he number of charges in the ring is theproduct of the surface area of the ring, 2πρdρ , and η , thenumber of charges per unit area. We have, then,
环中的电荷数量，就是环的表面面积2πρdρ，与η的积，η是每单位面积内的电荷数量。因此，我们就有：
(30.12)

We wish to evaluate this integral from ρ=0to ρ=∞ . The variable t , of course, is to be heldfixed while we do the integral, so the only varying quantities are ρand r . Leaving out all the constant factors, including thefactor eiωt , for the moment, the integral we wishis
我们希望估算从ρ=0到 ρ=∞ 的积分。当然，积分时，变量t，被固定下来，所以，剩下的变量就是ρ和 r。现在，不考虑所有的常数因子，包括因子eiωt，我们希望的积分就是：
(30.13)
To do this integral we need to use therelation between r and ρ :
要做这个积分，我们需要使用r和 ρ之间的关系：
r2=ρ2+z2. (30.14)
Since z is independent of ρ, when we take the differential of this equation, we get
由于z，独立于ρ，所以，当我们对此方程求微分时，我们得到：
2rdr=2ρdρ,
which is lucky, since in our integral wecan replace ρdρ by rdr and the r will cancel theone in the denominator. The integral we want is then the simpler one
这很幸运，由于在我们的积分中，我们可以用rdr，替换ρdρ，且 r会消去分母中的那个r。因此，我们想要的积分，就简化为：
(30.15)
To integrate an exponential is very easy.We divide by the coefficient of r in the exponent and evaluate theexponential at the limits. But the limits of r are not the same asthe limits of ρ . When ρ=0 , we have r=z , sothe limits of r are z to infinity. We get for theintegral
积分指数，非常容易。我们用指数中r的系数，来除，然后，估算指数的限制。但是，r的限制，与ρ的限制，不一样。当ρ=0时 , 我们有 r=z ，所以，r的限制，就是z到无穷。对于积分，我们得到：
(30.16)
where we have written ∞ for (ω/c)∞, since they both just mean a very large number!
这里，我们把(ω/c)∞，写作∞，由于它们两个，都只不过意味着一个非常大的数。

Now e−i∞ is amysterious quantity. Its real part, for example, is cos(−∞) , which,mathematically speaking, is completely indefinite (although we would expect itto be somewhere—or everywhere (?)—between +1 and −1 !). But in a physicalsituation, it can mean something quite reasonable, and usually can just betaken to be zero. To see that this is so in our case, we go back to consideragain the original integral (30.15).
现在，e−i∞是一个神秘的量。例如，其实部，是 cos(−∞)，从数学上讲，它是完全不确定的（虽然，我们可以期待，它是在某处--或者是在+1和-1之间的任何一个地方！）。但是，在一个物理情况中，它则可以意味着某个相当合理的东西，通常，只是被当做零而已。在我们的案例中，要看到它确实如此，那么，我们就要返回去，再次考虑原始积分（30.15）。

We can understand (30.15)as a sum of many small complex numbers, each of magnitude Δr , andwith the angle θ=−ωr/c in the complex plane. We cantry to evaluate the sum by a graphical method. In Fig. 30–11 wehave drawn the first five pieces of the sum. Each segment of the curve has thelength Δr and is placed at the angle Δθ=−ωΔr/cwith respect to the preceding piece. The sum for these first five pieces isrepresented by the arrow from the starting point to the end of the fifthsegment. As we continue to add pieces we shall trace out a polygon until we getback to the starting point (approximately) and then start around once more.Adding more pieces, we just go round and round, staying close to a circle whoseradius is easily shown to be c/ω . We can see now why theintegral does not give a definite answer!
我们可以把（30.15），理解为很多小的复数之和，每个复数的大小，都是Δr，在复数平面内的角度是θ=−ωr/c。我们可以通过图形方法，来尝试估算这个和。在图30-11中，我们已经画出了，此和的前五段。曲线的每一段，长度都是Δr，且与前一段，夹角为 Δθ=−ωΔr/c。前五段之和，通过一个箭头来表示：从起点开始，到第五段的终点结束。随着我们不断增加段，我们将描绘出一个多边形，直到我们回到起点（大约），然后，再接着转。增加更多的段，我们只是绕着一个圆，转了一圈又一圈，该圆半径，通过c/ω，很容易指出。现在，我们可以看出，为什么积分，不能给出一个确定的答案了！

