Invitation to a Contemporary Physics (2004)

Second Law simply tells us that the force exerted by the particle is the rate at which its momentum changes. It follows then: (1) the momentum remains constant in the absence of unbalanced forces, and (2) the total momentum of an isolated two-particle system is a constant of motion. We recognize in these results a simple restatement of the First and Third Laws, which, however, has the virtue of making a generalization to systems of more than two particles almost trivial: if a system is isolated so that there are no external forces, the total momentum of the system is constant (the law of conservation of momentum). If there are external forces, the rate of change of the momentum is vectorially equal to the total external force. The law of conservation of momentum has a more general validity and a deeper meaning than its derivation given here would indicate; it is a manifestation of the existence of a space symmetry (Chapter 1).

Angular Momentum

So far, we have discussed motion as if it involved only translational motion, i.e., a motion that keeps the relative orientation of di erent parts of the body unchanged. But pure translation rarely occurs in nature; there is always some degree of revolution or rotation in any motion. In revolution, a particle is acted on by a force that continually changes its direction of action, making it move along some curved path. In rotation, no straight line (except one) connecting any two points in the body remains parallel to itself; the line that does is called the body’s axis of rotation. In pure rotation, a body changes its orientation without changing its position and so must be acted on by something other than simple force.

Suppose we nail a long rod at one end loosely to the ground. If we push or pull the rod near its free end, the applied force would generate an equal and opposite reaction of the rod at the fixed end. Two equal and opposite forces, which are not on the same line and hence uncompensated, act on the rod and make it rotate about the fixed end. Thus, a correct measure of the e ects observed cannot be given by the applied force alone, but rather by that combination of forces, called a torque (meaning ‘twist’). Clearly the amount of torque depends on where the force is applied. It also depends on the orientation of the force with respect to the rod (or, generally, on the position vector that defines the location of the application point relative to a fixed point); in particular, it vanishes if the force is parallel to the rod. A non-vanishing torque is a vector perpendicular to both the force and the position vector. Just as force produces changes in (linear) momentum, torque produces changes in angular momentum; if no torque exists, the angular momentum does not change; it is said to be conserved.

The angular momentum (L) of a particle is a vector perpendicular to both the position vector (r) and the momentum (p) of the particle relative to some origin of coordinates and whose magnitude is rp sin θ, where θ is the angle between the two vectors r and p. As we have seen, in the absence of forces, a particle moves along a straight line at constant velocity. The direction of its angular momentum

426 General Concepts in Classical Physics

relative to an arbitrary reference point is perpendicular to the plane defined by the line of motion and the reference point. Its magnitude is simply the product of its momentum (p) and the shortest distance between the origin of coordinates and the line of motion (r sin θ). Thus, it is a constant vector for the chosen origin; in the absence of forces, both vectors p and L are constants of motion. If the origin changes, L changes to a new constant vector. In particular, we can make L vanish by locating the origin on the line of motion (r sin θ = 0). Now, take a particle in uniform circular motion, i.e., a motion with constant velocity along a circle. Its momentum is constant in magnitude but continuously changing in orientation. There is a force (centripetal force) directed toward the center of the circle continuously acting on the particle to keep it on its circular orbit. The angular momentum of the particle, defined with respect to the center of the circle taken as the origin of coordinates, is simply L = rp, the product of the circle radius and the particle momentum; it is perpendicular to the plane of the circle. So, again, there is no torque, and L is a conserved vector although the momentum is not. But, in contrast to the previous case, there is no way we can make L vanish by some choice of the reference point. Thus, angular momentum is a property characteristic of rotational motion.

Kepler’s Second Law (the radial line segment from the sun to a planet sweeps out equal areas in equal times) can be seen as an example of application of the principle of conservation of angular momentum. Since the gravitational force of the sun on a planet is along the sun–planet line, there is no torque, and the angular momentum of the planet (which is essentially equal to the area swept out in a unit time) is conserved. Just as the conservation of momentum implies a symmetry in the laws of nature, so does the conservation of angular momentum. This symmetry — the rotational symmetry — implies that space is isotropic: the geometry of space is the same in all directions.

