Fig. 14–4.Potential due to a sphericalshell of radius a . 图 14-4 可归于半径为a的球形壳的势。
In the last chapter we used this formula,that the potential is the sum of the potentials from all the different objects,to calculate the potential due to a spherical shell of matter by adding thecontributions to the potential at a point from all parts of the shell. Theresult of this calculation is shown graphically in Fig. 14–4. Itis negative, having the value zero at r=∞ and varying as 1/r down to the radius a , and then is constant inside the shell. Outside the shell thepotential is −Gm/r , where m is the mass of the shell, which is exactly the same as it would havebeen if all the mass were located at the center. But it is not everywhereexactly the same, for inside the shell the potential turns out to be −Gm/a, and is a constant! When the potential is constant, there is nofield, or when the potential energy is constant there is no force, becauseif we move an object from one place to another anywhere inside the sphere thework done by the force is exactly zero. Why? Because the work done in movingthe object from one place to the other is equal to minus the change in thepotential energy (or, the corresponding field integral is the change of thepotential). But the potential energy is the same at any two pointsinside, so there is zero change in potential energy, and therefore no work isdone in going between any two points inside the shell. The only way the workcan be zero for all directions of displacement is that there is no force atall.
再上一章，我们用过这个公式，势就是所有不同对象所产生的势的总和，要计算一个球形壳所产生的势，就是把壳上所有部分，对一个点的势的贡献，全加起来。这个计算的结果，如图14-4所示。在 r=∞处，它是负的，随着1/r变化，直到半径为a处，然后，在壳的内部，是常数。在壳的外面，势是−Gm/r ，这里m是壳的质量，这个势，与球壳质量全集中在中心时，是一样的。但是，并不是处处都完全一样，因为在壳的内部，势就变成了−Gm/a，是一个常数！当势是一个常数时，就没有场，或者，当势能是一个常数时，就没有力，因为，如果在球内，我们把一个对象，从一个地方挪到另外一个地方，力所做的功就是零。为什么？因为把一个对象，从一个地方挪到另外一个地方，所做的功，等于负的势能的变化（或者相关场的积分，就是势的变化）。但是，在内部，任意两点的势能都一样，所以，势能的变化就是零，因此，在球壳内，从一点到另一点，并不做功。对于任意方向的位移，功为零的唯一方法，就是根本没有力。

This gives us a clue as to how we canobtain the force or the field, given the potential energy. Let us suppose thatthe potential energy of an object is known at the position (x,y,z)and we want to know what the force on the object is. It will not do toknow the potential at only this one point, as we shall see; it requiresknowledge of the potential at neighboring points as well. Why? How can wecalculate the x -component of the force? (If we can do this, of course, we can alsofind the y - and z -components, and we will then know the whole force.) Now, if we wereto move the object a small distance Δx , the work done by the force on the object would be the x -component of the force times Δx , if Δx is sufficiently small, and this should equal the change in potentialenergy in going from one point to the other:
ΔW=−ΔU=FxΔx. (14.9)
We have merely used the formula ∫F⋅ds=−ΔU , but for a very short path. Now we divide by Δx and so find that the force is
Fx=−ΔU/Δx. (14.10)
这给了我们一条线索，即势能被给予了，如何得到力或场。我们假设，一个对象，在点(x,y,z)，其势能已知，我们想知道：作用于其上的力是什么。我们将会看到，只知道这一个点的势，是不行的；对于周围的点，也需要其势的知识。为什么？我们如何计算力的x分量呢？（如果我们可以做到这点，当然，我们就可以同样得到y方向和z方向的分量，然后，我们就可以知道整个力。）现在，如果我们把对象，移动一个小的距离Δx，那么，对于对象所做的功，将是x方向的力乘以Δx，如果Δx足够小，那么，这就应该等于：从一个点到另一个点的势能的变化：
ΔW=−ΔU=FxΔx. (14.9)
我们只是用了公式∫F⋅ds=−ΔU，但只是对一个非常短的距离。现在，除以Δx，就得到力：
Fx=−ΔU/Δx. (14.10)

Of course this is not exact. What we really want is the limitof (14.10)as Δx gets smaller and smaller, because it is only exactly right inthe limit of infinitesimal Δx . This we recognize as the derivative of U with respect to x , and we would be inclined, therefore, to write −dU/dx. But U depends on x , y , and z , and the mathematicians have invented a different symbol to remind usto be very careful when we are differentiating such a function, so as toremember that we are considering that only x varies, and y and z do not vary. Instead of a d they simply make a “backwards 6 ,” or ∂ . (A ∂ should have been used in the beginning of calculus because we alwayswant to cancel that d , but we never want to cancel a ∂ !) So they write ∂U/∂x , and furthermore, in moments of duress, if they want to be verycareful, they put a line beside it with a little yz at the bottom (∂U/∂x|yz ), which means “Take the derivative of U with respect to x , keeping y and z constant.” Most often we leave out the remark about what is keptconstant because it is usually evident from the context, so we usually do notuse the line with the y and z . However, always use a ∂ instead of a d as a warning that it is a derivative with some other variables keptconstant. This is called a partial derivative; it is a derivative inwhich we vary only x .
当然，这并不准确。我们真正想要的，是当Δx变得越来越小时，（14.10）的极限，因为，它只在间隔Δx取极限时，才完全正确。我们把这个，认为是U对x的导数，因此，我们倾向于写−dU/dx。但是, U 依赖于 x , y , 和z，所以，数学家们发明了一个不同的符号，提醒我们，当我们对这种函数求微分时，要非常小心，为的是记住，我们考虑的只是x的变化，而y和z，并没有变。为了代替d，他们简单地发明了一个“向后的6”，或者∂.（在微积分开始时，就应该用到∂，因为，我们总是想消去那个d，但是，我们永远也不会想着消去∂！{？}）于是，它们写出∂U/∂x,更进一步，在需要强迫时，如果他们想非常小心，他们会在底部，放一条线，和小的yz，(∂U/∂x|yz)，其意思是“求U对x 的导数，y和 z保持为常数。”大多数情况下，那些需要被保持为常数的项，我们不会写，因为，通常从上下文看，很明显，所以，我们通常不用线及y和 z。然而，总是用 ∂ 代替 d，作为一种警告：这个导数中，有些变量保持为常数。这被称为偏导数；在这个导数中，我们只变更x。

Therefore, we find that the force in the x-direction is minus the partial derivative of U with respect to x :
Fx=−∂U/∂x. (14.11)
In a similar way, the force in the y -direction can be found by differentiating U with respect to y , keeping x and z constant, and the third component, of course, is the derivative withrespect to z , keeping y and x constant:
Fy=−∂U/∂y, Fz=−∂U/∂z. (14.12)
This is the way to get from the potential energy to the force. We getthe field from the potential in exactly the same way:
Cx=−∂Ψ/∂x, Cy=−∂Ψ/∂y, Cz=−∂Ψ/∂z. (14.13)
因此，我们发现，x方向的力，就是负的 U对x的偏导数:
Fx=−∂U/∂x. (14.11)
类似地，y方向的力，就是负的 U对y的偏导数，x和z保持为常数，而第三个分量，当然就是对z的偏导数，y和x保持为常数：
Fy=−∂U/∂y, Fz=−∂U/∂z. (14.12)
这就是从势能出发，得到力的方法。我们从势出发，以完全同样的方式，得到场：
Cx=−∂Ψ/∂x, Cy=−∂Ψ/∂y, Cz=−∂Ψ/∂z. (14.13)

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。