We shall next discuss the characteristicsof molecular forces. These are forces between the atoms, and are the ultimateorigin of friction. Molecular forces have never been satisfactorily explainedon a basis of classical physics; it takes quantum mechanics to understand themfully. Empirically, however, the force between atoms is illustratedschematically in Fig. 12–2, wherethe force F between two atoms is plotted as a function of the distance rbetween them. There are different cases: in the water molecule, forexample, the negative charges sit more on the oxygen, and the mean positions ofthe negative charges and of the positive charges are not at the same point; consequently,another molecule nearby feels a relatively large force, which is called adipole-dipole force. However, for many systems the charges are very much betterbalanced, in particular for oxygen gas, which is perfectly symmetrical. In thiscase, although the minus charges and the plus charges are dispersed over the molecule,the distribution is such that the center of the minus charges and the center ofthe plus charges coincide. A molecule where the centers do not coincide iscalled a polar molecule, and charge times the separation between centers iscalled the dipole moment. A nonpolar molecule is one in which the centers ofthe charges coincide. For all nonpolar molecules, in which all the electricalforces are neutralized, it nevertheless turns out that the force at very largedistances is an attraction and varies inversely as the seventh power of thedistance, or F=k/r7 , where k is a constant that depends on the molecules. Why this is we shalllearn only when we learn quantum mechanics. When there are dipoles the forcesare greater. When atoms or molecules get too close they repel with a very largerepulsion; that is what keeps us from falling through the floor!
下面我们讨论分子力的特性。这些力是原子间的力，是摩擦的终极起源。分子力从来没有在经典物理学的基础上，得到过满意的解释。然而，根据经验，原子间的力，可以大致地图示，见图12-2，这里，两个原子之间的力F被当作它们之间距离r的函数。还有些不同的情况：例如，在水分子中，负电荷更偏于氧原子，负电荷与正电荷的平均位置，并不在同一个点上；所以，附近的另外一个分子，就会感到相对较大的力，这被称为偶极子-偶极子的力。然而，对于很多系统来说，电荷还是比较好地被平衡了，特别是对氧气，它是完全对称的。在这种情况下，虽然负电荷和正电荷被散布在分子中，这种分布是这样的，负电荷的中心与正电荷的中心，是重合的。一个分子，它的中心如果不重合，就被称为极性分子，而电荷乘以两个中心之间的距离，被称为偶极矩。一个非偶极的分子呢，就是中电荷中心重合的分子。一个非极性的分子，就是在其中，电荷的中心是重合的分子。对于所有的非极性分子来说，尽管在其中，所有的电力都被中和了，但结果则是，在很大的距离上的力，是吸引力，且反比于距离的7次方，或者F=k/r7，这里，k是一个常数，依赖于分子。为什么会是这样，只有当我们学习电动力学时，才能学到。当有偶极子时，力就会更大些。当原子或分子变得太近时，它们就会用一个很大的力，相互排斥；我们不会从地板上掉下去，就是因为它托着。

These molecular forces can be demonstratedin a fairly direct way: one of these is the friction experiment with a slidingglass tumbler; another is to take two very carefully ground and lapped surfaceswhich are very accurately flat, so that the surfaces can be brought very closetogether. An example of such surfaces is the Johansson blocks that are used inmachine shops as standards for making accurate length measurements. If one suchblock is slid over another very carefully and the upper one is lifted, theother one will adhere and also be lifted by the molecular forces, exemplifyingthe direct attraction between the atoms on one block for the atoms on the otherblock.
这些分子力，可以用一种相当直接的方式，来展示：其中之一，就是用一个滑动的无脚玻璃，所做的摩擦实验；另一个，就是两个磨得非常平的表面，非常仔细地叠加在一起，这样，表面就可以连得很近。一个这种表面的例子，就是约翰逊块，在机械工厂中，制造准确长度的仪器时，作为标准。如果一个这种块，被非常仔细地滑过另一个，且上面的块被抬了起来，那么，另一个块，会吸附上面的块，通过分子力，也会被抬起，这例示了：一个块中的原子，对另一块中的原子的直接吸引。

