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1 Electromagnetism Review: Chapter 12, Vol. I, Characteristics of Force 1–1 Electrical forces Consider a force like gravitation which varies predominantly inversely as the square of the distance, but which is about a billion-billion-billion-billion times stronger. And with another difference. There are two kinds of “matter,” which we can call positive and negative. Like kinds repel and unlike kinds attract—unlike gravity where there is only attraction. What would happen? A bunch of positives would repel with an enormous force and spread out in all directions. A bunch of negatives would do the same. But an evenly mixed bunch of positives and negatives would do something completely different. The opposite pieces would be pulled together by the enormous attractions. The net result would be that the terrific forces would balance themselves out almost perfectly, by forming tight, fine mixtures of the positive and the negative, and between two separate bunches of such mixtures there would be
2 Differential Calculus of Vector Fields Review: Chapter 11, Vol. I, Vectors 2–1 Understanding physics The physicist needs a facility in looking at problems from several points of view. The exact analysis of real physical problems is usually quite complicated, and any particular physical situation may be too complicated to analyze directly by solving the differential equation. But one can still get a very good idea of the behavior of a system if one has some feel for the character of the solution in different circumstances. Ideas such as the field lines, capacitance, resistance, and inductance are, for such purposes, very useful. So we will spend much of our time analyzing them. In this way we will get a feel as to what should happen in different electromagnetic situations. On the other hand, none of the heuristic models, such as field lines, is really adequate and accurate for all situations. There is only one precise way of presenting the laws, and that is by means of differential eq
3 Vector Integral Calculus 3–1 Vector integrals; the line integral of We found in Chapter 2 that there were various ways of taking derivatives of fields. Some gave vector fields; some gave scalar fields. Although we developed many different formulas, everything in Chapter 2 could be summarized in one rule: the operators , , and are the three components of a vector operator . We would now like to get some understanding of the significance of the derivatives of fields. We will then have a better feeling for what a vector field equation means. We have already discussed the meaning of the gradient operation ( on a scalar). Now we turn to the meanings of the divergence and curl operations. The interpretation of these quantities is best done in terms of certain vector integrals and equations relating such integrals. These equations cannot, unfortunately, be obtained from vector algebra by some easy substitution, so you will just have to learn them as something new. Of these integral formul
4 Electrostatics Review: Chapters 13, and 14 Vol. I, Work and Potential Energy 4–1 Statics We begin now our detailed study of the theory of electromagnetism. All of electromagnetism is contained in the Maxwell equations. Maxwell's equations: The situations that are described by these equations can be very complicated. We will consider first relatively simple situations, and learn how to handle them before we take up more complicated ones. The easiest circumstance to treat is one in which nothing depends on the time—called the static case. All charges are permanently fixed in space, or if they do move, they move as a steady flow in a circuit (so and are constant in time). In these circumstances, all of the terms in the Maxwell equations which are time derivatives of the field are zero. In this case, the Maxwell equations become: Electrostatics: Magnetostatics: You will notice an interesting thing about this set of four equations. It can be separated into two pairs. The electric f
5 Application of Gauss’ Law 5–1 Electrostatics is Gauss’ law plus … There are two laws of electrostatics: that the flux of the electric field from a volume is proportional to the charge inside—Gauss’ law, and that the circulation of the electric field is zero— is a gradient. From these two laws, all the predictions of electrostatics follow. But to say these things mathematically is one thing; to use them easily, and with a certain amount of ingenuity, is another. In this chapter we will work through a number of calculations which can be made with Gauss’ law directly. We will prove theorems and describe some effects, particularly in conductors, that can be understood very easily from Gauss’ law. Gauss’ law by itself cannot give the solution of any problem because the other law must be obeyed too. So when we use Gauss’ law for the solution of particular problems, we will have to add something to it. We will have to presuppose, for instance, some idea of how the field looks—based, for exa
6 The Electric Field in Various Circumstances Review: Chapter 23, Vol. I, Resonance 6–1 Equations of the electrostatic potential This chapter will describe the behavior of the electric field in a number of different circumstances. It will provide some experience with the way the electric field behaves, and will describe some of the mathematical methods which are used to find this field. We begin by pointing out that the whole mathematical problem is the solution of two equations, the Maxwell equations for electrostatics: In fact, the two can be combined into a single equation. From the second equation, we know at once that we can describe the field as the gradient of a scalar (see Section 3–7): We may, if we wish, completely describe any particular electric field in terms of its potential . We obtain the differential equation that must obey by substituting Eq. (6.3) into (6.1), to get The divergence of the gradient of is the same as operating on : so we write Eq. (6.4) as The operat
