Maxwell's Equations of Electrodynamics an Explanation [2012]

SPIE PRESS

Maxwell’s Equations of Electrodynamics: An Explanation is a concise discussion of Maxwell's four equations of electrodynamics—the fundamental theory of electricity, magnetism, and light. It guides readers step-by-step through the vector calculus and development of each equation. Pictures and diagrams illustrate what the equations mean in basic terms. The book not only provides a fundamental description of our universe but also explains how these equations predict the fact that light is better described as "electromagnetic radiation."

P.O. Box 10 Bellingham, WA 98227-0010 ISBN: 9780819494528 SPIE Vol. No.: PM232

Bellingham, Washington USA

Library of Congress Cataloging-in-Publication Data Ball, David W. (David Warren), 1962Maxwell’s equations of electrodynamics : an explanation / David W. Ball. pages cm Includes bibliographical references and index. ISBN 978-0-8194-9452-8 1. Maxwell equations. 2. Electromagnetic theory. I. Title. QC670.B27 2012 530.1401–dc23 2012040779 Published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: +1 360.676.3290 Fax: +1 360.647.1445 Email: [email protected] Web: http://spie.org c 2012 Society of Photo-Optical Instrumentation Engineers Copyright (SPIE) All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher. The content of this book reflects the work and thoughts of the author(s). Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. Printed in the United States of America. First printing

Dedication This book is dedicated to the following cadets whom I had the honor of teaching while serving as Distinguished Visiting Faculty at the US Air Force Academy in Colorado Springs, Colorado, during the 2011–12 academic year: Jessica Abbott, Andrew Alderman, Bentley Alsup, Ryan Anderson, Austin Barnes, Daniel Barringer, Anthony Bizzaco, Erin Bleyl, Nicholas Boardman, Natasha Boozell, Matthew Bowersox, Patrick Boyle, Andrew Burns, Spencer Cavanagh, Kyle Cousino, Erin Crow, Michael Curran, Chad Demers, Nicholas Fitzgerald, Kyle Gartrell, James Gehring, Nicholas Gibson, Ahmed Groce, Kassie Gurnell, Deion Hardy, Trevor Haydel, Aaron Henrichs, Clayton Higginson, Anthony Hopf, Christopher Hu, Vania Hudson, Alexander Humphrey, Spencer Jacobson, Stephen Joiner, Fedor Kalinkin, Matthew Kelly, Ye Kim, Lauren Linscott, Patrick Lobo, Shaun Lovett, James Lydiard, Ryan Lynch, Aaron Macy, Dylan Mason, Ryan Mavity, Payden McBee, Blake Morgan, Andrew Munoz, Patrick Murphy, David Myers, Kathrina Orozco, Nathan Orrill, Anthony Paglialonga, Adam Pearson, Emerald Peoples, Esteban Perez, Charles Perkins, Hannah Peterson, Olivia Prosseda, Victoria Rathbone, Anthony Rosati, Sofia Schmidt, Craig Stan, James Stofel, Rachele Szall, Kevin Tanous, David Tyree, Joseph Uhle, Tatsuki Watts, Nathanael Webb, Max Wilkinson, Kamryn Williams, Samantha Wilson, Trevor Woodward, and Aaron Wurster. May fortune favor them as they serve their country.

Contents Preface .................................................................................................... ix Chapter 1

1.1 1.2 1.3 Chapter 2

2.1 2.2 2.3 2.4 2.5 Chapter 3

3.1 3.2 3.3 Chapter 4

4.1 4.2 4.3 4.4

History.............................................................................

1

History (Ancient) ............................................................................. 2 History (More Recent).................................................................... 7 Faraday ................................................................................................ 11 First Equation of Electrodynamics ............................ 17

Enter Stage Left: Maxwell ............................................................ A Calculus Primer............................................................................ More Advanced Stuff ...................................................................... A Better Way ..................................................................................... Maxwell’s First Equation ..............................................................

17 19 26 35 40

Second Equation of Electrodynamics ....................... 47

Introduction ........................................................................................ 47 Faraday’s Lines of Force ............................................................... 48 Maxwell’s Second Equation ......................................................... 50 Third Equation of Electrodynamics ........................... 55

Beginnings .......................................................................................... Work in an Electrostatic Field...................................................... Introducing the Curl ........................................................................ Faraday’s Law ................................................................................... vii

55 59 67 70

viii

Contents

Chapter 5

5.1 5.2 5.3 Afterword

A.1 A.2 A.3

Fourth Equation of Electrodynamics ........................ 75

Ampère’s Law ................................................................................... 75 Maxwell’s Displacement Current ............................................... 78 Conclusion .......................................................................................... 82 Whence Light? ............................................................... 83

Recap: The Four Equations .......................................................... 83 Whence Light? .................................................................................. 84 Concluding Remarks....................................................................... 87

Bibliography .......................................................................................... 89 Index ....................................................................................................... 91

Preface As the contributing editor of “The Baseline” column in Spectroscopy magazine, I get a lot of leeway from my editor regarding the topics I cover in the column. For example, I once did a column on clocks, only to end with the fact that atomic clocks, currently our most accurate, are based on spectroscopy. But most of my topics are more obviously related to the title of the publication. In late 2010 or so, I had an idea to do a column on Maxwell’s equations of electrodynamics, since our understanding of light is based on them. It did not take much research to realize that a discussion of Maxwell’s equations was more than a 2000-word column could handle—indeed, whole books are written on them! (Insert premonitional music here.) What I proceeded to do was write about them in seven sequential installments over an almost two-year series of issues of the magazine. I’ve seldom had so much fun with, or learned so much from, one of my ideas for a column. Not long into writing it (and after getting a better understanding of how long the series would be), I thought that the columns might be collected together, revised as needed, and published as a book. There is personal precedent for this: In the early 2000s, SPIE Press published a collection of my “Spectroscopy” columns in a book titled The Basics of Spectroscopy, which is still in print. So I contacted then-acquisitions-editor at SPIE Press, Tim Lamkins, with the idea of a book on Maxwell’s equations. He responded in less than two hours. . . with a contract. (Note to budding authors: that’s a good sign.) Writing took a little longer than expected, what with having to split columns and a year-long professional sojourn to Colorado, but here it is. I hope the readers enjoy it. If, by any chance, you can think of a better way to explain Maxwell’s equations, let me know—the hope is that this will be one of the premiere explanations of Maxwell’s equations available. Thanks to Tim Lamkins of SPIE Press for showing such faith in my idea, and for all the help in the process; also to his colleagues, ix

x

Preface

Dara Burrows and Scott McNeill, who did a great job of converting manuscript to book. Thanks to the editor of Spectroscopy, Laura Bush, for letting me venture on such a long series of columns on a single topic. My gratitude goes to the College of Sciences and Health Professions, Cleveland State University (CSU), for granting me a leave of absence so I could spend a year at the US Air Force Academy, where much of the first draft was written, as well as to the staff in the Department of Chemistry, CSU, for helping me manage certain unrelinquishable tasks while I was gone. Thanks to Bhimsen Shivamoggi and Yakov Soskind for reading over the manuscript and making some useful suggestions; any remaining errors are the responsibility of the author. Never-ending appreciation goes to my family—Gail, Stuart, and Casey—for supporting me in my professional activities. Finally, thanks to John Q. Buquoi of the US Air Force Academy, Department of Chemistry, for his enthusiastic support and encouragement throughout the writing process. David W. Ball Cleveland, Ohio November 2012

Chapter 1

History Maxwell’s equations for electromagnetism are the fundamental understanding of how electric fields, magnetic fields, and even light behave. There are various versions, depending on whether there is vacuum, a charge present, matter present, or the system is relativistic or quantum, or is written in terms of differential or integral calculus. Here, we will discuss a little bit of historical development as a prelude to the introduction of the laws themselves.

One of the pinnacles of classical science was the development of an understanding of electricity and magnetism, which saw its culmination in the announcement of a set of mathematical relationships by James Clerk Maxwell in the early 1860s. The impact of these relationships would be difficult to minimize if one were to try. They provided a theoretical description of light as an electromagnetic wave, implying a wave medium (the ether). This description inspired Michelson and Morley, whose failure inspired Einstein, who was also inspired by Planck and who (among others) ushered in quantum mechanics as a replacement for classical mechanics. Again, it’s difficult to minimize the impact of these equations, which in modern form are known as Maxwell’s equations. In this chapter, we will discuss the history of electricity and magnetism, going through early 19th-century discoveries that led to Maxwell’s work. This historical review will set the stage for the actual presentation of Maxwell’s equations. 1

2

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1.1 History (Ancient) Electricity has been experienced since the dawn of humanity, and from two rather disparate sources: an aquatic animal and lightning. Ancient Greeks, Romans, Arabs, and Egyptians recorded the effect that electric eels caused when touched, and who could ignore the fantastic light shows that accompany certain storms (Fig. 1.1)? Unfortunately, although the

Figure 1.1 One of the first pictures of a lightning strike, taken by William N. Jennings around 1882.

History

3

ancients could describe the events, they could not explain exactly what was happening. The ancient Greeks noticed that if a sample of amber (Greek elektron) was rubbed with animal fur, it would attract small, light objects such as feathers. Some, like Thales of Miletos, related this property to that of lodestone, a natural magnetic material that attracted small pieces of iron. (A relationship between electricity and magnetism would be re-awakened over 2000 years later.) Since the samples of this rock came from the nearby town of Magnesia (currently located in southern Thessaly, in central Greece), these stones ultimately were called magnets. Note that it appears that magnetism was also recognized by the same time as static electricity effects (Thales lived in the 7th and 6th century BCE [before common era]), so both phenomena were known, just not understood. Chinese fortune tellers were also utilizing lodestones as early as 100 BCE. For the most part, however, the properties of rubbed amber and this particular rock remained novelties. By 1100 CE (common era), the Chinese were using spoon-shaped pieces of lodestone as rudimentary compasses, a practice that spread quickly to the Arabs and to Europe. In 1269, Frenchman Petrus Peregrinus used the word “pole” for the first time to describe the ends of a lodestone that point in particular directions. Christopher Columbus apparently had a simple form of compass in his voyages to the New World, as did Vasco de Gama and Magellan, so Europeans had compasses by the 1500s—indeed, the great ocean voyages by the explorers would likely have been extremely difficult without a working compass (Fig. 1.2). In books published in 1550 and 1557, the Italian mathematician Gerolamo Cardano argued that the attractions caused by lodestones and the attractions of small objects to rubbed amber were caused by different phenomena, possibly the first time this was explicitly pointed out. (His proposed mechanisms, however, were typical of much 16th-century physical science: wrong.) In 1600, English physician and natural philosopher William Gilbert published De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure (On the Magnet and Magnetic Bodies, and on the Great Magnet the Earth) in which he became the first to make the distinction between attraction due to magnetism and attraction due to rubbed elektron. For example, Gilbert argued that elektron lost its attracting ability with heat, but lodestone did not. (Here is another example of a person being correct, but for the wrong reason.) He proposed that rubbing removed a substance he termed effluvium, and it was the return of the lost effluvium to the

4

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Figure 1.2 Compasses like the one drawn here were used by 16th century maritime explorers to sail around the world. These compasses were most likely the first applied use of magnetism.

object that caused attraction. Gilbert also introduced the Latinized word electricus as meaning “like amber in its ability to attract.” Note also what the title of Gilbert’s book implies—the Earth is a “great magnet,” an idea that only arguably originated with him, as the book was poorly referenced. The word electricity itself was first used in 1646 in a book by English author Thomas Browne. In the late 1600s and early 1700s, Englishman Stephen Gray carried out some experiments with electricity and was among the first to distinguish what we now call conductors and insulators. In the course of his experiments, Gray was apparently the first to logically suggest that the sparks he was generating were the same thing as lightning, but on a much smaller scale. Interestingly, announcements of Gray’s discoveries were stymied by none other than Isaac Newton, who was having a dispute with other scientists (with whom Gray was associated, so Gray was guilty by association) and, in his position of president of the Royal Society, impeded these scientists’ abilities to publish their work. (Curiously, despite his

History

5

advances in other areas, Newton himself did not make any major and lasting contributions to the understanding of electricity or magnetism. Gray’s work was found documented in letters sent to the Royal Society after Newton had died, so his work has been historically documented.) Inspired by Gray’s experiments, French chemist Charles du Fay also experimented and found that objects other than amber could be altered. However, du Fay noted that in some cases, some altered objects attract, while other altered objects repel. He concluded that there were two types of effluvium, which he termed “vitreous” (because rubbed glass exhibited one behavior) and “resinous” (because rubbed amber exhibited the other behavior). This was the first inkling that electricity comes as two kinds. French mathematician and scientist René Descartes weighed in with his Principia Philosophiae, published in 1644. His ideas were strictly mechanical: magnetic effects were caused by the passage of tiny particles emanating from the magnetic material and passing through the luminiferous ether. Unfortunately, Descartes proposed a variety of mechanisms for how magnets worked, rather than a single one, which made his ideas arguable and inconsistent. The Leyden jar (Fig. 1.3) was invented simultaneously in 1745 by Ewald von Kleist and Pieter van Musschenbrök. Its name derives from the University of Leyden (also spelled Leiden) in the Netherlands, where Musschenbrök worked. It was the first modern example of a condenser or capacitor. The Leyden jar allowed for a significant (to that date) amount of charge to be stored for long periods of time, letting researchers experiment with electricity and see its effects more clearly. Advances were quick, given the ready source of what we now know as static electricity. Among other discoveries was the demonstration by Alessandro Volta that materials could be electrified by induction in addition to direct contact. This, among other experiments, gave rise to the idea that electricity was some sort of fluid that could pass from one object to another, in a similar way that light or heat (“caloric” in those days) could transfer from one object to another. The fact that many conductors of heat were also good conductors of electricity seemed to support the notion that both effects were caused by a “fluid” of some sort. Charles du Fay’s idea of “vitreous” and “resinous” substances, mentioned above, echoed the ideas that were current at the time. Enter Benjamin Franklin, polymath. It would take (and has taken) whole books to chronicle Franklin’s contributions to a variety of topics, but here we will focus on his work on electricity. Based on his own experiments,

6

Chapter 1

Figure 1.3 Example of an early Leyden jar, which was composed of a glass jar coated with metal foil on the outside (labeled A) and the inside (B).