Nevertheless these molecular forces ofattraction are still not fundamental in the sense that gravitation is fundamental;they are due to the vastly complex interactions of all the electrons and nucleiin one molecule with all the electrons and nuclei in another. Anysimple-looking formula we get represents a summation of complications, so we stillhave not got the fundamental phenomena.
尽管如此，万有引力是基础的，从这个意义看，这些分子的吸引力，并不是基础的；它们可归于，一个分子中的所有电子和原子核，与另一个分子中的所有电子和原子核，它们之间的巨大复杂的相互作用。任何我们得到的、看上去简单的公式，都代表着一种复杂性的总和，所以，我们尚未得到基础现象。

Since the molecular forces attract at largedistances and repel at short distances, as shown in Fig. 12–2, wecan make up solids in which all the atoms are held together by theirattractions and held apart by the repulsion that sets in when they are tooclose together. At a certain distance d (where the graph in Fig. 12–2crosses the axis) the forces are zero, which means that they are all balanced,so that the molecules stay that distance apart from one another. If themolecules are pushed closer together than the distance d they all show a repulsion, represented by the portion of the graphabove the r -axis. To push the molecules only slightly closer together requires agreat force, because the molecular repulsion rapidly becomes very great atdistances less than d . If the molecules are pulled slightly apart there is a slight attraction,which increases as the separation increases. If they are pulled sufficientlyhard, they will separate permanently—the bond is broken.
由于分子间的力，距离大时相吸，距离短时相斥，如图12-2，我们可以造出固体，在其中，所有原子被它们的吸引力抓在一起，被它们的排斥力保持一定距离，此排斥力是它们太近时所产生的。在某一距离d（那里图12-2中的曲线穿过轴），力是零，这意味着它们{吸引与排斥}全平衡了，于是，分子之间，就保持在那个距离。如果分子被压缩，之间的距离小于d，那么，它们都会显示出排斥，这由图中r -axis上的部分来代表。把分子推得靠近一点点，需要巨大的力，因为，当分子间的距离小于d时，分子间的斥力，很快就变得非常巨大。如果分子被轻轻地拉开，有轻微的吸引力，它随着分开距离的增大而增大。如果被足够硬地拉开，它们将永久性地分开—键断了。

If the molecules are pushed only a verysmall distance closer, or pulled only a very small distance fartherthan d , the corresponding distance along the curve of Fig. 12–2 isalso very small, and can then be approximated by a straight line. Therefore, inmany circumstances, if the displacement is not too great the force is proportionalto the displacement. This principle is known as Hooke’s law, or the law ofelasticity, which says that the force in a body which tries to restore the bodyto its original condition when it is distorted is proportional to thedistortion. This law, of course, holds true only if the distortion is relativelysmall; when it gets too large the body will be torn apart or crushed, dependingon the kind of distortion. The amount of force for which Hooke’s law is validdepends upon the material; for instance, for dough or putty the force is verysmall, but for steel it is relatively large. Hooke’s law can be nicelydemonstrated with a long coil spring, made of steel and suspended vertically. Asuitable weight hung on the lower end of the spring produces a tiny twist throughoutthe length of the wire, which results in a small vertical deflection in eachturn and adds up to a large displacement if there are many turns. If the totalelongation produced, say, by a 100 -gram weight, is measured, it is found that additional weights of 100 grams will each produce an additional elongation that is verynearly equal to the stretch that was measured for the first 100 grams. This constant ratio of force to displacement begins tochange when the spring is overloaded, i.e., Hooke’s law no longer holds.
如果分子被推近一个非常小的距离，或被拉开一个非常小的、比d大的距离，那么，沿着图12-2中的曲线的相应的距离，也会很小，因此，可以通过一个直线来近似。因此，在很多情形中，如果位移不是很大，力与位移成正比。这个原理，被称为胡克规律，或者弹性规律，它说，当一个物体，被扭曲之后，它有一个力，尝试把物体恢复到原始状态，这个力正比于此扭曲。当然，只有当扭曲较小时，这个力才正确；当扭曲太大时，物体就会被撕裂或者变形，这依赖于是扭曲是哪种。胡克规律的有效范围，依赖于材料；例如，对于面团或腻子，这个力非常小，对于钢就较大。胡克规律可以用一个线圈状的弹簧，非常好地演示，弹簧由钢做成，垂直悬挂。合适的重量，被挂在弹簧的低端，让整个弹簧，都产生一个轻微扭曲，这导致了弹簧的每一圈，都有一个小的偏移，如果有很多圈，就会产生一个大的位移。比如说通过100克的重量，所产生的总的拉长，被测量了，那么，就会发现，每增加100克所产生的拉长，都几乎与第一个100克所产生的拉长相等。当弹簧过载时，这个力对位移的恒定比率，就会开始改变，也就是说，胡克定律不再起作用了。

