In the same way, we cannot just call F=maa definition, deduce everything purely mathematically, and make mechanicsa mathematical theory, when mechanics is a description of nature. Byestablishing suitable postulates it is always possible to make a system ofmathematics, just as Euclid did, but we cannot make a mathematics of the world,because sooner or later we have to find out whether the axioms are valid forthe objects of nature. Thus we immediately get involved with these complicatedand “dirty” objects of nature, but with approximations ever increasing inaccuracy.
以同样的方式，我们不能把F=ma，只称为是一个定义，当力学是对自然的描述时，却把力学，变成了数学理论，纯粹从数学中推出每件事情。通过建立合适的公设，建立一个数学系统，总是可能的，正如欧几里得所做，但是，我们不能把世界，变成数学的，因为，迟早我们都需要去找出：对于自然的对象来说，公理是否有效。这样，我们立即就与自然的“肮脏”的对象，发生了关系，但是，对象的近似的准确性，是不断在提高的。

12–2Friction 12-2 摩擦力
The foregoing considerations show that atrue understanding of Newton’s laws requires a discussion of forces, and it isthe purpose of this chapter to introduce such a discussion, as a kind of completionof Newton’s laws. We have already studied the definitions of acceleration andrelated ideas, but now we have to study the properties of force, and thischapter, unlike the previous chapters, will not be very precise, because forcesare quite complicated.
前面的考虑，指出了，对牛顿规律的真正理解，要求一个对力的讨论，本章的目的，就是介绍这种讨论，以作为牛顿规律的某种完善。我们已经研究了加速度及相关想法的定义，但是现在，我们必须研究力的属性，与前面诸章不同，本章将不是很精确，因为力相当复杂。

To begin with a particular force, let usconsider the drag on an airplane flying through the air. What is the law forthat force? (Surely there is a law for every force, we must have a law!)One can hardly think that the law for that force will be simple. Try to imaginewhat makes a drag on an airplane flying through the air—the air rushing overthe wings, the swirling in the back, the changes going on around the fuselage,and many other complications, and you see that there is not going to be a simplelaw. On the other hand, it is a remarkable fact that the drag force on anairplane is approximately a constant times the square of the velocity, or F≈cv2.
要从一种具体的力开始，让我们考虑，空中飞行的飞机所受到的阻力。这个力的规律是什么呢？（当然，每种力都有其规律，我们必须有规律！）你很难想到，这个力很简单。想象一下，究竟是什么阻碍着空气中的飞机呢—流过机翼的空气，后部的涡旋，机身所发生的变化，及其他很多复杂的情况，你看，不会有一条简单的规律。另一方面，有个事实，值得注意，就是飞机上的阻力，大约是一个常数乘以矢速的平方，或 F≈cv2。

