Incidentally, we shall mention here another notation, which we shallnot actually use for quite a while: Since C is a vector and has x -, y -, and z -components, the symbolized ∂/∂x , ∂/∂y , and ∂/∂z which produce the x -, y -, and z -components are something like vectors. The mathematicians haveinvented a glorious new symbol, ∇ , called “grad” or “gradient”, which is not a quantity but an operatorthat makes a vector from a scalar. It has the following “components”: The x-component of this “grad” is ∂/∂x , the y -component is ∂/∂y , and the z -component is ∂/∂z , and then we have the fun of writing our formulas this way:
F=−∇U, C=−∇Ψ.(14.14)
Using ∇ gives us a quick way of testing whether we have a real vector equationor not, but actually Eqs. (14.14)mean precisely the same as Eqs. (14.11), (14.12)and (14.13);it is just another way of writing them, and since we do not want to write threeequations every time, we just write ∇U instead.
顺便说说，我们这里要提一下另外一种记号，虽然我们在相当长的时间内，不会实际地用它：由于C是一个矢量，有 x、y、和 z 分量，所以，∂/∂x 、 ∂/∂y、和 ∂/∂z，就是某种符号化的的东西，用来产生x、y、和 z 分量，类似于矢量。数学家们，发明了一个极好的新符号，∇，被称为“梯度”或“梯度变化”{？}，它不是一个量，而是一个运算符，从一个标量，造出一个矢量。它有如下“分量”：这个“梯度”的x分量就是 ∂/∂x ，y分量是 ∂/∂y , z分量是∂/∂z ，因此，把我们的公式，写成如下形式，将非常有趣：
F=−∇U, C=−∇Ψ.(14.14)
使用∇，给我们的提供了一种方法，可以快速地检查，我们的方程，是否是一个真的矢量方程，但实际上，方程（14.14），与方程（14.11）、（14.12）、（14.13）完全一样；只不过是另外一种写它们的方式罢了，由于我们不想每次都写三个方程，我们只写∇U来代替。

One more example of fields and potentialshas to do with the electrical case. In the case of electricity the force on astationary object is the charge times the electric field: F=qE. (In general, of course, the x -component of force in an electrical problem has also a part whichdepends on the magnetic field. It is easy to show from Eq. (12.11) that the force on a particle due to magnetic fieldsis always at right angles to its velocity, and also at right angles to thefield. Since the force due to magnetism on a moving charge is at right anglesto the velocity, no work is done by the magnetism on the moving chargebecause the motion is at right angles to the force. Therefore, in calculatingtheorems of kinetic energy in electric and magnetic fields we can disregard thecontribution from the magnetic field, since it does not change the kineticenergy.) We suppose that there is only an electric field. Then we can calculatethe energy, or work done, in the same way as for gravity, and calculate aquantity ϕ which is minus the integral of E⋅ds , from the arbitrary fixed point to the point where we make thecalculation, and then the potential energy in an electric field is just chargetimes this quantity ϕ :
ϕ(r)=−∫E⋅ds,U=qϕ.
场和势的另外一个例子，与电的案例有关。在电的案例中，作用于一个静止对象上的力，就是电荷乘以电场：F=qE。（当然，一般在电的问题中，力的x分量，部分地依赖于磁场。从方程（12.11）很容易指出，一个粒子上的、可归于磁场的力，总是垂直于矢速，也垂直于场。对于一个移动电荷，由于作用于其上的、可归于磁场的力，垂直于矢速，所以，磁场并没有对它做功，因为，运动垂直于力。因此，在电磁场中，对于计算动能的理论，我们可以忽略磁场的贡献，由于它没有改变动能。）我们假设，只有一个电场。因此，我们计算能量、或所做的功时，就可以像在万有引力场中那样做，另外，还要计算一个量ϕ，它是负的 E⋅ds的积分，积分路径为从任意固定点，到我们做此计算的那个点，因此，电场中的势能，就只是电荷乘以这个量ϕ ：
ϕ(r)=−∫E⋅ds, U=qϕ.