7 The Electric Field in Various Circumstances (Continued) 7–1 Methods for finding the electrostatic field This chapter is a continuation of our consideration of the characteristics of electric fields in various particular situations. We shall first describe some of the more elaborate methods for solving problems with conductors. It is not expected that these more advanced methods can be mastered at this time. Yet it may be of interest to have some idea about the kinds of problems that can be solved, using techniques that may be learned in more advanced courses. Then we take up two examples in which the charge distribution is neither fixed nor is carried by a conductor, but instead is determined by some other law of physics. As we found in Chapter 6, the problem of the electrostatic field is fundamentally simple when the distribution of charges is specified; it requires only the evaluation of an integral. When there are conductors present, however, complications arise because the charge
8 Electrostatic Energy Review: Chapter 4, Vol. I, Conservation of Energy Chapters 13, and 14 Vol. I, Work and Potential Energy 8–1 The electrostatic energy of charges. A uniform sphere In the study of mechanics, one of the most interesting and useful discoveries was the law of the conservation of energy. The expressions for the kinetic and potential energies of a mechanical system helped us to discover connections between the states of a system at two different times without having to look into the details of what was occurring in between. We wish now to consider the energy of electrostatic systems. In electricity also the principle of the conservation of energy will be useful for discovering a number of interesting things. The law of the energy of interaction in electrostatics is very simple; we have, in fact, already discussed it. Suppose we have two charges and separated by the distance . There is some energy in the system, because a certain amount of work was required to bring the
9 Electricity in the Atmosphere Reference: Chalmers, J. Alan, Atmospheric Electricity, Pergamon Press, London (1957). 9–1 The electric potential gradient of the atmosphere On an ordinary day over flat desert country, or over the sea, as one goes upward from the surface of the ground the electric potential increases by about volts per meter. Thus there is a vertical electric field of volts/m in the air. The sign of the field corresponds to a negative charge on the earth’s surface. This means that outdoors the potential at the height of your nose is volts higher than the potential at your feet! You might ask: “Why don’t we just stick a pair of electrodes out in the air one meter apart and use the volts to power our electric lights?” Or you might wonder: “If there is really a potential difference of volts between my nose and my feet, why is it I don’t get a shock when I go out into the street?” We will answer the second question first. Your body is a relatively good conductor. If yo
10 Dielectrics 10–1 The dielectric constant Here we begin to discuss another of the peculiar properties of matter under the influence of the electric field. In an earlier chapter we considered the behavior of conductors, in which the charges move freely in response to an electric field to such points that there is no field left inside a conductor. Now we will discuss insulators, materials which do not conduct electricity. One might at first believe that there should be no effect whatsoever. However, using a simple electroscope and a parallel-plate capacitor, Faraday discovered that this was not so. His experiments showed that the capacitance of such a capacitor is increased when an insulator is put between the plates. If the insulator completely fills the space between the plates, the capacitance is increased by a factor which depends only on the nature of the insulating material. Insulating materials are also called dielectrics; the factor is then a property of the dielectric, and i
11 Inside Dielectrics Review: Chapter 31, Vol. I, The Origin of the Refractive Index Chapter 40, Vol. I, The Principles of Statistical Mechanics 11–1 Molecular dipoles In this chapter we are going to discuss why it is that materials are dielectric. We said in the last chapter that we could understand the properties of electrical systems with dielectrics once we appreciated that when an electric field is applied to a dielectric it induces a dipole moment in the atoms. Specifically, if the electric field induces an average dipole moment per unit volume , then , the dielectric constant, is given by We have already discussed how this equation is applied; now we have to discuss the mechanism by which polarization arises when there is an electric field inside a material. We begin with the simplest possible example—the polarization of gases. But even gases already have complications: there are two types. The molecules of some gases, like oxygen, which has a symmetric pair of atoms in each mo