12–4Fundamental forces. Fields 12-4 基础力。场
We shall now discuss the only remainingforces that are fundamental. We call them fundamental in the sense that theirlaws are fundamentally simple. We shall first discuss electrical force. Objectscarry electrical charges which consist simply of electrons or protons. If anytwo bodies are electrically charged, there is an electrical force between them,and if the magnitudes of the charges are q1 and q2 , respectively, the force varies inversely as the square of the distancebetween the charges, or F=(const)q1q2/r2. For unlike charges, this law is like the law of gravitation, but forlike charges the force is repulsive and the sign (direction) isreversed. The charges q1 and q2 can be intrinsically either positive or negative, and in any specificapplication of the formula the direction of the force will come out right ifthe q ’s are given the proper plus or minus sign; the force is directedalong the line between the two charges. The constant in the formula depends, ofcourse, upon what units are used for the force, the charge, and the distance.In current practice the charge is measured in coulombs,the distance in meters, and the force in newtons. Then, in order to get theforce to come out properly in newtons, the constant (which for historicalreasons is written 1/4πϵ0 ) takes the numerical value
ϵ0=8.854×10−12 coul2/newton⋅m2
or
1/4πϵ0=8.99×109 N⋅m2/coul2.
Thus the force law for static charges is
F=q1q2r/4πϵ0r3. (12.2)
最后留下的力，就是基础性的力，我们现在讨论它们。我们称它们为基础性的，意思就是，它们的规律是基础性地简单。我们首先讨论电力。对象载有电荷，而电荷简单地构成了电子或质子。任何两个对象，如果充电了，那么，它们之间，就有电力，如果电荷的大小分别是q1和q2，那么，力就与两个电荷之间距离的平方成反比，或F=(const)q1q2/r2。与电荷不同的是，这个规律，像万有引力规律，而与电荷相同的则是，这个力是排斥的，且符号（方向）是相反的。电荷q1和q2，或为正或为负，这是内在固有的，在这个公式的任何具体的应用中，如果这些q的正号或符号，被给予了，那么，力的方向，就会直接得出；力的方向，就是沿着两个电荷之间的连线。公式中的常数，当然依赖于力、电荷、和距离所用的单位。在当前的实践中，电荷用库仑，距离用米，力用牛顿。因此，为了让力，能够用牛顿合适地表达，这个常数（由于历史的原因被写作1/4πϵ0），取下面的数值：
ϵ0=8.854×10−12 coul2/newton⋅m2
或：
1/4πϵ0=8.99×109 N⋅m2/coul2.
这样，对于静电荷，力的公式就是：
F=q1q2r/4πϵ0r3. (12.2)
{这个习惯可以保留，但应该加上括号}

