Chapter11

Chapter11 - CHAPTER GRAVITY ELECTRICITY AND MAGNETISM in...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CHAPTER GRAVITY ELECTRICITY, AND MAGNETISM in order to obtain physical ideas without adopting a physical theory we must make ourselves familiar with the existence of physical analogies. By a physical analogy I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other. Thus all the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. Passing from the most universal analogies to a very partial one, we find the same resemblance in mathematical form between two different phenomena giving rise to a physical theory of light. James Clerk Maxwell, “On Faraday‘s Lines of Force" (1855) 11.1 FINDING THE CONNECTION BETWEEN ELECTRICITY AND MAGNETISM The flood of forces identified and classified in the eighteenth century was reduced to a trickle after it was realized that electric forces were responsible for many phenomena. Nature, it seemed, acknowledged only a handful of forces. Each force had its own “universal constant.” For electric forces, it was Kc; for gravity, G; for magnetism, there was an additionaLconstant Km; and for light, there was the speed of light, known since 1630 to be 3 X 108 m/s. Surely many physicists wondered whether these constants and 217 218 GRAVITY, ELECTRICITY, AND MAGNETISM the forces they represent are somehow related. Similarities in mathematical forms of electric and magnetic forces, an 1820 lecture demonstration, and an experimental virtuoso led the way in the search for such relationships. In the days of the ancient Greeks, it was known that a certain substance found in the ground has the power of attracting pieces of iron. The name magnet was derived from the district in Greece, Magnesia, where the material was found in great quantities. As time went on many fabulous tales of the rare and magic properties of magnets arose. Certain varieties were recommended as love potions; in the presence of diamonds or of garlic, a magnet’s property was believed to be lost, but fortunately the attractive power could be restored by the timely use of goat’s blood. By the eleventh century magnets appeared as compass needles and became an in- dispensable tool in navigation. However, credit for the first scientific studies of magnets usually goes to William Gilbert, court physician to Queen Elizabeth I of England. Gilbert rejected the idle tales of magnetic virtues and determined the properties of magnets through experiments. He suggested that the earth itself is a large magnet, and he pointed out that every magnet has definite points in it, which he called the north pole and south pole, each of the same strength. By floating two magnets he showed that unlike poles attract and like poles repel each other. In addition, Gilbert found that if a magnet is cut in half, it acquires poles where it had been neutral. Although there is no doubting the importance of Gilbert’s contribution, much of what he wrote had already appeared in a thirteenth— century manuscript, written during a leisurely siege of a town in Southern Italy by a mysterious scholar named Pierre of Merricourt. In 1750 a young Cambridge theology student, John Michell, aided by a torsion balance like the one Cavendish had independently developed, measured the magnetic force between two magnet poles at different distances. From his experiments he concluded that the force follows an inverse square law similar to the force between two charges. If p1 and p2 are the pole strengths, we can describe the force between the two poles as P1172. F =Km--2-—r. (11.1) r This equation is the same as for electric or gravitational forces, except that now the force is between two poles and there is a new constant Km. Coulomb, like Michell, verified the inverse square law for magnetic poles, and observed that a magnet when broken in pieces, however small, still has poles at the broken ends. Magnetism, unlike electricity, apparently always comes in pairs of poles. Physicists today are still searching for single magnetic poles, called magnetic monopoles, because some unified theories of the forces of nature predict that magnetic monopoles were created in the early stages of the universe. If they exist, however, they are certainly rare, so we shall assume that every pole comes attached to an equal and opposite partner. By the end of the eighteenth century certain similarities between electric and magnetic phenomena were realized, yet there was no clearly established relationship between the two forces. Electricity and magnetism were regarded as forces obeying similar laws but as being fundamentally different in nature. However, in 1820 a Danish scientist, Hans Christian Oersted, not a consummate experimentalist, but rather nearsighted and bum- bling, nevertheless found a connection that opened a new epoch in physics. Oersted was an ardent popularizer of science whose public lectures attracted curious citizens of Co- 11.2 FARADAY’S FIELDS 219 penhagen. Folklore about his discovery places him in one such lecture on electricity, electric currents (known then as galvanism), and magnetism, during which he placed a wire conducting a current over a compass and at right angles to the needle. No effect was observed and obviously the audience was unimpressed. After the lecture he tried the experiment