12 Electrostatic Analogs 12–1 The same equations have the same solutions The total amount of information which has been acquired about the physical world since the beginning of scientific progress is enormous, and it seems almost impossible that any one person could know a reasonable fraction of it. But it is actually quite possible for a physicist to retain a broad knowledge of the physical world rather than to become a specialist in some narrow area. The reasons for this are threefold: First, there are great principles which apply to all the different kinds of phenomena—such as the principles of the conservation of energy and of angular momentum. A thorough understanding of such principles gives an understanding of a great deal all at once. Second, there is the fact that many complicated phenomena, such as the behavior of solids under compression, really basically depend on electrical and quantum-mechanical forces, so that if one understands the fundamental laws of electricity and qu
13 Magnetostatics Review: Chapter 15, Vol. I, The Special Theory of Relativity 13–1 The magnetic field The force on an electric charge depends not only on where it is, but also on how fast it is moving. Every point in space is characterized by two vector quantities which determine the force on any charge. First, there is the electric force, which gives a force component independent of the motion of the charge. We describe it by the electric field, . Second, there is an additional force component, called the magnetic force, which depends on the velocity of the charge. This magnetic force has a strange directional character: At any particular point in space, both the direction of the force and its magnitude depend on the direction of motion of the particle: at every instant the force is always at right angles to the velocity vector; also, at any particular point, the force is always at right angles to a fixed direction in space (see Fig. 13–1); and finally, the magnitude of the force is
14 The Magnetic Field in Various Situations 14–1 The vector potential In this chapter we continue our discussion of magnetic fields associated with steady currents—the subject of magnetostatics. The magnetic field is related to electric currents by our basic equations \begin{gather} \label{Eq:II:14:1} \FLPdiv{\FLPB}=0,\\[1ex] \label{Eq:II:14:2} c^2\FLPcurl{\FLPB}=\frac{\FLPj}{\epsO}. \end{gather} We want now to solve these equations mathematically in a general way, that is, without requiring any special symmetry or intuitive guessing. In electrostatics, we found that there was a straightforward procedure for finding the field when the positions of all electric charges are known: One simply works out the scalar potential by taking an integral over the charges—as in Eq. (4.25). Then if one wants the electric field, it is obtained from the derivatives of . We will now show that there is a corresponding procedure for finding the magnetic field if we know the current density of all movin
15 The Vector Potential 15–1 The forces on a current loop; energy of a dipole In the last chapter we studied the magnetic field produced by a small rectangular current loop. We found that it is a dipole field, with the dipole moment given by where is the current and is the area of the loop. The direction of the moment is normal to the plane of the loop, so we can also write where is the unit normal to the area . A current loop—or magnetic dipole—not only produces magnetic fields, but will also experience forces when placed in the magnetic field of other currents. We will look first at the forces on a rectangular loop in a uniform magnetic field. Let the -axis be along the direction of the field, and the plane of the loop be placed through the -axis, making the angle with the -plane as in Fig. 15–1. Then the magnetic moment of the loop—which is normal to its plane—will make the angle with the magnetic field. Fig. 15–1. A rectangular loop carrying the current sits in a uniform field
16 Induced Currents 16–1 Motors and generators The discovery in 1820 that there was a close connection between electricity and magnetism was very exciting—until then, the two subjects had been considered as quite independent. The first discovery was that currents in wires make magnetic fields; then, in the same year, it was found that wires carrying current in a magnetic field have forces on them. One of the excitements whenever there is a mechanical force is the possibility of using it in an engine to do work. Almost immediately after their discovery, people started to design electric motors using the forces on current-carrying wires. The principle of the electromagnetic motor is shown in bare outline in Fig. 16–1. A permanent magnet—usually with some pieces of soft iron—is used to produce a magnetic field in two slots. Across each slot there is a north and south pole, as shown. A rectangular coil of copper is placed with one side in each slot. When a current passes through the coil,
17 The Laws of Induction 17–1 The physics of induction In the last chapter we described many phenomena which show that the effects of induction are quite complicated and interesting. Now we want to discuss the fundamental principles which govern these effects. We have already defined the emf in a conducting circuit as the total accumulated force on the charges throughout the length of the loop. More specifically, it is the tangential component of the force per unit charge, integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels once around the circuit. We have also given the “flux rule,” which says that the emf is equal to the rate at which the magnetic flux through such a conducting circuit is changing. Let’s see if we can understand why that might be. First, we’ll consider a case in which the flux changes because a circuit is moved in a steady field. Fig. 17–1. An emf is induced in a loop if the flux