In nature, the most important charge of all is the charge on a singleelectron, which is 1.60×10−19 coulomb. In working with electrical forces between fundamentalparticles rather than with large charges, many people prefer thecombination (qel)2/4πϵ0 , in which qel is defined as the charge on an electron. This combination occursfrequently, and to simplify calculations it has been defined by thesymbol e2 ; its numerical value in the mks system of units turns out tobe (1.52×10−14)2 . The advantage of using the constant in this form is that the forcebetween two electrons in newtons can then be written simply as e2/r2, with r in meters, without all the individual constants. Electrical forces aremuch more complicated than this simple formula indicates, since the formulagives the force between two objects only when the objects are standing still.We shall consider the more general case shortly.
在自然中，所有电荷中最重要的电荷，就是在一个单独的电子上的电荷，它就是1.60×10−19 coulomb。在考虑基本粒子之间的电力，而不是大的电荷之间的电力时，很多人倾向于组合 (qel)2/4πϵ0，其中qel被定义为：一个电子上的电荷，这个组合，经常出现，为了简化计算，它被定义为符号e2；在mks（米千克秒）制中，其数值变为 (1.52×10−14)2。这种形式使用这个常数，好处就是，两个电子之间的力，在单位为牛顿时，可写作e2/r2，r是米，没有其他的常数了。电力比这个简单的形式所表示的，要复杂的多，此乃因为，此公式所给出的两个对象之间的力，只是对象静止时的。不久，我们将考虑更普遍的情形。

In the analysis of forces of the morefundamental kinds (not such forces as friction, but the electrical force or thegravitational force), an interesting and very important concept has been developed.Since at first sight the forces are very much more complicated than isindicated by the inverse-square laws and these laws hold true only when theinteracting bodies are standing still, an improved method is needed to deal withthe very complex forces that ensue when the bodies start to move in acomplicated way. Experience has shown that an approach known as the concept ofa “field” is of great utility for the analysis of forces of this type. To illustratethe idea for, say, electrical force, suppose we have two electrical charges, q1and q2 , located at points P and R respectively. Then the force between the charges is given by
F=q1q2r/4πϵ0r3. (12.3)
有几种力，更基础些（不是摩擦力这种，而是电力或万有引力这种），经过对这些力的分析，发展出了一个有趣的且非常重要的概念。因为，初看上去，这些力，比平方反比规律所表明的，要更复杂些，且这些规律，只有当交互的物体静止时，才成立；当物体以一种复杂的方式，开始运动时，就需要一种改进的方法，以保证能够处理这种非常复杂的力。经验指出，“场”的概念，是一种解决方式，对于分析这种类型的力来说，它是一个强有力的工具。为了说明这种想法，例如，为了对电力说明：假设我们有两个电荷q1和q2，分别位于P和 R处。那么这两个电荷之间的力，如此给出：
F=q1q2r/4πϵ0r3. (12.3)

To analyze this force by means of the field concept, we say that thecharge q1 at P produces a “condition” at R , such that when the charge q2 is placed at R it “feels” the force. This is one way, strange perhaps, of describingit; we say that the force F on q2 at R can be written in two parts. It is q2 multiplied by a quantity E that would be there whether q2 were there or not (provided we keep all the other charges in theirright places). E is the “condition” produced by q1 , we say, and F is the response of q2 to E . E is called an electric field, and it is a vector. The formulafor the electric field E that is produced at R by a charge q1 at P is the charge q1 times the constant 1/4πϵ0 divided by r2 (r is the distance from P to R ), and it is acting in the direction of the radius vector (the radius vector rdivided by its own length). The expression for E is thus
E=q1r/4πϵ0r3. (12.4)
We then write
F=q2E, (12.5)
which expresses the force, the field, and the charge in the field.What is the point of all this? The point is to divide the analysis into twoparts. One part says that something produces a field. The other partsays that something is acted on by the field. By allowing us to look atthe two parts independently, this separation of the analysis simplifies the calculationof a problem in many situations. If many charges are present, we first work outthe total electric field produced at R by all the charges, and then, knowing the charge that is placedat R , we find the force on it.
凭借场概念，来分析这个力，我们说P处的电荷q1，在R处产生了一个“条件”，这样，当电荷q2被放在R时，就会“感觉”到这个力。这是描述它的一种方式，或许有些奇怪。我们说，在R处作用于q2的力F，可被写成两个部分。它就是：q2乘以E，无论q2在不在那里，E总在那里（假设我们让所有其他的电荷，都在其正确的位置）。我们说, E就是由q1所产生的“条件”,而F就是 q2 对 E的反应。E被称为电场，它是一个矢量。由P处的电荷q1，在R处产生的电场为E，E的公式就是：q1乘以常数1/4πϵ0 ，除以r2 (r 是P到 R的距离)，它在径向矢量的方向上，起作用（径向矢量r除以其自己的长度）。这样，E的表达式就是：
E=q1r/4πϵ0r3. (12.4)
然后我们写：
F=q2E, (12.5)
它表达了力、场、和场中的电荷。所有这一切的意思是什么呢？意思就是，把分析，分成了两个部分。一部分说，某物产生了一个场。另一部分说，某物被这个场给作用了。通过允许我们分别地看待这两个部分，这个对分析的隔离，在很多情况下，简化了问题的计算。如果有很多电荷在场，那么首先，我们得出所有电荷在R处所产生的总电场，然后，知道了位于R处的电荷，我们就可以找出它所受的力。