Fig. 14–5.Field between parallel plates. 图14-5 平行板之间的场。
Let us take, as an example, the case of twoparallel metal plates, each with a surface charge of ±σ per unit area. This is called a parallel-plate capacitor. We foundpreviously that there is zero force outside the plates and that there is aconstant electric field between them, directed from + to − and of magnitude σ/ϵ0 (Fig. 14–5). Wewould like to know how much work would be done in carrying a charge from oneplate to the other. The work would be the (force)⋅(ds) integral, which can be written as charge times the potential value atplate 1 minus that at plate 2 :
We can actually work out the integral because the force is constant, andif we call the separation of the plates d , then the integral is easy:
The difference in potential, Δϕ=σd/ϵ0 , is called the voltage difference, and ϕ is measured in volts. When we say a pair of plates is charged to acertain voltage, what we mean is that the difference in electrical potential ofthe two plates is so-and-so many volts. For a capacitor made of two parallelplates carrying a surface charge ±σ , the voltage, or difference in potential, of the pair of plates is σd/ϵ0.
让我们看一个例子，两个平行金属板的案例，每个板上充电，表面的单位面积电量为 ±σ。它叫平行板电容器。前面我们已经得出，在板外，力为零，在板间，有一个电场，为常数，从+指向-，大小为σ/ϵ0（图14-5）。我们想希望知道，把一个电荷，从一个板载到另一个板，做了多少功。功应该是 (力)⋅(ds)的积分，它可被写为：电荷乘以，板1的势减去板2的势：

实际上，我们可以的得出积分，因为力是常数，如果我们把板间距离，称为d，那么，积分就很容易：

势的差，Δϕ=σd/ϵ0，被称为电位差，测量ϕ用伏特。当我们说，一对板子，被充电到一定伏特时，我们的意思是，两个板之间的电势差，就是如此这般的伏特数。一个电容器，由平行板所做，表面所带电荷为±σ，其伏特、或势能差，就是σd/ϵ0。

Chapter15.The Special Theory of Relativity第15章 相对论的特殊原理
15–1The principle of relativity 15-1 相对论原理
For over 200 years the equations of motionenunciated by Newton were believed to describe nature correctly, and the firsttime that an error in these laws was discovered, the way to correct it was alsodiscovered. Both the error and its correction were discovered by Einstein in1905.
牛顿所宣布的运动方程，在200多年中，都被认为，是正确地描述了自然，在这些方程中，第一次发现错误，并且更改错误的方式也被发现了。错误和更改，皆由爱因斯坦发现，在1905年。

Newton’s Second Law, which we haveexpressed by the equation
F=d(mv)/dt,
was stated with the tacit assumption that m is a constant, but we now know that this is not true, and that themass of a body increases with velocity. In Einstein’s corrected formula mhas the value
(15.1)
where the “rest mass” m0 represents the mass of a body that is not moving and c is the speed of light, which is about 3×105 km⋅sec−1 or about 186,000 mi⋅sec−1.
牛顿第二规律，我们通过下面方程表示：
F=d(mv)/dt,
它有一个心照不宣的假定，即m是常数，但我们现在知道，这并不正确，物体的质量随着矢速而增加。在爱因斯坦的更正后的公式中，m的值为：
(15.1)
这里“静态质量” m0，代表物体未运动时的质量，c 是光速，约为3×105 km⋅sec−1，或约 186,000 mi⋅sec−1。

For those who want to learn just enoughabout it so they can solve problems, that is all there is to the theory ofrelativity—it just changes Newton’s laws by introducing a correction factor to themass. From the formula itself it is easy to see that this mass increase is verysmall in ordinary circumstances. If the velocity is even as great as that of asatellite, which goes around the earth at 5 mi/sec, then v/c=5/186,000 : putting this value into the formula shows that the correction to themass is only one part in two to three billion, which is nearly impossible toobserve. Actually, the correctness of the formula has been amply confirmed bythe observation of many kinds of particles, moving at speeds ranging up topractically the speed of light. However, because the effect is ordinarily sosmall, it seems remarkable that it was discovered theoretically before it wasdiscovered experimentally. Empirically, at a sufficiently high velocity, theeffect is very large, but it was not discovered that way. Therefore it isinteresting to see how a law that involved so delicate a modification (at thetime when it was first discovered) was brought to light by a combination ofexperiments and physical reasoning. Contributions to the discovery were made bya number of people, the final result of whose work was Einstein’s discovery.
有些人学东西，能够解决问题就行了，对于这些人来说，关于相对论的东西，都在这里了—它只通过引入对质量的一个更改因子，改变了牛顿规律。从公式本身，很易看出，在通常情形下，这种质量增加，非常小。卫星绕地球的速度是5 mi/sec ，如果矢速是这个，那么，v/c=5/186,000：把这个值带入公式，将指出，对质量的更正，只是20到30亿分之一，这几乎无法观察到。实际上，此公式的更正，已经通过很多粒子的观察，被充分地证明了，这些粒子的移动速度，实际上可达到光速。然而，因为效果通常是如此小，似乎值得说明的是，它首先是理论上被发现，然后才是实验中被发现。根据经验，在足够高的矢速下，效果显著，但它不是这样被发现的。因此，一条规律，有如此精致的修改（在它首次被发现时），看到它，通过实验和物理理论推理，而被发现，是件有趣之事。对这个发现的贡献，由若干人完成，最终结果，是爱因斯坦的发现。

