Demystifying Electromagnetic Equations: A Complete Explanation of EM Unit Systems and Equation Transformations

345 Pages·2001·2.632 MB·English

by  Douglas L. Cohen

Demystifying
Electromagnetic

Equations

A Complete Explanation of EM Unit
Systems and Equation Transformations

Douglas L. Cohen
Demystifying
Electromagnetic
Equations
A Complete Explanation of EM Unit
Systems and Equation Transformations
Demystifying
Electromagnetic
Equations
A Complete Explanation of EM Unit
Systems and Equation Transformations
Douglas L. Cohen
Bellingham, Washington USA
Library of Congress Cataloging-in-Publication Data
Cohen, Douglas L.
Demystifying electromagnetic equations : a complete explanation of EM unit systems
and equation transformations / by Douglas L. Cohen.
p. cm.
Includes bibliographical references and index.
ISBN 0-8194-4234-8
1.Electromagnetic theory–Mathematics. 2. Electric units. I. Title.
QC670 .C49 2001
530.14'1'0151--dc21
2001032770
Published by
SPIE—The International Society for Optical Engineering
P.O. Box 10
Bellingham, Washington 98227-0010
Phone: [phone]
Fax: [phone]
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Copyright © 2001 The Society of Photo-Optical Instrumentation Engineers
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.
Printed in the United States of America.