Work and Energy

Let us now turn to the concepts of work and energy. To simplify the discussion, we take the case of one-dimensional motion. If a constant force F acting along the x-axis causes a particle to move some given distance x, then the work done is defined by W = F x. This is a reasonable definition, because it corresponds precisely to the e ort we would provide and what we would be paid for to push the object that distance, regardless of the nature of the object. If the force applied is not constant over the distance being covered, then a careful summation (integration) over all small segments composing the distance must be made to get the correct result. When we do work on a body, the body changes by acquiring ‘energy,’ a quantity defined such that the acquired energy equals the work done on the body by the force. If we apply a force to a particle of mass m at rest on a smooth (frictionless) surface in a vacuum, the particle will be moving at a definite speed v after the force

stops acting on it. The force applied is F = ma and the work done is W = 12 mv2, which we equate with the acquired energy. Similarly, if the particle has initial speed v1 and acquires a final speed v2, the work done is W = 12 mv12 − 12 mv22. This is the acquired energy. Since the energy appears in the form of motion, we call it the kinetic energy of the particle and define it as K.E. = 12 mv2.

But energy may appear in other forms as well. When it represents the capacity of the particle to do work by virtue of its position in space, it is called the potential energy. A simple example is that of a particle moving under the influence of a constant gravitational force (e.g., the earth’s). To lift vertically a body of mass m a height z above the surface of the earth, the work we must do on the particle is −mgz, where g is the gravitational acceleration and the minus sign comes from the downward direction of the force of gravity. We call the function V (z) = mgz the gravitational potential energy. It is defined relative to the surface of the earth, that is, the particle’s energy at height z is greater by mgz than it is on the earth’s surface. If we now release the particle to fall back freely from this height, it will reach a speed v when it hits the ground. If gravity is the only force acting on the particle, the work done by gravity on the particle must be the same in both directions. Thus, the sum E = K.E. + V (z), called the mechanical energy, has the same value at any height although the two energy components may vary separately. This is the law of conservation of mechanical energy.

So far we have ignored dissipative forces, e.g., friction. If now we attempt to push a heavy object on a rough surface, we must do more work to obtain the same result because not all work done will be expended into changing the body’s mechanical energy; some energy will be dissipated in heat, for which we do not usually get paid. In this case, the mechanical energy clearly is not conserved. However, the loss is only apparent because we are taking the macroscopic point of view, i.e., working on the level of the whole body, ignoring its microscopic components. As the body is rubbing against the surface, the energy lost by the body is in fact transmuted into disorderly kinetic and potential energy of the atoms in the body and the surface. This disorderly mechanical energy of atoms is what is meant by heat. No energy has been lost, only part of it has been transformed into another kind of energy. To be fair, we should be paid for work done on both the body and the atoms it contains.

Energy can manifest itself in many other ways as well. Some of the basic forms are the following. Chemical energy is the term often used for molecular energy or atomic energy, which are, respectively, kinetic and potential energy of atoms in molecules or electrons in atoms. Nuclear energy is kinetic and potential energy of particles (neutrons and protons) contained in atomic nuclei. Electromagnetic energy is the form of energy carried by all sorts of electromagnetic radiation, such as moonlight and TV signals. Another important form of energy is mass, as expressed by the famous equation discovered by Albert Einstein: E = mc2. This equation means that an amount m of mass is equivalent to an amount E of energy given numerically by product of m and the square of the speed of light c. The equation

can be read in either direction: energy can be transformed into mass, and mass can be converted into energy — energy and mass are equivalent.

It is a general property of our physical world that the total amount of energy of any isolated system never changes in time, although the form of energy may change. This is the general law of conservation of energy. Again, this conservation law arises from a symmetry of space, the irrelevance of the absolute measure of time.

To summarize the above, let us restate that the laws of classical mechanics depend neither on the origins of space and time coordinates nor on the orientations of the coordinate system. Also, they are insensitive to the state of motion of physical events, as long as this motion is uniform, rectilinear and free of rotation. All these invariance properties are products of experience, not a priori truths. It is only because regularities such as these exist in Nature that we can hope to discover its secrets.

As a simple application of the above discussion, consider a particle in uniform circular motion with constant speed v along a circular path of radius r under the influence of some force F . The situation just described can be seen as an idealization of the motion of an artificial satellite around the earth (F = gravitational force), or of an electron around the atomic nucleus (F = electric force). One can show (with the help of calculus) that the magnitude of the centripetal acceleration is a = v2/r. It follows from F = ma that mv2 = rF , which represents work done to move the particle to infinity (where it has zero potential energy by convention). Hence, the kinetic energy of the particle, K.E. = 12 mv2, satisfies the equation K.E. = −21 P.E., where P.E. stands for the potential energy of the particle. Although the derivation relies on a particular situation, the relation obtained gives the statement of an important general result of classical mechanics, the so-called virial theorem. The theorem is statistical in nature in the sense that it involves the time average of various mechanical quantities. It is applicable to a large class of physical systems (those in quasistationary motion, in which coordinates and momenta always remain within finite limits).

