So to allow for this time delay, we put alittle prime on r , meaning how far away it was when theinformation now arriving at P left q . Just for a momentsuppose that the charge carried a light, and that the light could only cometo P at the speed c . Then when we look at q, we would not see where it is now, of course, but where it was at someearlier time. What appears in our formula is the apparentdirection er′ —the direction itused to be—the so-called retarded direction—and at the retardeddistance r′ . That would be easy enough to understand, too, but itis also wrong. The whole thing is much more complicated.
所以，要允许这个时间延迟，我们在 r上面放一个小撇号，意思是，当到达P的信息，在离开q时，距离有多远？暂且让我们假设{？}，电荷带着光，而光只能以速度c，来到P。因此，当我们看q时，我们不能看到，现在，它在什么地方，当然，只能看到，早些时候，它在什么地方。在我们公式中出现的，是明显的方向er′--这是曾经的方向--被称为迟滞的方向—且在迟滞的距离 r′处。这已经很容易理解了，但它仍是错的。整个事情，要复杂地多。

There are several more terms. The next termis as though nature were trying to allow for the fact that the effect isretarded, if we might put it very crudely. It suggests that we should calculatethe delayed Coulomb field and add a correction to it, which is its rate of changetimes the time delay that we use. Nature seems to be attempting to guess whatthe field at the present time is going to be, by taking the rate of change andmultiplying by the time that is delayed. But we are not yet through. There is athird term—the second derivative, with respect to t , of the unitvector in the direction of the charge. Now the formula is finished, andthat is all there is to the electric field from an arbitrarily moving charge.
还有几项。下一项，就好像是，自然在尝试允许：影响可以被迟滞这一事实，如果我们可以这么粗略地表达的话。它提议，我们应该计算被延迟的库仑场，并对它，增加一个修正，那就是，我们所用的时间延迟的对时间的变化率。{？}自然似乎是在尝试去猜测，当前时间的场，会怎么变，{此猜测}通过取变化率，及乘以所延迟的时间。项还没完。还有第三项--一个二阶导数，即电荷方向的单位矢量，对时间的二阶导数。现在，公式讲完了，从任意一个移动中的电荷中，所形成的关于电场的东西，就是这些。

The magnetic field is given by
磁场由下式给出：
(28.4)
We have written these down only for thepurpose of showing the beauty of nature or, in a way, the power of mathematics.We do not pretend to understand why it is possible to write so much insuch a small space, but (28.3)and (28.4)contain the machinery by which electric generators work, how light operates,all the phenomena of electricity and magnetism. Of course, to complete thestory we also need to know something about the behavior of the materialsinvolved—the properties of matter—which are not described properly by (28.3).
我把这些写出，目的就是，指出自然的美，或者，以某种方式，指出数学的力量。我们并不假装理解了，为什么，在这么一个小的空间中，把这么多东西写出来，是可能的，但是，（28.3）和（28.4），包含了这样的机制，通过此机制，发电机可以工作，可以解释，光是如何运转的，及所有的电磁现象。要把这个故事讲完，我们还需要知道，所牵扯到的材料的表现，即物质的属性，（28.3）并未给出适当描述。

To finish with our description of the worldof the 19th century we must mention one other great synthesis whichoccurred in that century, one with which Maxwell had a great deal to do also,and that was the synthesis of the phenomena of heat and mechanics. We shallstudy that subject soon.
以上是19世纪对世界的描述，要结束这个话题，我们必须提到另外一个伟大的综合，它也出现在19世纪，对于它，麦克斯韦尔也做了很多工作，它就是热学和力学现象的综合。我们很快将会研究这个主题。

What had to be added in the20th century was that the dynamical laws of Newton were found to be allwrong, and quantum mechanics had to be introduced to correct them. Newton’slaws are approximately valid when the scale of things is sufficiently large.These quantum-mechanical laws, combined with the laws of electricity, have onlyrecently been combined to form a set of laws called quantum electrodynamics.In addition, there were discovered a number of new phenomena, of which thefirst was radioactivity, discovered by Becquerel in 1896—he just sneaked it inunder the 19th century. This phenomenon of radioactivity was followed upto produce our knowledge of nuclei and new kinds of forces that are notgravitational and not electrical, but new particles with differentinteractions, a subject which has still not been unravelled.
关于20世纪的知识，要增加的，就是牛顿的力学规律，被发现，全是错的，量子力学，必须被引入，以纠正它们。牛顿的规律，只在事物的尺度充分大时，才大约有效。这些量子力学的规律，只是在最近，才与电的规律，结合起来，形成了一组规律，被称为量子电动力学。另外，还发现了若干新的现象，其中第一个，就是放射性，它是由贝克勒尔(Becquerel)在1896年发现的--他只是偷偷地溜入了19世纪而已{1903诺奖}。追踪这个放射的现象，就将会产生我们关于原子核和新的力的知识，这些不是重力的电子的，而是带有不同交互作用的新粒子，这个主题，尚未被揭示。