History

7

Franklin rejected du Fay’s “two fluid” idea of electricity and instead proposed a “one fluid” idea, arbitrarily defining one object “negative” if it lost that fluid and “positive” if it gained fluid. (We now recognize his idea as prescient but backwards.) As part of this idea, Franklin enunciated an early concept of the law of conservation of charge. In noting that pointed conductors lose electricity faster than blunt ones, Franklin invented the lightning rod in 1749. Although metal had been used in the past to adorn the tops of buildings, Franklin’s invention appears to the first time such a construction was used explicitly to draw a lightning strike and spare the building itself. [Curiously, there was some ecclesiastical resistance to the lightning rod, as lightning was deemed by many to be a sign of divine intervention, and the use of a lightning rod was considered by some to be an attempt to control a deity! The danger of such attitudes was illustrated in 1769 when the rodless Church of San Nazaro in Brescia, Italy (in Lombardy in the northern part of the country) was struck by lightning, igniting approximately 100 tons of gunpowder stored in the church. The resulting explosion was said to have killed 3000 people and destroyed one-sixth of the city. Ecclesiastical opposition weakened significantly after that.] In 1752, Franklin performed his apocryphal experiment with a kite, a key, and a thunderstorm. The word “apocryphal” is used intentionally: different sources have disputed accounts of what really happened. (Modern analyses suggest that performing the elementary-school portrayal of the experiment would definitely have killed Franklin.) What is generally not in dispute, though, is the result: lightning was just electricity, not some divine sign. As mundane as it seems to us today, Franklin’s demonstration that lightning was a natural phenomenon was likely as important as Copernicus’ heliocentric ideas were in demonstrating the lack of privileged, heaven-mandated position mankind has in the universe.

1.2 History (More Recent) Up to this point, most demonstrations of electric and magnetic effects were qualitative, not quantitative. This began to change in 1779 when French physicist Charles-Augustin de Coulomb published Théorie des Machines Simples (Theory of Simple Machines). To refine this work, through the early 1780s Coulomb constructed a very fine torsion balance with which he could measure the forces due to electrical charges. Ultimately, Coulomb proposed that electricity behaves as if it were tiny particles of matter, acting

8

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in a way similar to Newton’s law of gravity, which had been known since 1687. We now know this idea as Coulomb’s law. If the charge on one body is q1 , and the charge on another body is q2 , and the bodies are a distance r apart, then the force F between the bodies is given by the equation F=k

q1 q2 , r

(1.1)

where k is a constant. In 1785 Coulomb also proposed a similar expression for magnetism but, curiously, did not relate the two phenomena. As seminal and correct as this advance was, Coulomb still adhered to a “two fluid” idea of electricity—and magnetism—as it turned out. (Interestingly, about 100 years later, James Clerk Maxwell, about whom we will have more to say, published papers from reclusive scientist Henry Cavendish, demonstrating that Cavendish had performed similar electrical experiments at about the same time but never published them. If he had, we might be referring to this equation as Cavendish’s law.) Nonetheless, through the 1780s, Coulomb’s work established all of the rules of static (that is, nonmoving) electrical charge. The next advances had to do with moving charges, but it was a while before people realized that movement was involved. Luigi Galvani was an Italian doctor who studied animal anatomy at the University of Bologna. In 1771, while dissecting a frog as part of an experiment in static electricity, he noticed a small spark between the scalpel and the frog leg he was cutting into, and the immediate jerking of the leg itself. Over the next few decades, Galvani performed a host of experiments trying to pin down the conditions and reasons for this muscle action. The use of electricity from a generating electric machine or a Leyden jar caused a frog leg to jerk, as did contact with two different metals at once, like iron and copper. Galvani eventually proposed that the muscles in the leg were acting like a “fleshy Leyden jar” that contained some residual electrical fluid he called animal electricity. In the 1790s, he printed pamphlets and passed them around to his colleagues, describing his experiments and his interpretation. One of the recipients of Galvani’s pamphlets was Alessandro Volta, a professor at the University of Pavia, about 200 km northwest of Bologna. Volta was skeptical of Galvani’s work but successfully repeated the frogleg experiments. Volta took the work further, positioning metallic probes exclusively on nerve tissue and bypassing the muscle. He made the crucial observation that some sort of induced activity was only produced if two different metals were in contact with the animal tissue. From

History

9

these observations, Volta proposed that the metals were the source of the electrical action, not the animal tissue itself; the muscle was serving as the detector of electrical effects, not the source. In 1800, Volta produced a stack of two different metals soaked in brine that supplied a steady flow of electricity (Fig. 1.4). The construction became known as a voltaic pile, but we know it better as a battery. Instead of a sudden spark, which was how electricity was produced in the past from Leyden jars, here was a construction that provided a steady flow, or current, of electricity. The development of the voltaic pile made new experiments in electricity possible. Advances came swiftly with this new, easily constructed pile. Water was electrolyzed, not for the first time but for the first time systematically, by Nicholson and Carlisle in England. English chemist Humphrey Davy, already a well-known chemist for his discoveries, proposed that if electricity were truly caused by chemical reactions (as many chemists at the time had perhaps chauvinistically thought), then perhaps electricity can cause chemical reactions in return. He was correct, and in 1807 he produced elemental potassium and sodium electrochemically for the first time. The electrical nature of chemistry was first realized then, and its ramifications continue today. Davy also isolated elemental chlorine in 1810, following up with isolation of the elements magnesium, calcium, strontium, and barium, all for the first time and all by electricity. (Davy’s greatest discovery, though, was not an element; arguably, Davy’s greatest discovery was Michael Faraday. But more on Faraday later.) Much of the historical development so far has focused on electrical phenomena. Where has magnetism been? Actually, it’s been here all along, but not much new has been developed. This changed in 1820. Hans Christian Ørsted was a Danish physicist who was a disciple of the metaphysics of Emmanual Kant. Given the recently demonstrated connection between electricity and chemistry, Ørsted was certain that there were other fundamental physical connections in nature. In the course of a lecture in April 1820, Ørsted passed an electrical current from a voltaic pile through a wire that was placed parallel to a compass needle. The needle was deflected. Later experiments (especially by Ampère) demonstrated that with a strong enough current, a magnetized compass needle orients itself perpendicular to the direction of the current. If the current is reversed, the needle points in the opposite direction. Here was the first definitive demonstration that electricity and magnetism affected each other: electromagnetism.

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Figure 1.4 Diagram of Volta’s first pile that produced a steady supply of electricity. We know it now as a battery.

History

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Detailed follow-up studies by Ampère in the 1820s quantified the relationship a bit. Ampère demonstrated that the magnetic effect was circular about a current-carrying wire and, more importantly, that two wires with current moving in them attracted and repelled each other as if they were magnets. Magnets, Ampère argued, are nothing more than moving electricity. The development of electromagnets at this time freed scientists from finding rare lodestones and allowed them to study magnetic effects using only coiled wires as the source of magnetism. Around this time, Biot and Savart’s studies led to the law named after them, relating the intensity of the magnetic effect to the square of the distance from the electromagnet: a parallel to Coulomb’s law. In 1827, Georg Ohm announced what became known as Ohm’s law, a relationship between the voltage of a pile and the current passing through the system. As simple as Ohm’s law is (V = IR, in modern usage), it was hotly contested in his native Germany, probably for political or philosophical reasons. Ohm’s law became a cornerstone of electrical circuitry. It’s uncertain how the term “current” became applied to the movement of electricity. Doubtless, the term arose during the time that electricity was thought to be a fluid, but it has been difficult to find the initial usage of the term. In any case, the term was well established by the time Sir William Robert Grove wrote The Correlation of Physical Forces in 1846, which was a summary of what was known about the phenomenon of electricity to date.

1.3 Faraday Probably no single person set the stage more for a mathematical treatment of electricity and magnetism than Michael Faraday (Fig. 1.5). Inspired by Ørsted’s work, in 1821 Faraday constructed a simple device that allowed a wire with a current running through it to turn around a permanent magnet, and conversely a permanent magnet to turn around a wire that had a current running through it. Faraday had, in fact, constructed the first rudimentary motor. What Faraday’s motor demonstrated experimentally is that the forces of interaction between the current-containing wire and the magnet were circular in nature, not radial like gravity. By now it was clear that electrical current could generate magnetism. But what about the other way around: could magnetism generate electricity? An initial attempt to accomplish this was performed in 1824 by French scientist François Arago, who used a spinning copper disk to

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Chapter 1

Figure 1.5 Michael Faraday was not well versed in mathematics, but he was a first-rate experimentalist who made major advances in physics and chemistry.

deflect a magnetized needle. In 1825, Faraday tried to repeat Arago’s work, using an iron ring instead of a metal disk and wrapping coils of wire on either side, one coil attached to a battery and one to a galvanometer. In doing so, Faraday constructed the first transformer, but the galvanometer did not register the presence of a current. However, Faraday did note a slight jump of the galvanometer’s needle. In 1831, Faraday passed a magnet through a coil of wire attached to a galvanometer, and the galvanometer registered, but only if the magnet was moving. When the

History

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magnet was halted, even when it was inside the coil, the galvanometer registered zero. Faraday understood that it wasn’t the presence of a magnetic field that caused an electrical current, it was a change in the magnetic field that caused a current. It did not matter what moved; a moving magnet can induce current in a stationary wire, or a moving wire can pick up a current induced by moving it across a stationary magnet. Faraday also realized that the wire wasn’t necessary. Taking a page from Arago, Faraday constructed a generator of electricity using a copper disk that rotated through the poles of a permanent magnet (Fig. 1.6). Faraday used this to generate a continuous source of electricity, essentially converting mechanical motion to electrical motion. Faraday invented the dynamo, the basis of the electric industry even today. After dealing with some health issues during the 1830s, in the mid1840s Faraday was back at work. Twenty years earlier, Faraday had investigated the effect of magnetic fields on light but got nowhere. Now, with stronger electromagnets available, Faraday went back to that investigation on the advice of William Thomson, Lord Kelvin. This time, with better equipment, Faraday noticed that the plane of plane-polarized light rotated when a magnetic field was applied to a piece of flint glass with the light passing through it. The magnetic field had to be oriented along the direction of the light’s propagation. This effect, known now as the Faraday effect or Faraday rotation, convinced Faraday of several things. First, light

Figure 1.6 Diagram of Faraday’s original dynamo, which generated electricity from magnetism.

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and magnetism are related. Second, magnetic effects are universal and not just confined to permanent or electromagnets; after all, light is associated with all matter, so why not magnetism? In late 1845, Faraday coined the term “diamagnetism” to describe the behavior of materials that are not attracted and are actually slightly repelled by a magnetic field. (The amount of repulsion toward a magnetic field is typically significantly less than the magnetic attraction by materials, now known as paramagnetism. This is one reason that magnetic repulsion was not widely recognized until then.) Finally, this finding reinforced in Faraday the concept of magnetic fields and their importance. In physics, a field is nothing more than a physical property whose value depends on its position in three-dimensional space. The temperature of a sample of matter, for example, is a field, as is the pressure of a gas. Temperature and pressure are examples of scalar fields, fields that have magnitude but no direction; vector fields, like magnetic fields, have magnitude and direction. This was first demonstrated in the 13th century, when Petrus Peregrinus used small needles to map out the lines of force around a magnet and was able to place the needles in curved lines that terminated in the ends of the magnet. That was how he devised the concept of “pole” that was mentioned above. (See Fig. 1.7 for an

Figure 1.7 An early photo of iron filings on a piece of paper that is placed over a bar magnet, showing the lines of force that make up the magnetic field. The magnetic field itself is not visible; the iron filings are simply lining up along the lines of force that define the field.

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example of such a mapping.) An attempt to describe a magnetic field in terms of “magnetic charges” analogous to electrical charges was made by the French mathematician/physicist Simeon-Denis Poisson in 1824, and while successful in some aspects, it was based on the invalid concept of such “magnetic charges.” However, development of the Biot–Savart law and some work by Ampère demonstrated that the concept of a field was useful in describing magnetic effects. Faraday’s work on magnetism in the 1840s reinforced his idea that the magnetic field was the important quantity, not the material—whether natural or electronic—that produced the field of forces that could act on other objects and produce electricity. As crucial as Faraday’s advances were for understanding electricity and magnetism, Faraday himself was not a mathematical man. He had a relatively poor grasp of the theory that modern science requires to explain observed phenomena. Thus, while he made crucial experimental advancements in the understanding of electricity and magnetism, he made little contribution to the theoretical understanding of these phenomena. That chore awaited others. This was the state of electricity and magnetism in the mid-1850s. What remained to emerge was the presence of someone who did have the right knowledge to be able to express these experimental observations as mathematical models.

Chapter 2

First Equation of Electrodynamics ∇ · E =

ρ ε0

In this chapter, we introduce the first equation of electrodynamics. A note to the reader: This is going to get a bit mathematical. It can’t be helped. Models of the physical universe, like Newton’s second law F = ma, are based in math. So are Maxwell’s equations.