As another application, let us consider how the law of universal gravitation could be derived. To begin with, we observe that planets must be under the action of some net (attractive) force, because otherwise they would be moving in straight lines instead of curved paths, as required by Newton’s First Law. Such a force acting on a planet must be directed at any instant toward the center of motion, otherwise Kepler’s empirical Second Law could not be satisfied. For an elliptical orbit, the center of motion is one of the foci of the ellipse — for a circular orbit, the center of the circle. Now, Newton proved mathematically that the centripetal force acting on a body revolving in an ellipse, circle or parabola must be proportional to the inverse square of the distance of the body to the focus.

What is the origin of that force? Newton suggested that the force governing the motion of the planets around the sun, or the revolution of the moon around the earth is of the very same nature as the gravitational attraction that makes an apple fall to the ground. Whatever their makeup, all heavy bodies experience the same

basic force. Consider any two objects, for example the earth (mass m1) and a stone (mass m2). We know by experience that the stone is pulled toward the earth by its weight, which is proportional to its own mass m2; but, similarly, there must be a weight of the earth pulling it toward the stone and proportional to m1. By the symmetry of action and reaction, we see that the gravitational force is proportional to both m1 and m2. Assuming this force to depend only on the masses of the two bodies in interaction and the distance between them, we come to the conclusion that the gravitational attraction between any two objects of masses m1 and m2 separated by a distance r is given in magnitude by F = Gm1m2/r2. It is a vector directed from one body’s center-of-mass to the other’s. This is Newton’s famous law of gravitation. The constant of proportionality, G, has come to be known as the universal gravitational constant.

A.3 Waves and Fields

The physical universe of the late nineteenth century was so dominated by Newton’s ideas that it must have appeared to be endowed with a superb unity. Not only could the motion of all bodies, whether on earth or in space, be explained by the same laws of motion, but the invisible world of the atoms was also made part of the realm of mechanics when heat was treated as a mere mechanical phenomenon by the kinetic theory of gases. Certainly, there would not be any problems, most physicists of the time must have thought, that could not be ultimately solved within the existing framework.

But problems there were, and not necessarily anodyne. First, what is the nature of light? Is light composed of something similar to microscopic particles, as Newton believed, or of something immaterial akin to wave-like impulse, as Christiaan Huygens, Thomas Young, and Augustin Fresnel advocated? Then, how to incorporate into the existing mechanical framework the increasingly large number of new electric and magnetic phenomena observed by Hans Oersted and Michael Faraday, among others? Finally, what to do with the clearly unacceptable action- at-a-distance interpretation — i.e., that two particles interact even though they are not touching — as suggested by Newton’s law of gravitation and the newly discovered Coulomb’s law of electric force? It turns out that the answers to these questions are not unrelated, all relying on two new concepts, waves and fields, which are to play key roles in the physics of the twentieth century.

Waves and Wave Propagation

Before discussing the nature of light, let us pause briefly to describe what turns out to be an analogous phenomenon, the vibrations of a string that is fixed at one end. If we take hold of the free end of the string and move it rhythmically up and down, imparting to it both motion and energy, successive segments of the string, from the

free end on down the line, also move up and down rhythmically. A crest (maximum displacement) is generated, then a valley, and both travel down the string, soon to be followed by another maximum, then another minimum, and so on. Thus, we have a periodic wave traveling along the string, carrying with it energy (due to the motions and positions of the particles) from one end to the other. The speed with which a crest travels may be taken as the speed of propagation of the wave, v; the distance between two successive crests is called the wavelength, λ; and the time needed for a crest to cover that distance is called the period of vibration, T . The three quantities are related by v = λ/T and the reciprocal of T gives the frequency of vibration. Here, the wave propagates horizontally while segments of the string move up and down; such a wave is called a traveling transverse wave.

Now, if we take a long rubber cord or a row of particles connected by massless springs and if we snap the free end of the string back and forth horizontally, a disturbance is forced on the string and is transmitted from one particle to the next. The disturbance begins traveling along the string and will produce successive variations of particle density — condensations and rarefactions of particles, which oscillate in the same direction as the motion of the wave itself. Such a wave is called a traveling longitudinal wave. Let us imagine we have a number of such identical strings, of either kind, fixed to the same wall and forced to vibrate together by the same disturbance. We may then define a wavefront as an imaginary surface, generally perpendicular to the direction of propagation of the wave, which represents the disturbance as it travels down all the strings.

