Maxwell's Equations of Electrodynamics an Explanation [2012]

7/25/2019 Maxwell's Equations of Electrodynamics an Explanation [2012]

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SPIE PRESS


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Bellingham, Washington USA


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Library of Congress Cataloging-in-Publication Data

Ball, David W. (David Warren), 1962-Maxwells equations of electrodynamics : an explanation / David W. Ball.

pages cmIncludes bibliographical references and index.

ISBN 978-0-8194-9452-81. Maxwell equations. 2. Electromagnetic theory. I. Title.QC670.B27 2012

530.14

1dc232012040779

Published by

SPIEP.O. Box 10Bellingham, Washington 98227-0010 USAPhone: +1 360.676.3290

Fax: +1 360.647.1445Email: [email protected]: http://spie.org

Copyright c 2012 Society of Photo-Optical Instrumentation Engineers

(SPIE)

All rights reserved. No part of this publication may be reproduced or

distributed in any form or by any means without written permission ofthe 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 informa-

tion or for any outcomes resulting from reliance thereon.

Printed in the United States of America.

First printing


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Dedication

This book is dedicated to the following cadets whom I had the honorof teaching while serving as Distinguished Visiting Faculty at the US

Air Force Academy in Colorado Springs, Colorado, during the 201112

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, ErinCrow, 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, AndrewMunoz, 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.


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Contents

Preface .................................................................................................... ix

Chapter 1 History............................................................................. 1

1.1 History (Ancient) ............................................................................. 2

1.2 History (More Recent).................................................................... 7

1.3 Faraday ................................................................................................ 11

Chapter 2 First Equation of Electrodynamics ............................ 17

2.1 Enter Stage Left: Maxwell ............................................................ 17

2.2 A Calculus Primer............................................................................ 19

2.3 More Advanced Stuff...................................................................... 26

2.4 A Better Way ..................................................................................... 35

2.5 Maxwells First Equation .............................................................. 40

Chapter 3 Second Equation of Electrodynamics ....................... 47

3.1 Introduction ........................................................................................ 47

3.2 Faradays Lines of Force ............................................................... 48

3.3 Maxwells Second Equation......................................................... 50

Chapter 4 Third Equation of Electrodynamics ........................... 55

4.1 Beginnings .......................................................................................... 55

4.2 Work in an Electrostatic Field...................................................... 59

4.3 Introducing the Curl ........................................................................ 67

4.4 Faradays Law ................................................................................... 70

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viii Contents

Chapter 5 Fourth Equation of Electrodynamics ........................ 75

5.1 Ampres Law ................................................................................... 75

5.2 Maxwells Displacement Current ............................................... 78

5.3 Conclusion.......................................................................................... 82

Afterword Whence Light?............................................................... 83

A.1 Recap: The Four Equations.......................................................... 83

A.2 Whence Light? .................................................................................. 84

A.3 Concluding Remarks....................................................................... 87

Bibliography.......................................................................................... 89

Index ....................................................................................................... 91


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Preface

As the contributing editor of The Baseline column in Spectroscopymagazine, 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 Maxwells equations

of electrodynamics, since our understanding of light is based on them.

It did not take much research to realize that a discussion of Maxwellsequations was more than a 2000-word column could handleindeed,

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. Ive 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 personalprecedent 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 Maxwells equations.

He responded in less than two hours. . . with a contract. (Note to budding

authors: thats agoodsign.)

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 Maxwells equations, let me knowthe hope is that this will be

one of the premiere explanations of Maxwells 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,

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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 myfamilyGail, Stuart, and Caseyfor 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


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Chapter 1

History

Maxwells 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 ClerkMaxwell 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, its difficult to minimize the impact of these equations,which in modern form are known as Maxwells equations.

In this chapter, we will discuss the history of electricity and magnetism,

going through early 19th-century discoveries that led to Maxwells work.

This historical review will set the stage for the actual presentation of

Maxwells equations.

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2 Chapter 1

1.1 History (Ancient)

Electricity has been experienced since the dawn of humanity, and from

two rather disparate sources: an aquatic animal and lightning. AncientGreeks, Romans, Arabs, and Egyptians recorded the effect that electriceels caused when touched, and who could ignore the fantastic light showsthat 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.


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History 3

ancients could describe the events, they could not explain exactly whatwas happening.