For those purists who know more (theprofessors who happen to be reading this), we should add that when we saythat (28.3)is a complete expression of the knowledge of electrodynamics, we are not beingentirely accurate. There was a problem that was not quite solved at the end ofthe 19th century. When we try to calculate the field from all the charges includingthe charge itself that we want the field to act on, we get into troubletrying to find the distance, for example, of a charge from itself, and dividingsomething by that distance, which is zero. The problem of how to handle thepart of this field which is generated by the very charge on which we want thefield to act is not yet solved today. So we leave it there; we do not have acomplete solution to that puzzle yet, and so we shall avoid the puzzle for aslong as we can.
所谓的纯粹主义者，就是知道的比较多的人（教授刚好在读这些东西），对于他们来说，我们应该增加的就是，当我们说，（28.3）是电动力学知识的一个完整的表达时，我们并非完全准确。在19世纪末，还有一个问题，尚未被完全解决。当我们尝试去计算，所有电荷所产生的场，包括那个我们想让这个场作用于其上的那个电荷，则这时，我们就陷入了一个麻烦，即尝试要找出距离的麻烦，例如，一个电荷，到它自己的距离，因为，我们要用某物，来除以这个距离，而它是零。对于一个电荷，我们想让一个场，作用于其上，但是，这个电荷，也会产生一个场，如何处理这个场，还是一个问题，今天仍未解决。所以，我们把这个问题挂起来；对于这个谜，我们还没有一个完整的解，所以，我们将避开它，能避多久算多久。

28–2Radiation 28-2 辐射
That, then, is a summary of the worldpicture. Now let us use it to discuss the phenomena called radiation. To discussthese phenomena, we must select from Eq. (28.3)only that piece which varies inversely as the distance and not as the square ofthe distance. It turns out that when we finally do find that piece, it is sosimple in its form that it is legitimate to study optics and electrodynamics inan elementary way by taking it as “the law” of the electric field produced by amoving charge far away. We shall take it temporarily as a given law which wewill learn about in detail next year.
因此，这就是对世界图像的一个总结。现在，让我们用它来讨论辐射现象。要讨论这些现象，我们就应该从方程（28.3）中，只选取与距离成反比的部分，而不是与距离的平方成反比的。结果就是，当我们最终找到这一部分，就会发现，它在形式上，是如此简单，以至于，对于一个遥远的移动电荷所产生的电场，通过把这一部分，当作此电场的“规律”，从而，以一种基本的方式，来研究光学和电动力学，是正当合理的。暂时，我们将只把它当作一个给定的规律，具体细节，明年再学。

Of the terms appearing in (28.3),the first one evidently goes inversely as the square of the distance, and thesecond is only a correction for delay, so it is easy to show that both of themvary inversely as the square of the distance. All of the effects we areinterested in come from the third term, which is not very complicated, afterall. What this term says is: look at the charge and note the direction of theunit vector (we can project the end of it onto the surface of a unit sphere).As the charge moves around, the unit vector wiggles, and the acceleration ofthat unit vector is what we are looking for. That is all. Thus
对于（28.3）中的各项，第一项，明显是与距离的平方成反比，第二项，只是对时间延迟的修正，所以，很容易指出，它们两个，都与距离的平方成反比。我们感兴趣的效果，都来自第三项，毕竟，它并不是很复杂。这一项告诉我们的是：看这个电荷，且注意单位矢量的方向（我们可以把它的尾端，投影到一个单位球的表面）。随着电荷的移动，单位矢量在摆动，这个单位矢量的加速度，正是我们所要寻找的东西。这就是一切。这样：
(28.5)
is a statement of the laws of radiation,because that is the only important term when we get far enough away that thefields are varying inversely as the distance. (The parts that go as the squarehave fallen off so much that we are not interested in them.)
此公式，就是辐射规律的声明，因为，当我们离得足够远时，场的变化，与距离成反比，它就是唯一重要的项（与距离的平方成反比的部分，减小的很快，所以，我们对它们不感兴趣。）