18 The Maxwell Equations 18–1 Maxwell’s equations In this chapter we come back to the complete set of the four Maxwell equations that we took as our starting point in Chapter 1. Until now, we have been studying Maxwell’s equations in bits and pieces; it is time to add one final piece, and to put them all together. We will then have the complete and correct story for electromagnetic fields that may be changing with time in any way. Anything said in this chapter that contradicts something said earlier is true and what was said earlier is false—because what was said earlier applied to such special situations as, for instance, steady currents or fixed charges. Although we have been very careful to point out the restrictions whenever we wrote an equation, it is easy to forget all of the qualifications and to learn too well the wrong equations. Now we are ready to give the whole truth, with no qualifications (or almost none). The complete Maxwell equations are written in Table 18–1, in words
19 The Principle of Least Action 19–1 A special lecture—almost verbatim1 “When I was in high school, my physics teacher—whose name was Mr. Bader—called me down one day after physics class and said, ‘You look bored; I want to tell you something interesting.’ Then he told me something which I found absolutely fascinating, and have, since then, always found fascinating. Every time the subject comes up, I work on it. In fact, when I began to prepare this lecture I found myself making more analyses on the thing. Instead of worrying about the lecture, I got involved in a new problem. The subject is this—the principle of least action. Fig. 19–1. “Mr. Bader told me the following: Suppose you have a particle (in a gravitational field, for instance) which starts somewhere and moves to some other point by free motion—you throw it, and it goes up and comes down (Fig. 19–1). It goes from the original place to the final place in a certain amount of time. Now, you try a different motion. Suppose that
20 Solutions of Maxwell’s Equations in Free Space Review: Chapter 47, Vol. I, The Wave Equation Chapter 28, Vol. I, Electromagnetic Radiation 20–1 Waves in free space; plane waves In Chapter 18 we had reached the point where we had the Maxwell equations in complete form. All there is to know about the classical theory of the electric and magnetic fields can be found in the four equations: When we put all these equations together, a remarkable new phenomenon occurs: fields generated by moving charges can leave the sources and travel alone through space. We considered a special example in which an infinite current sheet is suddenly turned on. After the current has been on for the time , there are uniform electric and magnetic fields extending out the distance from the source. Suppose that the current sheet lies in the -plane with a surface current density going toward positive . The electric field will have only a -component, and the magnetic field, only a -component. The field compone
21 Solutions of Maxwell's Equations with Currents and Charges Review: Chapter 28, Vol. I, Electromagnetic Radiation Chapter 31, Vol. I, The Origin of the Refractive Index Chapter 34, Vol. I, Relativistic Effects in Radiation 21–1 Light and electromagnetic waves We saw in the last chapter that among their solutions, Maxwell's equations have waves of electricity and magnetism. These waves correspond to the phenomena of radio, light, x-rays, and so on, depending on the wavelength. We have already studied light in great detail in Vol. I. In this chapter we want to tie together the two subjects—we want to show that Maxwell's equations can indeed form the base for our earlier treatment of the phenomena of light. When we studied light, we began by writing down equations for the electric and magnetic fields produced by a charge which moves in any arbitrary way. Those equations were and [See Eqs. (28.3) and (28.4), Vol. I. As explained below, the signs here are the negatives of the old ones.] I
22 AC Circuits Review: Chapter 22, Vol. I, Algebra Chapter 23, Vol. I, Resonance Chapter 25, Vol. I, Linear Systems and Review 22–1 Impedances Most of our work in this course has been aimed at reaching the complete equations of Maxwell. In the last two chapters we have been discussing the consequences of these equations. We have found that the equations contain all the static phenomena we had worked out earlier, as well as the phenomena of electromagnetic waves and light that we had gone over in some detail in Volume I. The Maxwell equations give both phenomena, depending upon whether one computes the fields close to the currents and charges, or very far from them. There is not much interesting to say about the intermediate region; no special phenomena appear there. There still remain, however, several subjects in electromagnetism that we want to take up. We want to discuss the question of relativity and the Maxwell equations—what happens when one looks at the Maxwell equations with re