The ancient Greeks noticed that if a sample of amber (Greekelektron)was rubbed with animal fur, it would attract small, light objects such asfeathers. Some, like Thales of Miletos, related this property to that oflodestone, a natural magnetic material that attracted small pieces of iron.(A relationship between electricity and magnetism would be re-awakenedover 2000 years later.) Since the samples of this rock came from the nearbytown of Magnesia (currently located in southern Thessaly, in centralGreece), these stones ultimately were called magnets. Note that it appears

that magnetism was also recognized by the same time as static electricityeffects (Thales lived in the 7th and 6th century BCE [before common era]),so both phenomena were known, just not understood. Chinese fortunetellers were also utilizing lodestones as early as 100 BCE. For the mostpart, however, the properties of rubbed amber and this particular rockremained novelties.

By 1100 CE (common era), the Chinese were using spoon-shapedpieces of lodestone as rudimentary compasses, a practice that spreadquickly to the Arabs and to Europe. In 1269, Frenchman Petrus Peregrinusused the word pole for the first time to describe the ends of a lodestonethat point in particular directions. Christopher Columbus apparently had asimple form of compass in his voyages to the New World, as did Vasco deGama and Magellan, so Europeans had compasses by the 1500sindeed,the great ocean voyages by the explorers would likely have been extremelydifficult without a working compass (Fig. 1.2). In books published in 1550

and 1557, the Italian mathematician Gerolamo Cardano argued that theattractions caused by lodestones and the attractions of small objects torubbed amber were caused by different phenomena, possibly the first timethis was explicitly pointed out. (His proposed mechanisms, however, weretypical of much 16th-century physical science: wrong.)

In 1600, English physician and natural philosopher William GilbertpublishedDe Magnete, Magneticisque Corporibus, et de Magno MagneteTellure (On the Magnet and Magnetic Bodies, and on the Great Magnetthe Earth) in which he became the first to make the distinction betweenattraction due to magnetism and attraction due to rubbed elektron. Forexample, 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 substancehe termed effluvium, and it was the return of the lost effluvium to the


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4 Chapter 1

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 wordelectricusas meaning like amber in its ability to attract. Note also what

the title of Gilberts book impliesthe Earth is a great magnet, an ideathat only arguably originated with him, as the book was poorly referenced.The word electricity itself was first used in 1646 in a book by Englishauthor Thomas Browne.

In the late 1600s and early 1700s, Englishman Stephen Gray carried outsome experiments with electricity and was among the first to distinguishwhat we now call conductors and insulators. In the course of hisexperiments, Gray was apparently the first to logically suggest that thesparks he was generating were the same thing as lightning, but on a muchsmaller scale. Interestingly, announcements of Grays discoveries werestymied by none other than Isaac Newton, who was having a dispute withother scientists (with whom Gray was associated, so Gray was guilty byassociation) and, in his position of president of the Royal Society, impededthese scientists abilities to publish their work. (Curiously, despite his


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History 5

advances in other areas, Newton himself did not make any major andlasting contributions to the understanding of electricity or magnetism.

Grays work was found documented in letters sent to the Royal Societyafter Newton had died, so his work has been historically documented.)Inspired by Grays experiments, French chemist Charles du Fay alsoexperimented 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 typesof effluvium, which he termed vitreous (because rubbed glass exhibitedone 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 Ren Descartes weighed in with

his Principia Philosophiae, published in 1644. His ideas were strictlymechanical: magnetic effects were caused by the passage of tinyparticles emanating from the magnetic material and passing throughthe luminiferous ether. Unfortunately, Descartes proposed a variety ofmechanisms for how magnets worked, rather than a single one, whichmade his ideas arguable and inconsistent.

The Leyden jar (Fig. 1.3) was invented simultaneously in 1745 byEwald von Kleist and Pieter van Musschenbrk. Its name derives fromthe University of Leyden (also spelled Leiden) in the Netherlands, whereMusschenbrk worked. It was the first modern example of a condenser orcapacitor. The Leyden jar allowed for a significant (to that date) amount ofcharge to be stored for long periods of time, letting researchers experimentwith electricity and see its effects more clearly. Advances were quick,

given the ready source of what we now know as static electricity. Amongother discoveries was the demonstration by Alessandro Volta that materialscould be electrified by induction in addition to direct contact. This, amongother experiments, gave rise to the idea that electricity was some sort offluid that could pass from one object to another, in a similar way thatlight or heat (caloric in those days) could transfer from one object toanother. The fact that many conductors of heat were also good conductorsof electricity seemed to support the notion that both effects were caused by

a fluid of some sort. Charles du Fays idea of vitreous and resinoussubstances, mentioned above, echoed the ideas that were current at thetime.