2.1 Enter Stage Left: Maxwell James Clerk Maxwell (Fig. 2.1) was born in 1831 in Edinburgh, Scotland. His unusual middle name derives from his uncle, who was the 6th Baronet Clerk of Penicuik (pronounced “penny-cook”), a town not far from Edinburgh. Clerk was, in fact, the original family name; Maxwell’s father, John Clerk, adopted the surname Maxwell after receiving a substantial inheritance from a family named Maxwell. By most accounts, James Clerk Maxwell (hereafter referred to as simply Maxwell) was an intelligent but relatively unaccomplished student. However, Maxwell began blossoming intellectually in his early teens, becoming interested in mathematics (especially geometry). He eventually attended the University of Edinburgh and, later, Cambridge University, where he graduated in 1854 with a degree in mathematics. He stayed on for a few years as a Fellow, then moved to Marischal College in Aberdeen. When Marischal merged with another college to form the University of Aberdeen in 1860, Maxwell was laid off (an action for which the University of Aberdeen should still be kicking themselves, but who can foretell the future?), and he found another position at King’s College London (later the University of London). He returned to Scotland in 1865, 17

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Figure 2.1

James Clerk Maxwell as a young man and an older man.

only to go back to Cambridge in 1871 as the first Cavendish Professor of Physics. He died of abdominal cancer in November 1879 at the relatively young age of 48; curiously, his mother died of the same ailment and at the same age, in 1839. Though he had a relatively short career, Maxwell was very productive. He made contributions to color theory and optics (indeed, the first photo in Fig. 2.1 shows Maxwell holding a color wheel of his own invention), and actually produced the first true color photograph as a composite of three images. The Maxwell–Boltzmann distribution is named partially after him, as he made major contributions to the development of the kinetic molecular theory of gases. He also made major contributions to thermodynamics, deriving the relations that are named after him and devising a thought experiment about entropy that was eventually called “Maxwell’s demon.” He demonstrated mathematically that the rings of Saturn could not be solid, but must instead be composed of relatively tiny (relative to Saturn, of course) particles—a hypothesis that was supported spectroscopically in the late 1800s but finally directly observed for the first time when the Pioneer 11 and Voyager 1 spacecrafts passed through the Saturnian system in the early 1980s (Fig. 2.2). Maxwell also made seminal contributions to the understanding of electricity and magnetism, concisely summarizing their behaviors with four mathematical expressions known as Maxwell’s equations of

First Equation of Electrodynamics

19

Figure 2.2 Maxwell proved mathematically that the rings of Saturn couldn’t be solid objects but were likely an agglomeration of smaller bodies. This image of a back-lit Saturn is a composite of several images taken by the Cassini spacecraft in 2006. Depending on the reproduction, you may be able to make out a tiny dot in the 10 o’clock position just inside the second outermost diffuse ring—that’s Earth. If you can’t see it, look for high-resolution pictures of “pale blue dot” on the Internet.

electromagnetism. He was strongly influenced by Faraday’s experimental work, believing that any theoretical description of a phenomenon must be grounded in experimental observations. Maxwell’s equations essentially summarize everything about classical electrodynamics, magnetism, and optics, and were only supplanted when relativity and quantum mechanics revised our understanding of the natural universe at certain of its limits. Far away from those limits, in the realm of classical physics, Maxwell’s equations still rule just as Newton’s equations of motion rule under normal conditions.

2.2 A Calculus Primer Maxwell’s laws are written in the language of calculus. Before we move forward with an explicit discussion of the first law, here we deviate to a review of calculus and its symbols. Calculus is the mathematical study of change. Its modern form was developed independently by Isaac Newton and German mathematician Gottfried Leibnitz in the late 1600s. Although Newton’s version was used heavily in his influential Principia Mathematica (in which Newton used calculus to express a number of fundamental laws of nature), it is

20

Chapter 2

Leibnitz’s notations that are commonly used today. An understanding of calculus is fundamental to most scientific and engineering disciplines. Consider a car moving at constant velocity. Its distance from an initial point (arbitrarily set as a position of 0) can be plotted as a graph of distance from zero versus time elapsed. Commonly, the elapsed time is called the independent variable and is plotted on the x axis of a graph (called the abscissa), while distance traveled from the initial position is plotted on the y axis of the graph (called the ordinate). Such a graph is plotted in Fig. 2.3. The slope of the line is a measure of how much the ordinate changes as the abscissa changes; that is, slope m is defined as m=

∆y . ∆x

(2.1)

For the straight line shown in Fig. 2.3, the slope is constant, so m has a single value for the entire plot. This concept gives rise to the general formula for any straight line in two dimensions, which is y = mx + b,

(2.2)

where y is the value of the ordinate, x is the value of the abscissa, m is the slope, and b is the y intercept, which is where the plot would intersect with the y axis. Figure 2.3 shows a plot that has a positive value of m. In a plot with a negative value of m, the slope would be going down, not up, moving

Figure 2.3 A plot of a straight line, which has a constant slope m, given by ∆y/∆x.


21

from left to right. A horizontal line has a value of 0 for m; a vertical line has a slope of infinity. Many lines are not straight. Rather, they are curves. Figure 2.4 gives an example of a plot that is curved. The slope of a curved line is more difficult to define than that of a straight line because the slope is changing. That is, the value of the slope depends on the point (x, y) where you are on the curve. The slope of a curve is the same as the slope of the straight line that is tangent to the curve at that point (x, y). Figure 2.4 shows the slopes at two different points. Because the slopes of the straight lines tangent to the curve at different points are different, the slopes of the curve itself at those two points are different. Calculus provides ways of determining the slope of a curve in any number of dimensions [Figure 2.4 is a two-dimensional plot, but we recognize that functions can be functions of more than one variable, so plots can have more dimensions (a.k.a. variables) than two]. We have already seen that the slope of a curve varies with position. That means that the slope of a curve is not a constant; rather, it is a function itself. We are not concerned about the methods of determining the functions for the slopes of curves here; that information can be found in a calculus text. Here, we are concerned with how they are represented. The word that calculus uses for the slope of a function is derivative. The derivative of a straight line is simply m, its constant slope. Recall that we mathematically defined the slope m above using ∆ symbols, where ∆ is the Greek capital letter delta. ∆ is used generally to represent change, as in ∆T

Figure 2.4 A plot of a curve showing (with the thinner lines) the different slopes at two different points. Calculus helps us determine the slopes of curved lines.

22

Chapter 2

(change in temperature) or ∆y (change in y coordinate). For straight lines and other simple changes, the change is definite; in other words, it has a specific value. In a curve, the change ∆y is different for any given ∆x because the slope of the curve is constantly changing. Thus, it is not proper to refer to a definite change because (to overuse a word) the definite change changes during the course of the change. What we need here is a thought experiment: Imagine that the change is infinitesimally small over both the x and y coordinates. This way, the actual change is confined to an infinitesimally small portion of the curve: a point, not a distance. The point involved is the point at which the straight line is tangent to the curve (Fig. 2.4). Rather than using ∆ to represent an infinitesimal change, calculus starts by using d. Rather than using m to represent the slope, calculus puts a prime on the dependent variable as a way to represent a slope (which, remember, is a function and not a constant). Thus, the definition of the slope y0 of a curve is y0 =

dy . dx

(2.3)

We hinted earlier that functions may depend on more than one variable. If that is the case, how do we define the slope? First, we define a partial derivative as the derivative of a multivariable function with respect to only one of its variables. We assume that the other variables are held constant. Instead of using d to indicate a partial derivative, we use a symbol based on the lowercase Greek delta δ known as Jacobi’s delta. It is also common to explicitly list the variables being held constant as subscripts to the derivative, although this can be omitted because it is understood that a partial derivative is a one-dimensional derivative. Thus, we have f x0

∂f = ∂x

!

y,z,...

=

∂f , ∂x

(2.4)

spoken as “the partial derivative of the function f (x, y, z, . . .) with respect to x.” Graphically, this corresponds to the slope of the multivariable function f in the x dimension, as shown in Fig. 2.5. The total differential of a function, df, is the sum of the partial derivatives in each dimension; that is, with respect to each variable


23

Figure 2.5 For a function of several variables, a partial derivative is a derivative in only one variable. The line represents the slope in the x direction.

individually. For a function of three variables, f (x, y, z), the total differential is written as df =

∂f ∂f ∂f dx + dy + dz, ∂x ∂y ∂z

(2.5)

where each partial derivative is the slope with respect to each individual variable, and dx, dy, and dz are the finite changes in the x, y, and z directions. The total differential has as many terms as the overall function has variables. If a function is based in three-dimensional space, as is commonly the case for physical observables, then there are three variables and so three terms in the total differential. When a function typically generates a single numerical value that is dependent on all of its variables, it is called a scalar function. An example of a scalar function might be F(x, y) = 2x − y2 .

(2.6)

According to this definition, F(4, 2) = 2 · 4 − 22 = 8 − 4 = 4. The final value of F(x, y), 4, is a scalar: it has magnitude but no direction. A vector function is a function that determines a vector, which is a quantity that has magnitude and direction. Vector functions can be easily expressed using unit vectors, which are vectors of length 1 along each dimension of the space involved. It is customary to use the

24

Chapter 2

representations i, j, and k to represent the unit vectors in the x, y, and z dimensions, respectively (Fig. 2.6). Vectors are typically represented in print as boldfaced letters. Any random vector can be expressed as, or decomposed into, a certain number of i vectors, j vectors, and k vectors as is demonstrated in Fig. 2.6. A vector function in two dimensions might be as simple as F = xi + yj,

(2.7)

as illustrated in Fig. 2.7 for a few discrete points. Although only a few discrete points are shown in the figure, understand that the vector function is continuous. That is, it has a value at every point in the graph. One of the functions of a vector that we will evaluate is called a dot product. The dot product between two vectors a and b is represented and

Figure 2.6 The definition of the unit vectors i, j, and k, and an example of how any vector can be expressed in terms of the number of each unit vector.


25

Figure 2.7 An example of a vector function F = xi + yj. Each point in two dimensions defines a vector. Although only 12 individual values are illustrated here, in reality, this vector function is a continuous, smooth function on both dimensions.

defined as a · b = |a||b| cos θ,

(2.8)

where |a| represents the magnitude (that is, length) of a, |b| is the magnitude of b, and cos θ is the cosine of the angle between the two vectors. The dot product is sometimes called the scalar product because the value is a scalar, not a vector. The dot product can be thought of physically as how much one vector contributes to the direction of the other vector, as shown in Fig. 2.8. A fundamental definition that uses the dot product is that for work w, which is defined in terms of the force vector F and the displacement vector of a moving object s, and the angle between these two vectors: w = F · s = |F||s| cos θ.

(2.9)

Thus, if the two vectors are parallel (θ = 0 deg, so cos θ = 1) the work is maximized, but if the two vectors are perpendicular to each other

26

Chapter 2

Figure 2.8 Graphical representation of the dot product of two vectors. The dot product gives the amount of one vector that contributes to the other vector. An equivalent graphical representation would have the b vector projected into the a vector. In both cases, the overall scalar results are the same.

(θ = 90 deg, so cos θ = 0), the object does not move, and no work is done (Fig. 2.9).

2.3 More Advanced Stuff Besides taking the derivative of a function, the other fundamental operation in calculus is integration, whose representation is called an integral, which is represented as Z

a

f (x) dx,

(2.10)

b

R where the symbol is called the integral sign and represents the integration operation, f (x) is called the integrand and is the function to be integrated, dx is the infinitesimal of the dimension of the function, and a and b are the limits between which the integral is numerically evaluated, if it is to be numerically evaluated. [If the integral sign looks like an elongated “s”, it should be numerically evaluated. Leibniz, one of the cofounders of calculus (with Newton), adopted it in 1675 to represent sum, since an integral is a limit of a sum.] A statement called the fundamental


27

Figure 2.9 Work is defined as a dot product of a force vector and a displacement vector. (a) If the two vectors are parallel, they reinforce, and work is performed. (b) If the two vectors are perpendicular, no work is performed.

theorem of calculus establishes that integration and differentiation are the opposites of each other, a concept that allows us to calculate the numerical value of an integral. For details of the fundamental theorem of calculus, consult a calculus text. For our purposes, all we need to know is that the two are related and calculable. The most simple geometric representation of an integral is that it represents the area under the curve given by f (x) between the limits a and b and bound by the x axis. Look, for example, at Fig. 2.10(a). It is a graph of the line y = x or, in more general terms, f (x) = x. What is the area under this function but above the x axis, shaded gray in Fig. 2.10(a)? Simple geometry indicates that the area is 1/2 units; the box defined by x = 1 and y = 1 is 1 unit (1 × 1), and the right triangle that is shaded gray is one-half of that total area, or 1/2 unit in area. Integration of the function f (x) = x gives us the same answer. The rules of integration will not be discussed here; it is assumed that the reader can perform simple integration: Z

0

1

f (x) dx =

Z

0

1

1 1 2 1 1 2 1 2 1 x dx = x = (1) − (0) = − 0 = . (2.11) 2 0 2 2 2 2

28

Chapter 2

Figure 2.10 The geometric interpretation of a simple integral is the area under a function and bounded on the bottom by the x axis (that is, y = 0). (a) For the function f (x) = x, the areas as calculated by geometry and integration are equal. (b) For the function f (x) = x2 , the approximation from geometry is not a good value for the area under the function. A series of rectangles can be used to approximate the area under the curve, but in the limit of an infinite number of infinitesimally narrow rectangles, the area is equal to the integral.

It is a bit messier if the function is more complicated. But, as first demonstrated by Reimann in the 1850s, the area can be calculated geometrically for any function in one variable (most easy to visualize, but in theory this can be extended to any number of dimensions) by using rectangles of progressively narrower widths, until the area becomes a limiting value as the number of rectangles goes to infinity and the width of each rectangle becomes infinitely narrow. This is one reason a good calculus course begins with a study of infinite sums and limits! But, I digress. For the function in Fig. 2.10(b), which is f (x) = x2 , the area under the curve, now poorly approximated by the shaded triangle, is calculated exactly with an integral: Z

0

1

f (x) dx =

Z

0

1

1 1 3 1 1 x dx = x = − 0 = . 3 0 3 3 2

(2.12)

As with differentiation, integration can also be extended to functions of more than one variable. The issue to understand is that when visualizing functions, the space you need to use has one more dimension than variables because the function needs to be plotted in its own dimension. Thus, a plot of a one-variable function requires two dimensions, one to represent the variable and one to represent the value of the function. Figures 2.3 and 2.4, thus, are two-dimensional plots. A two-variable function needs to be


29

Figure 2.11 A multivariable function f (x, y) with a line paralleling the y axis. The equation of the line is represented by P.

plotted or visualized in three dimensions, like Figs. 2.5 or 2.6. Looking at the two-variable function in Fig. 2.11, we see a line across the function’s values, with its projection in the (x, y) plane. The line on the surface is parallel to the y axis, so it is showing the trend of the function only as the variable x changes. If we were to integrate this multivariable function with respect only to (in this case) x, we would be evaluating the integral only along this line. Such an integral is called a line integral (also called a path integral). One interpretation of this integral would be that it is simply the part of the “volume” under the overall surface that is beneath the given line; that is, it is the area under the line. If the surface represented in Fig. 2.11 represents a field (either scalar or vector), then the line integral represents the total effect of that field along the given line. The formula for calculating the “total effect” might be unusual, but it makes sense if we start from the beginning. Consider a path whose position is defined by an equation P, which is a function of one or more variables. What is the distance of the path? One way of calculating the distance s is velocity v times time t, or s = v × t.

(2.13)

But velocity is the derivative of position P with respect to time, or dP/dt. Let us represent this derivative as P0 . Our equation now becomes s = P0 × t.