Waves have a characteristic property that distinguishes them from particles. When two or more waves propagate in the same medium, they fuse together to form a new wave. We call this phenomenon an interference; it can be either constructive (if the incident waves reinforce each other) or destructive (if the incoming waves tend to cancel out). Streams of particles do not, apparently, exhibit such behavior under normal conditions.

Nature of Light

To the question, Is light particle-like or wave-like?, experiments have given a clearcut answer: transmission of light exhibits all the properties characteristic of the propagation of waves. One of the first decisive experiments which helped to establish the wave theory of light is the famous Young’s double-slit interference experiment, described in Chapter 2. But we will find it instructive to repeat the historical argument here. It hinges on explaining why a light ray crossing the interface between a rare and a denser medium is refracted, that is, deflected toward the normal to the interface. If light is particle-like, one would naturally expect the light ray to be slowed down by the denser medium (more precisely, its velocity’s component perpendicular to the interface to be reduced and the component parallel to the interface unchanged) and hence to be deflected away from the normal, contrary

Figure A.2: Refraction of a light pulse going from air to water; wavefronts are at right angles to the light ray and delineate successive maxima. In (a) it is assumed that the speed of light is smaller in water than in air, and so wavefronts travel closer together in water; in (b) light is assumed to move faster in water.

to observations. However, if the ‘light particles’ are accelerated at the interface by some unknown force, then their path would be bent toward the normal, as observed.

On the other hand, if light is wave-like, one would expect that when a light pulse strikes an interface at an oblique angle, the wavefront is split into two, one traveling in the first medium with the old speed, the other in the second medium with a new speed. From Fig. A.2, one sees that light will be bent toward the normal if its speed is smaller in the new medium and will be deflected away from the normal if its speed is greater. This means that if light is wave-like, it is slowed down in going from air to water, just the opposite of the behavior of a light particle. The predictions of two competing theories could not be more unambiguous. To decide between the two theories, it su ces to compare the speed of light in air with that in water or glass. It was not until the mid-nineteenth century that such a delicate experiment could be carried out, with the result by now well-known: light travels faster in air than in water.

Electric and Magnetic Fields

The electric force between two stationary charged particles was experimentally discovered by Charles-Augustin de Coulomb in 1785. Its magnitude is given by F = Kqq /r2, where q and q are the two interacting charges, r is their separation distance and K a numerical constant; its direction is along the line joining the two particles. The electric force is repulsive if the charges are of like signs and attractive if the charges are of opposite signs. When the particles are set in motion, there is an extra force acting on these particles; this extra force is called the magnetic force. The first concrete evidence that moving charges induce a magnetic force was given by Hans Oersted in 1820 when he showed that a strong electric current (flow of charges) sent through a wire aligned along the north–south direction caused a magnetic compass needle, originally set parallel to the wire, to rotate by 90◦ and settle in the east–west direction. When the direction of flow of the current was

reversed, the needle immediately rotated by 180◦ and aligned itself perpendicular to the wire. The behavior of the needle indicated that it was acted on by a force quite unlike the electric force, emanating from the current and perpendicular to it.

Like every physics student, anybody who has seen the regular alignment of small bits of thread between a pair of charged plates or the pattern of iron filings near a magnet, is convinced that space is modified by the presence of charges or magnets. Something new must have appeared in the intervening space.

To discover what it is, let us perform this experiment. Let us attach a small body to the end of a non-conducting massless string and give it a small charge q, so small that it will not a ect any system of charges we want to study (we assume the charges to be at rest for the moment). As we place this device at every point in space, we realize that our probe experiences an electric force which depends on the location of the body, the source charges that set up the action and the probe charge q itself. To obtain a quantity intrinsic to the system, independent of q, we define at any given point P the electric field strength (call it E) as the net force F on charge q at point P divided by q. E is a vector, as is F . We could fill the whole three-dimensional space with imaginary arrows to a given scale to represent all such vectors. We call the full set of the E-vectors in a given region the electric field in that region. As an aid to visualization, we may represent the electric field by electric field lines. At every point along such a line, the electric field is tangential to the line. Also, the density of the lines is proportional to the field strength. When lines crowd together, we have a relatively stronger field; when they spread out, we have a relatively weaker field (Fig. A.3).