Enter Benjamin Franklin, polymath. It would take (and has taken) wholebooks to chronicle Franklins contributions to a variety of topics, but herewe will focus on his work on electricity. Based on his own experiments,


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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).


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History 7

Franklin rejected du Fays two fluid idea of electricity and insteadproposed a one fluid idea, arbitrarily defining one object negative if it

lost that fluid and positive if it gained fluid. (We now recognize his ideaas prescient but backwards.) As part of this idea, Franklin enunciated anearly concept of the law of conservation of charge. In noting that pointed

conductors lose electricity faster than blunt ones, Franklin invented thelightning rod in 1749. Although metal had been used in the past to adorn

the tops of buildings, Franklins invention appears to the first time such aconstruction was used explicitly to draw a lightning strike and spare thebuilding itself. [Curiously, there was some ecclesiastical resistance to the

lightning rod, as lightning was deemed by many to be a sign of divineintervention, 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 illustratedin 1769 when the rodless Church of San Nazaro in Brescia, Italy (inLombardy in the northern part of the country) was struck by lightning,

igniting approximately 100 tons of gunpowder stored in the church. Theresulting explosion was said to have killed 3000 people and destroyed

one-sixth of the city. Ecclesiastical opposition weakened significantly afterthat.]

In 1752, Franklin performed his apocryphal experiment with a kite, akey, and a thunderstorm. The word apocryphal is used intentionally:

different sources have disputed accounts of what really happened. (Modernanalyses suggest that performing the elementary-school portrayal of theexperiment 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, Franklinsdemonstration that lightning was a natural phenomenon was likely asimportant as Copernicus heliocentric ideas were in demonstrating the lackof 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 Frenchphysicist Charles-Augustin de Coulomb published Thorie des MachinesSimples (Theory of Simple Machines). To refine this work, through theearly 1780s Coulomb constructed a very fine torsion balance with which

he could measure the forces due to electrical charges. Ultimately, Coulombproposed that electricity behaves as if it were tiny particles of matter, acting


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8 Chapter 1

in a way similar to Newtons law of gravity, which had been known since1687. We now know this idea as Coulombs law. If the charge on one body

isq1, and the charge on another body isq2, and the bodies are a distancerapart, then the forceFbetween the bodies is given by the equation

F =kq1q2

r, (1.1)

wherekis a constant. In 1785 Coulomb also proposed a similar expressionfor magnetism but, curiously, did not relate the two phenomena. Asseminal and correct as this advance was, Coulomb still adhered toa two fluid idea of electricityand magnetismas it turned out.(Interestingly, about 100 years later, James Clerk Maxwell, about whomwe will have more to say, published papers from reclusive scientist HenryCavendish, demonstrating that Cavendish had performed similar electricalexperiments at about the same time but never published them. If he had,we might be referring to this equation as Cavendishs law.) Nonetheless,through the 1780s, Coulombs 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 anItalian doctor who studied animal anatomy at the University of Bologna. In1771, 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 cuttinginto, 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 generatingelectric machine or a Leyden jar caused a frog leg to jerk, as did contactwith two different metals at once, like iron and copper. Galvani eventuallyproposed that the muscles in the leg were acting like a fleshy Leyden jarthat contained some residual electrical fluid he called animal electricity. Inthe 1790s, he printed pamphlets and passed them around to his colleagues,describing his experiments and his interpretation.

One of the recipients of Galvanis pamphlets was Alessandro Volta, aprofessor at the University of Pavia, about 200 km northwest of Bologna.Volta was skeptical of Galvanis work but successfully repeated the frog-leg experiments. Volta took the work further, positioning metallic probesexclusively on nerve tissue and bypassing the muscle. He made thecrucial observation that some sort of induced activity was only producedif two different metals were in contact with the animal tissue. From


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History 9

these observations, Volta proposed that the metals were the source of theelectrical 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 constructionbecame known as a voltaic pile, but we know it better as a battery. Insteadof a sudden spark, which was how electricity was produced in the pastfrom Leyden jars, here was a construction that provided a steady flow,or current, of electricity. The development of the voltaic pile made newexperiments in electricity possible.

