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10.1 Magnetic Force on Particles 10.2 Electric and Magnetic There are relatively few problems in electromagnetics that can be solved without some sort of.
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Instructor's Manual to accompany. ELECTROMAGNETIC FIELD THEORY: A PROBLEM SOLVING APPROACH by. Markus Zahn. Professor of Electrical Engineering and Computer
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Concrete examples are liberally used and numerous graphs and sketches are given. I have found in many years of teaching that the solution of most problems
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Problems & Solutions on Electromagnetism. 4περ τ2. $₁E.α = $1/12/(1-Var) dr. *[-e()+2vf(六)]。 =0. From Coulomb's law F124or2. 9192 Cria we can obtain the
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This series of physics problems and solutions which consists of seven volumes Mechanics
Solved Problems in Electromagnetics
Problems of type A are thought for beginners in electricity and magnetism or for lectures on General Physics where definitions and concepts about this subject
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First edition published as 2000 Solved Problems in Electromagnetics copyright 1992 by McGraw-Hill. Printed in the United States of America. ISBN: 1891121464.
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This solution can then be used to solve various composite problems. In particular the E- plane step is solved using this representation as opposed to a
Solved Problems in Electromagnetics
solved problems the majority of which have been proposed by the Department of Physics II and Electromagnetism and Waves in the Mining and Energy School.
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PUBLICATIONSA11100989856
Montgomery,JamesPa/Electromagnetlcbou
QC100.U556V164;1979C.1NBS-PUB-C1979
QNBSMONOGRAPH164
ElectromagneticBoundary-ValueProblems
BasedUponaModificationof
ResidueCalculusand
FunctionTheoreticTechniques
QC. 100NATIONALBUREAUOFSTANDARDS
AbsolutePhysicalQuantities' - RadiationResearch - Thermodynamicsand MolecularScience - AnalyticalChemistry - MaterialsScience. AppliedMathematics - ElectronicsandElectricalEngineering^ - Mechanical EngineeringandProcessTechnology' - BuildingTechnology - FireResearch -ConsumerProductTechnology - FieldMethods.
SystemsandSoftware - ComputerSystemsEngineering - InformationTechnology. mailingaddressWashington,D.C.20234.ElectromagneticBoundary-ValueProblems
andFunctionTheoreticTechniques©QJCOJamesPatrickMontgomery^'^^'^
DavidC.ChangC^
CenterforElectronics
andElectricalEngineeringNationalEngineeringLaboratory
NationalBureauofStandards
Boulder,Colorado80302
IssuedJune1979
NationalBureauofStandardsMonograph164
CODEN:NBSMA6
U.S.GOVERNMENTPRINTINGOFFICE
WASHINGTON:1979
StockNo.OO3-O03-O2O75-3Price$5.00
IPREFACE
need. thematerialbeingmeasured. materials iscurrentlybeingactivelypursued.W.E.Little
NationalEngineeringLaboratory
NationalBiureauofStandards
Boulder,Colorado80302
