Fundamental forces. Fields
We shall now discuss the only remaining forces that are fundamental. We call them fundamental in the sense that their laws are fundamentally simple. We shall first discuss electrical force. Objects carry electrical charges which consist simply of electrons or protons. If any two bodies are electrically charged, there is an electrical force between them, and if the magnitudes of the charges are $q_1$ and $q_2$, respectively, the force varies inversely as the square of the distance between the charges, or $F=(\text{const}) q_1q_2/r^2$. For unlike charges, this law is like the law of gravitation, but for like charges the force is repulsive and the sign (direction) is reversed. The charges $q_1$ and $q_2$ can be intrinsically either positive or negative, and in any specific application of the formula the direction of the force will come out right if the $q$’s are given the proper plus or minus sign; the force is directed along the line between the two charges. The constant in the formula depends, of course, 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 the force to come out properly in newtons, the constant (which for historical reasons is written $1/4\pi\epsO$) takes the numerical value \begin{equation*} \epsO=8.854\times10^{-12}\text{ coul}^2/\text{newton}\cdot\text{m}^2 \end{equation*} or \begin{equation*} 1/4\pi\epsO=8.99\times10^9\text{ N}\cdot\text{m}^2/\text{coul}^2. \end{equation*} Thus the force law for static charges is \begin{equation} \label{Eq:I:12:2} \FLPF=q_1q_2\FLPr/4\pi\epsO r^3. \end{equation} In nature, the most important charge of all is the charge on a single electron, which is $1.60\times10^{-19}$ coulomb. In working with electrical forces between fundamental particles rather than with large charges, many people prefer the combination $(q_{\text{el}})^2/4\pi\epsO$, in which $q_{\text{el}}$ is defined as the charge on an electron. This combination occurs frequently, and to simplify calculations it has been defined by the symbol $e^2$; its numerical value in the mks system of units turns out to be $(1.52\times10^{-14})^2$. The advantage of using the constant in this form is that the force between two electrons in newtons can then be written simply as $e^2/r^2$, with $r$ in meters, without all the individual constants. Electrical forces are much more complicated than this simple formula indicates, since the formula gives the force between two objects only when the objects are standing still. We shall consider the more general case shortly.  In the analysis of forces of the more fundamental kinds (not such forces as friction, but the electrical force or the gravitational force), an interesting and very important concept has been developed. Since at first sight the forces are very much more complicated than is indicated by the inverse-square laws and these laws hold true only when the interacting bodies are standing still, an improved method is needed to deal with the very complex forces that ensue when the bodies start to move in a complicated way. Experience has shown that an approach known as the concept of a “field” is of great utility for the analysis of forces of this type. To illustrate the idea for, say, electrical force, suppose we have two electrical charges, $q_1$ and $q_2$, located at points $P$ and $R$ respectively. Then the force between the charges is given by \begin{equation} \label{Eq:I:12:3} \FLPF=q_1q_2\FLPr/4\pi\epsO r^3. \end{equation} To analyze this force by means of the field concept, we say that the charge $q_1$ at $P$ produces a “condition” at $R$, such that when the charge $q_2$ is placed at $R$ it “feels” the force. This is one way, strange perhaps, of describing it; we say that the force $\FLPF$ on $q_2$ at $R$ can be written in two parts. It is $q_2$ multiplied by a quantity $\FLPE$ that would be there whether $q_2$ were there or not (provided we keep all the other charges in their right places). $\FLPE$ is the “condition” produced by $q_1$, we say, and $\FLPF$ is the response of $q_2$ to $\FLPE$. $\FLPE$ is called an electric field, and it is a vector. The formula for the electric field $\FLPE$ that is produced at $R$ by a charge $q_1$ at $P$ is the charge $q_1$ times the constant $1/4\pi\epsO$ divided by $r^2$ ($r$ is the distance from $P$ to $R$), and it is acting in the direction of the radius vector (the radius vector $\FLPr$ divided by its own length). The expression for $\FLPE$ is thus \begin{equation} \label{Eq:I:12:4} \FLPE=q_1\FLPr/4\pi\epsO r^3. \end{equation} We then write \begin{equation} \label{Eq:I:12:5} \FLPF=q_2\,\FLPE, \end{equation} 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 two