(2.14)

30

Chapter 2

This is for finite values of distance and time, and for that matter, for constant P0 . (Example: total distance at 2.0 m/s for 4.0 s = 2.0 m/s × 4.0 s = 8.0 m. In this example, P0 is 2.0 m/s and t is 4.0 s.) For infinitesimal values of distance and time, and for a path whose value may be a function of the variable of interest (in this case, time), the infinitesimal form is ds = P0 dt.

(2.15)

To find the total distance, we integrate between the limits of the initial position a and the final position b: s=

Z

a

P0 dt.

(2.16)

b

The point is, it’s not the path P that we need to determine the line integral; it’s the change in P, denoted as P0 . This seems counterintuitive at first, but hopefully the above example makes the point. It’s also a bit overkill when one remembers that derivatives and integrals are opposites of each other; the above analysis has us determine a derivative and then take the integral, undoing our original operation, to obtain the answer. One might have simply kept the original equation and determined the answer from there. We’ll address this issue shortly. One more point: it doesn’t need to be a change with respect to time. The derivative involved can be a change with respect to a spatial variable. This allows us to determine line integrals with respect to space as well as time. Suppose that the function for the path P is a vector. For example, consider a circle C in the (x, y) plane having radius r. Its vector function is C = r cos θi + r sin θj + 0k (see Fig. 2.12), which is a function of the variable θ, the angle from the positive x axis. What is the circumference of the circle?; that is, what is the path length as θ goes from 0 to 2π, the radian measure of the central angle of a circle? According to our formulation above, we need to determine the derivative of our function. But for a vector, if we want the total length of the path, we care only about the magnitude of the vector and not its direction. Thus, we’ll need to derive the change in the magnitude of the vector. We start by defining the magnitude: the magnitude |m| of a three- (or lesser-) magnitude vector is the Pythagorean combination of its components: |m| =

q

x2 + y2 + z2 .

(2.17)


31

Figure 2.12 How far is the path around the circle? A line integral can tell us, and this agrees with what basic geometry predicts (2πr).

For the derivative of the path/magnitude with respect to time, which is the velocity, we have |m0 | =

q

(x0 )2 + (y0 )2 + (z0 )2 .

(2.18)

For our circle, we have the magnitude as simply the i, j, and/or k terms of the vector. These individual terms are also functions of θ. We have d(r cos θi) = −r sin θi, dθ d(r sin θj) = r cos θj, y0 = dθ z0 = 0. x0 =

(2.19)

From this we have (x0 )2 = r2 sin2 θi2 , (y0 ) = r2 cos2 θj2 ,

(2.20)

32

Chapter 2

and we will ignore the z part, since it’s just zero. For the squares of the unit vectors, we have i2 = j2 = i · i = j · j = 1. Thus, we have s=

Z

a

0

P dt =

Z

2π

0

b

p

r2

2

sin θ +

r2

cos2

θ dθ.

(2.21)

We can factor out the r2 term from each term and then factor out the r2 term from the square root to obtain s=

Z

2π

0

p r sin2 θ + cos2 θdθ.

(2.22)

Since, from elementary trigonometry, sin2 θ + cos2 θ = 1, we have s=

Z

0

2π

Z √ (r 1) dθ =

0

2π

2π r dθ = r · θ = r(2π − 0) = 2πr. (2.23) 0

This seems like an awful lot of work to show what we all know, that the circumference of a circle is 2πr. But hopefully it will convince you of the propriety of this particular mathematical formulation. Back to “total effect.” For a line integral involving a field, there are two expressions we need to consider: the definition of the field F[x(q), y(q), z(q)] and the definition of the vector path p(q), where q represents the coordinate along the path. (Note that, at least initially, the field F is not necessarily a vector.) In that case, the total effect s of the field along the line is given by s=

Z

p

F[x(q), y(q), z(q)] · |p0 (q)| dq.

(2.24)

The integration is over the path p, which needs to be determined by the physical nature of the system of interest. Note that in the integrand, the two functions F and |p0 | are multiplied together. If F is a vector field over the vector path p(q), denoted F[p(q)], then the line integral is defined similarly: s=

Z

p

F[p(q)] · p0 (q) dq.

(2.25)


33

Here, we need to take the dot product of the F and p0 vectors. A line integral is an integral over one dimension that gives, effectively, the area under the function. We can perform a two-dimensional integral over the surface of a multidimensional function, as pictured in Fig. 2.13. That is, we want to evaluate the integral Z

g(x, y, z) dS ,

(2.26)

S

where g(x, y, z) is some scalar function on a surface S . Technically, this expression is a double integral over two variables. This integral is generally called a surface integral. The mathematical tactic for evaluating the surface integral is to project the functional value into the perpendicular plane, accounting for the variation of the function’s angle with respect to the projected plane (the region R in Fig. 2.13). The proper variation is the cosine function, which gives a relative contribution of 1 if the function and the plane are parallel [i.e., cos(0 deg) = 1] and a relative contribution of 0 if the function and the plane are perpendicular [i.e., cos(90 deg) = 0]. This automatically makes us think of a dot product. If the space S is being projected into the (x, y) plane, then the dot product will involve the unit vector in the z direction, or k. (If the space is projected into other planes, other unit vectors are involved, but the concept is the same.) If n(x, y, z) is the unit vector that

Figure 2.13 A surface S over which a function f (x, y) will be integrated. R represents the projection of the surface S in the (x, y) plane.

34

Chapter 2

defines the line perpendicular to the plane marked out by g(x, y, z) (called the normal vector), then the value of the surface integral is given by " R

g(x, y, z) dx dy, n(x, y, z) · k

(2.27)

where the denominator contains a dot product, and the integration is over the x and y limits of the region R in the (x, y) plane of Fig. 2.13. The dot product in the denominator is actually fairly easy to generalize. When that happens, the surface integral becomes " R

g(x, y, z) ·

s

∂f 1+ ∂x

!2

∂f + ∂y

!2

dx dy,

(2.28)

where f represents the function of the surface, and g represents the function you are integrating over. Typically, to make g a function of only two variables, you let z = f (x, y) and substitute the expression for z into the function g, if z appears in the function g. If, instead of a scalar function g we had a vector function F, the above equation becomes a bit more complicated. In particular, we are interested in the effect that is normal to the surface of the vector function. Since we previously defined n as the vector normal to the surface, we’ll use it again; we want the surface integral involving F · n, or Z

S

F · n dS .

(2.29)

For a vector function F = F x i + Fy j + Fz k and a surface given by the expression f (x, y) ≡ z, this surface integral is Z

S

# " " ∂f ∂f − Fy + Fz dx dy. F · n dS = −F x ∂x ∂y R

(2.30)

This is a bit of a mess! Is there a better, easier, more concise way of representing this?


35

2.4 A Better Way There is a better way to represent this last integral, but we need to back up a bit. What exactly is F · n? Actually, it’s just a dot product, but the integral Z

S

F · n dS

(2.31)

is called the flux of F. The word “flux” comes from the Latin word fluxus, meaning “flow.” For example, suppose you have some water flowing through the end of a tube, as represented in Fig. 2.14(a). If the tube is cut straight, the flow is easy to calculate from the velocity of the water (given by F) and the geometry of the tube. If you want to express the flow in terms of the mass of water flowing, you can use the density of the water as a conversion. But what if the tube is not cut straight, as shown in Fig. 2.14(b)? In this case, we need to use some morecomplicated geometry—vector geometry—to determine the flux. In fact, the flux is calculated using the last integral in the previous section. So, flux is calculable. Consider an ideal cubic surface with the sides parallel to the axes (as shown in Fig. 2.15) that surround the point (x, y, z). This cube represents our function F, and we want to determine the flux of F. Ideally, the flux at any point can be determined by shrinking the cube until it arrives at a single point. We will start by determining the flux for a finite-sized side, then take the limit of the flux as the size of the size goes to zero. If we look at the top surface, which is parallel to the (x, y) plane, it should be obvious

Figure 2.14 Flux is another word for amount of flow. (a) In a tube that is cut straight, the flux can be determined from simple geometry. (b) In a tube cut at an angle, some vector mathematics is needed to determine flux.

36

Figure 2.15 small?

Chapter 2

What is the surface integral of a cube as the cube gets infinitely

that the normal vector is the same as the k vector. For this surface by itself, the flux is then Z F · k dS . (2.32) S

If F is a vector function, its dot product with k eliminates the i and j parts (since i · k = j · k = 0; recall that the dot product a · b = |a||b| cos θ, where |a| represents the magnitude of vector a) and only the z component of F remains. Thus, the integral above is simply Z

S

Fz dS .

(2.33)

If we assume that the function Fz has some average value on that top surface, then the flux is simply that average value times the area of the


37

surface, which we will propose is equal to ∆x · ∆y. We need to note, though, that the top surface is not located at z (the center of the cube), but at z + ∆z/2. Thus, we have for the flux at the top surface: ! ∆z · ∆x∆y, top flux ≈ Fz x, y, z + 2

(2.34)

where the symbol ≈ means “approximately equal to.” It will become “equal to” when the surface area shrinks to zero. The flux of F on the bottom side is exactly the same except for two small changes. First, the normal vector is now – k, so there is a negative sign on the expression. Second, the bottom surface is lower than the center point, so the function is evaluated at z – ∆z/2. Thus, we have bottom flux ≈ −Fz x, y, z −

! ∆z · ∆x∆y. 2

(2.35)

The total flux through these two parallel planes is the sum of the two expressions: ! ! ∆z ∆z · ∆x∆y − Fz x, y, z − · ∆x∆y. flux ≈ Fz x, y, z + 2 2

(2.36)

We can factor the ∆x∆y out of both expressions. Now, if we multiply this expression by ∆z/∆z (which equals 1), we have "

! !# ∆z ∆z ∆z flux ≈ Fz x, y, z + − Fz x, y, z − · ∆x∆y . 2 2 ∆z

(2.37)

We rearrange as follows: flux ≈

h

Fz x, y, z +

∆z 2

− Fz x, y, z −

∆z 2

∆z

i

· ∆x∆y∆z,

(2.38)

and recognize that ∆x∆y∆z is the change in volume of the cube ∆V: flux ≈

h

Fz x, y, z +

∆z 2

− Fz x, y, z −

∆z

∆z 2

i

· ∆V.

(2.39)

38

Chapter 2

As the cube shrinks, ∆z approaches zero. In the limit of infinitesimal change in z, the first term in the product above is simply the definition of the derivative of Fz with respect to z! Of course, it’s a partial derivative because F depends on all three variables, but we can write the flux more simply as flux =

∂ Fz · ∆V. ∂z

(2.40)

A similar analysis can be performed for the two sets of parallel planes; only the dimension labels will change. We ultimately obtain ∂ Fy ∂ Fx ∂ Fz ∆V + ∆V + ∆V ∂x ∂y ∂z ! ∂ F x ∂ Fy ∂ Fz ∆V. + + = ∂x ∂y ∂z

total flux =

(2.41)

(Of course, as ∆x, ∆y, and ∆z go to zero, so does ∆V, but this does not affect our end result.) The expression in the parentheses above is so useful that it is defined as the divergence of the vector function F: divergence of F ≡

∂ F x ∂ Fy ∂ Fz + + (where F = F x i + Fy j + Fz k). (2.42) ∂x ∂y ∂z

Because divergence of a function is defined at a point, and the flux (two equations above) is defined in terms of a finite volume, we can also define the divergence as the limit as volume goes to zero of the flux density (defined as flux divided by volume): 1 ∂ F x ∂ Fy ∂ Fz + + = lim (total flux) ∆V→0 ∆V ∂x ∂y ∂z Z 1 = lim F · ndS . (2.43) ∆V→0 ∆V S

divergence of F =

There are two abbreviations to indicate the divergence of a vector function. One is to simply use the abbreviation “div” to represent


39

divergence: div F =

∂ F x ∂ Fy ∂ Fz + + . ∂x ∂y ∂z

(2.44)

The other way to represent the divergence is with a special function. The function ∇ (called “del”) is defined as ∇=i

∂ ∂ ∂ +j +k . ∂x ∂y ∂z

(2.45)

If we were to take the dot product between ∇ and F, we would obtain the following result: ! ∂ ∂ ∂ · F x i + Fy j + Fz k ∇·F= i +j +k ∂x ∂y ∂z ∂ F x ∂ Fy ∂ Fz = + + , ∂x ∂y ∂z

(2.46)

which is the divergence! Note that, although we expect to obtain nine terms in the dot product above, cross terms between the unit vectors (such as i · k or k · j) all equal zero and cancel out, while like terms (that is, j · j) all equal 1 because the angle between a vector and itself is zero and cos 0 = 1. As such, our nine-term expansion collapses to only three nonzero terms. Alternately, one can think of the dot product in terms of its other definition, a · b = Σai bi = a1 b1 + a2 b2 + a3 b3 ,

(2.47)

where a1 , a2 , etc., are the scalar magnitudes in the x, y, etc., directions. So, the divergence of a vector function F is indicated by divergence of F = ∇ · F.

(2.48)

What does the divergence of a function mean? First, note that the divergence is a scalar, not a vector, field. No unit vectors remain in the expression for the divergence. This is not to imply that the divergence is a constant; it may in fact be a mathematical expression whose value varies

40

Chapter 2

in space. For example, for the field F = x3 i,

(2.49)

∇ · F = 3x2 ,

(2.50)

the divergence is

which is a scalar function. Thus, the divergence changes with position. Divergence is an indication of how quickly a vector field spreads out at any given point; that is, how fast it diverges. Consider the vector field F = xi + yj,

(2.51)

which we originally showed in Fig. 2.7 and are reshowing in Fig. 2.16. It has a constant divergence of 2 (easily verified), indicating a constant “spreading out” over the plane. However, for the field F = x2 i,

(2.52)

whose divergence is 2x, the vectors grow farther and farther apart as x increases (see Fig. 2.17).

2.5 Maxwell’s First Equation If two electric charges were placed in space near each other, as shown in Fig. 2.18, there would be a force of attraction between the two charges; the charge on the left would exert a force on the charge on the right, and vice versa. That experimental fact is modeled mathematically by Coulomb’s law, which in vector form is F=

q1 q2 r, r2

(2.53)

where q1 and q2 are the magnitudes of the charges (in elementary units, where the elementary unit is equal to the charge on the electron), and r is the scalar distance between the two charges. The unit vector r represents the line between the two charges q1 and q2 . The modern version of Coulomb’s law includes a conversion factor between charge units


41

Figure 2.16 The divergence of the vector field F = xi + yj is 2, indicating a constant divergence and, thus, a constant spreading out, of the field at any point in the (x, y) plane.