again with the wire parallel to the compass needle as shown in Fig. 11.1. This time the compass needle moved. <— Direction of Current N S CL—I— compass needle Figure 11.1 Oersted’s experiment relating electricity and magnetism. What Oersted discovered was that electric charges in motion — electric current — can exert forces on a compass needle. The news of Oerstcd’s discovery spread throughout Europe like wildfire, and dozens of more dexterous experimentalists explored the effect and derived the laws describing it. Although Oersted did not quantitatively investigate the phenomenon he had discovered, he did speculate about its ultimate cause. Somehow, he believed, the medium surrounding the conducting wire plays a role in the transmission of force from a wire to a magnet. Oersted’s discovery showed that electricity and magnetism are not separate, inde- pendent forces, and it fueled the search for unification. It showed that a force exists between magnetic poles and an electric current. But the direction of the force depends in an unexpected way on the direction of the magnet and the direction of their current. Electromagnetism seemed very complicated. 11.2 FARADAY’S FIELDS No one more vigorously explored the connection between electricity and magnetism than Michael Faraday, a nineteenth—century English chemist. Faraday, whose passion for science started from reading an article on electricity in the Encyclopaedia Britannica, had little formal education, never attended a university, and never learned mathematics. Yet he consistently made the most important discoveries of his time, because he had an intuition that grasped the essence of things far better than the formulas of the most powerful mathematician. It had been long known that if iron filings are sprinkled on a sheet of paper and a magnet is held underneath, the filings will arrange themselves in definite curves, as shown in Fig. 11.2a. Faraday, of course, had noticed this effect and imagined the curves of iron 220 GRAVITY, ELECTRICITY, AND MAGNETISM .37“ M” X y / ¢A-,., ' ' x "n a Figure 11.2 (a) Iron filings form curves around a magnet. (b) Magnetic field lines around a magnet. filings as following lines of force that spread out from the north pole to the south pole. In his mind’s eye, he saw all space as filled with such lines, never crossing or tangling, so that at every point there was a potential force ready to act on a magnet pole if one were placed there. From these speculations the idea of the magnetic field, a new vector quantity, was born. The direction of the field at any point is the direction of the line, and the strength of the field depends on how densely packed the lines are. The field is strong where the lines are closely packed, as near the poles of a magnet, and weaker where they are spread apart, as in the center, as Fig. 11.2b illustrates. The force a magnetic field B exerts on a magnetic pole with strength p0 can be described by F = pOB. (11.2) Since magnetic poles come in pairs, the magnetic field pushes a north pole one way and a south pole the opposite way, as shown in Fig. 11.3a. The result of the forces is to rotate the magnet so that it lines up with the field, as Fig. 11.3b illustrates. (a) (b) Figure 11.3 (a) Forces on poles of a magnet from a magnetic field B. (b) Forces acting on a magnet align it with the field. 11.2 FARADAY’S FIELDS 221 To explain Oersted’s discovery, Faraday conjectured that a current-carrying wire creates a magnetic field around itself. This field, in turn, causes a magnet to become aligned. Using a small compass needle, Faraday mapped out the magnetic field around a current-carrying wire. As Fig. 11.4 indicates, the field characteristically forms circles about the wire. The direction of the field can be found by pointing the thumb of your right hand in the direction of the current; your fingers then curl in the direction of B. (b) Figure 11.4 Magnetic field due to a current-carrying wire. Not only did Faraday imagine that magnets and currents create fields, but also that electric charge creates an electric field. Electric field lines reach out from negative charges and end on positive charges, as shown in Fig. 11.5. The direction of the force on a small charge at any point is the same as the way in which the field line is directed. We write the force on a charge go in an electric field E as F = qOE. (11.3) Figure 11.5 Electric field around a negatively charged sphere. In the same way, we can imagine a gravitational field G created by mass at each point in space. If a mass m0 is placed in this field, the force due to the gravitational field G is F = moG. (11.4) 222 GRAVITY, ELECTRICITY, AND MAGNETISM Knowing that the gravitational force between two masses is moM . F = — G — r, 8.1 r2 ( ) we can mathematically describe the gravitational field around a mass M by GM G = ——f‘. ,2 Figure 11.6a illustrates the gravitational field of the earth, where the lines of force are radially inward. If we look very close to the surface, all the lines go straight down, as shown in Fig. 11.6b. A mass placed in this field feels a force which causes it to fall vertically downward. By Eq. (11.4) the magnitude of the force is proportional to the mass of the object, so the larger the mass of the object, the greater the force on it. Consequently, all objects fall