In the case of gravitation, we can doexactly the same thing. In this case, where the force F=−Gm1m2r/r3, we can make an analogous analysis, as follows: the force on a bodyin a gravitational field is the mass of that body times the field C. The force on m2 is the mass m2 times the field C produced by m1 ; that is, F=m2C . Then the field C produced by a body of mass m1 is C=−Gm1r/r3and it is directed radially, as in the electrical case.
在万有引力的情况下，我们可以做完全同样的事情。在这种情况下，力是 F=−Gm1m2r/r3，我们可以做类似的分析如下：在一个万有引力场中，作用于一个物体上的力，就是那个物体的质量，乘以场C。作用于m2上的力，就是质量 m2，乘以m1所产生的场； 也就是说，, F=m2C。因此，由质量为 m1的物体所产生的场C，就是 C=−Gm1r/r3，它直接就是径向的，就像电场那样。

In spite of how it might at first seem, thisseparation of one part from another is not a triviality. It would be trivial,just another way of writing the same thing, if the laws of force were simple,but the laws of force are so complicated that it turns out that the fields havea reality that is almost independent of the objects which create them. One can dosomething like shake a charge and produce an effect, a field, at a distance; ifone then stops moving the charge, the field keeps track of all the past,because the interaction between two particles is not instantaneous. It isdesirable to have some way to remember what happened previously. If the forceupon some charge depends upon where another charge was yesterday, which itdoes, then we need machinery to keep track of what went on yesterday, and thatis the character of a field. So when the forces get more complicated, the fieldbecomes more and more real, and this technique becomes less and less of anartificial separation.
把一部分，与另一部分分开，这件事，不论最初看上去如何，都不是一件小事。如果力的规律是简单的，那么，它就只不过是同一件事的另一种写法，如此，它就是一件小事；但是，力的规律，是如此复杂，结果就是，场有一种现实性：几乎是独立于创造它的对象的。某人可以做一件事，如摇动一个电荷，从而在某距离处，产生一种作用、一个场；如果然后，他停止移动电荷，这个场保持所有过去的轨迹，因为，两个粒子之间的交互作用，不是瞬时的。这就要求，有某种方式，来记忆前面发生了什么。如果某些电荷上的力，依赖于另外一个电荷昨天在哪里，且确有其事，那么，我们就需要一种机制，跟踪昨天究竟发生了什么，这就是场的特性。于是，当力变得更加复杂时，场就变得越来越真实，而这个技术，就变得越来越不像是一种人工分开。

In analyzing forces by the use of fields,we need two kinds of laws pertaining to fields. The first is the response to afield, and that gives the equations of motion. For example, the law of responseof a mass to a gravitational field is that the force is equal to the mass timesthe gravitational field; or, if there is also a charge on the body, theresponse of the charge to the electric field equals the charge times theelectric field. The second part of the analysis of nature in these situationsis to formulate the laws which determine the strength of the field and how itis produced. These laws are sometimes called the field equations. We shalllearn more about them in due time, but shall write down a few things about themnow.
通过场的使用，来分析力，我们需要两类适用于场的规律。第一个，是对场的反应，这给出了运动方程。例如，质量对万有引力场的反应规律，就是力等于：质量乘以万有引力场；或者，如果物体上有一个电荷，那么，这个电荷对电场的反应，就等于：电荷乘以电场。在这些情形中，对自然分析的第二部分，就是详细地制定规律，这些规律规定了场的强度，及它是如何产生的。这些规律有时也被称为场方程。对于这些规律，在合适的时候，我们将会更多地学习，但现在，我们将只写下关于它们的几件事情。