Thus Eq. (28.5)is the complete and correct formula for radiation; even relativity effects areall contained in it. However, we often want to apply it to a still simplercircumstance in which the charges are moving only a small distance at arelatively slow rate. Since they are moving slowly, they do not move anappreciable distance from where they start, so that the delay time ispractically constant. Then the law is still simpler, because the delay time isfixed. Thus we imagine that the charge is executing a very tiny motion at aneffectively constant distance. The delay at the distance r is r/c. 这样，方程（28.5）就是关于辐射的完整的和正确的公式，甚至相对论的效果，也包含在其中。然而，通常我们只想把它，应用到更简单的情形，在此情形中，电荷只是以一个相对较慢的速率，移动了一个小的距离。由于它们移动慢，所以，它们所移动的距离，从其起点处看，并不明显可观。所以，延迟时间，是一个实践上的常数。因此，这个规律，仍是比较简单，因为延迟时间是固定的。这样，我们就可以想象，电荷实际上是在一个常数距离上，做了一个非常小的运动。距离 r处的延迟，就是 r/c。Then our rule becomes the following: If the charged object is movingin a very small motion and it is laterally displaced by the distance x(t), then the angle that the unit vector e r′is displaced is x/r , and since r is practicallyconstant, the x -component of d2er′/dt2is simply the acceleration of x itself at an earlier time dividedby r , and so finally we get the law we want, which is
因此，我们的规则，就变为如下：如果带电的对象，是在一个小的运动中运动，并且，它是被距离x(t)，横向地取代，那么，单位向量e r′被取代的角度，就是 x/r，由于r，实际上是一个常数，所以，d2er′/dt2的x分量，就是x本身在较早时间的一个加速度，除以r，于是，最终我们就得到了我们想要的规律，它就是
(28.6)
Only the component ax, perpendicular to the line of sight, is important. Let us see why that is.Evidently, if the charge is moving in and out straight at us, the unit vectorin that direction does not wiggle at all, and it has no acceleration. So it isonly the sidewise motion which is important, only the acceleration that we seeprojected on the screen.
分量ax，垂直于视线，只有它是重要的。让我们看为何如此。很明显，如果电荷直对着我们，移进移出，那么，单位矢量在此方向上，根本没有摇摆，所以，它就没有加速度。于是，只有侧向的运动，是重要的，只有我们看到的、投影到屏幕上的加速度，是重要的。{？}

28–3The dipole radiator 28-3 偶极子辐射体
As our fundamental “law” of electromagneticradiation, we are going to assume that (28.6)is true, i.e., that the electric field produced by an accelerating charge whichis moving nonrelativistically at a very large distance r approachesthat form. The electric field varies inversely as r and isproportional to the acceleration of the charge, projected onto the “plane ofsight,” and this acceleration is not today’s acceleration, but the accelerationthat it had at an earlier time, the amount of delay being a time, r/c. In the remainder of this chapter we shall discuss this law so that we canunderstand it better physically, because we are going to use it to understandall of the phenomena of light and radio propagation, such as reflection,refraction, interference, diffraction, and scattering. It is the central law,and is all we need. All the rest of Eq. (28.3)was written down only to set the stage, so that we could appreciatewhere (28.6)fits and how it comes about.
作为我们电磁辐射的基础规律，我们将假设（28.6）为真，也就是说，在非常大的距离 r处，有一个正在加速的电荷，它以非性对论的方式在移动，它所产生的电场，在接近于这个形式{？方程的形式}。电场的变化，与r成反比，且正比于电荷的加速度，{加速度}被投射到“视觉平面”，且这个加速度，并不是今天的加速度，而是电荷在早些时候所拥有的加速度，所延迟的量，就是一个时间r/c。在本章的剩余部分，我们将讨论这条规律，这样，我们就可以从物理上，对它有更好的理解，因为，我们将利用它，来理解所有光和无线电波传播的现象，例如反射、折射、干涉、衍射、和散射。它是中心规律，是我们所需要的一切。方程（28.3）的所有其他部分，只是用来把故事发生的环境，讲全了，这样，我们就可以知道，（28.6）适合于何处，及这个事情是如何发生的。

Because the room in which the waves we aremeasuring has other objects in it, our electric field will shake electrons inthese other objects; the electric field makes these other charges go up anddown, and in going up and down, these also produce an effect on our probe. Thusfor a successful experiment we must hold things fairly close together, so thatthe influences from the walls and from ourselves—the reflected waves—arerelatively small. So the phenomena will not turn out to appear to be preciselyand perfectly in accord with Eq. (28.6),but will be close enough that we shall be able to appreciate the law.
我们是在一个房间中，测量波，房间中还有其他的对象，我们的电场，也会让这些对象中的电子摇动；电场会让这些其他的电荷，上下走动，在上下走动时，这些电荷，也会对我们的探头，产生影响。这样，对于一个成功的实验来说，我们应该把所有的事物，紧密地拢在一起，这样，从墙壁和从我们来的影响--反射的波--，就会相对较小。所以，表现出来的现象。将不会像方程（28.6）所描述的那样精确和完美，但是，已经非常接近了，足以让我们赞赏这个规律。

Fig. 28–1.A high-frequency signal generatordrives charges up and down on two wires. 图28-1 一个高频信号产生器，驱动电荷在两条线中上下走。