We have just discussed two cases of friction,resulting from fast movement in air and slow movement in honey. There isanother kind of friction, called dry friction or sliding friction, which occurswhen one solid body slides on another. In this case a force is needed tomaintain motion. This is called a frictional force, and its origin, also, is avery complicated matter. Both surfaces of contact are irregular, on an atomiclevel. 我们已经讨论了两种摩擦情况，一个是空气中的快速运动，另一个是在蜂蜜中的慢速运动。还有另外一种摩擦，被称为干摩擦或滑动摩擦，是指一个固体，在另外一个固体上滑动时，出现的摩擦。在这种情况下，要保证运动，就需要一个力。这个力被称为摩擦力，其起源也是一个非常复杂的事情。两个接触的表面，从原子层面看，都是不规则的。There are many points of contact where the atoms seem to clingtogether, and then, as the sliding body is pulled along, the atoms snap apartand vibration ensues; something like that has to happen. Formerly the mechanismof this friction was thought to be very simple, that the surfaces were merelyfull of irregularities and the friction originated in lifting the slider overthe bumps; but this cannot be, for there is no loss of energy in that process,whereas power is in fact consumed. 在很多接触点上，原子似乎是粘在一起的，然后，随着物体被拖着走，原子们断裂开，因而产生了振动；诸如此类。以前，这种摩擦的机制，被认为很简单，即表面只是不规则的，在上面的滑动块，移过下面的凹凸时，会产生摩擦；但不会是这样，因为，在这个过程中，没有能量的损失，因为能量确实被消耗了。The mechanism of power loss is that as the slider snaps over thebumps, the bumps deform and then generate waves and atomic motions and, after awhile, heat, in the two bodies. Now it is very remarkable that again, empirically,this friction can be described approximately by a simple law. This law is thatthe force needed to overcome friction and to drag one object over anotherdepends upon the normal force (i.e., perpendicular to the surface) between the twosurfaces that are in contact. Actually, to a fairly good approximation, thefrictional force is proportional to this normal force, and has a more or lessconstant coefficient; that is, 能量损耗的机制就是，当滑块滑过这些凹凸时，凹凸变形，然后产生波和原子运动，过一会儿，会在两个物体中产生热。现在，值得说明的是，这个摩擦，可以经验性地被描述为一个简单的规律。这个规律就是：要克服摩擦，并拖着一个对象，滑过另外一个对象，需要一个力，此力依赖于两个正在接触的表面之间的法向力（亦即，垂直于表面的）。实际上，一个比较好的近似说法就是，摩擦力正比于这个法向力，且有一个或多或少可为常数的系数，也就是说：
F=μN,(12.1)
where μ is called the coefficient of friction (Fig. 12–1). Although this coefficient is not exactlyconstant, the formula is a good empirical rule for judging approximately theamount of force that will be needed in certain practical or engineeringcircumstances. If the normal force or the speed of motion gets too big, the lawfails because of the excessive heat generated. It is important to realize thateach of these empirical laws has its limitations, beyond which it does not reallywork.
这里μ被称为摩擦系数（图12-1）。虽然这个系数，并不是一个完全的常数，但是，在某些实践的或工程的环境中，为了大约估计需要多大的力，这个公式，还是一个好的经验性的规则。如果法向力或运动的速度，变得太大，这个规律就会失效，因为过多的热就会产生。有一点，很重要，每种经验性的规律，都有其限制，超出此限，规律就不会真正有效。

Fig. 12–1.The relation between frictionalforce and the normal force for sliding contact. 图12-1 滑动接触中的摩擦力和法向力。

That the formula F=μN is approximately correct can be demonstrated by a simple experiment.We set up a plane, inclined at a small angle θ , and place a block of weight W on the plane. We then tilt the plane at a steeper angle, until theblock just begins to slide from its own weight. The component of the weightdownward along the plane is Wsinθ , and this must equal the frictional force F when the block is sliding uniformly. The component of the weightnormal to the plane is Wcosθ , and this is the normal force N . With these values, the formula becomes Wsinθ=μWcosθ, from which we get μ= sinθ/cosθ= tanθ . If this law were exactly true, an object would start to slide atsome definite inclination. If the same block is loaded by putting extra weighton it, then, although W is increased, all the forces in the formula are increased in the sameproportion, and W cancels out. If μ stays constant, the loaded block will slide again at the same slope.When the angle θ is determined by trial with the original weight, it is found that withthe greater weight the block will slide at about the same angle. This will betrue even when one weight is many times as great as the other, and so we concludethat the coefficient of friction is independent of the weight.
公式F=μN 只是近似地正确，可通过一个简单的实验来演示。我们用一块平板，搭一个斜坡，有一个小的角度θ，把重量为W的物体，放在上面。然后，我们把平板变得更陡些，直到这个物体开始滑动。沿着平板方向的，物体重量的分量，是Wsinθ，这应该等于物体开始均匀滑动时的摩擦力F。垂直于平板方向，物体重量的分量，是Wcosθ，这就是法向力N。由这些值，公式就变成了Wsinθ=μWcosθ ，由此得出μ= sinθ/cosθ= tanθ 。如果这条规律确实为真，那么在某个确定的倾斜角度，物体就应该开始滑动。如果给同一个物体，增加额外重量，那么，虽然 W增加了，但公式中的所有力都同比地增加了，所以 W消除了。如果μ保持为常数，那么，加重了的物体，到了同样的斜度时，将也会开始滑动。当角度θ，是用原始重量，通过实验来决定时，结果发现，更重的物体，也会在几乎相同的角度，开始滑动。甚至当一个重量，是另一个的数倍时，这也成立，所以，我们得出结论：摩擦系数独立于重量。