parts. One part says that something produces a field. The other part says that something is acted on by the field. By allowing us to look at the two parts independently, this separation of the analysis simplifies the calculation of a problem in many situations. If many charges are present, we first work out the total electric field produced at $R$ by all the charges, and then, knowing the charge that is placed at $R$, we find the force on it. In the case of gravitation, we can do exactly the same thing. In this case, where the force $\FLPF=-Gm_1m_2\FLPr/r^3$, we can make an analogous analysis, as follows: the force on a body in a gravitational field is the mass of that body times the field $\FLPC$. The force on $m_2$ is the mass $m_2$ times the field $\FLPC$ produced by $m_1$; that is, $\FLPF=m_2\FLPC$. Then the field $\FLPC$ produced by a body of mass $m_1$ is $\FLPC=-Gm_1\FLPr/r^3$ and it is directed radially, as in the electrical case. In spite of how it might at first seem, this separation 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 have a reality that is almost independent of the objects which create them. One can do something like shake a charge and produce an effect, a field, at a distance; if one then stops moving the charge, the field keeps track of all the past, because the interaction between two particles is not instantaneous. It is desirable to have some way to remember what happened previously. If the force upon some charge depends upon where another charge was yesterday, which it does, then we need machinery to keep track of what went on yesterday, and that is the character of a field. So when the forces get more complicated, the field becomes more and more real, and this technique becomes less and less of an artificial separation. In analyzing forces by the use of fields, we need two kinds of laws pertaining to fields. The first is the response to a field, and that gives the equations of motion. For example, the law of response of a mass to a gravitational field is that the force is equal to the mass times the gravitational field; or, if there is also a charge on the body, the response of the charge to the electric field equals the charge times the electric field. The second part of the analysis of nature in these situations is to formulate the laws which determine the strength of the field and how it is produced. These laws are sometimes called the field equations. We shall learn more about them in due time, but shall write down a few things about them now. First, the most remarkable fact of all, which is true exactly and which can be easily understood, is that the total electric field produced by a number of sources is the vector sum of the electric fields produced by the first source, the second source, and so on. In other words, if we have numerous charges making a field, and if all by itself one of them would make the field $\FLPE_1$, another would make the field $\FLPE_2$, and so on, then we merely add the vectors to get the total field. This principle can be expressed as \begin{equation} \label{Eq:I:12:6} \FLPE=\FLPE_1+\FLPE_2+\FLPE_3+\dotsb \end{equation} or, in view of the definition given above, \begin{equation} \label{Eq:I:12:7} \FLPE=\sum_i\frac{q_i\FLPr_i}{4\pi\epsO r_i^3}. \end{equation} Can the same methods be applied to gravitation? The force between two masses $m_1$ and $m_2$ was expressed by Newton as $\FLPF=-Gm_1m_2\FLPr/r^3$. But according to the field concept, we may say that $m_1$ creates a field $\FLPC$ in all the surrounding space, such that the force on $m_2$ is given by \begin{equation} \label{Eq:I:12:8} \FLPF=m_2\FLPC. \end{equation} By complete analogy with the electrical case, \begin{equation} \label{Eq:I:12:9} \FLPC_i=-Gm_i\FLPr_i/r_i^3 \end{equation} and the gravitational field produced by several masses is \begin{equation} \label{Eq:I:12:10} \FLPC=\FLPC_1+\FLPC_2+\FLPC_3+\dotsb \end{equation} 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). Equations (12.6) and (12.10) express what is known as the principle of superposition of fields. This principle states that the total field due to all the sources is the sum of the fields due to each source. So far as we know today, for electricity this is an absolutely guaranteed law, which is true even when the force law is complicated because of the motions of the charges. There are apparent violations, but more careful analysis has always shown these to be due to the overlooking of certain moving charges. However, although the principle of superposition applies exactly for electrical forces, it is not exact for gravity if the field is too strong, and Newton’s equation (12.10) is only