iiiCONTENTS
PARTISOLUTIONOFCLOSEDREGIONPROBLEMS
PAGE1.INTRODUCTION(PARTI)1
1.Introduction4
2.TheCanonicalProblem4
3.THETRIFURCATEDWAVEGUIDE15
1.Introduction15
2.FormulationoftheEquations15
3.Asymptotics18
4.TruncationoftheEquations21
5.DielectricLoading22
6.TheScatteredFields25
7.NumericalResults26
7.1Introduction26
7.2TheTrifurcatedWaveguide26
4.THEN-FURCATEDWAVEGUIDE34
1.Introduction34
2.FormulationoftheEquations34
3.Asymptotics37
4.TruncationoftheEquations42
5.DielectricLoading45
6.TheScatteredFields48
7.NumericalResults48
7.1Introduction48
7.2TheN-FurcatedWaveguide
'495.OTHERCLOSEDREGIONPROBLEMS53
1.Introduction53
2.TEEigenvaluesofRidgedWaveguide53
3.ScatteringbyaDielectricStep58
4.DielectricLoadedN-FurcatedWaveguide58
6.CONCLUSIONS(PARTI)60
PARTII
SOLUTIONOFOPENREGIONPROBLEMS
CHAPTERPAGE
7.INTRODUCTION(PARTII)64
1.Introduction65
2.TheCanonicalProblem65
vCHAPTERPAGE
2.1Introduction65
2.2TheElectricWallCase67
2.3TheMagneticWallCase.80
1.Introduction88
2.FormulationoftheEquations88
3.NumericalSolution91
4.NumericalResults92
10.AFINITEPHASEDARRAY.102
1.Introduction102
2.FormulationoftheEquations102
2.1Introduction102
2.2TheElectricSymmetryWall102
2.3TheMagneticSymmetryWall111
3.NumericalResults115
3.1TheElectricWallCase115
3.2TheMagneticWallCase121
3.3SuperpositionoftheResultsforanElectricandaMagneticWall - 126
1.Introduction132
2.FormulationoftheEquations133
2.1Introduction133
2.2TheElectricSymmetryCase133
2.3TheMagneticSymmetryCase136
3.NumericalResults138
3.1Introduction138
3.2TheElectricWallCase138
3.3TheMagneticWallCase143
3.4SuperpositionoftheResults143
12.OTHEROPENREGIONPROBLEMS148
1.Introduction148
4.RadiationfromaSlotinaWaveguideWall157
13.CONCLUSIONS(PARTII)157
14.COMMENTSANDFINALSUMMARY159
BIBLIOGRAPHY161
APPENDIXB:AsymptoticBehaviorofthePerturbationSumfortheE-PlaneStep - 165Waveguide169
viJamesPatrickMontgomery*
DavidC.Chang**
ABSTRACT
techniquesispresented. slotinawaveguidewall. alsoincluded. PARTISOLUTIONOFCLOSEDREGIONPROBLEMS
CHAPTER1.INTRODUCTION
secondpartofthismonograph. strictlyTEorTMsolutionispossible. I nonsolublegeometry. theGSMTsolution. 2 guide. 31.Introduction
2.TheCanonicalProblem
mannerandwillnotbegiven. aregivenby[MittraandLee,1971] In=0 /NYz-Yz.(o)na..naAe+Aenn mr,,cos - (x-x")aU In=0B<°)en
~YuZY, z 'nb^^'nb+Ben n-rr,.cos - (x-x") =In=0 /N-YzC^°)e+Cenn
Yzncmr.cos - (x-x,
c1V(mr/h)^-,n7r/h>k
'nh -nTTmr/hY^Bbee""^"=2jkbA^"-*e°°6mbmmoomo
y\J-mbo YZ ,(o)niT.mrbnao°°A - sme
+(.1)^+1I^§S n=lY~Ymbna mrCOA - sm - YZmrbnao m=0,1,2, n=lYu+Ymbna (2.1)Y^B^°)beembmm
-yi,zmbo-jkz =2jkbAe°°&oomo Yz .(o)niT.mrbnao00A - sme
n=lYu+Ymbna .mrooA - sm _l_ ,.,m+lynaaY~Yn=lmbna
- YZmrbnao m=0,1,2,(2.2)Yz,,jkz
YCcee=2ikcAeomcmmoomo
YZ .(o)nir.mrbnaoA - smenaaY~Yn=lmcna
I.-Yznir.riTTDnao00A - sine
+I"%+^ ;m=0.1,2,...,(2.3) n=lmcna YZ .(o)niT.nirbnaoA - sine1Y+Yn=lmcna
~YZnirJnirbnao00A - sinnaa+I^^;m=0,1,2,....(2.4)Y~Yn=lmcna where .2,m=0 m1,m>_1 and6istheKroneckerdelta,mn toBandC.nn eliminatethesummationsandfindA=^c^°^+^b(°^(2.5)oaoaoK^'-iJ
whichisauniquefeatureoftheTMsolution. j>,(2:6)2Triw-Yi_2iTjoj-Y'mbmc