Figure 2.17 A nonconstant divergence is illustrated by this one-dimensional field F = x2 i whose divergence is equal to 2x. The arrowheads, which give the divergence and not the vector field itself, represent lengths of the vector field at values of x = 1, 2, 3, 4, etc. The greater the value of x, the farther apart the vectors become; that is, the greater the divergence.

42

Chapter 2

Figure 2.18 It is an experimental fact that charges exert forces on each other. That fact is modeled by Coulomb’s law.

(coulombs, C) and force units (newtons, N), and is written as F=

q1 q2 r, 4πε0 r2

(2.54)

where ε0 is called the permittivity of free space and has an approximate value of 8.854 . . . × 10−12 C2 /N · m2 . How does a charge cause a force to be felt by another charge? Michael Faraday suggested that a charge has an effect in the surrounding space called an electric field, a vector field, labeled E. The electric field is defined as the Coulombic force felt by another charge divided by the magnitude of the original charge, which we will choose to be q2 : E=

q1 F = r, q2 4πε0 r2

(2.55)

where in the second expression we have substituted the expression for F. Note that E is a vector field (as indicated by the bold-faced letter) and is dependent on the distance from the original charge. E, too, has a unit vector that is defined as the line between the two charges involved, but, in this case, the second charge has yet to be positioned, so, in general, E can be thought of as a spherical field about the charge q1 . The unit for an electric field is newton per coulomb, or N/C. Since E is a field, we can pretend it has flux; that is, something is “flowing” through any surface that encloses the original charge. What is flowing? It doesn’t matter; all that matters is that we can define the flux mathematically. In fact, we can use the definition of flux given earlier. The electric flux Φ is given by Z Φ= E · n dS , (2.56) S

which is perfectly analogous to our previous definition of flux.


43

Let us consider a spherical surface around our original charge that has some constant radius r. The normal unit vector n is simply r, the radius unit vector, since the radius unit vector is perpendicular to the spherical surface at any of its points (Fig. 2.19). Since we know the definition of E from Coulomb’s law, we can substitute into the expression for electric flux: Φ=

Z

S

q1 r · r dS . 4πε0 r2

(2.57)

Figure 2.19 A charge in the center of a spherical shell with radius r has a normal unit vector equal to r in the radial direction and with unit length at any point on the surface of the sphere.

44

Chapter 2

The dot product r · r is simply 1, so this becomes Φ=

Z

S

q1 dS . 4πε0 r2

(2.58)

If the charge q1 is constant, 4 is constant, π is constant, the radius r is constant, and the permittivity of free space is constant, these can all be removed from the integral to obtain q1 Φ= 4πε0 r2

Z

dS .

(2.59)

S

What is this integral? Well, we defined our system as a sphere, so the surface integral above is the surface area of a sphere. The surface area of a sphere is known: 4πr2 . Thus, we have Φ=

q1 · 4πr2 . 4πε0 r2

(2.60)

The 4, the π, and the r2 terms cancel. What remains is Φ=

q1 . ε0

(2.61)

Recall, however, that we previously defined the divergence of a vector function as 1 ∂ F x ∂ Fy ∂ Fz + + = lim (total flux) ∆V→0 ∆V ∂x ∂y ∂z Z 1 F · ndS . = lim ∆V→0 ∆V S

div F =

(2.62)

Note that the integral in the definition has exactly the same form as the electric field flux Φ. Therefore, in terms of the divergence, we have for E: 1 div E = lim ∆V→0 ∆V

Z

S

1 1 q1 Φ = lim , (2.63) ∆V→0 ∆V ∆V→0 ∆V ε0

E · ndS = lim


45

where we have made the appropriate substitutions to obtain the final expression. We rewrite this last expression as div E =

lim q1 ∆V→0 ∆V ε0

.

(2.64)

The expression q1 /∆V is simply the charge density at a point, which we will define as ρ. This last expression becomes simply div E =

ρ . ε0

(2.65)

This equation is Maxwell’s first equation. It is also written as ∇·E=

ρ . ε0

(2.66)

Maxwell’s first equation is also called Gauss’ law, after Carl Friedrich Gauss, the German polymath who first determined it but did not publish it. [It was finally published in 1867 (after Gauss’ death) by his colleague William Weber; Gauss had a habit of not publishing much of his work, and his many contributions to science and mathematics were realized only posthumously.] What does this first equation mean? It means that a charge puts out a field whose divergence is constant and depends on the charge density and a universal constant. In words, it says that the electric flux (the left side of the equation) is proportional to the charge inside the closed surface. This may not seem like much of a statement, but then, we’re only getting started.

Chapter 3

Second Equation of Electrodynamics ∇ · B = 0 Maxwell’s first equation dealt only with electric fields, saying nothing about magnetism. That changes with the introduction of Maxwell’s second equation. A review of the mathematical development of the divergence, covered in detail in Chapter 2, may be helpful.

3.1 Introduction A magnet is any object that produces a magnetic field. That’s a rather circular definition (and saying such is a bit of a pun, when you understand Maxwell’s equations!), but it is a functional one; a magnet is most simply defined by how it functions. As mentioned in previous chapters, technically speaking, all matter is affected by magnets. It’s just that some objects are affected more than others, and we tend to define “magnetism” in terms of the more obvious behavior. An object is magnetic if it attracts certain metals such as iron, nickel, or cobalt, and if it attracts and repels (depending on its orientation) other magnets. The earliest magnets occurred naturally and were called lodestones, a name that apparently comes from the Middle English “leading stone,” suggesting an early recognition of the rock’s ability to point in a certain direction when suspended freely. Lodestone, by the way, is simply a magnetic form of magnetite, an ore whose name comes from the Magnesia region of Greece, which is itself a part of Thessaly in central eastern Greece bordering the Aegean Sea. Magnetite’s chemical formula is Fe3 O4 , and the ore is actually a mixed FeO-Fe2 O3 mineral. Magnetite itself is not uncommon, although the permanently magnetized 47

48

Chapter 3

form is, and how it becomes permanently magnetized is still an open question. (The chemists among us also recognize Magnesia as giving its name to the element magnesium. Ironically, the magnetic properties of Mg are about 1/5000th that of Fe.) Magnets work by setting up a magnetic field. What actually is a magnetic field? To be honest, it may be difficult to put into words, but effects of a magnet can be measured all around the object. It turns out that these effects exert forces that have magnitude and direction; that is, the magnetic field is a vector field. These forces are most easily demonstrated by objects that have either a magnetic field themselves or are moving and have an electrical charge on them, as the exerted force accelerates (changes the magnitude and/or direction of the velocity of) the charge. The magnetic field of a magnet is represented as B, and again is a vector field. (The symbol H is also used to represent a magnetic field, although in some circumstances there are some subtle differences between the definitions of the B field and the of the H field. Here we will use B.)

3.2 Faraday’s Lines of Force When Michael Faraday (see Chapter 1) was investigating magnets starting in the early 1830s, he invented a description that was used to visualize magnets’ actions: lines of force. There is some disagreement as to whether Faraday thought of these lines as being caused by the emanation of discrete particles or not, but no matter. The lines of force are what is visualized when fine iron filings are sprinkled over a sheet of paper that is placed over a bar magnet, as shown in Fig. 3.1. The filings show some distinct “lines” about which the iron pieces collect, although this is more a physical effect than it is a representation of a magnetic field. There are several features that can be noted from the positions of the iron filings in Fig. 3.1. First, the field seems to emanate from two points in the figure, where the iron filings are most concentrated. These points represent poles of the magnet. Second, the field lines exist not only between the poles but arc above and below the poles in the plane of the figure. If this figure extended to infinity in any direction, you would still see evidence (albeit less and less as you proceed farther away from the magnet) of the magnetic field. Third, the strength of the field is indicated by the density of lines in any given space; the field is stronger near the poles and directly between the poles, and the field is weaker farther away from the poles. Finally, we note that the magnetic field is three-dimensional. Although most of the figure shows iron filings

Second Equation of Electrodynamics

49

Figure 3.1 Photographic representation of magnetic lines of force. Here, a magnetic stir bar was placed under a sheet of paper, and fine iron filings were carefully sprinkled onto the paper. While the concept of “lines of force” is a useful one, magnetic fields are continuous and are not broken down into discrete lines as is pictured here. (Photo by author, with assistance from Dr. Royce W. Beal, Mr. Randy G. Ramsden, and Dr. James Rohrbough of the US Air Force Academy Department of Chemistry.)

on a flat plane, around the two poles the iron filings are definitely out of the plane of the figure, pointing up. (The force of gravity is keeping the filings from piling too high, but the visual effect is obvious.) For the sake of convention, the lines are thought of as “coming out” of the north pole of a magnet and “going into” the south pole of the magnet, although in Fig. 3.1 the poles are not labeled. Faraday was able to use the concept of lines of force to explain attraction and repulsion by two different magnets. He argued that when the lines of force from opposite poles of two magnets interact, they join together in such a way as to try to force the poles together, accounting for attraction of opposites [Fig. 3.2(a)]. However, if lines of force from similar poles of two magnets interact, they interfere with each other in such a way as to repel [Fig. 3.2(b)]. Thus, the lines of force were useful constructs to describe the known behavior of magnets. Faraday could also use the lines-of-force concept to explain why some materials were attracted by magnets (“paramagnetic” materials, or in their extreme, “ferromagnetic” materials) or repelled by magnets (“diamagnetic” materials). Figure 3.3 illustrates that materials attracted by a magnetic field concentrate the lines of force inside the material, while

50

Chapter 3

Figure 3.2 Faraday used the concept of magnetic lines of force to describe attraction and repulsion. (a) When opposite poles of two magnets interact, the lines of force combine to force the two poles together, causing attraction. (b) When like poles of two magnets interact, the lines of force resist each other, causing repulsion.

materials repelled by a magnetic field exclude the lines of force from the material. As useful as these descriptions were, Faraday was not a theorist. He was a very phenomenological scientist who mastered experiments but had little mathematical training with which to model his results. Others did that— others in Germany and France—but none more so than in his own Great Britain.

3.3 Maxwell’s Second Equation Two British scientists contributed to a better theoretical understanding of magnetism: William Thomson (also known as Lord Kelvin) and James Clerk Maxwell. However, it was Maxwell who did the more complete job. Maxwell was apparently impressed with the concept of Faraday’s lines of force. In fact, the series of four papers in which he described what was to become Maxwell’s equations were titled On Physical Lines of Force. Maxwell was a very geometry-oriented person; he felt that the behavior of the natural universe could, at the very least, be represented by a drawing or picture. So, consider the lines of force pictured in Fig. 3.1. Figure 3.4 shows one ideal line of force for a bar magnet in two dimensions. Recall that this is a thought experiment—a magnetic field is not composed of individual lines; rather, it is a continuous vector field. And it is a vector field, so the field lines have some direction as well as magnitude. By convention, the magnetic field vectors are thought of as emerging from the north pole


51

Figure 3.3 Faraday used the lines-of-force concept to explain how objects behave in a magnetic field. (a) Most substances (such as glass, water, or elemental bismuth) actually slightly repel a magnetic field; Faraday explained that they excluded the magnetic lines of force from themselves. (b) Some substances (such as aluminum) are slightly attracted to a magnetic field; Faraday suggested that they include magnetic lines of force within themselves. (c) Some substances (such as iron) are very strongly attracted to a magnetic field, including (according to Faraday) a large density of lines of force. Such materials can be turned into magnets themselves under the proper conditions.

52

Chapter 3

Figure 3.4 Hypothetical line of force about a magnet. Compare this to the photo in Fig. 2.19.

of the magnet and entering the south pole of the magnet. This vector scheme allows us to apply the right-hand rule when describing the effects of the magnetic field on other objects, such as charged particles and other magnetic phenomena. Consider any box around the line of force. In Fig. 3.4, the box is shown by the dotted rectangle. What is the net change of the magnetic field through the box? By focusing on the single line of force drawn, we can conclude that the net change is zero: there is one line entering the box on its left side, and one line leaving the box on its right side. This is easily seen in Fig. 3.4 for one line of force and in two dimensions, but now let’s expand our mental picture to include all lines of force and all three dimensions; there will always be the same number of lines of force going into any arbitrary volume about the magnet as there are coming out. There is no net change in the magnetic field in any given volume. This concept holds no matter how strong the magnetic field and no matter what size the volume considered. How do we express this mathematically? Why, using vector calculus, of course. In the previous discussion of Maxwell’s first law, we introduced the divergence of a vector function F as divergence of F ≡

∂ F x ∂ Fy ∂ Fz + + , ∂x ∂y ∂z

(3.1)

where F = F x i + Fy j + Fz k. Note what the divergence really is: it is the change in the x-dimensional value of the function F across the x dimension plus the change in the y-dimensional value of the function F across the y dimension plus the change in the z-dimensional value of the function F across the z dimension. But we have already argued form our lines-of-force illustration that the magnetic field coming in a volume equals the magnetic


53

field going out of the volume, so that there is no net change. Thus, using B to represent our magnetic field, ∂ Bx ∂ By ∂ Bz = = = 0. ∂x ∂y ∂z

(3.2)

That means that the divergence of B can be written as div B =

∂ Bx ∂ By ∂ Bz = = =0 ∂x ∂y ∂z

(3.3)

or simply div B = 0.

(3.4)

This is Maxwell’s second equation of electromagnetism. It is sometimes called Gauss’ law for magnetism. Since we can also write the divergence as the dot product of the del operator (∇) with the vector field, Maxwell’s second equation is ∇ · B = 0.

(3.5)

What does Maxwell’s second equation mean? Because the divergence is an indicator of the presence of a source (a generator) or a sink (a destroyer) of a vector field, it implies that a magnetic field has no separate generator or destroyer points in any definable volume. Contrast this with an electric field. Electric fields are generated by two different particles, positively charged particles and negatively charged particles. By convention, electric fields begin at positive charges and end at negative charges. Since electric fields have explicit generators (positively charged particles) and destroyers (negatively charged particles), the divergence of an electric field is nonzero. Indeed, by Maxwell’s first equation, the divergence of an electric field E is ∇·E=

ρ , ε0

(3.6)

which is zero only if the charge density ρ is zero; if the charge density is not zero, then the divergence of the electric field is also not zero. Further,

54

Chapter 3

the divergence can be positive or negative depending on whether the charge density is a source or a sink. For magnetic fields, however, the divergence is exactly zero, which implies that there is no discrete source (“positive” magnetic particle) or sink (“negative” magnetic particle). One implication of this is that magnetic field sources are always dipoles; there is no such thing as a magnetic “monopole.” This mirrors our experience when we break a magnet in half, as shown in Fig. 3.5. We do not end up with two separated poles of the original magnet. Rather, we have two separate magnets, complete with north and south poles. In the next chapter, we will continue our discussion of Maxwell’s equations and see how E and B are related to each other. The first two of Maxwell’s equations deal with E and B separately; we will see, however, that they are anything but separate.