Advances came swiftly with this new, easily constructed pile. Waterwas 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 ifelectricity were truly caused by chemical reactions (as many chemists atthe time had perhaps chauvinistically thought), then perhaps electricitycan cause chemical reactions in return. He was correct, and in 1807 heproduced elemental potassium and sodium electrochemically for the firsttime. The electrical nature of chemistry was first realized then, and itsramifications continue today. Davy also isolated elemental chlorine in1810, following up with isolation of the elements magnesium, calcium,strontium, and barium, all for the first time and all by electricity. (Davysgreatest discovery, though, was not an element; arguably, Davys greatestdiscovery 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, its been here allalong, but not much new has been developed. This changed in 1820.Hans Christian rsted was a Danish physicist who was a disciple ofthe metaphysics of Emmanual Kant. Given the recently demonstratedconnection between electricity and chemistry, rsted was certain that therewere other fundamental physical connections in nature.

In the course of a lecture in April 1820, rsted passed an electricalcurrent from a voltaic pile through a wire that was placed parallel to a

compass needle. The needle was deflected. Later experiments (especiallyby Ampre) demonstrated that with a strong enough current, a magnetizedcompass 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 magnetismaffected each other: electromagnetism.


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Figure 1.4 Diagram of Voltas first pile that produced a steady supply of electricity.

We know it now as a battery.


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History 11

Detailed follow-up studies by Ampre in the 1820s quantified therelationship a bit. Ampre demonstrated that the magnetic effect was

circular about a current-carrying wire and, more importantly, that twowires with current moving in them attracted and repelled each other asif they were magnets. Magnets, Ampre argued, are nothing more thanmoving electricity. The development of electromagnets at this time freedscientists from finding rare lodestones and allowed them to study magnetic

effects using only coiled wires as the source of magnetism. Around thistime, Biot and Savarts studies led to the law named after them, relatingthe intensity of the magnetic effect to the square of the distance from the

electromagnet: a parallel to Coulombs law.In 1827, Georg Ohm announced what became known as Ohms law,

a relationship between the voltage of a pile and the current passingthrough the system. As simple as Ohms law is (V = IR, in modernusage), it was hotly contested in his native Germany, probably for political

or philosophical reasons. Ohms law became a cornerstone of electricalcircuitry. Its uncertain how the term current became applied to the

movement of electricity. Doubtless, the term arose during the time thatelectricity was thought to be a fluid, but it has been difficult to find theinitial usage of the term. In any case, the term was well established by thetime Sir William Robert Grove wrote The Correlation of Physical Forcesin 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 treatmentof electricity and magnetism than Michael Faraday (Fig. 1.5). Inspired byrsteds work, in 1821 Faraday constructed a simple device that allowed

a wire with a current running through it to turn around a permanentmagnet, and conversely a permanent magnet to turn around a wire thathad a current running through it. Faraday had, in fact, constructed the first

rudimentary motor. What Faradays motor demonstrated experimentally is

that the forces of interaction between the current-containing wire and themagnet werecircularin 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 1824by French scientist Franois Arago, who used a spinning copper disk to


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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 Aragos 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. Indoing so, Faraday constructed the first transformer, but the galvanometerdid not register the presence of a current. However, Faraday did note

a slight jump of the galvanometers needle. In 1831, Faraday passed

a magnet through a coil of wire attached to a galvanometer, and thegalvanometer registered, but only if the magnet was moving. When the


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History 13

magnet was halted, even when it was inside the coil, the galvanometerregistered zero.

Faraday understood that it wasnt the presenceof a magnetic field thatcaused an electrical current, it was a change in the magnetic field thatcaused a current. It did not matter what moved; a moving magnet caninduce current in a stationary wire, or a moving wire can pick up a currentinduced by moving it across a stationary magnet. Faraday also realized thatthe wire wasnt necessary. Taking a page from Arago, Faraday constructeda generator of electricity using a copper disk that rotated through thepoles of a permanent magnet (Fig. 1.6). Faraday used this to generate a

continuous source of electricity, essentially converting mechanical motionto electrical motion. Faraday invented thedynamo, the basis of the electric

industry even today.

