light were like a lot of arrows shooting along, howcould other arrows go through them so easily? Such philosophical arguments arenot of much weight. One could always say that light is made up of arrows whichgo through each other!
这个理论，曾经是针对微粒子理论的最强有力的论证；惠更斯使用过它。如果光像很多箭头在射，那么，其它箭头，通过它们，为什么会这么容易呢？这种哲学的论证，并没有多少份量。总有人会说，光是由箭头构成的，相互之间，可以穿过。

Fig. 26–1.The angle of incidence is equalto the angle of reflection. 图26-1 入射角等于反射角。
The discussion above gives enough of thebasic idea of geometrical optics—now we have to go a little further intothe quantitative features. Thus far we have light going only in straight linesbetween two points; now let us study the behavior of light when it hits variousmaterials. The simplest object is a mirror, and the law for a mirror is thatwhen the light hits the mirror, it does not continue in a straight line, butbounces off the mirror into a new straight line, which changes when we changethe inclination of the mirror. The question for the ancients was, what is therelation between the two angles involved? This is a very simple relation,discovered long ago. The light striking a mirror travels in such a way that thetwo angles, between each beam and the mirror, are equal. For some reason it iscustomary to measure the angles from the normal to the mirror surface. Thus theso-called law of reflection is
上面的讨论，关于几何光学，已经给出了足够多的基本概念--现在，我们要更深入一点，讨论定量特征。到目前为止，我们的光，都是在两点之间，走直线，现在，让我们研究，当光打到不同材料上时，表现如何。最简单的对象，就是镜子，对于镜子的规律就是：当光打到镜子上时，它并不是继续走一条直线，而是从镜子上弹开，走一条新的直线，当我们改变镜子的倾斜角时，这条线，也跟着变。对于古人的问题就是，所牵扯到的两个角度，其关系是什么？这个关系，非常简单，很久之前，就已发现。光打到镜子上，运动方式是这样：每束光，与镜子之间的夹角，是相等的。由于某种原因，测量这个角度，是从法线，到镜子的平面。这样，所谓的反射规律就是：
θi = θr. (26.1)