The magnetic field (B) can be similarly represented by magnetic field lines. Repeating Oersted’s experiment by placing a compass needle at various points around a current-carrying wire, we discover that the force exerted by the current on the magnet is circular. We can mentally picture the wire as surrounded by concentric lines of force. The tangent to a field line indicates the direction of the field B and the density of field lines, the strength of the field.

What Oersted showed was that a steady electric current produced a constant magnetic field around the current-carrying circuit. Inversely, one may ask, could

Figure A.3: (a) Representative electric field lines generated by two charges equal in magnitude and opposite in sign. (b) Representative magnetic field lines around a straight wire carrying an electric current. The current emerges perpendicular to the plane of the page.

a constant magnetic field or a steady current flowing in one circuit generate a current in another circuit nearby? The answer was given by Michael Faraday, who discovered the general principle of electromagnetic induction: (1) a constant magnetic field or a steady current in one wire cannot induce a current in another wire; (2) only changing lines of magnetic force can induce a current in a loop; the change of the force lines can be caused either by moving a magnet relative to the loop, or by suddenly varying the current in a wire nearby.

What now transpires from these results is that: (1) the electric and magnetic fields are sensitive to both space and time; and (2) they are interdependent; if one changes in any way, so does the other. It remained for James Clerk Maxwell to discover the exact physical laws governing the electric and magnetic fields, which he stated in the form of a set of di erential equations for E and B. These equations, which form the basis of his electromagnetic theory, describe how the space and time variations of the electric and magnetic fields can be determined for a given electromagnetic source, or distribution of charge and current.

At points far removed from a localized source, Maxwell’s equations are reduced to two separate equations, one for E and one for B. They are called the wave equations because they show that the two fields propagate together as periodic oscillations, both perpendicular to each other and perpendicular to the direction of propagation of the waves. Thus, the electric and magnetic fields behave as transverse waves. They cause any electric charges or magnetic poles found anywhere in their region of action to oscillate with a characteristic frequency, in the same manner as a wave transmitted along a string would force particles on its path to fluctuate.

Maxwell’s theory predicts, remarkably, that electromagnetic waves propagate with the same speed as light. Generally, a wave equation relates the spatial variations of a field amplitude, A, to its time variations. The spatial variations of A are essentially A divided by the square of a small distance, symbolically A/d2, and the time variations are essentially A divided by the square of a small time interval, or A/t2. These two quantities are dimensionally di erent, one has the dimension of the reciprocal of a squared length, the other the dimension of the reciprocal of a squared time. They cannot enter as two terms in the same equation unless the denominator of the second term is multiplied by the square of a speed. This speed is precisely the speed of propagation of the wave, which Maxwell showed to be numerically equal to the speed of light. It was a most remarkable result with far-reaching implications. His theory also predicted that a beam of electromagnetic waves would be reflected o metallic surfaces and that it would be refracted on entering a layer of glass. In other words, electromagnetic radiation is expected on theoretical grounds to behave in every way like light.

Maxwell had thus unified not only electricity and magnetism but also light in a single theory, the electromagnetic theory of radiation, which ultimately encompasses the whole electromagnetic spectrum, from the longest radio waves to the shortest gamma rays. The correctness of his views was eventually confirmed by Heinrich

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Hertz, who demonstrated by experiment that an oscillating current indeed sent out electromagnetic waves of the same frequency as the emitting oscillations, and that these waves carried momentum and energy that could induce a fluctuating current in a wire nearby. Maxwell gave the electromagnetic theory mathematical rigor, Hertz gave it physical reality.

To summarize, the concept of force can be conveniently replaced by the concept of field of force. Instead of saying that a particle exerts a force on another, we may say that a particle creates a field all around itself which acts on any other particles placed in its zone of action. While in non-relativistic physics, the field is merely another mode of describing interactions of particles, in relativistic mechanics, where the speed of light is considered finite, the concept of field takes on a fundamental importance and, in fact, acquires a physical reality of its own. The picture of particles interacting at a distance gives place to a picture of interaction by contact, in which a particle interacts with a field, and the field in turn acts on another particle, such that there is overall conservation of energy and momentum. Applied to electricity and magnetism, the concept of field is essential to a unified treatment of these two phenomena. The unified field thus introduced — the electromagnetic field — generates electromagnetic radiation, which behaves in every way like transverse waves in regions far from the source that produces it. Light is but one form of such radiation. The field concept, introduced by physicists of the nineteenth century, will blossom to full significance in the physics of the twentieth century.