7 sothattheintegralsexistandarezero.27rj(o-Yn=ly-Yn=ly+Ymcnamcnamc
and ,,vm+l,,00RES[T,Y](-1)^T(a))da)^Vna''2TTj^w-Y,~Y-Y-umbn=lnamb
ooRES[T,-Y] (_l)ynaY+Yn=lnamb
+(-1)^"'^T(y^^)=0.(2.8) ofequationsareidenticalprovidedthat f.yTiTiorm1,(o)mir.mirb^ma^o^"(i)RES[T,-YJ=-A - smem=1,2,•••mamaa u-Yz (ii)RES[T,Y^^]=-A^ - sm - em=l,2,...Wecanalsoconsidertheintegrals
i-]J^^TUOdo)1T(a))da)^29)277j^(jj+Y^2T:j^co+Ymbmc
(2.1)and(2.3) Yz (vi)T(-Y)=CYce™'^°m=1,2,...mcmmc jkz (viii)T(-jk^)=2jk^c[C^-A^°^]e°° Alternatively,(ix)canbereplacedbyA^°^= - C+ - Bwhichisthenanalogoustotheoaoao (x)T((o)=0(0)-^/^),|a)| n(a),Y)n((i),Y)T(a))=KH(u))^7^-71(2.10)
a where coh/nirn(w,Yj^)=n n=l 1- ^nh e ,a)/ir[bInJ-+cInr]T(a))=KH(a))w""^/'^e^^,03|a)|
Inorderforcondition(x)tobemetwemusthave
H(a))=e"'^'^^'-^ln(b/a)+cln(c/a)](2.11)
ii(jk,v)n(jkY J oa -jkZ =2jk^ce°° 9Hence,thesolutionis
T(a))=2ik^e"^''^'"H(a))F(a.)(2.12)
whereF(w)= n(a),Y^)n(a),Y^) n(a),Y^) ciThereflectioncoefficientisgivenby(ix)
-j2kzH(-jk)F(-jk)-c00oo o"a^H(jk^)F(jk^)(2.13) (Marcuvitz,1964). rT(a))=H(a))F(a3)K-(a)-jk)H - no'••111 -
V (b) n=lnb (c)(a) 'w-Y^1w+Yn=lncn=lna (2.14) whereH(a))=e-co/TT±bInb/a+cInc/al
-jkz,,H(-Y)F(-Y)(y+jk)^nananaon
(a) YZ .(o)niT.nirbnao=-A - sinenaan=1,2,(2.16) "/N,.(n),^^^ric(c) "(\c)^^\c^y^"nc ^\-yz =-C^^^Yce° nncn-1,2,(2.17) ^^(^>r^'-^nbj^o^(b) "^^nb^^^^nb^77^nnb =(-1)'^B^°^YKbennb 'nbon=1,2,(2.18) 10 indicatedanumberofsuchproblems. (c)^ (2.14)wehave fT(a))=H(a))F(a))
oogK-(w-jk)In=lnc
(3.1) expansion.Using(2.17)wehavethatSn=\'n^(3-2)
dielectricas "rz (\,=JT)ecos - (x-b)(3.3) yn=0^ 11 T b i±z (a)E-PlaneStep (b)AuxiliaryGeometryFig.2.3.1:TheE-PlaneStepandtheAuxiliary
Problem
12 where nc mr O wecaneasilyfindthatC^°^=RC=nnn
ey-rncnc ey+rncnc -2y& eCn(3.4) and r6+Y52eY..D=encnc^^(o)
ney-rn(3.5) ncnc for6=01/2g=0(n'D),n->oo
''nnD=0(n"^^^"^),n-^n(3.6)
where hence1,-1(e-1)A= - sin";,.,
IT[2(e+l)
g^=0(n),n^ (3.7) (3.8) ^oorrCOT(w)^H((i))F(a))K-((o-jk)iI+iI^
VJ (3.9) II -A, dominate.Hence,wemusthaveK-Ig"-gIn^=0.(3.10)
n=ln=N+lThisargumenthasassumedthat
13 n=N+l"^nc (3.11) as|aj|->ThisisshowninAppendixB.T(-Y)=YcCmcmcm
Using(3.2)and(3.4)wehave
mcmcmmm ,Ng_"H(-Y)F(-Y)[K-(Y+jk){I+gImcmc'omco-,Y+Yt
•n=lncmcn=N n-l-A],,,-1^.-1 ;N=YcRKgY+Y;mcmmnN+1ncmc (3.12) form=1,2,...N,KF(jk)H(jk)=2jk -
ooooa ,(o)T(-jk)2jkco(3.13) properties(v)and(iv)ofsection2, (seeRoyerandMittra,1972). 14CHAPTER3.THETRIFURCATEDWAVEGUIDE
1.Introduction
2.FormulationoftheEquations
problem.T^(a))=H^(a))F^(a))oo(jj+Yn=lno
(2) oo^-co-Yn=lnc (2.1) (2.2) whereH^(a))=e
-a)/Tr[b^ln(b^/c)+b2ln(b2/c)]E^i^)=e
-aj/Tr[bln(b/a)+cInCc/a)]oo n(a),Y,)n(a),Y,)bb "i'"' =nt...) n(co,Y^)n(to,Y^) 15 r T (a)TrifurcatedWaveguide T t: X z=0z=A t 1 i [1 1 T ("b)AuxiliaryGeometryFig.3.2.1:TheTrifurcatedWaveguideandthe
AuxiliaryProblem.
16 /\/o\nn canonicalproblem(seeFig.2.2.1). section2,Chapter2as .(1) K o (2)2jkbJC+-B^°Jo21o0,2
2jkcoA-C+oo
/(H^ak^)F^ak^)H2(jk^)F2(jk^)
(2.3) (2.^) whereC+=b,/cB^°J+b,/cB^°;o2o,21o,l
A=c/aC"^+b/aB^°^oooo,o
Chapter2.
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