with the same acceleration. (b) Figure 11.6 (a) Gravitational field G of the earth. (b) Field lines near the surface of the earth. What we’ve just found out is not new; we discovered why all bodies fall with the same acceleration in Chapter 8. What is new is the idea of a field, which was Faraday’s answer to action-at-a-distance forces, for which Newton made no hypothesis. In Faraday’s vivid imagination, the space surrounding charges, masses, or magnets is filled with a field which transmits forces to other charges, masses, or magnets. The fertile idea of fields led Faraday to unravel the connection between electricity and magnetism. We still have one puzzle to unravel: How is an electric current related to a magnetic pole? Since we know how a pole reacts to a magnetic field, let’s ask how an electric current reacts to a magnetic field. Suppose we have a charge q, moving with velocity v (that constitutes an electric current), in a magnetic field B. What is the force on the charge? Experimentally the magnitude of the force on a moving charge is simply found to be F = qu sin 6, where 6 is the angle between v and B. The direction of this force, however, is not so simple. The direction of F depends upon the direction of v as well as the direction of B. 11.2 FARADAY'S FIELDS 223 Furthermore, the force is in neither of these directions; it is perpendicular to both. Therefore the cross product (introduced in Chapter 5) provides a natural mathematical tool to help us describe the situation. In terms of the cross product, the force F on a charge q moving with a velocity v through a magnetic field B is F = qv X B. (11.5) Using this force and the idea of fields, we can predict what will happen when two current- carrying wires are placed near each other. As shown in Fig. 11.7, we can think of one wire creating a magnetic field B in circles around it. The current in the second wire consists of charges moving in the magnetic field of the first wire. From Eq. (11.5), the direction of the force on any charge in the second wire is directly toward the first wire. Consequently, the wire is attracted. The same thing happens to the first wire: the second wire creates a magnetic field, which exerts a force on the charges moving in the first wire; the result is that the wires are attracted. Experimentally this is precisely what happens: wires carrying currents in the same direction are attracted. current direction Figure 11.7 Two parallel current—carrying wires attract each other. Questions 1. Knowing that the magnetic field around a current-carrying wire forms circles about the wire, explain why Oersted’s first experiment did not produce a deflection of the compass needle. 2. Using the idea of a gravitational field, show why all bodies fall with the same ac- celeration. 3. According to Faraday’s field idea, the strength of a field is indicated by the number of field lines in a region. Using Fig. 11.6, discuss how the strength of the gravita- tional field is indicated. 4. What is the net force on a magnet placed in a uniform (constant everywhere in space) magnetic field? Does the magnet accelerate? 5. Suppose you are given two identical metal bars, of which one is a magnet. How can you tell which one is the magnet, using only the two bars? 224 GRAVITY, ELECTR|C|TY, AND MAGNETISM 6. The diagram below represents the tracks of three particles that passed through a region where there is a uniform magnetic field perpendicular to the plane of the paper and indicated by the x’s. What is the sign of the charge of each particle? 11.3 A PREDICTION FROM ELECTROMAGNETISM The researches of Faraday into electromagnetic effects excited nineteenth-century phy- sicists. One person stimulated by this work was James Clerk Maxwell, a Scottish student at Cambridge. Like Faraday, Maxwell possessed a keen power of visualizing physical ideas, but in addition he had mathematical prowess. His first contributions to electro- magnetic theory amounted to a translation of Faraday’s ideas into mathematical language. Like Faraday, he abandoned the notion of action at a distance and sought to interpret electric and magnetic phenomena in terms of fields. Maxwell succeeded in expressing Faraday’s ideas in a compact set of equations. Maxwell’s equations unite electric and magnetic phenomena in one common description. But the success was not simply mathematical cunning. Based upon his theory, Maxwell found a connection between electricity and magnetism and something else — light. If electricity and magnetism are related, Maxwell thought, then there should exist a relationship between the constant Kc, which pertains to electric phenomena, and Km, the magnetic constant. The constant Ke had been determined from Coulomb’s experiment by measuring the force between two charges, F = Keqqu (101) ,2 The result of measurements is 1re = 9 x 109N mz/Cz. Can we repeat the same measuring process for the force between magnetic poles, F=Kmp—1§2f? (11.1) r The problem here is that somehow magnetic poles are created by moving electric charges. We simply can’t adopt a standard pole, lock it away in a vault in Paris, and compare all other poles to it. That was the whole point of Oersted’s experiment. There is, however, a way of defining a unit pole. For magnets, we found that the force exerted on a pole by a magnetic field is