Fig. 30–11.Graphical solution of ∫z∞e−iωr/c dr . 图30-11 ∫z∞e−iωr/c dr的图形解决方案。

But now we have to go back to the physicsof the situation. In any real situation the plane of charges cannot beinfinite in extent, but must sometime stop. If it stopped suddenly, and wasexactly circular in shape, our integral would have some value on the circle inFig. 30–11.If, however, we let the number of charges in the plane gradually taper off atsome large distance from the center (or else stop suddenly but in an irregularshape so for larger ρ the entire ring of width dρ no longercontributes), then the coefficient η in the exact integral woulddecrease toward zero. Since we are adding smaller pieces but still turningthrough the same angle, the graph of our integral would then become a curvewhich is a spiral. The spiral would eventually end up at the center of ouroriginal circle, as drawn in Fig. 30–12.The physically correct integral is the complex number A inthe figure represented by the interval from the starting point to the center ofthe circle, which is just equal to
但是现在，我们必须回到此情况的物理中。在任何真实的情况中，电荷平面，就面积而言，不可能无限，而是有停止之时。如果它突然停止了，而且是一个准确的圆，那么，我们的积分，在图30-11的圆上，应该就有某些值。然而，如果我们让平面内电荷的数目，从到中心较大的距离处，逐渐减少（或者，突然停止，但形状不规则，这样，对于较大的ρ，整个宽度为dρ的圆环，将没有贡献），因此，在准确积分中的系数η，将会减少到零。由于我们不断增加的，是较小的段，并且是以同样的角度在转，因此，我们的积分图形，将会变成一条螺旋线。此螺旋线，最终会在我们初始圆的中心，终止，如图30-12所示。物理上正确的积分，就是图形中的复数A，被从开始点到圆中心的间歇，给代表了，它等于：
(30.17)
as you can work out for yourself. This isthe same result we would get from Eq. (30.16)if we set e−i∞=0 .
这你可以自己去求证。在方程（30.16）中，如果我们设 e−i∞=0，那么，就可得到同样的结果。

Fig. 30–12.Graphical solution of ∫z∞ηe−iωr/cdr. 图30-12 ∫z∞ηe−iωr/cdr图形解决方案。

(There is also another reason why thecontribution to the integral tapers off for large values of r , andthat is the factor we have omitted for the projection of the acceleration onthe plane perpendicular to the line PQ .)
（对于大的 r值来说，为什么对积分的贡献，逐渐减小，还有另外一个原因，那就是，在垂直于线PQ的平面上，有一个加速度的投影因子{？}，我们把此因子，忽略了。）

(There is also another reason why thecontribution to the integral tapers off for large values of r , andthat is the factor we have omitted for the projection of the acceleration onthe plane perpendicular to the line PQ .)
（对于大的 r值来说，为什么对积分的贡献，逐渐减小，还有另外一个原因，那就是，在垂直于线PQ的平面上，有一个加速度的投影因子{？}，我们把此因子，给忽略了。）
We are, of course, interested only inphysical situations, so we will take e−i∞ equal tozero. Returning to our original formula (30.12)for the field and putting back all of the factors that go with the integral, wehave the result
当然，我们感兴趣的，只是物理情况，所以，我们将取e−i∞等于零。回到我们最初的关于场的公式（30.12），且把所有与积分有关的因子，全部放回，我们就有结果：
(30.18)
(remembering that 1/i=−i ).
（记住1/i=−i ）
It is interesting to note that (iωx0eiωt) is just equal to the velocity of the charges, so that we can alsowrite the equation for the field as
注意到下面一点，很有趣，即(iωx0eiωt)，正好等于电荷的矢速，于是，我们就可以把场的方程，写为：
(30.19)
which is a little strange, because theretardation is just by the distance z , which is the shortestdistance from P to the plane of charges. But that is the way it comesout—fortunately a rather simple formula. (We may add, by the way, that althoughour derivation is valid only for distances far from the plane of oscillatorycharges, it turns out that the formula (30.18)or (30.19)is correct at any distance z , even for z<λ .)
这略微有点奇怪，因为，迟滞正是通过距离z，它是从P到电荷平面的最短距离。但是，最终的结果，就是如此--幸运的是，此公式，相当简单。（还要增加一点，这是顺便说的，虽然我们的推导，只是对‘到震荡电荷的平面很远的距离’，才有效，但结果则是，公式（30.18）或（30.19），对任何距离z，都正确，甚至包括z<λ。）

