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。

The first part of the experiment is toapply a negative voltage to the lower plate, which means that extra electrons havebeen placed on the lower plate. Since like charges repel, the light spot on thescreen instantly shifts upward. (We could also say this in another way—that theelectrons “felt” the field, and responded by deflecting upward.) We nextreverse the voltage, making the upper plate negative. The light spot onthe screen now jumps below the center, showing that the electrons in the beam wererepelled by those in the plate above them. (Or we could say again that theelectrons had “responded” to the field, which is now in the reverse direction.)
实验的第一部分，是让下面的板子，带负电，意思是说，额外的电子，被放在下面的板子上了。由于电荷排斥，屏幕上的光点，立即就会向上移。（对此，我们也可以用另外一种方式来说：电子感觉到了电场，故通过向上偏移，做出反应。）下面，我们的反转电极，让上面的板子，带负电。屏幕上的光点，现在就会跳到中心以下，显示出，光速中的电子，被上面的电子排斥了。（或者，我们又可以说，电子对电场，做出了“反应”，电场的方向，现在是相反的。）

The second part of the experiment is to disconnectthe voltage from the plates and test the effect of a magnetic field on theelectron beam. This is done by means of a horseshoe magnet, whose poles are farenough apart to more or less straddle the tube. Suppose we hold the magnetbelow the tube in the same orientation as the letter U, with its poles up andpart of the tube in between. We note that the light spot is deflected, say,upward, as the magnet approaches the tube from below. So it appears that the magnetrepels the electron beam. However, it is not that simple, for if we invert themagnet without reversing the poles side-for-side, and now approach the tube fromabove, the spot still moves upward, so the electron beam is notrepelled; instead, it appears to be attracted this time. Now we start again,restoring the magnet to its original U orientation and holding it below thetube, as before. Yes, the spot is still deflected upward; but now turn themagnet 180 degrees around a vertical axis, so that it is still in the U positionbut the poles are reversed side-for-side. Behold, the spot now jumps downward,and stays down, even if we invert the magnet and approach from above, asbefore.
实验的第二部分，是去掉板上的电压，测试磁场对电子束的影响。这是通过一个马蹄形的磁铁来完成的，磁铁的两极，距离足够远，足以横跨在管子上。假设我们在管子下方，举着磁铁，就像举着字母U一样，让磁极向上，管子的一部分在磁极之间。我们注意到，光点被偏移了，比如说，向上，由于磁铁是从下方，接近管子。所以，似乎磁铁是在排斥电子束。然而，因为，如果我们倒转磁铁，而不颠倒磁极，那么现在，磁铁是从上方接近管子，光点依然向上移动，于是，电子束并没有被排斥；这次好像是被吸引了。现在，我们再次开始，把磁铁恢复到最初的方向，就像以前一样，在管子下方举着它。是的，光点仍然向上偏转；但是现在，我们绕一个垂直轴，把磁铁转180度，这样，它仍处于U型位置，但是，磁极颠倒了。看那，现在光点向下跳了，就算我们像刚才那样，反转磁铁，从上方接近管子，它也是保持向下。

3247322242@qq.com谢谢大佬！

To understand this peculiar behavior, wehave to have a new combination of forces. We explain it thus: Across the magnetfrom one pole to the other there is a magnetic field. This field has adirection which is always away from one particular pole (which we could mark)and toward the other. Inverting the magnet did not change the direction of thefield, but reversing the poles side-for-side did reverse its direction. Forexample, if the electron velocity were horizontal in the x -direction and the magnetic field were also horizontal but in the y-direction, the magnetic force on the moving electrons would bein the z -direction, i.e., up or down, depending on whether the field was inthe positive or negative y -direction.
要理解这一独特的表现，我们必须有一个新的力的组合。我们这样解释它：在磁铁的两极之间，有一个磁场。这个场有一个方向，总是从一个磁极（我们可以标出）到另一个。反转磁铁，并不能改变磁场的方向，但是，颠倒磁极，则可颠倒其方向。例如，如果电子的矢速，在x方向是水平的，磁场也是水平的，但却是在y方向，那么，作用于电子的磁力，就是在z方向，亦即，上方或下方，这依赖于磁场是在y的正方向，还是y的负方向。

Although we shall not at the present timegive the correct law of force between charges moving in an arbitrary manner,one relative to the other, because it is too complicated, we shall give one aspectof it: the complete law of the forces if the fields are known.两个电荷，相互之间，任意移动，对于此情，我们现在，还不会给出正确的力的规律，因为太复杂，我们只给出其一个方面：场已知时，完整的力的规律。 The force on a charged object depends upon its motion; if, when theobject is standing still at a given place, there is some force, this is takento be proportional to the charge, the coefficient being what we call the electricfield. 一个被充电的对象，作用于其上的力，依赖于它的运动；如果，当对象在给定地方静止时，有某些力，这些力被认为与电荷成正比，系数就是我们称为电场的东西。When the object moves the force may be different, and thecorrection, the new “piece” of force, turns out to be dependent exactly linearlyon the velocity, but at right angles to v and to another vector quantity which we call the magnetic induction B.当对象移动时，力可能会不同，{对力的}修正、或新的力，就会发现是准确地线性地依赖于矢速，但却是：垂直于v、及另外一个矢速量，我们称之为磁感应B。If the componentsof the electric field E and the magnetic induction B are, respectively, (Ex,Ey,Ez)and (Bx,By,Bz), and if the velocity v has the components (vx,vy,vz), then the total electric and magnetic force on a moving charge qhas the components 如果电场E和磁感应B的分量，分别是，(Ex,Ey,Ez)和(Bx,By,Bz)，且如果矢速v有分量 (vx,vy,vz)，那么，作用于一个移动电荷q上的总的电磁力，就有分量：
Fx= q (Ex +vyBz − vzBy),
Fy=q (Ey +vzBx − vxBz), (12.11)
Fz=q (Ez+ vxBy − vyBx).
If, for instance, the only component of the magnetic field were Byand the only component of the velocity were vx , then the only term left in the magnetic force would be a force inthe z -direction, at right angles to both B and v .
例如，如果磁场的唯一分量是By ，速度的唯一分量是vx，那么，磁力就只剩下一项，即z方向的力，与B和 v垂直。

12–5Pseudo forces 12-5 伪力
The next kind of force we shall discussmight be called a pseudo force. In Chapter 11we discussed the relationship between two people, Joe and Moe, who usedifferent coordinate systems. Let us suppose that the positions of a particleas measured by Joe are x and by Moe are x′ ; then the laws are as follows:
x=x′+s, y=y′, z=z′,
where s is the displacement of Moe’s system relative to Joe’s. If we supposethat the laws of motion are correct for Joe, how do they look for Moe? We findfirst, that
dx/dt=dx′/dt+ds/dt.
我们下面将要讨论的一种力，可被称为伪力。在第11章，我们讨论了Joe 和Moe两人之间的关系，他们使用不同的坐标系。我们假设，一个粒子的位置，由Joe来测是x，由Moe来测是x′；那么，规律就是：
x=x′+s, y=y′, z=z′,
这里s，就是Moe的系统相对于Joe的系统的位移。如果我们假设，运动规律，对于Joe来说是正确的，那么，对于Moe来说，它们看上去会是什么样子呢？我们首先发现：
dx/dt=dx′/dt+ds/dt。