In performing this experiment it isnoticeable that when the plane is tilted at about the correct angle θ, the block does not slide steadily but in a halting fashion. At oneplace it may stop, at another it may move with acceleration. This behaviorindicates that the coefficient of friction is only roughly a constant, andvaries from place to place along the plane. The same erratic behavior is observedwhether the block is loaded or not. Such variations are caused by differentdegrees of smoothness or hardness of the plane, and perhaps dirt, oxides, orother foreign matter. The tables that list purported values of μ for “steel on steel,” “copper on copper,” and the like, are all false,because they ignore the factors mentioned above, which really determine μ. The friction is never due to “copper on copper,” etc., but to theimpurities clinging to the copper.
在做这一实验时，有一点，要注意，就是当平板被倾斜，快接近正确的角度θ时，物体并不是开始稳定地滑动，而是时断时续地滑动。在一个地方它可能会停止，在另一地方，它可能会加速。不论物体加重还是不加重，都能观察到这一不稳定的表现。这种变化，是因为平板的不同的光滑度和硬度所引起的，或许是脏东西，氧化物，或其他什么东西。有些表，据称是列出了“钢在钢上”、“铜在铜上”等的μ值，全是错的，因为它们忽略了上面提到的因素，这些因素确实影响着μ。摩擦永远也不会归于“铜在铜上”等，而是归于铜上的杂质。

It was pointed out above that attempts tomeasure μ by sliding pure substances such as copper on copper will lead tospurious results, because the surfaces in contact are not pure copper, but aremixtures of oxides and other impurities. If we try to get absolutely pure copper,if we clean and polish the surfaces, outgas the materials in a vacuum, and takeevery conceivable precaution, we still do not get μ . For if we tilt the apparatus even to a vertical position, the sliderwill not fall off—the two pieces of copper stick together! Thecoefficient μ , which is ordinarily less than unity for reasonably hard surfaces,becomes several times unity! The reason for this unexpected behavior is thatwhen the atoms in contact are all of the same kind, there is no way for the atomsto “know” that they are in different pieces of copper. When there are otheratoms, in the oxides and greases and more complicated thin surface layers of contaminantsin between, the atoms “know” when they are not on the same part. When weconsider that it is forces between atoms that hold the copper together as asolid, it should become clear that it is impossible to get the right coefficientof friction for pure metals.
上面已经指出，尝试通过让一个实质体在另一个实质体上滑动--例如铜对铜，来测量 μ，会导致虚假的结果，因为，相接触的表面，并非纯粹的铜，而是氧和其他杂质的混合物。如果我们想得到绝对纯粹的铜，如果我们清理并抛光表面，在真空中除掉表面的空气泡，且采取所有的预防措施，我们仍得不到μ。因为，如果我们倾斜这个仪器，甚至到达垂直的位置，滑动块也不会掉下来，因为两个铜块已经粘在了一起。系数μ，通常对于合理硬度的表面来说，比一个单位小，而现在，它变成了单位的好几倍。这一表现，没有料到，其原因就是，当相接触的原子，都是同类时，就没有任何方式，可以让原子们“知道”，它们是在不同的铜块中。当有其他的原子时，即在两个铜块之间，有氧化物、润滑油或更复杂的薄的表面层时，那么，原子就会“知道”，什么时候，它们不是在同一个部分。当我们认为，正是原子之间的力，把铜抓在了一起，形成一个固体，那么，下面一点，就很清楚了：要得到纯金属的正确的摩擦力系数，是不可能的。

The same phenomenon can be observed in asimple home-made experiment with a flat glass plate and a glass tumbler. If thetumbler is placed on the plate and pulled along with a loop of string, itslides fairly well and one can feel the coefficient of friction; it is a littleirregular, but it is a coefficient. If we now wet the glass plate and the bottomof the tumbler and pull again, we find that it binds, and if we look closely weshall find scratches, because the water is able to lift the grease and theother contaminants off the surface, and then we really have a glass-to-glasscontact; this contact is so good that it holds tight and resists separation somuch that the glass is torn apart; that is, it makes scratches.
可以在家里做一个实验，用一个平的玻璃板和一个无脚玻璃杯，也可以观察到同样的现象。把玻璃杯放在玻璃板上，用一个绳子来回拉着它，那么，它就会滑动地相当好，这样就可以感觉到摩擦力系数，这是一个小的、不太规则的系数，但确实是一个系数。现在，我们把玻璃板和玻璃杯都弄湿，然后再拉这个玻璃杯，我们就会发现了，它粘住了，如果我们更仔细地观察，我们就会发现刮痕，因为，水可以把表面的油脂和其他物质抬起来，然后，我们就会有一个玻璃对玻璃的接触；这种接触，非常紧，以至于难以分开，玻璃就好像被撕开一样；也就是说，会产生刮痕。

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。