After dealing with some health issues during the 1830s, in the mid-1840s Faraday was back at work. Twenty years earlier, Faraday hadinvestigated 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-polarizedlight rotated when a magnetic field was applied to a piece of flint glass withthe light passing through it. The magnetic field had to be oriented along thedirection of the lights propagation. This effect, known now as the Faradayeffect or Faraday rotation, convinced Faraday of several things. First, light

Figure 1.6 Diagram of Faradays original dynamo, which generated electricity

from magnetism.


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14 Chapter 1

and magnetism are related. Second, magnetic effects are universal and notjust confined to permanent or electromagnets; after all, light is associated

with all matter, so why not magnetism? In late 1845, Faraday coinedthe term diamagnetism to describe the behavior of materials that arenot attracted and are actually slightly repelled by a magnetic field. (Theamount 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 untilthen.)

Finally, this finding reinforced in Faraday the concept of magnetic fields

and their importance. In physics, a field is nothing more than a physicalproperty 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 magneticfields, 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 curvedlines 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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History 15

example of such a mapping.) An attempt to describe a magnetic field interms of magnetic charges analogous to electrical charges was made by

the French mathematician/physicist Simeon-Denis Poisson in 1824, andwhile successful in some aspects, it was based on the invalid concept ofsuch magnetic charges. However, development of the BiotSavart lawand some work by Ampre demonstrated that the concept of a field wasuseful in describing magnetic effects. Faradays work on magnetism inthe 1840s reinforced his idea that the magnetic field was the importantquantity, not the materialwhether natural or electronicthat producedthe field of forces that could act on other objects and produce electricity.

As crucial as Faradays advances were for understanding electricityand magnetism, Faraday himself was not a mathematical man. He hada relatively poor grasp of the theory that modern science requires toexplain observed phenomena. Thus, while he made crucial experimentaladvancements in the understanding of electricity and magnetism, he madelittle 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 theright knowledge to be able to express these experimental observations asmathematical models.


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Chapter 2

First Equation of

Electrodynamics E =

0

In this chapter, we introduce the first equation of

electrodynamics. A note to the reader: This is going to

get a bit mathematical. It cant be helped. Models of the

physical universe, like Newtons second lawF = ma,are

based in math. So are Maxwells 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; Maxwells father,

John Clerk, adopted the surname Maxwell after receiving a substantialinheritance 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 Kings College

London (later the University of London). He returned to Scotland in 1865,

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18 Chapter 2

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 ofPhysics. 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 MaxwellBoltzmann 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 Maxwells 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) particlesa 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 Maxwells equations of


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First Equation of Electrodynamics 19

Figure 2.2 Maxwell proved mathematically that the rings of Saturn couldnt 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 oclock position just inside the second outermost diffuse ringthats Earth.

If you cant see it, look for high-resolution pictures of pale blue dot on the Internet.

electromagnetism. He was strongly influenced by Faradays experimental

work, believing that any theoretical description of a phenomenon must be

grounded in experimental observations. Maxwells 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, Maxwellsequations still rule just as Newtons equations of motion rule under normal

conditions.

2.2 A Calculus Primer

Maxwells 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 Newtons 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


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20 Chapter 2

Leibnitzs 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 initialpoint (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

yaxis 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, slopem 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)

wherey is the value of the ordinate, xis the value of the abscissa, m is the

slope, andbis theyintercept, which is where the plot would intersect with

theyaxis. Figure 2.3 shows a plot that has a positive value ofm. In a plot

with a negative value ofm, 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 byy/x.


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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 anexample 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 slopemabove using symbols, where is the

Greek capital letter delta. is used generally to representchange, 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.


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22 Chapter 2

(change in temperature) or y(change inycoordinate). 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

slopeyof a curve is

y = 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 dto indicate a partial derivative, we use a symbol basedon the lowercase Greek deltaknown as Jacobis 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

fx = 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

fin the xdimension, 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


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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 xdirection.

individually. For a function of three variables, f(x,y,z), the total

differential is written as

d f = f

xdx +

f

ydy +

f

zdz, (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 variablesand 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) = 2422 = 84 = 4. The finalvalue ofF(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


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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, ordecomposed 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 aandbis 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.


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Figure 2.7 An example of a vector function F = xi + yj. Each point in twodimensions 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, (2.8)

where |a| represents the magnitude (that is, length) ofa, |b| is the magnitudeofb, andcos 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 workw, which

is defined in terms of the force vector Fand the displacement vector of a

moving objects, and the angle between these two vectors:

w =F s =|F||s| cos . (2.9)

Thus, if the two vectors are parallel ( = 0 deg, so cos = 1) the

work is maximized, but if the two vectors are perpendicular to each other


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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.