Figure 3.5 If you break a magnet, you don’t get two separate magnetic poles (“monopoles,” top), but instead you get two magnets, each having north and south poles (bottom). This is consistent with Maxwell’s second law of electromagnetism.

Chapter 4

Third Equation of Electrodynamics ∇ × E = − ∂t∂ B Maxwell’s equations are expressed in the language of vector calculus, so a significant portion of the previous chapters has been devoted to explaining vector calculus, not Maxwell’s equations. For better or worse, that’s par for the course, and it’s going to happen again in this chapter. The old adage “the truth will set you free” might be better stated, for our purposes, as “the math will set you free.” And that’s the truth.

4.1 Beginnings In mid-1820, Danish physicist Hans Christian Ørsted discovered that a current in a wire can affect the magnetic needle of a compass. His experiments were quickly confirmed by François Arago and, more exhaustively, by André Marie Ampère. Ampère’s work demonstrated that the effects generated by the current, which defined a so-called “magnetic field” (labeled B in Fig. 4.1), were centered on the wire, were perpendicular to the wire, and were circularly symmetric about the wire. By convention, the vector component of the field had a direction given by the right-hand rule: if the thumb of the right hand were pointing in the direction of the current, the curve of the fingers on the right hand gives the direction of the vector field. Other careful experiments by Jean-Baptiste Biot and Félix Savart established that the strength of the magnetic field was directly related to the current I in the wire and inversely related to the radial distance from 55

56

Figure 4.1 through it.

Chapter 4

The “shape” of a magnetic field about a wire with a current running

the wire r. Thus, we have 1 B∝ , r

(4.1)

where “∝” means proportional to. To make a proportionality into an equality, we introduce a proportionality constant. However, because of the axial symmetry of the field, we typically include a factor of 2π (the radian angle of a circle) in the denominator of any arbitrary proportionality constant. As such, our new equation is B=

µ I , 2π r

(4.2)

where the constant µ is our proportionality constant and is called the permeability of the medium that the magnetic field is in. In a vacuum, the permeability is labeled µ0 and, because of how the units of B and I are defined, is equal to exactly 4π × 10−7 T · m/A (tesla × meters per ampere). Not long after the initial demonstrations, Ampère had another idea: curve the wire into a circle. Sure enough, inside the circle, the magnetic field increased in strength as the concentric circles of the magnetic field overlapped on the inside (Fig. 4.2). Biot and Savart found that the magnetic field B created by the loop was related to the current I in the loop and the radius of the loop R by the following: B=

µI . 2R

(4.3)

Multiple loops can be joined in sequence to increase B, and from 1824 to 1825 English inventor William Sturgeon wrapped loops around a piece

58

Chapter 4

Figure 4.3 (a) A magnet inside a coil of wire does not generate a current. (b) A magnet moving through a coil of wire does generate a current.

Actually, this is not far from the truth (it would have become another of Maxwell’s equations if it were the truth), but the more complete truth is expressed in a different, more applicable form.

Third Equation of Electrodynamics

59

4.2 Work in an Electrostatic Field The simple physical definition of work w is force F times displacement ∆s: w = F × ∆s.

(4.5)

This is fine for straight-line motion, but what if the motion occurs on a curve (Fig. 4.4 in two dimensions) with perhaps a varying force? Then calculating the work is not as straightforward, especially since force and displacement are both vectors. However, it can be easily justified that the work is the integral, from initial point to final point, of the dot product force vector F with the unit vector tangent to the curve, which we will label t: Z final point w= F · t ds. (4.6) initial point

Because of the dot product, only the force component in the direction of the tangent to the curve contributes to the work. This makes sense if you

Figure 4.4 If the force F is not parallel to the displacement s (shown here as variable, but F can be vary, too), then the work performed is not as straightforward to calculate.

60

Chapter 4

remember the definition of the dot product, a · b = |a||b| cos θ; if the force is parallel to the displacement, work is maximized [because the cosine of the angle between the two vectors is cos(0 deg) = 1], while if the force is perpendicular to the displacement, work is zero [because now the cosine is cos(90 deg) = 0]. Now consider two random points inside of an electrostatic field E (Fig. 4.5). Keep in mind that we have defined E as static; that is, not moving or changing. Imagine that an electric particle with charge q were to travel from P1 to P2 and back again along the paths s1 and s2 , as indicated. Since the force F on the particle is given by qE (from Coulomb’s law), we have for an imagined two-step process w=

Z

P2 P1

qE · t ds1 +

Z

P1

qE · t ds2 .

(4.7)

P2

Each integral covers one pathway, but eventually you end up where you started.

Figure 4.5 Two arbitrary points in an electric field. The relative strength of the field is indicated by the darkness of the color.


61

This last statement is a crucial one: eventually you end up where you started. According to Coulomb’s law, the only variable that the force or electric field between the two particles depends on is the radial distance r. This further implies that the work w depends only on the radial distance between any two points in the electric field. Even further still, this implies that if you start and end at the same point, as in our example, the overall work is zero because you are starting and stopping at the same radial point r. Thus, the equation above must be equal to zero: Z

P2

qE · t ds1 +

P1

Z

P1

qE · t ds2 = 0.

(4.8)

P2

Since we are starting and stopping at the same point, the combined paths s1 and s2 are termed a closed path. Notice, too, that, other than being closed, we have not imposed any requirement on the overall path itself; it can be any path. We say that this integral, which must equal zero, is path independent. H The symbol for an integral over a closed path is . Thus, we have I

qE · t ds = 0.

(4.9)

We can divide by the constant q to obtain something slightly more simple: I

E · t ds = 0.

(4.10)

This is one characteristic of an electrostatic field: the path-independent integral over any closed path in an electrostatic field is exactly zero. The key word in the above statement is “any.” You can select any random closed path in an electric field, and the integral of E · t over that path is exactly zero. How can we generalize this for any closed path? Let us start with a closed path in one plane, as shown by Fig. 4.6. The complete closed path has four parts, labeled T, B, L, and R for top, bottom, left, and right sides, respectively, and it surrounds a point at some given coordinates (x, y, z). T and B are parallel to the x axis, while L and R are parallel to the y axis. The dimensions of the path are ∆x by ∆y (these will be useful shortly). Right now the area enclosed by the path is arbitrary, but later on we will want to shrink the closed path down so that the area goes

62

Chapter 4

Figure 4.6

A closed, two-dimensional path around a point.

to zero. Finally, the path is immersed in a three-dimensional field F whose components are F x , Fy , and Fz . That is, F = iF x + jFy + kFz

(4.11)

in terms of the three unit vectors i, j, and k in the x, y, and z dimension, respectively. Let us evaluate the work of each straight segment of the path separately, starting with path B. The work is wB =

Z F · t ds.

(4.12)

B

The tangent vector t is simply the unit vector i, since path B points along the positive x axis. When you take the dot product of i with F [see Eq. (4.12)], the result is simply F x . (Can you verify this?) Finally, since the displacement s is along the x axis, ds is simply dx. Thus, we have wB =

Z F x dx. B

(4.13)


63

Although the value of F x can vary as you move across path B—in fact, it is better labeled as F x (x, y, z)—let us assume some average value of F x as indicated by its value at a y-axis position of y − ∆y/2, which is the y value that is one-half of the height of the box below the point in the center. Thus, we have ! ∆y , Z · ∆x, x, y − z

Z

wB =

F x dx ≈ F x B

(4.14)

where we have replaced the infinitesimal dx with the finite ∆x. We can do the same for the work at the top of the box, which is path T. There are only two differences: first, the tangent vector is −i because the path is moving in the negative direction, and second, the average value of F x is judged at y + ∆y/2, which is one-half of the height of the box above the center point. Hence we can simply write wT ≈ −F x

! ∆y x, y + , z · ∆x. 2

(4.15)

The sum of the work on the top and bottom are thus ! ! ∆y ∆y x, y − , z · ∆x − F x x, y + , z · ∆x. 2 2

WT+B ≈ F x

(4.16)

Rearranging Eq. (4.16) so that it is in the form “top minus bottom” and factoring out ∆x, this becomes " WT+B ≈ − F x

! !# ∆y ∆y , z − F x x, y − , z ∆x. x, y + 2 2

(4.17)

Let us multiply this expression by 1, in the form of ∆y/∆y. We now have h WT+B ≈ −

F x x, y +

∆y 2 ,z

− F x x, y −

∆y

∆y 2 ,z

i ∆x∆y.

(4.18)

Recall that this work is actually a sum of two integrals involving, originally, the integrand F · t. With this in mind, Eq. (4.18) can be written

64

Chapter 4

as WT+B =

h

Z F · t ds ≈ −

F x x, y +

∆y 2 ,z

− F x x, y −

∆y 2 ,z

i

∆y

T+B

∆x∆y. (4.19)

The term ∆x∆y is the area A of the path. Dividing by the area on the left side of the equation, we have 1 A

h

Z F · t ds

F x x, y +

∆y 2 ,z

− F x x, y −

∆y

T+B

∆y 2 ,z

i .

(4.20)

Suppose that we take the limit of this expression as ∆x = ∆y = A → 0. What we would have is the amount of work done over any infinitesimal area defined by any random path; the only restriction is that the path is in the (x, y) plane. Equation (4.20) above becomes " lim

A→0

1 A

Z T+B

 h #  F x x, y + F · t ds = lim − ∆y→0

∆y 2 ,z

− F x x, y −

∆y

∆y 2 ,z

i    .  (4.21)

Looking at the second limit in Eq. (4.21) and recalling our basic calculus, that limit defines a derivative with respect to y! But because F x is a function of three variables, this is better defined as the partial derivative with respect to y. Thus, we have " Z # 1 ∂ Fx F · t ds = − . A→0 A T+B ∂y lim

(4.22)

Note the retention of the minus sign. We can do the same for paths L and R. The analysis is exactly the same; only the variables that are affected change. What we obtain is (you are welcome to verify the derivation) "

1 lim A→0 A

# ∂ Fy F · t ds = . ∂x L+R

Z

(4.23)


65

Now, we combine the two parts. The work done over an infinitesimally small closed path in the (x, y) plane is given by " I # ∂ Fy ∂ F x 1 w F · t ds = − . lim = lim A→0 A A→0 A ∂x ∂y

(4.24)

Now isn’t that a rather simple result? Let us see an example of this result so we can understand what it means. Consider a two-dimensional sink in the (x, y) plane, as diagrammed in Fig. 4.7. A thin film of water is going down the central drain, and in this case it is spinning in a counter-clockwise direction at some constant angular velocity. The vector field for the velocity of the spinning water is v=i

∂x ∂y +j . ∂t ∂t

(4.25)

In terms of the angular velocity ω, this can be written as v = ω(−iy + jx) = −iωy + jωx.

(4.26)

Figure 4.7 A two-dimensional sink with a film of water rotating counterclockwise as it goes down the drain.

66

Chapter 4

(A conversion to polar coordinates was necessary to go to this second expression for v, in case you need to do the math yourself.) In this vector field, F x = −ωy, and Fy = ωx. To determine the limit of the work per unit area, we evaluate the expression ∂ ∂ (ωx) − (−ωy). ∂x ∂y

(4.27)

= ω − (−ω) = 2ω.

(4.28)

This is easy to evaluate:

Suppose that we stand up a piece of cardboard on the sink, centered at the drain. Experience suggests to us that the cardboard piece will start to rotate, with the axis of rotation perpendicular to the flat sink. In this particular case, the axis of rotation will be in the z dimension, and in order to be consistent with the right-hand rule, we submit that in this case the axis points in the positive z direction. If this axis is considered to be a vector, then the unit vector in this case is (positive) k. Thus, vectorally speaking, the infinitesimal work per unit area is actually "

# ∂ Fy ∂ F x k. − ∂x ∂y

(4.29)

Thus, the closed loop in the (x, y) plane is related to a vector in the z direction. In the case of a vector field, the integral over the closed path is referred to as the circulation of the vector field. As a counterexample, suppose that water in our two-dimensional sink is flowing from left to right at a constant velocity, as shown in Fig. 4.8. In this case, the vector function is v = Ki,

(4.30)

where K is a constant. If we put a piece of cardboard in this sink, centered on the drain, does the cardboard rotate? No, it does not. If we evaluate the partial-derivative expression from above (in this case, F x = K and Fy = 0): "

# ∂ ∂ (K) − (0) k = 0k = 0. ∂x ∂y

(4.31)


Figure 4.8 velocity.

67

Water flowing in a two-dimensional sink with a constant left-to-right

(Recall that the derivative of a constant is zero.) This answer implies that no rotation is induced by the closed loop.