First, the most remarkable fact of all,which is true exactly and which can be easily understood, is that the totalelectric field produced by a number of sources is the vector sum of the electricfields produced by the first source, the second source, and so on. In otherwords, if we have numerous charges making a field, and if all by itself one ofthem would make the field E1 , another would make the field E2 , and so on, then we merely add the vectors to get the total field. Thisprinciple can be expressed as
E=E1+E2+E3+⋯ (12.6)
or, in view of the definition given above,
E=∑iqiri4πϵ0r3i. (12.7)
首先，最值得注意的一个事实就是，由一系列源所产生的总的电场，就是由每个源所产生的电场的总和；这一事实，实际上是真实的，也容易理解。换句话说，如果我们让许多电荷，来形成电场，如果其中一个电荷，完全凭其自己，可以形成电场E1，另外一个形成E2，如此等等，那么，我们只需把这些矢量加起来，就可得到总的场，这个原理，可被表达为：
E=E1+E2+E3+⋯ (12.6)
或者，从上面所给定义的视角看，就是：
(12.7)

Can the same methods be applied to gravitation? The force between two massesm1 and m2 was expressed by Newton as F=−Gm1m2r/r3. But according to the field concept, we may say that m1creates a field C in all the surrounding space, such that the force on m2is given by
F=m2C. (12.8)
By complete analogy with the electrical case,
Ci=−Gmiri/r3i (12.9)
and the gravitational field produced by several masses is
C=C1+C2+C3+⋯ (12.10)
In Chapter 9,in working out a case of planetary motion, we used this principle in essence.We simply added all the force vectors to get the resultant force on a planet.If we divide out the mass of the planet in question, we get Eq. (12.10).
同样的方法，能用于万有引力吗？质量m1和 m2之间的力，被牛顿规律表达为F=−Gm1m2r/r3。但是，依据场的概念，我们可以说，m1在周围空间中，创建了一个场 C，这样，作用于m2的力就是：
F=m2C. (12.8)
通过与电场的完全类比就有：
Ci=−Gmiri/r3i (12.9)
由若干质量所产生的重力场就是：
C=C1+C2+C3+⋯ (12.10)
在第9章，做出了一个行星运动的案例，本质上就是用了这个原理。我们简单地把所有力的矢量，加在一起，得到了行星上的合力。如果我们把问题中行星的质量约去，就得到方程（12.10）。

Equations (12.6)and (12.10)express what is known as the principle of superposition of fields. Thisprinciple states that the total field due to all the sources is the sum of thefields due to each source. So far as we know today, for electricity this is anabsolutely guaranteed law, which is true even when the force law is complicatedbecause of the motions of the charges. There are apparent violations, but morecareful analysis has always shown these to be due to the overlooking of certainmoving charges. However, although the principle of superposition appliesexactly for electrical forces, it is not exact for gravity if the field is toostrong, and Newton’s equation (12.10)is only approximate, according to Einstein’s gravitational theory.
场的叠加原理，闻名遐迩，方程（12.6）和（12.10），表示了它。这个原理说，一个总的场，若可归于所有源，那么，它就是每一个源所产生的场的总和。就目前我们所知，对于电来说，这是一个绝对被认可的规律，甚至，当电荷在运动，从而导致电的规律很复杂时，它也为真。有明显的例外，但是，更仔细的分析，总是会指出，这些例外，可归于忽视了某些运动着的电荷。然而，虽然重叠原理，对于电力，应用地很准确，但对于重力，如果场太强的话，它就不是很准确，{因为}依据爱因斯坦万有引力的理论，牛顿方程（12.10）只是一个近似。