( =90 deg, socos =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 isintegration, whose representation is called anintegral, whichis represented as

ab

f(x) dx, (2.10)

where the symbol

is called the integral sign and represents the

integration operation, f(x) is called the integrandand is the function to

be integrated,dxis the infinitesimalof the dimension of the function, and

aandb 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


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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 itrepresents the area under the curve given by f(x) between the limits a

andband 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 xaxis, 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 (11), and the right triangle that is shadedgray is one-half of that total area, or 1/2 unit in area. Integration of the

function f(x) = xgives us the same answer. The rules of integration willnot be discussed here; it is assumed that the reader can perform simple

integration:

10

f(x) dx =

10

x dx = 1

2x2

10

= 1

2(1)2 1

2(0)2 =

1

2 0 = 1

2. (2.11)


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28 Chapter 2

Figure 2.10 The geometric interpretation of a simple integral is the area undera function and bounded on the bottom by the x axis (that is, y = 0). (a) Forthe function f(x) = x, the areas as calculated by geometry and integration areequal. (b) For the function f(x) = x2, the approximation from geometry is not agood 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 firstdemonstrated 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 underthe curve, now poorly approximated by the shaded triangle, is calculated

exactly with an integral:

10

f(x) dx =

10

x2 dx = 1

3x3

1

0

= 1

3 0 = 1

3. (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


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Figure 2.11 A multivariable function f(x,y)with a line paralleling theyaxis. Theequation 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 functions

values, with its projection in the (x,y) plane. The line on the surface isparallel to theyaxis, so it is showing the trend of the function only as the

variablexchanges. 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 apath

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 scalaror 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 equationP, which is a function of one

or more variables. What is the distance of the path? One way of calculating

the distance sis velocityv times timet, or

s =v t. (2.13)

But velocity is the derivative of position Pwith respect to time, or dP/dt.

Let us represent this derivative asP. Our equation now becomes

s = P t. (2.14)


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30 Chapter 2

This is for finite values of distance and time, and for that matter, for

constant P. (Example: total distance at 2.0 m/s for 4.0 s = 2.0 m/s4.0s =8.0 m. In this example, P is 2.0 m/s andtis 4.0 s.) For infinitesimalvalues 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 = Pdt. (2.15)

To find the total distance, we integrate between the limits of the initial

positiona and the final positionb:

s =

ab

Pdt. (2.16)

The point is, its not the pathPthat we need to determine the line integral;

its the change in P, denoted as P. This seems counterintuitive at first,but hopefully the above example makes the point. Its 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 theintegral, undoing our original operation, to obtain the answer. One might

have simply kept the original equation and determined the answer from

there. Well address this issue shortly. One more point: it doesnt 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 = rcosi +rsinj +0k (see Fig. 2.12), which is a function of the

variable, the angle from the positivexaxis. What is the circumference of

the circle?; that is, what is the path length as goes from 0 to2, 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, well need to derive the changein the magnitude of the vector. We start by defining the magnitude: the

magnitude|m|of a three- (or lesser-) magnitude vector is the Pythagoreancombination of its components:

|m| =x2 +y2 +z2. (2.17)


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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(2r).

For the derivative of the path/magnitude with respect to time, which is the

velocity, we have

|m| =

(x)2 + (y)2 + (z)2. (2.18)

For our circle, we have the magnitude as simply thei,j, and/ork terms ofthe vector. These individual terms are also functions of. We have

x =d(rcosi)

d =rsini,

y =d(rsinj)

d =rcosj,

z =

0.

(2.19)

From this we have

(x)2 =r2 sin2 i2, (2.20)

(y) =r2 cos2 j2,


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32 Chapter 2

and we will ignore thezpart, since its just zero. For the squares of the unit

vectors, we havei2 =j2 =i i =j j =1. Thus, we have

s =

ab

Pdt = 2

0

r2 sin2 + r2 cos2

d. (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 = 2

0r

sin2 + cos2 d. (2.22)

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

s =

20

(r

1) d =

20

r d =r 2

0

=r(2 0) =2r. (2.23)

This seems like an awful lot of work to show what we all know, that the

circumference of a circle is2r. 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

fieldFis not necessarily a vector.) In that case, the total effects of the fieldalong the line is given by

s =

p

F[x(q),y(q),z(q)] |p(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 functionsFand |p| are multiplied together.IfFis a vector field over the vector pathp(q), denotedF[p(q)], then the

line integral is defined similarly:

s =

p

F[p(q)] p(q) dq. (2.25)


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Here, we need to take the dot product of the F andpvectors.A line integral is an integral over one dimension that gives, effectively,

the area under the function. We can perform a two-dimensional integralover the surface of a multidimensional function, as pictured in Fig. 2.13.