F = pOB. (11.2) For a moving charge, the force exerted by a magnetic field is F = qv x B. (11.5) 11.3 A PREDICTION FROM ELECTROMAGNETISM 225 The situation is complicated by the strange vector character of the relation between charges and magnets. Nevertheless, one thing is clear: a pole has the units of a charge multiplied by a velocity. If v is perpendicular to B, the force is qu, while the force on a pole is p08. Therefore we can define one pole as charge times speed: lpole =1Cm/s. If we measure the force between two unit poles, then we can determine the constant Km. In practice this is not done. Instead we measure the force between two wires carrying current which we discussed earlier. The idea, however, is the same: we measure a force to determine a constant, and the result is Km = 1.00 X 10‘7 N mz/(pole)2 = 1.00 X 10‘7 N sZ/CZ. Looking at the units of the two constants K6 and Km, we see Ke ~ N mZ/Cz, Km ~ N sZ/Cz, and so the ratio has units of K, 93 Km 52' In other words, the ratio is the square of a speed. Inserting the values of the constants we find that K 9><109 m m d= —e= ————=3x108—. Spec \/Km 1><10'7 s 5 That is the speed of light! When Maxwell discovered that the speed of light was buried in the forces between charges and magnets, he knew that he had in his hands a discovery of vast importance, or as he put it in 1865, “hold to be great guns.” By purely theoretical investigations he deduced that an electromagnetic disturbance — electric disturbances which create magnetic disturbances and vice versa — travels with the speed of light. By the time he finished his investigation, he realized that visible light was only a tiny portion of the kinds of dis- turbances that could occur in the electromagnetic field, and that all would travel at the same speed, 3 X 108 m/s. Maxwell’s theory is beautiful, perfect, and complete, and stands unchallenged to this day. Maxwell’s theory was not widely accepted by the more conservative physicists during his own lifetime. But ten years after Maxwell’s early death from stomach cancer at the age of 48, electromagnetic waves were actually produced and detected, their speed was measured, and the result was found to agree with the prediction of his theory. Almost immediately thereafter, the radio was invented; radio waves are only one of the many examples of Maxwell’s electromagnetic radiation. Radio, television, the total world of communications can be traced to Maxwell’s stunning discovery that the speed of light is buried in the forces between charges and magnets. 226 GRAVITY, ELECTRICITY. AND MAGNETISM Questions 7. A wire carries a current as shown and is immersed in a uniform magnetic field B. What is the direction of the force on the wire? B current _. 8. If the direction of current in a wire is reversed, does the magnetic field due to that current also reverse direction? 9. If two parallel wires carry currents in opposite directions, will they be attracted to each other or repelled? 11.4 A FINAL WORD Maxwell’s research into the fundamental nature of electricity and magnetism led directly to the invention of radio, television, and the whole revolution in long-range communication that has changed all our lives so much. But suppose Maxwell had started out with the specific goal of improving long-distance communication. Then he wouldn’t have fiddled around with the fundamental nature of electricity and magnetism. Instead he would have conducted experiments with giant megaphones and relay stations and the like. Or would he? History does hold some surprises, because the fact is Maxwell had been interested in long-distance communications since he was a young man. At that time, the 18503, the world was undertaking a great technological adventure, comparable to our own voyages to the moon. The first means of instantaneous long-distance communication was the telegraph, which was invented in 1838. By the 1850s an attempt was being made to lay a cable under the Atlantic Ocean. When completed, it would reduce the time for com- munication between New York and London from a period of weeks to a fraction of a second. This first attempt to lay a cable failed, however. The cable snapped somewhere in the middle of the Atlantic. The young James Clerk Maxwell marked that sad occasion by writing a poem about it, in a letter to a friend. The poem is called “Song of the Atlantic Telegraph Company: ” Under the sea, under the sea Something has surely gone wrong. What is the cause of it does not transpire But something has broken the telegraph wire Or else they’ve been pulling too strong. Under the sea, under the sea So many hundred miles long How could they spin out such durable stuff, Line all wiry, elastic and tough In the salt water so strong. 11.4 A FINAL WORD 227 Under the sea, under the sea There’ll be lots of cables ’ere long For they’ll spin a new cable and try it again And settle our bargains of cotton and grain With a line that will never go wrong. Figure 11.8 James Clerk Maxwell. (Courtesy of the Archives, California Institute of Technology.) ...
View Full Document

This note was uploaded on 03/30/2010 for the course PH 1a taught by Professor Goodstein during the Fall '07 term at Caltech.

Page1 / 11

Chapter11 - CHAPTER GRAVITY ELECTRICITY AND MAGNETISM in...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online