23 Cavity Resonators Review: Chapter 23, Vol. I, Resonance Chapter 49, Vol. I, Modes 23–1 Real circuit elements When looked at from any one pair of terminals, any arbitrary circuit made up of ideal impedances and generators is, at any given frequency, equivalent to a generator in series with an impedance . That comes about because if we put a voltage across the terminals and solve all the equations to find the current , we must get a linear relation between the current and the voltage. Since all the equations are linear, the result for must also depend only linearly on . The most general linear form can be expressed as In general, both and may depend in some complicated way on the frequency . Equation (23.1), however, is the relation we would get if behind the two terminals there was just the generator in series with the impedance . There is also the opposite kind of question: If we have any electromagnetic device at all with two terminals and we measure the relation between and
24 Waveguides 24–1 The transmission line In the last chapter we studied what happened to the lumped elements of circuits when they were operated at very high frequencies, and we were led to see that a resonant circuit could be replaced by a cavity with the fields resonating inside. Another interesting technical problem is the connection of one object to another, so that electromagnetic energy can be transmitted between them. In low-frequency circuits the connection is made with wires, but this method doesn’t work very well at high frequencies because the circuits would radiate energy into all the space around them, and it is hard to control where the energy will go. The fields spread out around the wires; the currents and voltages are not “guided” very well by the wires. In this chapter we want to look into the ways that objects can be interconnected at high frequencies. At least, that’s one way of presenting our subject. Another way is to say that we have been discussing the behavior
25 Electrodynamics in Relativistic Notation Review: Chapter 15, Vol. I, The Special Theory of Relativity Chapter 16, Vol. I, Relativistic Energy and Momentum Chapter 17, Vol. I, Space-Time Chapter 13, Vol. II, Magnetostatics In this chapter: 25–1 Four-vectors We now discuss the application of the special theory of relativity to electrodynamics. Since we have already studied the special theory of relativity in Chapters 15 through 17 of Vol. I, we will just review quickly the basic ideas. It is found experimentally that the laws of physics are unchanged if we move with uniform velocity. You can't tell if you are inside a spaceship moving with uniform velocity in a straight line, unless you look outside the spaceship, or at least make an observation having to do with the world outside. Any true law of physics we write down must be arranged so that this fact of nature is built in. The relationship between the space and time of two systems of coordinates, one, , in uniform motion in the -di
26 Lorentz Transformations of the Fields Review: Chapter 20, Vol. II, Solution of Maxwell’s Equations in Free Space In this chapter: 26–1 The four-potential of a moving charge Fig. 26–1. Finding the fields at due to a charge moving along the -axis with the constant speed . The field “now” at the point can be expressed in terms of the “present” position , as well as in terms of , the “retarded” position (at ). We saw in the last chapter that the potential is a four-vector. The time component is the scalar potential , and the three space components are the vector potential . We also worked out the potentials of a particle moving with uniform speed on a straight line by using the Lorentz transformation. (We had already found them by another method in Chapter 21.) For a point charge whose position at the time is , the potentials at the point are Equations (26.1) give the potentials at , , and at the time , for a charge whose “present” position (by which we mean the position at the t
27 Field Energy and Field Momentum 27–1 Local conservation It is clear that the energy of matter is not conserved. When an object radiates light it loses energy. However, the energy lost is possibly describable in some other form, say in the light. Therefore the theory of the conservation of energy is incomplete without a consideration of the energy which is associated with the light or, in general, with the electromagnetic field. We take up now the law of conservation of energy and, also, of momentum for the fields. Certainly, we cannot treat one without the other, because in the relativity theory they are different aspects of the same four-vector. Very early in Volume I, we discussed the conservation of energy; we said then merely that the total energy in the world is constant. Now we want to extend the idea of the energy conservation law in an important way—in a way that says something in detail about how energy is conserved. The new law will say that if energy goes away from a regi
28 Electromagnetic Mass 28–1 The field energy of a point charge In bringing together relativity and Maxwell’s equations, we have finished our main work on the theory of electromagnetism. There are, of course, some details we have skipped over and one large area that we will be concerned with in the future—the interaction of electromagnetic fields with matter. But we want to stop for a moment to show you that this tremendous edifice, which is such a beautiful success in explaining so many phenomena, ultimately falls on its face. When you follow any of our physics too far, you find that it always gets into some kind of trouble. Now we want to discuss a serious trouble—the failure of the classical electromagnetic theory. You can appreciate that there is a failure of all classical physics because of the quantum-mechanical effects. Classical mechanics is a mathematically consistent theory; it just doesn’t agree with experience. It is interesting, though, that the classical theory of electro