1. In our case T=Δ/c=mnλ/c , where cis the speed of light. The frequency ν=c/λ , so Δν=cΔλ/λ2. ↩
2. This is because Rayleigh’s criterion is a rough idea in the firstplace. It tells you where it begins to get very hard to tell whether the imagewas made by one or by two stars. Actually, if sufficiently careful measurementsof the exact intensity distribution over the diffracted image spot can be made,the fact that two sources make the spot can be proved even if θ is lessthan λ/L .
脚注：
1、在我们的案例中，T=Δ/c=mnλ/c，这里c是光速。频率ν=c/λ , 所以 Δν=cΔλ/λ2。
2、这是因为，雷利判据，最初只是一个粗糙的想法。它告诉你，要区分图像究竟是由一个恒星、还是有两个恒星形成的，在什么时候，开始变得困难起来。实际上，对于衍射图像点，如果关于其强度的准确分布，能够得到，那么，这个图像点，是由两个源形成的，可被证明，即便θ是小于 λ/L的。

Chapter31.The Origin of the Refractive Index 第31章 折射率的起源
31–1The index of refraction 31-1 折射率
We have said before that light goes slowerin water than in air, and slower, slightly, in air than in vacuum. This effectis described by the index of refraction n . Now we would like tounderstand how such a slower velocity could come about. In particular, weshould try to see what the relation is to some physical assumptions, orstatements, we made earlier, which were the following:
a. That the total electric field in any physical circumstance canalways be represented by the sum of the fields from all the charges in theuniverse.
b. That the field from a single charge is given by its accelerationevaluated with a retardation at the speed c , always (forthe radiation field).
我们前面说过，光在水中走的，比在空气中慢，而在空气中，又比在真空中，要稍微慢点儿。这个效果，通过折射率n来描述。现在，我们想理解，这样一个较慢的矢速，是如何产生的。特别是，我们想看一看，这与我们前面所做的一些假设、声明，是什么关系，这些假设、声明就是：
a.任何情形下的总的电场，总是可以被：宇宙中所有电荷的场之和，来代表。
b.一个单独的电荷所产生的场，总是可以通过其加速度，来给予，估算加速度时，要考虑到迟滞，其速度为 c，（对于辐射场来说）总是如此。

But, for a piece of glass, you might think:“Oh, no, you should modify all this. You should say it is retarded at thespeed c/n .” That, however, is not right, and we have tounderstand why it is not.
但是，对于一片玻璃，你可能会说：“奥，不，你应该修改这一切。你应该说，它是以速度c/n而被迟滞的。然而，这并不对，且我们应该去理解：这为什么不对。

It is approximately true that lightor any electrical wave does appear to travel at the speed c/nthrough a material whose index of refraction is n , but the fieldsare still produced by the motions of all the charges—including the chargesmoving in the material—and with these basic contributions of the fieldtravelling at the ultimate velocity c . Our problem is tounderstand how the apparently slower velocity comes about.
一个材料，折射率为n，光或任何电波，在此材料中传播时，速度似乎是c/n，这一点，似乎为真，但是，场，仍是由所有电荷的运动而产生的，包括在材料中运动的电荷，在这些场的基础性贡献下，传播的极限矢速，是c。我们的问题，就是去理解：矢速明显变慢，是如何发生的。

Fig. 31–1.Electric waves passing through alayer of transparent material. 图 31-1 电波穿过一个透明材料板。
We shall try to understand the effect in avery simple case. A source which we shall call “the external source” isplaced a large distance away from a thin plate of transparent material, sayglass. We inquire about the field at a large distance on the opposite side ofthe plate. The situation is illustrated by the diagram of Fig. 31–1,where S and P are imagined to be very far away from theplate. According to the principles we have stated earlier, an electric fieldanywhere that is far from all moving charges is the (vector) sum of the fieldsproduced by the external source (at S ) and the fieldsproduced by each of the charges in the plate of glass, every one withits proper retardation at the velocity c . Remember that thecontribution of each charge is not changed by the presence of the othercharges. These are our basic principles. The field at P can bewritten thus:
我们将尝试，在一个非常简单的案例中，理解这个效果。设有一个薄的透明材料板，比如说玻璃，距板很远处，有一个源，我们称之为“外部源”。我们就是要查出，在板的另外一侧，很远处，场是什么。这个情况，如图31-1所示，S和P，都被认为，距板很远。依据我们以前陈述过的原理，对于所有运动着的电荷，距其很远处的电场，就是下面两个场的（矢量）和，一、由外部源（在S）所产生的场；二、玻璃板中的每个电荷所产生的场；每个电场，都有合适的、矢速为c的迟滞。要记住，每个电荷的贡献，并不会由于其他电荷的在场，而发生改变。这些是我们的基本原理。这样，点P的场，就可写为：
(31.1)
or
或
(31.2)
where Es is the fielddue to the source alone and would be precisely the field at P ifthere were no material present. We expect the field at P to bedifferent from Es if there are any other moving charges.
这里，Es就是只归于源的场，并且，如果没有其他材料在场的话，就正是在点P的场。如果还有任何其他运动着的电荷，那么，我们期待，点P的场，与Es不同。