4.3 Introducing the Curl For a vector function F = F x i + Fy j + Fz k, we define the function "

# ∂ Fy ∂ F x k − ∂x ∂y

(4.32)

as the one-dimensional curl of F. Possibly improperly, it is designated “one dimensional” because the result is a vector in one dimension, in this case the z dimension. The analysis we performed in the earlier section (defining a closed path in a single plane and taking the limit of the path integral) can be performed for the (x, z) and (y, z) planes. When we do that, we obtain the following analogous results: # ∂ F x ∂ Fz j; − (x, z) plane : ∂z ∂x "

68

Chapter 4

# ∂ Fz ∂ Fy i. − (y, z) plane : ∂y ∂z "

(4.33)

The combination of all three expressions in Eqs. (4.32) and (4.33) gives us a general expression for the curl of F: " curl F =

# " # " # ∂ Fy ∂ F x ∂ F x ∂ Fz ∂ Fz ∂ Fy i+ j+ k. (4.34) − − − ∂y ∂z ∂z ∂x ∂x ∂y

This expression allows us to determine " lim

A→0

1 A

I

# F · t ds

(4.35)

for any vector function F in any plane. But what does the curl of a vector function mean? One way of thinking about it is that it is a variation in the vector function F that causes a rotational effect about a perpendicular axis. [Indeed, “curl F” is sometimes still designated “rot F,” and a vector function whose curl equals zero (see Fig. 4.8) is termed “irrotational.”] Also, a vector function with a nonzero curl can be thought of as curving around a particular axis, with that axis being normal to the plane of the curve. Thus, the rotating water in Fig. 4.7 has a nonzero curl, while the linearly flowing water in Fig. 4.8 has a zero curl. A mnemonic (that is, a memory aid) for the general expression for curl F takes advantage of the structure of a 3 × 3 determinant: i j ∂ ∂ curl F = ∂ x ∂y F x Fy

k ∂ . ∂z Fz

(4.36)

Understand that curl F is NOT a determinant; a determinant is a number that is a characteristic of a square matrix of numerical values. However, the expression for curl F can be constructed by performing the same operations on the expressions as one would perform with numbers to determine the value of a 3 × 3 determinant: constructing the diagonals, adding the right-downward diagonals, and subtracting the left-upwards


69

diagonals. In case you have forgotten how to do this, Fig. 4.9 shows how to determine the expression for the curl. The determinental form of the curl can be expressed in terms of the del operator ∇. Recall from Chapter 2 that the del operator is ∇≡i

∂ ∂ ∂ +j +k . ∂x ∂y ∂z

(4.37)

Recall from vector calculus that the cross product of two vectors A ≡ iA x + jAy + kAz and B defined analogously is written A × B and is given by the expression i j k A × B = A x Ay Az . Bx By Bz

(4.38)

By comparing this expression to the determinental form of the curl, it should be easy to see that the curl of a vector function F can be written as a cross product: curl F ≡ ∇ × F.

(4.39)

Figure 4.9 To determine the expression using a determinant, multiply the three terms on each arrow and apply the positive or negative sign to that product, as indicated. Combining all terms yields the proper expression for the curl of a vector function F having components F x , Fy , and Fz .

70

Chapter 4

Like the fact that curl is not technically a determinant, the curl of a function is technically not a cross product, as del is an operator, not a vector. The parallels, however, make it easy to gloss over this technicality and use the “del cross F” symbolism to represent the curl of a vector function. Because the work integral over a closed path through an electrostatic field E is zero, it is a short logical step to state that, therefore, ∇ × E = 0.

(4.40)

This is one more property of an electrostatic field: the field is not rotating about any point in space. Rather, an electrostatic field is a purely radial field, with all field “lines” going from a point in space straight to the electric charge.

4.4 Faraday’s Law An electrostatic field caused by a charged particle is thought of as beginning at a positive charge and ending at a negative charge. Since overall, matter is electrically neutral, every electric field emanating from a positive charge eventually ends at a negative charge. This is illustrated in Fig. 4.10(a). However, when a changing magnetic field creates a current in a conductor, this current is the product of an induced electric field. In the case of a bar-type magnet, the magnetic field is axially symmetric about the length of the bar, so the induced electric field is axially symmetric as well; that is, it is circular. This is illustrated in Fig. 4.10(b). The induced, circular electric field caused by a moving magnet causes charges to move in that circle. The circulation of the induced electric field vector can be constructed from our definition of “circulation” above; it is I circulation = E · t ds, (4.41) s

where E is the induced electric field, t is the tangent vector along the path, and s is the infinitesimal amount of path. In this case, the “circulation” is defined as the “electromotive force,” or EMF. What is this force doing? Why, causing charges to move, of course! As such, it is doing work, and our arguments using work in the sections above are all valid here. What Faraday found experimentally is that a changing magnetic field induced an electric field (which then forced a current). If you imagine that


71

Figure 4.10 (a) In an electrostatic field, the field lines go from the positive charge to the negative charge. (b) A moving magnetic field induces an electric field, but in this case the electric field is in a circle, following the axial nature of the magnetic field lines.

a magnetic field is composed of discrete field lines, what is happening is that as the number of magnetic field lines in a given area changes with time, an electric field is induced. Figure 4.11 illustrates this. Consider the loop of area outlined by the black line. As the magnet is moved to the right, the number of magnetic field lines that intersect the loop changes. It is this change that induces the electric field. The number of field lines per area is called the magnetic flux. In Chapter 2, we presented how to determine the flux of a vector field. For a changing vector field F having a unit vector perpendicular (or normal) to

72

Chapter 4

Figure 4.11 As the magnet is moved farther from the loop, the number of imaginary magnetic field lines intersect the loop changes [here, from (a) seven lines to (b) three lines]. It is this change that induces an electric field in the loop.

its direction of motion n over some surface S, the flux is defined as Z F · n dS . (4.42) flux = S

For our magnetic field B, this becomes magnetic flux =

Z B · n dS . S

(4.43)


73

However, the induced electric field is related to the change in magnetic flux with time. Thus, we are actually interested in the time derivative of the magnetic flux: ∂ ∂t

Z B · n dS .

(4.44)

S

At this stage, we bring everything together by citing the experimental facts as determined by Faraday and others: the electromotive force is equal to the change in the magnetic flux over time. That is, I S

∂ E · t ds = ∂t

Z B · n dS .

(4.45)

S

Let us divide each side of this equation by the area A of the circular path of the induced current. This area also corresponds to the surface S that the magnetic field flux is measured over, so we divide one side by A and one side by S to obtain 1 A

I

E · t ds = s

1 ∂ S ∂t

Z B · n dS .

(4.46)

S

Suppose that we want to consider the limit of this expression as the area of the paths shrink to zero size; that is, as A → 0. We would have 1 A→0 A

I

1 ∂ S →0 S ∂t

E · t ds = lim

lim

s

Z B · n dS .

(4.47)

S

The left side of Eq. (4.47) is, by definition, the curl of E. What about the right side? Rather than proving it mathematically, let’s consider the following argument. As the surface S goes to zero, the limit of the magnetic flux ultimately becomes one magnetic flux line. This single line will be perpendicular to the infinitesimal surface. Look at the rendering of the magnetic field lines in Fig. 4.1 if you need to convince yourself of this. Thus, the dot product B·n is simply B, and the infinite sum of infinitesimal pieces (which is what an integral is) degenerates to a single value of B. We therefore argue that ∂ lim S →0 ∂t

Z S

B · ndS =

∂ B. ∂t

(4.48)

74

Chapter 4

So what we now have is ∇×E=

∂ B. ∂t

(4.49)

We are almost done. The law of conservation of energy must be satisfied. Although it appears that we are getting an induced current from nowhere, understand that this induced current also generates a magnetic field. In order for the law of conservation of energy to be satisfied, the new magnetic flux must oppose the original magnetic flux (this concept is known as Lenz’s law after Henrich Lenz, a Russian physicist who discovered it). To represent this mathematically, a negative sign must be included in Eq. (4.49). By convention, the minus sign is put on the right side, so our final equation is ∇×E=−

∂ B. ∂t

(4.50)

This expression is known as Faraday’s law of induction, given that Michael Faraday discovered (or rather, first announced) magnetic induction of current. It is considered the third of Maxwell’s equations: a changing magnetic flux induces an electromotive force, which in a conductor will promote a current. Not meaning to minimize the importance of Maxwell’s other equations, but the impact of what this equation embodies is huge. Electric motors, electrical generators, and transformers are all direct applications of a changing magnetic field being related to an electromotive force. Given the electrified nature of modern society and the machines that make it that way, we realize that Maxwell’s equations have a huge impact on our everyday lives.

Chapter 5

Fourth Equation of Electrodynamics

∇ × B = µ0(J + ε0 ∂E ∂t ) 5.1 Ampère’s Law One of the giants in the development of the modern understanding of electricity and magnetism was the French scientist André-Marie Ampère (1775–1836). He was one of the first to demonstrate conclusively that electrical current generates magnetic fields. For a straight wire, Ampère demonstrated that (1) the magnetic field’s effects were centered on the wire carrying the current, (2) were perpendicular to the wire, and (3) were symmetric about the wire. Though illustrated in an earlier chapter, this is illustrated again in Fig. 5.1. This figure is strangely reminiscent of Fig. 4.7, reproduced here as Fig. 5.2, which depicts water circulating around a drain in a

Figure 5.1 through it.

The “shape” of a magnetic field about a wire with a current running 75

76

Chapter 5

Figure 5.2

Water going in a circular path about a drain (center).

counterclockwise fashion. But now, let’s put a paddle wheel in the drain, with its axis sticking in the drain as shown in Fig. 5.3. We see that the paddle wheel will rotate about an axis that is perpendicular to the plane of flow of the water. Rotate our water-and-paddle-wheel figure by 90 deg counterclockwise, and you have an exact analogy to Fig. 5.1. We argued in the previous chapter that water circulating in a sink as shown in Figs. 5.1 and 5.2 represents a function that has a nonzero curl. Recall that the curl of a vector function F designated “curl F” or “∇ × F” is defined as # " # " # " ∂ Fy ∂ F x ∂ Fz ∂ Fy ∂ F x ∂ Fz curl F = ∇ × F = i+ j+ k, (5.1) − − − ∂y ∂z ∂z ∂x ∂x ∂x where F x , Fy , and F x are the x, y, and z magnitudes of F, and i, j, and k are the unit vectors in the x, y, and z dimensions, respectively. In admitting the similarity between Figs. 5.1 and 5.2, we suggest that the curl of the magnetic field B is related to the current in the straight wire. There is a formal way to derive this. Recall from the last chapter that the curl of a


77

Figure 5.3 The paddle wheel rotates about a perpendicular axis when placed in circularly flowing water. We say that the water has a nonzero curl. The axis of the paddle wheel is consistent with the right-hand rule, as shown by the inset.

function F as 1 curl F = ∇ × F = lim A→0 A

I

S

E · t ds.

(5.2)

For the curl of the magnetic field, we thus have 1 ∇ × B = lim A→0 A

I

S

B · t ds.

(5.3)

Recall that S is the surface about which the line s is tracing, t is the tangent vector on the field line, and A is the area of the surface. This simplifies easily when one remembers that we have a formula for B in terms of the distance from the wire r; it was presented in Chapter 3 and is B=

µI , 2πr

(5.4)

where I is the current, r is the radial distance from the wire, and µ is the constant known as the permeability of the medium; for vacuum, the symbol µ0 is used, and its value is defined as 4π × 10−7 T · m/A (tesla × meters per ampere). We also know that the magnetic field paths are circles; thus, as we integrate about the surface, the integral over ds becomes simply the circumference of a surface, 2πr. Substituting these expressions into the

78

Chapter 5

curl of B, we obtain ∇ × B = lim

A→0

1 µI (2πr). A 2πr

(5.5)

The 2πr terms cancel, and we are left with ∇ × B = lim

A→0

µI . A

(5.6)

The constant µ can be taken out of the limit. What we have left to interpret is lim

A→0

I . A

(5.7)

This is the limit of the current I flowing through an area A of the wire as the area grows smaller and smaller, ultimately approaching zero. This infinitesimal current per area is called the current density and is designated by J; it has units of coulombs per square meter, or C/m2 . Since current is technically a vector, so is current density J. Thus, we have ∇ × B = µJ.

(5.8)

This expression is known as Ampère’s circuital law. In a vacuum, the expression becomes ∇ × B = µ0 J.

(5.9)

This is not one of Maxwell’s equations; it is incomplete. It turns out that there is another source of a magnetic field.

5.2 Maxwell’s Displacement Current The basis of Ampère’s circuital law was discovered in 1826 (although its modern mathematical formulation came more than 35 years later). By the time Maxwell was studying electromagnetic phenomena in the 1860s, something new had been discovered: magnetic fields from capacitors. Here’s how to think of this new development. A capacitor is a device that stores electrical charge. The earliest form of a capacitor was the


79

Leyden jar, described in Chapter 1, and a picture of which is shown in Fig. 5.4. Although the engineering of modern capacitors can vary, a simple capacitor can be thought of as two parallel metal plates separated by a vacuum or some other nonconductor, called a dielectric. Figure 5.5 is a diagram of a parallel-plate capacitor.

Figure 5.4 A series of four Leyden jars in a museum in Leiden, The Netherlands. This type of jar was to be filled with water. The apparatus on the bottom side is a simple electrometer, meant to give an indication of how much charge was stored in these ancient capacitors.

80

Chapter 5

Figure 5.5

Diagram of a simple parallel-plate capacitor.

A capacitor works because of the gap between the plates; in an electrical circuit, current enters a plate on one side of the capacitor. However, because of the gap between the plates, the current builds up on one side, ultimately causing an electric field to exist between the plates. We know now that current is electrons, so in modern terms, electrons build up on one side of the plate. However, electrons have a negative charge, which repel other electrons residing on the other plate. These electrons get forced away, resulting in the other plate building up an overall positive charge. These electrons that are forced away represent a current on the other side of the plate that continues until the maximum charge has built up on the other plate. This process is illustrated in Fig. 5.6. Although electrons are not flowing from one side of the capacitor to the other, during the course of charging the capacitor, a current flows and generates a magnetic field. This magnetic field, caused by the changing electric field, is not accounted for by Ampère’s circuital law because it is the result of a changing electric field, not a constant current. Maxwell was concerned about this new source of a magnetic field. He called this new type of current “displacement current” and set about integrating it into Ampère’s circuital law. Because this magnetic field was proportional to the development of an electric field; that is, the change in


81

Figure 5.6 Charging a capacitor: (1) Current enters one plate. (2) Electrons build up on the plate. (3) Electrons on the other plate are repelled, causing (4) a shortlived current to leave the other plate.

E with respect to time, we have ∇×B∝

∂E , ∂t

(5.10)

The proportionality constant needed to make this an equality is the permittivity of free space, symbolized as ε0 . This fundamental constant has a value of about 8.854 × 10−12 C2 /J · m. Since an electric field has units of volts per meter (V/m), the combined terms ε0 (∂E/∂t) have units of C2 /(J · m) × V/(m · s), where the “s” unit comes from the ∂t in the derivative. A volt is equal to a joule/coulomb, and a coulomb/second is equal to an ampere, so the combined units reduce to A/m2 , which is a unit of current density! Thus, we can add the displacement current term ε0 (∂E/∂t) to the original current density J, yielding Maxwell’s fourth equation of electrodynamics: ∇ × B = µ0 J + ε0

! ∂E . ∂t

This equation is sometimes called the Ampère–Maxwell law.