29 The Motion of Charges in Electric and Magnetic Fields Review: Chapter 30, Vol. I, Diffraction 29–1 Motion in a uniform electric or magnetic field We want now to describe—mainly in a qualitative way—the motions of charges in various circumstances. Most of the interesting phenomena in which charges are moving in fields occur in very complicated situations, with many, many charges all interacting with each other. For instance, when an electromagnetic wave goes through a block of material or a plasma, billions and billions of charges are interacting with the wave and with each other. We will come to such problems later, but now we just want to discuss the much simpler problem of the motions of a single charge in a given field. We can then disregard all other charges—except, of course, those charges and currents which exist somewhere to produce the fields we will assume. We should probably ask first about the motion of a particle in a uniform electric field. At low velocities, the motion
30 The Internal Geometry of Crystals Reference: C. Kittel, Introduction to Solid State Physics, John Wiley and Sons, Inc., New York, 2nd ed., 1956. 30–1 The internal geometry of crystals We have finished the study of the basic laws of electricity and magnetism, and we are now going to study the electromagnetic properties of matter. We begin by describing solids—that is, crystals. When the atoms of matter are not moving around very much, they get stuck together and arrange themselves in a configuration with as low an energy as possible. If the atoms in a certain place have found a pattern which seems to be of low energy, then the atoms somewhere else will probably make the same arrangement. For these reasons, we have in a solid material a repetitive pattern of atoms. In other words, the conditions in a crystal are this way: The environment of a particular atom in a crystal has a certain arrangement, and if you look at the same kind of an atom at another place farther along, you will fin
31 Tensors Review: Chapter 11, Vol. I, Vectors Chapter 20, Vol. I, Rotation in Space 31–1 The tensor of polarizability Physicists always have a habit of taking the simplest example of any phenomenon and calling it “physics,” leaving the more complicated examples to become the concern of other fields—say of applied mathematics, electrical engineering, chemistry, or crystallography. Even solid-state physics is almost only half physics because it worries too much about special substances. So in these lectures we will be leaving out many interesting things. For instance, one of the important properties of crystals—or of most substances—is that their electric polarizability is different in different directions. If you apply a field in any direction, the atomic charges shift a little and produce a dipole moment, but the magnitude of the moment depends very much on the direction of the field. That is, of course, quite a complication. But in physics we usually start out by talking about the sp
32 Refractive Index of Dense Materials Review: See Table 32–1. 32–1 Polarization of matter We want now to discuss the phenomenon of the refraction of light—and also, therefore, the absorption of light—by dense materials. In Chapter 31 of Volume I we discussed the theory of the index of refraction, but because of our limited mathematical abilities at that time, we had to restrict ourselves to finding the index only for materials of low density, like gases. The physical principles that produced the index were, however, made clear. The electric field of the light wave polarizes the molecules of the gas, producing oscillating dipole moments. The acceleration of the oscillating charges radiates new waves of the field. This new field, interfering with the old field, produces a changed field which is equivalent to a phase shift of the original wave. Because this phase shift is proportional to the thickness of the material, the effect is equivalent to having a different phase velocity in the m
33 Reflection from Surfaces Review: Chapter 33, Vol. I, Polarization 33–1 Reflection and refraction of light The subject of this chapter is the reflection and refraction of light—or electromagnetic waves in general—at surfaces. We have already discussed the laws of reflection and refraction in Chapters 26 and 33 of Volume I. Here’s what we found out there: The angle of reflection is equal to the angle of incidence. With the angles defined as shown in Fig. 33-1, The product is the same for the incident and transmitted beams (Snell's law): The intensity of the reflected light depends on the angle of incidence and also on the direction of polarization. For perpendicular to the plane of incidence, the reflection coefficient is For parallel to the plane of incidence, the reflection coefficient is For normal incidence (any polarization, of course!), (Earlier, we used for the incident angle and for the refracted angle. Since we can’t use for both “refracted” and “reflected” angles, we
34 The Magnetism of Matter Review: Section 15-1, Vol. II, “The forces on a current loop; energy of a dipole.” 34–1 Diamagnetism and paramagnetism In this chapter we are going to talk about the magnetic properties of materials. The material which has the most striking magnetic properties is, of course, iron. Similar magnetic properties are shared also by the elements nickel, cobalt, and—at sufficiently low temperatures (below C)—by gadolinium, as well as by a number of peculiar alloys. That kind of magnetism, called ferromagnetism, is sufficiently striking and complicated that we will discuss it in a special chapter. However, all ordinary substances do show some magnetic effects, although very small ones—a thousand to a million times less than the effects in ferromagnetic materials. Here we are going to describe ordinary magnetism, that is to say, the magnetism of substances other than the ferromagnetic ones. This small magnetism is of two kinds. Some materials are attracted toward magn