Contents
Preface vii
1 OutlineofNon-ElectromagneticSystemsofUnits . . . . . . . . . . . 1
1.1 Thebasicideaofaunit. . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Fundamentalandderivedunits . . . . . . . . . . . . . . . . . . . 4
1.3 Analysisofequationsandformulas . . . . . . . . . . . . . . . . . 9
1.4 Dimensionlessparameters . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Thecgsandmksmechanicalsystemsofunits . . . . . . . . . . . 14
1.6 TheUandNoperators . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Temperatureunits . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.8 Dimensionlessunits . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9 Removaloftheuniversalgasconstantfromtheidealgaslaw . . . 29
1.10 Removalofthespeedoflightfromrelativisticequations . . . . . 37
1.11 Invariantunits,connectingunits,andadditionofextradimensions 50
1.12 Simultaneousremovalofh¯,c,andk . . . . . . . . . . . . . . . . 56
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2 UnitsAssociatedwithNineteenth-CenturyElectromagneticTheory 65
2.1 Electricfields,magneticfields,andCoulomb’slaw . . . . . . . . 66
2.2 Combinedsystemsofelectricandmagneticunits . . . . . . . . . 70
2.3 Theesuandemusystemsofunits . . . . . . . . . . . . . . . . . . 75
2.4 TheDandBfields . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.5 Theelectricandmagneticpotentials . . . . . . . . . . . . . . . . 89
2.6 Thesystemofpracticalunits . . . . . . . . . . . . . . . . . . . . 91
2.7 The“ab-”and“stat-”prefixes . . . . . . . . . . . . . . . . . . . . 96
2.8 Theesuqandemuqsystemsofunits . . . . . . . . . . . . . . . . 104
2.9 The esuq and emuq connectionwith the esu and emu systems
ofunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.10 Directconversionbetweentheesuandemusystemsofunits . . . 125
2.11 TheBandH fieldsatthestartofthetwentiethcentury . . . . . . 128
2.12 Electromagneticconceptsusedtoanalyzebulkmatter. . . . . . . 133
Appendix 2.A: Magnetic-field measurementin the early nine-
teenthcentury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Appendix2.B:Dimensionlessvectorderivatives . . . . . . . . . . 139
v
vi CONTENTS
3 UnitsAssociatedwithTwentieth-CenturyElectromagneticTheory . 149
3.1 Maxwell’sequations . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.2 TheGaussiansystemofunits . . . . . . . . . . . . . . . . . . . . 151
3.3 RationalizationandtheHeaviside-Lorentzsystem . . . . . . . . . 166
3.4 Gaussian and Heaviside-Lorentz systems with c=1 and h¯ =
c=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.5 Equivalenceoftheesu,emu,andGaussiansystemswhenc=1 . 180
3.6 Rationalizedandunrationalizedmkssystems . . . . . . . . . . . 183
3.7 Conversionofequationstoandfromtheunrationalizedmkssystem190
3.8 Conversionofequationstoandfromtherationalizedmkssystem 204
3.9 Evaluationoftherationalizedmkssystem . . . . . . . . . . . . . 223
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4 Two Standard Shortcuts Used to Transform Electromagnetic
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
4.1 Thefree-parametermethod . . . . . . . . . . . . . . . . . . . . . 225
4.2 Basicequationsusingthefreeparametersk ,(cid:1)µ,(cid:1)ε,and(cid:6) . . . . 236
0
4.3 Understandingthesubstitutiontables . . . . . . . . . . . . . . . . 274
4.4 Usingthesubstitutiontables . . . . . . . . . . . . . . . . . . . . . 278
4.5 Problemswiththefree-parametermethodandsubstitutiontables 291
Appendix.Substitutiontables . . . . . . . . . . . . . . . . . . . . 292
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
Bibliography 325
Index 329
Preface
In classical Newtonian mechanics, equations and formulas never change form.
However,thesamethingcannotbesaidabouttheequationsandformulasofelec-
tromagnetictheory,which oftenchangeformwhenconvertedfromonesystemof
unitstoanother.Forthisreasonelectromagnetictextbooksarealmostalwayswrit-
ten using a single systemof units, and the technicalprofessionals who read them
end up being comfortable in only that system. When they encounter a new and
important formula in unfamiliar units later on, they must either use a conversion
table to change the formula to their preferred system of units or try to become
familiarwiththeformula’sunits.Althoughconversiontablesusuallygivethecor-
rect answer, they turn their users into computers who must push around numbers
andvariableswithoutanytrueunderstandingofwhatisbeingdone.Itisprobably
unwisetorelyblindlyonconversiontablesifonemustbeabsolutelysurethetrans-
formed formula is correct. Thatleavesthe secondoption: becomingfamiliar with
theformula’sunits.Thedrawbackhereisthatevenifatextbookcanbefoundthat
usestheformula’sunits,ithasbeenwrittentoteachthebasicprinciplesofelectro-
magnetismratherthanwhatthetechnicalprofessionalislookingfor,i.e.,adetailed
explanationofhowtoconvertequationsfromonesystemofunitstoanother.This
bookprovidesexactlythat,whileatthesametimeassumingagood—butnotnec-
essarilyadvanced—understandingofelectricityandmagnetism.
There are five widely recognized systems of electromagnetic units; four are
connected to the centimeter-gram-second (cgs) system of mechanical units and
oneis connectedto themeter-kilogram-second(mks)systemofmechanicalunits.
The four connected to the cgs mechanical units are the cgs Gaussian system, the