There are really two Einstein theories ofrelativity. This chapter is concerned with the Special Theory of Relativity, whichdates from 1905. In 1915 Einstein published an additional theory, called theGeneral Theory of Relativity. This latter theory deals with the extension ofthe Special Theory to the case of the law of gravitation; we shall not discussthe General Theory here.
爱因斯坦的相对论，分两种。本章关心的，是狭义相对论，它可追溯到1905。在1915，爱因斯坦发布了另外的理论，被称为广义相对论。这后一理论，处理：狭义相对论扩展到万有引力规律时的情况；我们这里不讨论广义相对论。

Fig. 15–1.Two coordinate systems in uniformrelative motion along their x -axes. 图15-1 两个坐标系，沿着x轴，做匀速相对运动。
Suppose that Moe is moving in the x -direction with a uniform velocity u , and he measures the position of a certain point, shown in Fig. 15–1. Hedesignates the “x -distance” of the point in his coordinate system as x′ . Joe is at rest, and measures the position of the same point,designating its x -coordinate in his system as x . The relationship of the coordinates in the two systems is clear fromthe diagram. After time t Moe’s origin has moved a distance ut , and if the two systems originally coincided,
x′=x−ut,
y′′=y,
z′=z,
t=t. (15.2)
假设Moe 沿x方向，以匀速u移动，他在测量某点的位置，如图15-1。他在他的坐标系中，指定这个点的“x距离”为x′。Joe是静止的，他也在测量同一个点，他在他的坐标系中，把此点的x坐标指定为x。如图，两个坐标系的关系，很清楚。在时间t之后，Moe的原点，移动了距离ut，如果这两个坐标系，最初是重叠的，那么：
x′=x−ut，
y′′=y，
z′=z，
t=t。 (15.2)

If we substitute this transformation of coordinates into Newton’s lawswe find that these laws transform to the same laws in the primed system; thatis, the laws of Newton are of the same form in a moving system as in astationary system, and therefore it is impossible to tell, by making mechanicalexperiments, whether the system is moving or not.
如果我们把这个坐标变换，带入牛顿规律，我们发现，这些规律，会变得与原先系统中的规律一样；也就是说，牛顿规律，在一个移动的系统中，与在一个静止的系统中，形式一样，因此，要通过做力学实验，来得到这个系统是否在移动，是不可能的。

15–2The Lorentz transformation 15-2 洛伦兹变换
When the failure of the equations of physicsin the above case came to light, the first thought that occurred was that thetrouble must lie in the new Maxwell equations of electrodynamics, which wereonly 20 years old at the time. It seemed almost obvious that theseequations must be wrong, so the thing to do was to change them in such a waythat under the Galilean transformation the principle of relativity would besatisfied. When this was tried, the new terms that had to be put into theequations led to predictions of new electrical phenomena that did not exist atall when tested experimentally, so this attempt had to be abandoned. Then itgradually became apparent that Maxwell’s laws of electrodynamics were correct,and the trouble must be sought elsewhere.
在上面的情况中，物理学的方程失败了，此事曝光后，第一个想法就是，问题肯定出在麦克斯韦尔的电动力学方程中，那时候，方程只有20岁。这些方程肯定错了，这一点，似乎很明显，于是，要做的事情，就是按如下方式来改变它们：即在伽利略变换下，也可以让相对论的原理，能被满足。这样试了之后，方程中，出现了新的项，导致了对新的电现象的预告，但经过实验检测，新现象根本不存在，所以，这个尝试，必须放弃。随后，麦克斯韦尔的电动力学规律是正确的，逐渐明显；问题应该出在其他地方。

In the meantime, H. A. Lorentz noticeda remarkable and curious thing when he made the following substitutions in theMaxwell equations:
(15.3)
namely, Maxwell’s equations remain in the same form when this transformationis applied to them! Equations (15.3)are known as a Lorentz transformation. Einstein, following a suggestionoriginally made by Poincaré, then proposed that all the physical lawsshould be of such a kind that they remain unchanged under a Lorentz transformation.In other words, we should change, not the laws of electrodynamics, but the lawsof mechanics. How shall we change Newton’s laws so that they will remainunchanged by the Lorentz transformation? If this goal is set, we then have torewrite Newton’s equations in such a way that the conditions we have imposedare satisfied. As it turned out, the only requirement is that the mass min Newton’s equations must be replaced by the form shown in Eq. (15.1).When this change is made, Newton’s laws and the laws of electrodynamics willharmonize. Then if we use the Lorentz transformation in comparing Moe’s measurementswith Joe’s, we shall never be able to detect whether either is moving, becausethe form of all the equations will be the same in both coordinate systems!
同时，当洛伦兹对麦克斯韦方程组，做如下替换时，注意到了一件引人注目且古怪的事情：
（15.3）
即对于麦克斯韦方程组，当这个变换，被应用于其上时，其形式，保持不变。方程组（15.3）以洛伦兹变换而闻名。爱因斯坦，遵循庞加莱提出的建议，提议说，所有物理规律，都应该是这种类型：它们在洛伦兹变换下，保持不变。换句话说，我们应该改变力学的规律，而不是电动力学的规律。我们应该如何改变牛顿规律，以让它们，在洛伦兹变换下，仍保持不变呢？如果这个目标定了，那么，我们就必须以这种方式，重写牛顿方程组，已让我们强制的条件，得以满足。由于结果就是，唯一的要求就是，牛顿方程组中的质量m，应该被方程（15.1）中的形式替代。做了这个改变之后，牛顿规律与电动力学的规律，就一致了。因此，在把Moe的测量，与Joe的测量相比较时，如果我们使用的是洛伦兹变换，则永远也不能探测出，哪一个坐标系在移动，因为，两个坐标系中的所有方程，形式都一样。