That is, we want to evaluate the integral

S

g(x,y,z) dS, (2.26)

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 generallycalled asurface 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 functions angle with respect to the projected plane (the

region Rin 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.) Ifn(x,y,z) is the unit vector that

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


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34 Chapter 2

defines the line perpendicular to the plane marked out by g(x,y,z)(called

thenormal vector), then the value of the surface integral is given by

R

g(x,y,z)

n(x,y,z) kdxdy, (2.27)

where the denominator contains a dot product, and the integration is over

the x andylimits of the regionRin 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)

1 +

f

x

2+

f

y

2dxdy, (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 onlytwo variables, you let z = f(x,y) and substitute the expression for z into

the functiong, ifz appears in the function g.

If, instead of a scalar function gwe had a vector functionF, 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 definednas the vector normal to the surface, well use it again;

we want the surface integral involvingFn, or

S

F n dS. (2.29)

For a vector function F = Fxi + Fyj + Fzk and a surface given by the

expression f(x,y)z, this surface integral is

S

F n dS =

R

Fx

f

x Fy

f

y+ Fz

dxdy. (2.30)

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

representing this?


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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 isF n? Actually, its just a dot product, but the integralS

F n dS (2.31)

is called theflux ofF. 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 iscut 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 more-

complicated geometryvector geometryto 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 ofF. 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.


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36 Chapter 2

Figure 2.15 What is the surface integral of a cube as the cube gets infinitely

small?

that the normal vector is the same as the kvector. For this surface by itself,

the flux is then S

F k dS. (2.32)

IfF is a vector function, its dot product with k eliminates thei andj parts

(sincei k =j k =0; recall that the dot product a b =|a||b| cos, where|a| represents the magnitude of vector a) and only the z component ofFremains. Thus, the integral above is simply

S

FzdS. (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


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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 atz + z/2. Thus, we have for the flux at the top surface:

top flux Fzx,y,z +

z

2

xy, (2.34)

where the symbol means approximately equal to. It will becomeequal to when the surface area shrinks to zero.

The flux ofF 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 Fzx,y,z z

2

xy. (2.35)

The total flux through these two parallel planes is the sum of the twoexpressions:

flux Fzx,y,z +

z

2

xy Fz

x,y,z z

2

xy. (2.36)

We can factor the xyout of both expressions. Now, if we multiply this

expression by z/z(which equals 1), we have

fluxFz

x,y,z +

z

2

Fz

x,y,z z

2

xyz

z. (2.37)

We rearrange as follows:

flux Fz x,y,z + z2

Fz x,y,z z

2 z xyz, (2.38)

and recognize that xyzis the change in volume of the cube V:

fluxFz

x,y,z + z

2

Fz

x,y,z z

2

z

V. (2.39)


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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 ofFz with respect to z! Of course, its a partial derivativebecauseF depends on all three variables, but we can write the flux more

simply as

flux = Fz

z V. (2.40)

A similar analysis can be performed for the two sets of parallel planes;only the dimension labels will change. We ultimately obtain

total flux = Fx

xV +

Fy

yV +

Fz

zV

=

Fx

x+

Fy

y+

Fz

z

V. (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 divergenceof the vector functionF:

divergence ofF Fxx

+Fy

y+

Fz

z(whereF = Fxi + Fyj + Fzk). (2.42)

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):

divergence ofF =Fx

x

+Fy

y

+ Fz

z

= limV0

1

V

(total flux)

= limV0

1

V

S

F ndS. (2.43)

There are two abbreviations to indicate the divergence of a vector

function. One is to simply use the abbreviation div to represent


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divergence:

divF = Fxx

+ Fyy

+ Fzz

. (2.44)

The other way to represent the divergence is with a special function. The

function (called del) is defined as

=i

x+j

y+ k

z. (2.45)