approximate, according to Einstein’s gravitational theory. Closely related to electrical force is another kind, called magnetic force, and this too is analyzed in terms of a field. Some of the qualitative relations between electrical and magnetic forces can be illustrated by an experiment with an electron-ray tube (Fig. 12–3). At one end of such a tube is a source that emits a stream of electrons. Within the tube are arrangements for accelerating the electrons to a high speed and sending some of them in a narrow beam to a fluorescent screen at the other end of the tube. A spot of light glows in the center of the screen where the electrons strike, and this enables us to trace the electron path. On the way to the screen the electron beam passes through a narrow space between a pair of parallel metal plates, which are arranged, say, horizontally. A voltage can be applied across the plates, so that either plate can be made negative at will. When such a voltage is present, there is an electric field between the plates. The first part of the experiment is to apply a negative voltage to the lower plate, which means that extra electrons have been placed on the lower plate. Since like charges repel, the light spot on the screen instantly shifts upward. (We could also say this in another way—that the electrons “felt” the field, and responded by deflecting upward.) We next reverse the voltage, making the upper plate negative. The light spot on the screen now jumps below the center, showing that the electrons in the beam were repelled by those in the plate above them. (Or we could say again that the electrons had “responded” to the field, which is now in the reverse direction.) The second part of the experiment is to disconnect the voltage from the plates and test the effect of a magnetic field on the electron beam. This is done by means of a horseshoe magnet, whose poles are far enough apart to more or less straddle the tube. Suppose we hold the magnet below the tube in the same orientation as the letter U, with its poles up and part 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 magnet repels the electron beam. However, it is not that simple, for if we invert the magnet without reversing the poles side-for-side, and now approach the tube from above, the spot still moves upward, so the electron beam is not repelled; instead, it appears to be attracted this time. Now we start again, restoring the magnet to its original U orientation and holding it below the tube, as before. Yes, the spot is still deflected upward; but now turn the magnet $180$ degrees around a vertical axis, so that it is still in the U position but 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, as before. To understand this peculiar behavior, we have to have a new combination of forces. We explain it thus: Across the magnet from one pole to the other there is a magnetic field. This field has a direction 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 the field, but reversing the poles side-for-side did reverse its direction. For example, 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 be in the $z$-direction, i.e., up or down, depending on whether the field was in the positive or negative $y$-direction. Although we shall not at the present time give 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 aspect of it: the complete law of the forces if the fields are known. The force on a charged object depends upon its motion; if, when the object is standing still at a given place, there is some force, this is taken to be proportional to the charge, the coefficient being what we call the electric field.  When the object moves the force may be different, and the correction, the new “piece” of force, turns out to be dependent exactly linearly on the velocity, but at right angles to $\FLPv$ and to another vector quantity which we call the magnetic induction $\FLPB$. If the components of the electric field $\FLPE$ and the magnetic induction $\FLPB$ are, respectively, $(E_x,E_y,E_z)$ and ($B_x,B_y,B_z)$, and if the velocity $\FLPv$ has the components ($v_x,v_y,v_z)$, then the total electric and magnetic force on a moving charge $q$ has the components \begin{equation} \begin{alignedat}{7} &F_x&&=q(E_x&&+v_y&&B_z&&-v_z&&B_y&&),\\[.5ex] &F_y&&=q(E_y&&+v_z&&B_x&&-v_x&&B_z&&),\\[.5ex] &F_z&&=q(E_z&&+v_x&&B_y&&-v_y&&B_x&&). \end{alignedat} \label{Eq:I:12:11} \end{equation} If, for instance, the only component of the magnetic field were $B_y$ and the only component of the velocity were $v_x$, then the only term left in the magnetic force would be a force in the $z$-direction, at right angles to both $\FLPB$ and $\FLPv$.