Fig. 28–2.The instantaneous electric fieldon a sphere centered at a localized, linearly oscillating charge. 图28-2一个中心处于本地的、线性震荡着的电荷，在一个球面上的瞬时电场。
Now we turn the generator on and hear theaudio signal. We find a strong field when the detector D isparallel to the generator G at point 1 (Fig. 28–2). Wefind the same amount of field also at any other azimuth angle about the axisof G , because it has no directional effects. On the other hand,when the detector is at 3 the field is zero. That is all right, becauseour formula said that the field should be the acceleration of the charge projectedperpendicular to the line of sight. Therefore when we look down on G, the charge is moving toward and away from D , and there is no effect.So that checks the first rule, that there is no effect when the charge ismoving directly toward us. 现在，我们把发电机打开，听这个声音信号。当探测器D在点1，平行于发电机 G时，我们发现了一个强场（图28-2）。在关于轴G的任何方位角处，我们也找到的场，具有同样的量。因为，这里没有直接的影响。另一方面。当探测器在3时，场是零。这很正常，因为，我们的公式说，场应该是电荷的加速度，垂直地投射于视线。因此，当我们向下看G时，电荷的移动，就是向着D、或离开D，所以没有影响。于是，这就检验了第一条规则：当电荷直接地向着我们移动时，没有影响。Secondly, the formula says that the electric field should beperpendicular to r and in the plane of G and r ;so if we put D at 1 but rotate it 90∘ , we should get no signal. And this is just what we find, theelectric field is indeed vertical, and not horizontal. When we move D tosome intermediate angle, we see that the strongest signal occurs when it isoriented as shown, because although G is vertical, it does not produce afield that is simply parallel to itself—it is the projection of theacceleration perpendicular to the line of sight that counts. The signal isweaker at 2 than it is at 1 , because of the projection effect.
第二，这个公式说，电场应该是垂直于r，且在G和r的平面内；于是，如果我们把D放在1，但是，把它旋转90度，那么，我们就应该得不到任何信号。这正是我们所发现的，电场确实是垂直的，而非水平。当我们把D挪到某个中间的角度时{？}，我们就会看到，当其方向，如图所示时，最强信号，就会出现，因为，虽然G是垂直的，但是，它不会产生一个只是简单地平行于其自己的场—这个场，是加速度，在垂直于视线方向的投影。信号在2，比在1弱，这是因为投影的影响。

28–4Interference 28-4 干涉
Next, we may test what happens when we havetwo sources side by side several wavelengths apart (Fig. 28–3). Thelaw is that the two sources should add their effects at point 1 when bothof the sources are connected to the same generator and are both moving up anddown the same way, so that the total electric field is the sum of the two andis twice as strong as it was before.
下面，我们来测试，当两个源，并排放在一起，相距几个波长时，会发生什么（图28-3）。规律就是，当两个源，都连到同一个发电机、且都以同样的方式上下移动时，它们就应该把其影响，加到点1，这样，总的电场，就是这两个之和，且是以前强度的两倍。

Fig. 28–3.Illustration of interference ofsources. 图28-3 源的干涉的图示。

Now comes an interesting possibility.Suppose we make the charges in S1 and S2both accelerate up and down, but delay the timing of S2so that they are 180∘ out ofphase. Then the field produced by S1 will be in onedirection and the field produced by S2 will be in theopposite direction at any instant, and therefore we should get no effectat point 1 . The phase of oscillation is neatly adjustable by means of apipe which is carrying the signal to S2 . By changingthe length of this pipe we change the time it takes the signal to arriveat S2 and thus we change the phase of that oscillation.By adjusting this length, we can indeed find a place where there is no moresignal left, in spite of the fact that both S1 and S2are moving! The fact that they are both moving can be checked, because if wecut one out, we can see the motion of the other. So the two of them togethercan produce zero if everything is adjusted correctly.
现在，一个有趣的可能性，来了。假设我们把S1和S2处的电荷，都加速，让其上下移动，但是，把S2的时间延迟，让它们有180度的相位差。这样，在任一瞬间，S1产生的电场，将会在一个方向，而S2产生的电场，则将会在相反的方向，因此，在点1，我们将不会得到任何效果。震荡的相位，可以通过一个管道，精细地调整，此管道把信号带到S2。通过改变管道的长度，我们可以改变：它把信号带到S2所花的时间，这样，我们就可以改变振荡的相位。通过调整这个长度，我们确实可以找到一个位置，在那里，没有更多的信号剩下来，尽管S1和S2，都在运动。它们两个都在运动这一事实，可以检查到，因为，如果我们把一个停掉，那么，我们就可以看到：另外一个的运动。所以，如果所有的事情，都调整到位的话，那么，它们两个在一起，就能产生零。