(5.11)

82

Chapter 5

5.3 Conclusion The presentation of Maxwell’s equations, in their modern differential form and in a vacuum, is complete. Collectively, they summarize all of the properties of electric and magnetic fields. However, there is one more startling conclusion to Maxwell’s equations that we will defer to an Afterword.

Afterword Whence Light? A.1 Recap: The Four Equations In a vacuum, Maxwell’s four equations of electrodynamics, as expressed in the previous four chapters, are: ρ Gauss’ law, ε0 ∇ · B = 0 Gauss’ law of electromagnetism, ∂B ∇×E=− Faraday’s law, and ∂t ! ∂E ∇ × B = µ0 J + ε0 Ampère–Maxwell law. ∂t ∇·E=

(A.1) (A.2) (A.3) (A.4)

A few comments are in order. First, Maxwell did not actually present the four laws in this form in his original discourse. His original work, detailed in a four-part series of papers titled On Physical Lines of Force contained dozens of equations. It remained to others, especially English scientist Oliver Heaviside, to reformulate Maxwell’s derivations into four concise equations using modern terminology and symbolism. We owe almost as much a debt to the scientists who took over after Maxwell’s untimely death in 1879 as we do to Maxwell himself for these equations. Second, note that the four equations have been expressed in differential forms. (Recall that the divergence and curl operations, ∇· and ∇× respectively, are defined in terms of derivatives.) There are other forms of Maxwell’s equations, including forms for inside matter (as opposed to a vacuum, which is what is considered exclusively in this book), integral forms, so-called “macroscopic” forms, relativistic forms, even forms that 83

Afterword

84

assume the existence of magnetic monopoles (likely only of interest to theoretical physicists and science fiction writers). The specific form you might want to use depends on the quantities you know, the boundary conditions of the problem, and what you want to predict. Persons interested in these other forms of Maxwell’s equations are urged to consult the technical literature.

A.2 Whence Light? We began Chapter 1 by claiming that light itself is explained by Maxwell’s equations. How? Actually, it comes from an analysis of Faraday’s law and the Ampère–Maxwell law, as these are the two of Maxwell’s equations that involve both E and B. Among the theorems of vector calculus is the proof (not given here) that the curl of a curl of a function is related to the divergence. For a given vector function F, the curl of the curl of F is given by ∇ × (∇ × F) = ∇(∇ · F) − ∇2 F.

(A.5)

Hopefully, you already recognize ∇ · F as the divergence of the vector function F. There is one other type of function present on the right side of the equation, the simple ∇ by itself (without a dot or a cross). Unadorned by the dot or cross, ∇ represents something called the gradient, which is nothing more than the three-dimensional slope of a vector function, itself expressed as a vector in terms of the unit vectors i, j, and k: ∇F = i

∂F ∂F ∂F +j +k . ∂x ∂y ∂z

(A.6)

The first term on the right of a curl of a curl [Eq. (A.5)], then, is the gradient of the divergence of F. The gradient can also be applied twice. (The gradient is the last term on the right-hand side, as seen by the ∇2 .) When this happens, what initially seems complicated simplifies quite a bit: ∂F ∂F ∂F +j +k ∇ F= i ∂x ∂y ∂z 2

!

! ∂F ∂F ∂F i , +j +k ∂x ∂y ∂z

(A.7)

Whence Light?

85

which simplifies to ∇2 F =

∂ 2F ∂ 2F ∂ 2F + + . ∂ 2 x ∂ 2y ∂ 2z

(A.8)

Note that there are now no i, j, or k vectors in these terms, and that there are no cross terms between x, y, and z. This is because the i, j, and k vectors are orthonormal: n1 · n2 = 1 if n1 and n2 are the same (that is, both are i or both are j), while n1 · n2 = 0 if n1 and n2 are different (for example, n1 represents i and n2 represents k). What we do is take the curl of both sides of Faraday’s law: ∇ × (∇ × E) = ∇ × −

! ∂B . ∂t

(A.9)

Because the curl is simply a group of spatial derivatives, it can be brought inside the derivative with respect to time on the right side of the equation: ! ∂(∇ × B) , ∇ × (∇ × E) = − ∂t

(A.10)

and we can substitute the expression for what the curl of a curl is on the left side: ! ∂(∇ × B) 2 ∇(∇ · E) − ∇ E = − . (A.11) ∂t The expression ∇ × B is defined by the Ampère–Maxwell law (Maxwell’s fourth equation), so we can substitute for ∇ × B: i   h   ∂ µ0 J + ε0 ∂E ∂t  . ∇(∇ · E) − ∇ E = −  ∂t 2

(A.12)

Faraday’s law (or Maxwell’s first equation) tells us what ∇ · E is: it equals ρ/ε0 . We substitute this into the first term on the left side: i   h !   ∂ µ0 J + ε0 ∂E ρ ∂t  . ∇ − ∇2 E = −  ε0 ∂t

(A.13)

Afterword

86

Now we will rewrite the right side by separating the two terms to obtain two derivatives with respect to time. Note that the second term becomes a second derivative with respect to time, and that µ0 , the permeability of a vacuum, distributes through to both terms. We obtain ∇

! ∂ 2E ∂(µ0 J) ρ − µ0 ε0 2 . − ∇2 E = − ε0 ∂t ∂t

(A.14)

In the absence of charge, ρ = 0, and in the absence of a current, J = 0. Under these conditions, the first terms on both sides are zero, and the negative signs on the remaining terms cancel. What remains is ∇2 E = µ0 ε0

∂ 2E . ∂t2

(A.15)

This is a second-order differential equation that relates an electric field that varies in space and time. That is, it describes a wave, and this differential equation is known in physics as the wave equation. The general form of the wave equation is ∇2 F =

1 ∂ 2F , v2 ∂t2

(A.16)

where v is the velocity of the wave. The function F can be expressed in terms of sine and cosine functions or as an imaginary exponential function; the exact expression for an E wave (light) depends on the boundary conditions and the initial value at some point in space. The wave equation implies that 1 vlight = √ . µ0 ε0

(A.17)

This can be easily demonstrated: 1 vlight = q −12 C2 (4π × 10−7 T·m A ) 8.8541878 × 10 J·m = 2.9979246 . . . × 108 m/s.

(A.18)

Whence Light?

87

(You need to decompose the tesla unit T into its fundamental units kg/A·s2 to see how the units work out. Remember also that J = kg · m2 /s2 , and that A = C/s, and everything works out naturally, as it should with units.) Even by the early 1860s, experimental determinations of the speed of light were around that value, leading Maxwell to conclude that light was a wave of an electric field that had a velocity of (1/µ0 ε0 )1/2 . Light is also a magnetic wave. How do we know? Because we can take the curl of the Ampère–Maxwell law and perform similar substitutions. This exercise is left to the reader, but the conclusion is not. Ultimately, you will obtain ∇2 B = µ0 ε0

∂ 2B . ∂t2

(A.19)

This is the same form of the wave equation, so we have the same conclusions: light is a wave of a magnetic field having a velocity of (1/µ0 ε0 )1/2 . However, because of Faraday’s law (Maxwell’s third equation), the electric field and the magnetic field associated with a light wave are perpendicular to each other. A modern depiction of what we now call electromagnetic waves is shown in Fig. A.1.

A.3 Concluding Remarks Along with the theory of gravity, laws of motion, and atomic theory, Maxwell’s equations were triumphs of classical science. Although

Figure A.1 light.

A modern depiction of the electromagnetic waves that we know as

88

Afterword

ultimately updated by Planck and Einstein’s quantum theory of light, Maxwell’s equations are still indispensable when dealing with everyday phenomena involving electricity and magnetism and, yes, light. They help us understand the natural universe better—after all, isn’t that what good scientific models should do?

Bibliography Baigrie, B., Electricity and Magnetism, Greenwood Press, Westport, CT (2007). Darrigol, O., Electrodynamics from Ampere to Einstein, Oxford University Press, Oxford, UK (2000). Halliday, D., R. Resnick, and J. Walker, Fundamentals of Physics, 6th edition, John Wiley and Sons, New York (2001). Hecht, E., Physics, Brooks-Cole Publishing Co., Pacific Grove, CA (1994). Marsden, J. E. and A. J. Tromba, Vector Calculus 2nd edition, W. H. Freeman and Company, New York (1981). Reitz, J. R., F. J. Milford, and R. W. Christy, Foundations of Electromagnetic Theory, Addison-Wesley Publishing Co., Reading, MA (1979). Schey, H. M., Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, 4th edition, W. W. Norton and Co., New York (2005).

89

Index A abscissa, 20 amber, 3 Ampère, 9, 15 Ampère’s circuital law, 78 Ampère, André Marie, 55, 75 Ampère–Maxwell law, 81, 84, 85, 87 Arago, François, 11, 55

color theory, 18 Columbus, Christopher, 3 compasses, 3 Copernicus, 7 Coulomb’s law, 8, 40, 61 curl, 67, 68, 73, 76, 83 D Davy, Humphrey, 9 de Coulomb, Charles-Augustin, 7 de Gama, Vasco, 3 del, 39, 53, 69 delta, 21, 22 derivative, 21, 38, 64 Descartes, René, 5 determinant, 68 diamagnetic, 49 diamagnetism, 14 dielectric, 79 displacement current, 80 divergence, 38, 39, 83 dot product, 24, 36, 73 du Fay, Charles, 5 dynamo, 13

B battery, 9 Biot and Savart, 11 Biot, Jean-Baptiste, 55 Biot–Savart law, 15 Browne, Thomas, 4 C calculus, 19 Cambridge, 18 capacitors, 78 Cardano, Gerolamo, 3 Cavendish, Henry, 8 changing electric field, 80 charge density, 45 Charles du Fay, 5 Church of San Nazaro, 7 circulation, 66, 70 closed path, 61

E Earth, 4 effluvium, 3, 5 electric eels, 2 91

92

electric field, 42, 80 electric motors, 74 electricity, 2, 4 electrons, 80 elektron, 3 F Faraday effect, 13 Faraday’s law, 74, 84, 85, 87 Faraday, Michael, 9, 11, 48, 57, 70, 74 ferromagnetic, 49 flux, 35, 42 Franklin, Benjamin, 5 G Galvani, Luigi, 8 Gauss’ law, 45 Gauss, Carl Friedrich, 45 generators, 74 Gilbert, William, 3 gradient, 84 Gray, Stephen, 4 Grove, William Robert, 11 H Heaviside, Oliver, 83 Henry, Joseph, 57 I induced electric field, 70 infinitesimal, 26 integral, 26 integral sign, 26 integrand, 26 integration, 26, 28 K kinetic molecular theory of gases, 18

Index

King’s College London, 17 L law of conservation of energy, 74 Leibnitz, Gottfried, 19, 26 Lenz’s law, 74 Lenz, Henrich, 74 Leyden jar, 5, 9, 79 lightning rod, 7 line integral, 29, 33 lines of force, 48, 49 lodestone, 3, 47 luminiferous ether, 5 M Magellan, 3 Magnesia, 3, 47 magnesium, 48 magnet, 47 magnetic field, 14, 47, 48, 55, 75, 76 magnetic flux, 71 magnetic monopole, 54 magnetite, 47 magnitude, 30 Marischal College, 17 Maxwell’s demon, 18 Maxwell’s equations of electrodynamics, 19 Maxwell’s first equation of electromagnetism, 45 Maxwell’s fourth equation of electrodynamics, 81 Maxwell’s second equation of electrodynamics, 53 Maxwell’s third equation of electrodynamics, 74 Maxwell, James Clerk, 8, 17, 50, 78, 83

93

Index

Maxwell–Boltzmann distribution, 18 N Newton, Isaac, 4, 19, 26 normal vector, 34 O Ohm’s law, 11 Ohm, Georg, 11 ordinate, 20 Ørsted, Hans Christian, 9, 55 other forms of Maxwell’s equations, 83 P paramagnetic, 49 paramagnetism, 14 partial derivative, 22 path independent, 61 path integral, 29 Peregrinus, Petrus, 3, 14 permeability, 56 permeability of a vacuum, 86 permittivity of free space, 42, 81 Poisson, Simeon-Denis, 15 pole, 3, 48

scalar, 39 Savart, Félix, 55 slope, 20 speed of light, 87 Sturgeon, William, 56 surface integral, 33 T Thales of Miletos, 3 thermodynamics, 18 Thomson, William, 50 Thomson, William, Lord Kelvin, 13 total differential, 22 transformers, 74 U unit vectors, 24, 32, 76 University of Aberdeen, 17 University of Leyden, 5

Q quantum mechanics, 19

V van Musschenbrök, Pieter, 5 vector, 48 vector function, 23 volt, 81 Volta, Alessandro, 5, 8 voltaic pile, 9 volume, 37 von Kleist, Ewald, 5

R Reimann, Bernhard, 28 relativity, 19 right-hand rule, 55 Royal Society, 4

W Water, 9 wave equation, 86, 87 Weber, William, 45 work, 59

S Saturn, 18

Y y intercept, 20

David W. Ball is a Professor in the Department of Chemistry at Cleveland State University (CSU) in Ohio. He received a Ph.D. in chemistry from Rice University in 1987 and, after post-doctoral research at Rice University and at Lawrence Berkeley Laboratory in Berkeley, California, joined the faculty at CSU in 1990, rising to the rank of Professor in 2002. He has authored more than 200 publications, equally split between research papers and works of a more educational bent, including eight books currently in print. Dr. Ball’s research interests include low-temperature infrared spectroscopy and computational chemistry. His wife, Gail, is an engineer who keeps him on his toes. His two sons, Stuart and Casey, also keep him on his toes. Professor Ball has a doppelgänger, David W. Ball, who writes historical fiction and lives in the Rocky Mountains. David (the chemistry professor) has read some of David’s (the real author’s) books and has enjoyed them. There is no word if the favor has been returned.

SPIE PRESS

Maxwell’s Equations of Electrodynamics: An Explanation is a concise discussion of Maxwell's four equations of electrodynamics—the fundamental theory of electricity, magnetism, and light. It guides readers step-by-step through the vector calculus and development of each equation. Pictures and diagrams illustrate what the equations mean in basic terms. The book not only provides a fundamental description of our universe but also explains how these equations predict the fact that light is better described as "electromagnetic radiation."

P.O. Box 10 Bellingham, WA 98227-0010 ISBN: 9780819494528 SPIE Vol. No.: PM232

Maxwell's Equations of Electrodynamics an Explanation [2012]

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