It is interesting to discuss what it meansthat we replace the old transformation between the coordinates and time with anew one, because the old one (Galilean) seems to be self-evident, and the newone (Lorentz) looks peculiar. We wish to know whether it is logically andexperimentally possible that the new, and not the old, transformation can be correct.To find that out, it is not enough to study the laws of mechanics but, asEinstein did, we too must analyze our ideas of space and time inorder to understand this transformation. We shall have to discuss these ideas andtheir implications for mechanics at some length, so we say in advance that theeffort will be justified, since the results agree with experiment.
在两个坐标系及时间之间，用一个新的变换，来代替旧的变换，讨论这种变换究竟意味着什么，是很有意思的问题，因为，旧的变换（伽利略），似乎是显而易见的，而新的变换（洛伦兹）看上去有点奇怪。我们希望知道，新的变换（非旧的变换）是正确的这一点，是逻辑上或实验上是可能的。要证明此点，只研究力学规律，远远不够，而是也应该像爱因斯坦那样，去分析我们关于空间和时间的想法，以便理解这个变换。这些想法及其在力学中的实现，我们将详细讨论，由于结果与实验，是一致的，所以，我们要提前说一下，这些努力，是有正当理由的。

15–3The Michelson-Morley experiment 15-3迈克耳孙--莫雷实验
As mentioned above, attempts were made todetermine the absolute velocity of the earth through the hypothetical “ether”that was supposed to pervade all space. The most famous of these experiments isone performed by Michelson and Morley in 1887. It was 18 years later before thenegative results of the experiment were finally explained, by Einstein.
正如上面提到的，曾经有个假设，认为有“以太”，弥漫于所有空间中，为了得出地球经过以太的绝对矢速，做了许多尝试。这些实验中最著名的一个，是由迈克耳孙和莫雷，在1887年做的。关于此实验的负面结果，18年后，才由爱因斯坦，给出了解释。

666

Fig. 15–2.Schematic diagram of the Michelson-Morleyexperiment. 图15-2 迈克耳孙--莫雷实验的规划图
The Michelson-Morley experiment was performedwith an apparatus like that shown schematically in Fig. 15–2. Thisapparatus is essentially comprised of a light source A , a partially silvered glass plate B , and two mirrors C and E , all mounted on a rigid base. The mirrors are placed at equaldistances L from B . The plate B splits an oncoming beam of light, and the two resulting beams continuein mutually perpendicular directions to the mirrors, where they are reflectedback to B . On arriving back at B , the two beams are recombined as two superposed beams, D and F . If the time taken for the light to go from B to E and back is the same as the time from B to C and back, the emerging beams D and F will be in phase and will reinforce each other, but if the two timesdiffer slightly, the beams will be slightly out of phase and interference willresult. If the apparatus is “at rest” in the ether, the times should beprecisely equal, but if it is moving toward the right with a velocity u, there should be a difference in the times. Let us see why.
迈克耳孙--莫雷实验，所用仪器的规划，如图15-2所示。这个仪器的核心组成为：光源 A, 部分镀银的玻璃板 B , 两个镜子C和E，都安装在一个固定的基座上。镜子到B的距离，都是L。板B把打过来的光束，劈成两份，它们以互相垂直的方向，继续走到镜子上，在那里，被反射回B。返回B后，这两束光，被当作两个重叠的光束D和 F ，重新结合。如果光从B到E再回来所化的时间，与光从B到C再回来所化的时间，相等，那么，显现出来的光束D和 F，就将是同相位的，且将互相加强，但是，如果时间有轻微差异，那么，光束的相位，将略有不同，则干涉将会发生。如果仪器在以太中是“静止的”，那么，时间应该是精确相等的，但是，如果它是以矢速u，向右移动，那么，时间就应有所不同。我们来看为什么。