35 Paramagnetism and Magnetic Resonance Review: Chapter 11, Inside Dielectrics 35–1 Quantized magnetic states In the last chapter we described how in quantum mechanics the angular momentum of a thing does not have an arbitrary direction, but its component along a given axis can take on only certain equally spaced, discrete values. It is a shocking and peculiar thing. You may think that perhaps we should not go into such things until your minds are more advanced and ready to accept this kind of an idea. Actually, your minds will never become more advanced—in the sense of being able to accept such a thing easily. There isn't any descriptive way of making it intelligible that isn't so subtle and advanced in its own form that it is more complicated than the thing you were trying to explain. The behavior of matter on a small scale—as we have remarked many times—is different from anything that you are used to and is very strange indeed. As we proceed with classical physics, it is a good idea
36 Ferromagnetism Review: Chapter 10, Dielectrics Chapter 17, The Laws of Induction 36–1 Magnetization currents In this chapter we will discuss some materials in which the net effect of the magnetic moments in the material is much greater than in the case of paramagnetism or diamagnetism. The phenomenon is called ferromagnetism. In paramagnetic and diamagnetic materials the induced magnetic moments are usually so weak that we don’t have to worry about the additional fields produced by the magnetic moments. For ferromagnetic materials, however, the magnetic moments induced by applied magnetic fields are quite enormous and have a great effect on the fields themselves. In fact, the induced moments are so strong that they are often the dominant effect in producing the observed fields. So one of the things we will have to worry about is the mathematical theory of large induced magnetic moments. That is, of course, just a technical question. The real problem is, why are the magnetic moments
37 Magnetic Materials References: Bozorth, R. M., “Magnetism,” Encyclopaedia Britannica, Vol. 14, 1957, pp. 636–667. Kittel, C., Introduction to Solid State Physics, John Wiley and Sons, Inc., New York, 2nd ed., 1956. 37–1 Understanding ferromagnetism In this chapter we will discuss the behavior and peculiarities of ferromagnetic materials and of other strange magnetic materials. Before proceeding to study magnetic materials, however, we will review very quickly some of the things about the general theory of magnets that we learned in the last chapter. First, we imagine the atomic currents inside the material that are responsible for the magnetism, and then describe them in terms of a volume current density . We emphasize that this is not supposed to represent the actual currents. When the magnetization is uniform the currents do not really cancel out precisely; that is, the whirling currents of one electron in one atom and the whirling currents of an electron in another atom do not ov
38 Elasticity Review: Chapter 47, Vol. I, Sound; the Wave Equation 38–1 Hooke’s law The subject of elasticity deals with the behavior of those substances which have the property of recovering their size and shape when the forces producing deformations are removed. We find this elastic property to some extent in all solid bodies. If we had the time to deal with the subject at length, we would want to look into many things: the behavior of materials, the general laws of elasticity, the general theory of elasticity, the atomic machinery that determine the elastic properties, and finally the limitations of elastic laws when the forces become so great that plastic flow and fracture occur. It would take more time than we have to cover all these subjects in detail, so we will have to leave out some things. For example, we will not discuss plasticity or the limitations of the elastic laws. (We touched on these subjects briefly when we were talking about dislocations in metals.) Also, we will n
39 Elastic Materials Reference: C. Kittel, Introduction to Solid State Physics, John Wiley and Sons, Inc., New York, 2nd ed., 1956. 39–1 The tensor of strain In the last chapter we talked about the distortions of particular elastic objects. In this chapter we want to look at what can happen in general inside an elastic material. We would like to be able to describe the conditions of stress and strain inside some big glob of jello which is twisted and squashed in some complicated way. To do this, we need to be able to describe the local strain at every point in an elastic body; we can do it by giving a set of six numbers—which are the components of a symmetric tensor—for each point. Earlier, we spoke of the stress tensor (Chapter 31); now we need the tensor of strain. Imagine that we start with the material initially unstrained and watch the motion of a small speck of “dirt” embedded in the material when the strain is applied. A speck that was at the point located at moves to a new po