If we were to take the dot product betweenand F, we would obtain thefollowing result:

F =i

x+j

y+ k

z

Fxi + Fyj + Fzk

=

Fx

x +

Fy

y + Fz

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 kor kj) all equal zero and cancel out, while like terms (that is, jj) allequal 1 because the angle between a vector and itself is zero andcos 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 = aibi =a1b1 + a2b2 + a3b3, (2.47)

wherea1,a2, etc., are the scalar magnitudes in the x,y, etc., directions. So,

the divergence of a vector functionF is indicated by

divergence ofF = 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


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40 Chapter 2

in space. For example, for the field

F = x3

i, (2.49)

the divergence is

F =3x2, (2.50)

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 = x2i, (2.52)

whose divergence is 2x, the vectors grow farther and farther apart as x

increases (see Fig. 2.17).

2.5 Maxwells 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 Coulombs

law, which in vector form is

F =q1q2

r2 r, (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 chargesq1andq2. The modern version

of Coulombs law includes a conversion factor between charge units


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Figure 2.16 The divergence of the vector field F = xi + yj is 2, indicating aconstant 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 = x2i whose divergence is equal to 2x. The arrowheads, which give thedivergence 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 thevectors become; that is, the greater the divergence.


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42 Chapter 2

Figure 2.18 It is an experimental fact that charges exert forces on each other.

That fact is modeled by Coulombs law.

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

F =q1q2

40r2r, (2.54)

where 0 is called the permittivity of free space and has an approximate

value of8.854 . . . 1012 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 = F

q2=

q1

40r2r, (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 doesnt 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 is given by

=

S

E n dS, (2.56)

which is perfectly analogous to our previous definition of flux.


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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 sphericalsurface at any of its points (Fig. 2.19). Since we know the definition of

E from Coulombs law, we can substitute into the expression for electric

flux:

=

S

q1

40r2r rdS. (2.57)

Figure 2.19 A charge in the center of a spherical shell with radiusrhas a normalunit vector equal to r in the radial direction and with unit length at any point on the

surface of the sphere.


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44 Chapter 2

The dot productr ris simply 1, so this becomes

=S

q140r2

dS. (2.58)

If the charge q1 is constant, 4 is constant, 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

= q140r2

S

dS. (2.59)

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:4r2. Thus, we have

=

q1

40r2 4r2. (2.60)

The 4, the, and ther2 terms cancel. What remains is

= q1

0. (2.61)

Recall, however, that we previously defined the divergence of a vector

function as

divF = Fx

x+

Fy

y+

Fz

z= lim

V01

V(total flux)

= limV0

1

VS F ndS. (2.62)Note that the integral in the definition has exactly the same form as the

electric field flux . Therefore, in terms of the divergence, we have for E:

divE = limV0

1

V

S

E ndS = limV0

1

V = lim

V01

V

q1

0, (2.63)


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where we have made the appropriate substitutions to obtain the final

expression. We rewrite this last expression as

divE =

limV0

q1V

0. (2.64)

The expression q1/V is simply the charge density at a point, which we

will define as. This last expression becomes simply

divE =

0 . (2.65)

This equation isMaxwells first equation. It is also written as

E = 0

. (2.66)

Maxwells first equation is also called Gauss law, after Carl Friedrich

Gauss, the German polymath who first determined it but did not publishit. [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 sideof the equation) is proportional to the charge inside the closed surface.

This may not seem like much of a statement, but then, were only getting

started.


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Chapter 3

Second Equation of

Electrodynamics B =0

Maxwells first equation dealt only with electric fields,

saying nothing about magnetism. That changes with the

introduction of Maxwells second equation. A review of

the mathematical development of the divergence, covered

in detail inChapter 2,may be helpful.

3.1 Introduction

A magnet is any object that produces a magnetic field. Thats a rather

circular definition (and saying such is a bit of a pun, when you understand

Maxwells 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 matteris affected by magnets. Its 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 rocks

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. Magnetites chemical

formula is Fe3O4, and the ore is actually a mixed FeO-Fe2O3 mineral.

Magnetite itself is not uncommon, although the permanently magnetized

47


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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 Mgare 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

effectsof 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 avectorfield. 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 (changesthe 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 Faradays 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 linesabout 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 representpoles 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


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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 AcademyDepartment 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

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Maxwell's Equations of Electrodynamics an Explanation [2012]