Heaviside-Lorentz system, the cgs electrostatic system, and the cgs electromag-
netic system. The system connected to the mks mechanical units is the Système
International or rationalized mks system. The units of the Système International
orrationalized mkssystemare often called SIunits. Thecgselectrostatic andcgs
electromagneticsystemsofunitsweredevelopedfirst.Thesearetheunitsinwhich
Maxwell’s equations—the foundation of classical electromagnetic theory—were
firstproposedduringthemiddleofthenineteenthcentury.TheHeaviside-Lorentz
and cgs Gaussian systems were introduced at the end of the nineteenth century,
followed almost immediately at the beginning of the twentieth century by the ra-
tionalizedmkssystem(SIunits).Therationalizedmkssystemisthemostpopular
electromagneticsysteminusetoday;almostallintroductorytextbooksuseSIunits
to explain the principles of electricity and magnetism. This book explains all five
systems in depth, along with two systems of mostly historical interest; the nine-
teenthcenturysystemof“practical”unitsandtheunrationalizedmkssystem.
vii
viii PREFACE
Onechronicproblemfoundin manyarticlesandbooksaboutsystemsofunits
is that the customary language of physics and engineering can permit ambiguity
whilesoundingexact.Suppose,forexample,wesay
“The electric-current unit in the cgs electrostatic system is the statamp and the
electric-currentunitinthecgselectromagneticsystemistheabamp,with
1abamp=c·statamp
wherec isthespeedoflightincgsunits.”
This seems clear enough, but notice that c =2.99792·1010cm/sec in cgs units.
In the above equation, should we take “c” to be “2.99792·1010” or “2.99792×
1010cm/sec?”Anaivestudentmightassumecwasthepurenumber2.99792·1010
becauseobviously all electric current is the same sort of thing and must have the
same type of unit; but later on, possiblyin anotherbook,that same studentmight
discover the cgs electrostatic unit of current is gm1/2 ·cm3/2·sec−2 and the cgs
electromagnetic unit of current is gm1/2 ·cm1/2 ·sec−1. At this point confusion
setsin,becausethisisnotcompatiblewiththeequation1abamp=c·statamp,no
matterhowitisinterpreted.
Toavoidthissortofambiguity,weintroduceheretheideaofUandNoperators,
with a U operator returning just the units associatedwith a physicalquantity and
an N operator returning just the pure number, or numeric component, associated
withaphysicalquantity.Inthecgssystem,forexample,wehave
N(c)=2.99792·1010
cgs
and
U(c)=cm/sec.
cgs
Authorswhoputtheequation1abamp=c·statampintheirbooksandarticlesare
usingitto saythat N(c)istheconversionfactorbetweenthenumericcomponent
cgs
ofthecurrentI incgselectrostaticunits, N(I),andthenumericcomponentofthe
esu
currentI incgselectromagneticunits, N (I).
emu
N(I)= N(c)· N (I).
esu cgs emu
TheUoperatorcanbeusedtoemphasizethattheunitofcurrentintheelectrostatic
systemisnotthesameastheunitofcurrentintheelectromagneticsystem.
gm1/2·cm3/2 gm1/2·cm1/2
= U(I)(cid:5)= U (I)=
sec2 esu emu sec
PREFACE ix
TheUandNoperatorsmakeiteasytobepreciseaboutthemathematicalrelation-
shipsbetweendifferentsystemsofunits.
TheabbreviationsoftheSIunitsare,unfortunately,anotherpossiblesourceof
confusionwhen separating equationsinto numeric componentsand units. For ex-
ample,thestandardabbreviationfortheSIunitofcharge,the coulomb, is C. The
capacitanceofacircuitelementisalsotraditionallyrepresentedasC,andwehave
already seenthat c is usedto representthe speedoflight. If all three quantities—
the coulomb, the speed of light, and the capacitance—have to be included in the
sameequation,therewillbeproblems.Toavoidthissourceofconfusion,wehave
lengthened the standard abbreviations for the electromagnetic units, representing
coulomb by coul,ampere by amp, and so on. This makes the notation less con-
fusing,butthereadershouldnotethattheabbreviationsusedhere,althougheasily
understandable, are not the official, internationally approved symbols for the SI
units.Theseinternationalsymbolsare,inanycase,offairlyrecentvintageandcan
befoundinvirtuallyallmoderntextbooksonelectromagnetictheory.
One final point worth mentioning is how we treat rationalization of electro-
magnetic equations. During the middle of the twentieth century it became clear
that there were two different schools of thought concerning the rationalization of
electromagneticequations:onethatitwasarescalingoftheelectromagneticunits,
and the other that it was a rescaling of the electromagneticquantities themselves.
Bothviewscanbeusedtodeducethesamesystemsofelectromagneticequations,
and both views allow engineers and scientists to transform electromagnetic mea-
surementsfromonesystemtoanothercorrectly.Intheend,neithersideconvinced
the other of the correctness of its views and the controversy faded away. For the
purposes of this book, we take the position that rationalization is a rescaling of
electromagneticphysicalquantitiesratherthanachangeofunits,notonlybecause
it is then easier to describe the units of the rationalized and unrationalized elec-
tromagnetic systems but also becauseit makes the transformation of equations to
andfromrationalizedelectromagneticsystemsa straightforward process.Theop-
posite position, that rationalization just involves rescaled units, is not necessarily
incorrect—thatis,afterall,howtheideaofrationalizationwasfirstproposedinthe
nineteenthcentury—butitcaneasilybecomeconfusinginabookofthissort.