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。

Now, it is very interesting to show thatthe addition of the two fields is in fact a vector addition. We havejust checked it for up and down motion, but let us check two nonparalleldirections. First, we restore S1 and S2to the same phase; that is, they are again moving together. But now we turn S1through 90∘ , as shownin Fig. 28–4.Now we should have at point 1 the sum of two effects, one of which is verticaland the other horizontal. The electric field is the vector sum of these twoin-phase signals—they are both strong at the same time and go through zerotogether; the total field should be a signal R at 45∘ . If we turn D to get the maximum noise, it should be atabout 45∘ , and notvertical. And if we turn it at right angles to that direction, we should getzero, which is easy to measure. Indeed, we observe just such behavior!
现在，我们来看一个非常有趣的事情：两个场相加，其实就是两个矢量相加。对于上下运动，我们已经检查了这一点。但是，让我们检查两个非平行方向的运动。首先，我们把S1和S2，恢复到同样的相位，也就是说，它们又一起运动了。但是，现在我们把S1转动了90度，如图28-4所示。现在，我们在点1，应该有它们两个的效果之和，其中之一，是垂直的，另一个，是水平的。这两个信号，是同相位的--它们同时变强、同时经过零，它们的矢量和，就是电场；总的场，应该是在45度处的一个信号R。如果我们转动D，以得到最大的噪音，那么，应该大约是在45度方向，而不是垂直方向。如果我们把它转一个直角，成为垂直的，我们就应该得到零，这很容易测量。确实，我们观察到的，正是如此。

Fig. 28–4.Illustration of the vectorcharacter of the combination of sources. 图28-4 两个源，组合在一起，其矢量特性之图示。

Now, how about the retardation? How can wedemonstrate that the signal is retarded? We could, with a great deal ofequipment, measure the time at which it arrives, but there is another, verysimple way. Referring again to Fig. 28–3,suppose that S1 and S2 are in phase.They are both shaking together, and they produce equal electric fields atpoint 1 . But suppose we go to a certain place 2 which is closerto S2 and farther from S1 . Then,in accordance with the principle that the acceleration should be retarded by anamount equal to r/c , if the retardations are not equal, thesignals are no longer in phase. Thus it should be possible to find a positionat which the distances of D from S1 and S2differ by some amount Δ , in such a manner that there is no net signal. 现在，迟滞又如何呢？我们如何演证，信号被迟滞了呢？我们可以用大量的设备，来测量信号到达的时间，但是，还有另外一个非常简单的方法。参考图28-3，假设S1与S2是同相位的。它们在一起摇动，在点1，产生的电场相等。但是，假设我们到某确定位置2，它离近S2，离S1远。因此，依据原理，加速度应该被迟滞一个量 r/c，如果迟滞不相等，那么，信号就不再是一个相位的了。这样，应该可能找到一个位置，在那里，D到S1的距离，与D到S2的距离，相差一个量 Δ，在这种方式下，净的信号，就没有了。That is, the distance Δ is to be the distance light goes inone-half an oscillation of the generator. We may go still further, and find apoint where the difference is greater by a whole cycle; that is to say, thesignal from the first antenna reaches point 3 with a delay in time that isgreater than that of the second antenna by just the length of time it takes forthe electric current to oscillate once, and therefore the two electric fieldsproduced at 3 are in phase again. At point 3 the signal is strongagain.
也就是说，距离Δ就是，光在发电机的震荡的半周期中，所走的距离。我们还可以走的更远，找到一个点，在那里，要相差一个完整的周期；也就是说，从第一个天线到点3，有一个时间延迟，从第二个天线到点3，又有一个时间延迟，两者之差，可以让电流，震荡一次，因此，在点3所产生的两个电场，就又是同相的了。在点三，信号又变强了。

This completes our discussion of theexperimental verification of some of the important features of Eq. (28.6).Of course we have not really checked the 1/r variation of theelectric field strength, or the fact that there is also a magnetic field thatgoes along with the electric field. To do so would require rather sophisticatedtechniques and would hardly add to our understanding at this point. In anycase, we have checked those features that are of the greatest importance forour later applications, and we shall come back to study some of the otherproperties of electromagnetic waves next year.
方程（28.6）有一些重要的特性，对于其中一些，我们做了实验性的验证，现在，已经讨论完毕。当然，我们并没真正去检查：电场强度随着1/r的变化，或者如下事实：跟电场在一起的，还有一个磁场。要做这些，要求相当老练成熟的技术，且在此时，也很难进一步加深我们对物理的理解。无论如何，我们已经检查的那些特性，对我们后面的应用来说，具有最重要的意义。明年，我们将回来学习：电磁波的另外一些属性。

Fig. 29–1.The electric field Edue to a positive charge whose retarded acceleration is a′ . 图29-1 电场 E，可归于一个正电荷，该电荷被迟滞的加速度是a′。
We have already physically analyzed themeaning of formula (28.6) quite satisfactorily, but there are a few pointsto be made about it mathematically. In the first place, if a charge isaccelerating up and down along a line, in a motion of very small amplitude, thefield at some angle θ from the axis of the motion is in a directionat right angles to the line of sight and in the plane containing both theacceleration and the line of sight (Fig. 29–1). Ifthe distance is called r , then at time t the electricfield has the magnitude
对于公式（28.6）的意义，我们已经从物理上，进行了分析，非常令人满意，但是，从数学上，还要再说几点。首先，如果一个电荷，正在沿着一条直线，被加速，而在做上下运动，且运动的幅度，非常小，那么，在运动的轴，成θ角的方向上，形成一个电场，其方向，垂直于视线的角度，且其所在平面，包含着加速度和视线（图29-1）。如果距离被称为r，那么，在时间t，电场大小就是：
(29.1)
where a(t−r/c)is the acceleration at the time (t−r/c) , called the retardedacceleration.
这里，a(t−r/c) ，就是在时间(t−r/c)的加速度，被称为迟滞的加速度。

Stated another way: if we add a littletime Δt , we can restore a(t−r/c) toits former value by adding a little distance Δr=cΔt. That is, as time goes on the field moves as a wave outward from the source.That is the reason why we sometimes say light is propagated as waves. It isequivalent to saying that the field is delayed, or to saying that the electricfield is moving outward as time goes on.
换种方式说：如果我们增加一个小的时间Δt，那么，通过增加一个小的距离Δr=cΔt，我们可以把a(t−r/c)，重建到其前一个值。也就是说，随着时间的过去，场从源处，作为一个波，向外移动。这也就是为什么我们有时候会说:光是以波的形式在传播。它等于是说：场被迟滞了，或者说：随着时间的过去，电场向外移动。

An interesting special case is that wherethe charge q is moving up and down in an oscillatory manner. Thecase which we studied experimentally in the last chapter was one in which thedisplacement x at any time t was equal to a certainconstant x0 , the magnitude of the oscillation,times cosωt . Then the acceleration is
一种有趣的特殊情况，就是电荷q，以一种的震荡的方式，上下移动。我们上一章，以实验的方式，研究过此情况，它就是，任意时间t的位移x，等于某个常数x0、即震荡的的大小，乘以cosωt。因此，加速度就是：
a=−ω2x0cosωt=a0cosωt, (29.2)
where a0 is the maximumacceleration, −ω2x0 . Putting this formulainto (29.1),we find
这里，a0是最大加速度，−ω2x0。把这个公式，代入（29.1），我们发现：
(29.3)
Now, ignoring the angle θ andthe constant factors, let us see what that looks like as a function of positionor as a function of time.
现在，忽略角度θ和常数因子，让我们看看，作为位置的函数，或作为时间的函数，它看上去像什么。