40 The Flow of Dry Water 40–1 Hydrostatics The subject of the flow of fluids, and particularly of water, fascinates everybody. We can all remember, as children, playing in the bathtub or in mud puddles with the strange stuff. As we get older, we watch streams, waterfalls, and whirlpools, and we are fascinated by this substance which seems almost alive relative to solids. The behavior of fluids is in many ways very unexpected and interesting—it is the subject of this chapter and the next. The efforts of a child trying to dam a small stream flowing in the street and his surprise at the strange way the water works its way out has its analog in our attempts over the years to understand the flow of fluids. We have tried to dam the water up—in our understanding—by getting the laws and the equations that describe the flow. We will describe these attempts in this chapter. In the next chapter, we will describe the unique way in which water has broken through the dam and escaped our attempts to
41 The Flow of Wet Water 41–1 Viscosity In the last chapter we discussed the behavior of water, disregarding the phenomenon of viscosity. Now we would like to discuss the phenomena of the flow of fluids, including the effects of viscosity. We want to look at the real behavior of fluids. We will describe qualitatively the actual behavior of the fluids under various different circumstances so that you will get some feel for the subject. Although you will see some complicated equations and hear about some complicated things, it is not our purpose that you should learn all these things. This is, in a sense, a “cultural” chapter which will give you some idea of the way the world is. There is only one item which is worth learning, and that is the simple definition of viscosity which we will come to in a moment. The rest is only for your entertainment. In the last chapter we found that the laws of motion of a fluid are contained in the equation In our “dry” water approximation we left out the
42 Curved Space 42–1 Curved spaces with two dimensions According to Newton everything attracts everything else with a force inversely proportional to the square of the distance from it, and objects respond to forces with accelerations proportional to the forces. They are Newton’s laws of universal gravitation and of motion. As you know, they account for the motions of balls, planets, satellites, galaxies, and so forth. Einstein had a different interpretation of the law of gravitation. According to him, space and time—which must be put together as space-time—are curved near heavy masses. And it is the attempt of things to go along “straight lines” in this curved space-time which makes them move the way they do. Now that is a complex idea—very complex. It is the idea we want to explain in this chapter. Our subject has three parts. One involves the effects of gravitation. Another involves the ideas of space-time which we already studied. The third involves the idea of curved space-time. W
Preface to the New Millennium Edition Nearly fifty years have passed since Richard Feynman taught the introductory physics course at Caltech that gave rise to these three volumes, The Feynman Lectures on Physics. In those fifty years our understanding of the physical world has changed greatly, but The Feynman Lectures on Physics has endured. Feynman's lectures are as powerful today as when first published, thanks to Feynman's unique physics insights and pedagogy. They have been studied worldwide by novices and mature physicists alike; they have been translated into at least a dozen languages with more than 1.5 millions copies printed in the English language alone. Perhaps no other set of physics books has had such wide impact, for so long. This New Millennium Edition ushers in a new era for The Feynman Lectures on Physics (FLP): the twenty-first century era of electronic publishing. FLP has been converted to eFLP, with the text and equations expressed in the LaTeX electronic typesettin
Feynman's Preface These are the lectures in physics that I gave last year and the year before to the freshman and sophomore classes at Caltech. The lectures are, of course, not verbatim—they have been edited, sometimes extensively and sometimes less so. The lectures form only part of the complete course. The whole group of 180 students gathered in a big lecture room twice a week to hear these lectures and then they broke up into small groups of 15 to 20 students in recitation sections under the guidance of a teaching assistant. In addition, there was a laboratory session once a week. The special problem we tried to get at with these lectures was to maintain the interest of the very enthusiastic and rather smart students coming out of the high schools and into Caltech. They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many would be very discouraged becau
Foreword For some forty years Richard P. Feynman focussed his curiosity on the mysterious workings of the physical world, and bent his intellect to searching out the order in its chaos. Now, he has given two years of his ability and his energy to his Lectures on Physics for beginning students. For them he has distilled the essence of his knowledge, and has created in terms they can hope to grasp a picture of the physicist's universe. To his lectures he has brought the brilliance and clarity of his thought, the originality and vitality of his approach, and the contagious enthusiasm of his delivery. It was a joy to behold. The first year's lectures formed the basis for the first volume of this set of books. We have tried in this the second volume to make some kind of a record of a part of the second year's lectures—which were given to the sophomore class during the 1962–1963 academic year. The rest of the second year's lectures will make up Volume III. Of the second year of lectures, the
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