Why should there be charges moving in theglass? We know that all material consists of atoms which contain electrons.When the electric field of the source acts on these atoms it drives theelectrons up and down, because it exerts a force on the electrons. And movingelectrons generate a field—they constitute new radiators. These new radiatorsare related to the source S , because they are driven by the fieldof the source. The total field is not just the field of the source S, but it is modified by the additional contribution from the other movingcharges. This means that the field is not the same as the one which was therebefore the glass was there, but is modified, and it turns out that it ismodified in such a way that the field inside the glass appears to be moving ata different speed. That is the idea which we would like to work outquantitatively.
为什么在玻璃中，有电荷在运动？我们知道，所有的材料，都是由原子组成，而原子包含着电子。当源的电场，作用于这些原子时，它会驱动电子，上下运动，因为它有一个力，作用于电子。移动的电子，会产生场—它们会组成一个新的辐射器。这些新的辐射器，与源S有关，因为，它们是被源的场，所驱动。总的场，不仅只是源S的场，它也被来自移动电荷的贡献，给修改了。这就意味着，这个场，与没有玻璃时的场，不一样，而是被修改了，结果就是，它是以如下方式被修改的：玻璃中的场，运动速度，似乎不同。对此想法，我们希望，能够定量给出。

Now this is, in the exact case, prettycomplicated, because although we have said that all the other moving chargesare driven by the source field, that is not quite true. If we think of aparticular charge, it feels not only the source, but like anything else in theworld, it feels all of the charges that are moving. It feels, inparticular, the charges that are moving somewhere else in the glass. So thetotal field which is acting on a particular charge is a combination ofthe fields from the other charges, whose motions depend on what thisparticular charge is doing! You can see that it would take a complicated setof equations to get the complete and exact formula. It is so complicated thatwe postpone this problem until next year.
现在，在这个例子中，这是相当复杂的，因为，虽然我们说过，所有其他移动着的电荷，都是被源的场，所驱动，但是，这并不完全正确。如果我们考虑一个具体的电荷，就会感觉到，不仅是源，甚至世界上的任何其他事物，所有运动着的电荷{都在起作用}。特别是，感觉到，在玻璃中运动着的电荷。所以，作用于一个具体电荷上的总场，就是来自其他电荷的场的组合，这些其他电荷的运动，也依赖于这个具体电荷，在做什么！你可看到，这将需要一组非常复杂的方程，才能得到完整和准确的公式。这个问题，如此复杂，我们将把它推迟，明年再说。

Instead we shall work out a very simplecase in order to understand all the physical principles very clearly. We take acircumstance in which the effects from the other atoms are very small relativeto the effects from the source. In other words, we take a material in which thetotal field is not modified very much by the motion of the other charges. Thatcorresponds to a material in which the index of refraction is very closeto 1 , which will happen, for example, if the density of the atoms is verylow. Our calculation will be valid for any case in which the index is for anyreason very close to 1. In this way we shall avoid the complications ofthe most general, complete solution.
相反，我们将做出一个非常简单的案例，为的是，清楚地理解所有的物理原理。在我们所取的情形中，来自其他原子的效果，与源的效果相比，非常小。换句话说，我们取一种材料，在其中，其他电荷的运动，对总的场，并无多少改变。相当于有这样一种材料，在其中，折射率非常接近于1，例如，如果原子的密度非常低的话，这种情况就可能发生。对于任何情况，不论原因为何，只要其折射率非常接近于1，那么，我们的计算，就会有效。更普遍的、完整的解决方案，会有很多复杂性，以这种方式，我们将避免之。