1 MAKERERE UNIVERSITY DEPARTMENT OF ELECTRICAL ENGINEERNG ELE2211: ELECTROMAGNETIC FILEDS CLASS NOTES 2Table of Contents CHAPTER 1: RELATIONSHIP BETWEEN FIELD AND CIRCUIT THEORY .....................................3 1.1.INTRODUCTION ........................................................................................................................... 3 1.2.CIRCUIT LAWS OBTAINED USING FIELD QUANTITIES ...................................................... 3 1.3.MAXWELLS EQUATIONS AS GENERALISATIONS OF CIRCUIT EQUATIONS .............. 5 1.4.BREAK DOWN OF SIMPLE CIRCUIT THEORY IN PROBLEM ANALYSIS ......................... 8 CHAPTER TWO: UNBOUNDED WAVE PROPAGATION ............................................................... 10 2.1.THE WAVE EQUATION IN A PERFECT DIELECTRIC ......................................................... 10 2.2.UNIFORM PLANE WAVES ........................................................................................................ 12 2.3.FREQUENCY DEPENDENCE OF THE CLASSIFICATION MATERIALS .......................... 14 2.4.WAVE PROPAGATION IN A CONDUCTIVE MEDIUM .......................................................... 15 2.5.POWER FLOW IN ELECTROMAGNETIC FIELDS ................................................................ 18 2.6.REFLECTION AND REFRACTION OF UNIFORM PLANE WAVES .................................... 21 2.7.POLARISATION .......................................................................................................................... 24 CHAPTER 3: WAVE PROPAGATION IN TRANSMISSION LINES .................................................. 30 3.1.INTRODUCTION ............................................................................. ERROR! BOOKMARK NOT DEFINED. 3.2.TRANSMISSION LINE EQUATIONS (DISTRIBUTED CIRCUIT ANALYSIS) .................... 30 3.3.STANDING WAVES ON TRANSMISSION LINES .................................................................. 35 3.4.TRANSMISSION LINES MATCHING CONSIDERATIONS................................................... 38 3.5.GRAPHICAL AIDS TO TRANSMISSION LINE CALCULATIONS ........................................ 44 CHAPTER 4: ELECTROMAGNETIC WAVE PROPAGATION IN WAVEGIDES . ERROR! BOOKMARK NOT DEFINED. 4.1.THE INFINITE PLANE WAVEGUIDE ........................................... ERROR! BOOKMARK NOT DEFINED. 4.2.THE RECTANGULAR WAVEGUIDE. ........................................... ERROR! BOOKMARK NOT DEFINED. 4.3.CIRCULAR WAVEGIDES ............................................................... ERROR! BOOKMARK NOT DEFINED. CHAPTER 5: WAVE ROPAGATION IN OTHER SYSTEMS ........... ERROR! BOOKMARK NOT DEFINED. 6.1.PLASMAS ......................................................................................... ERROR! BOOKMARK NOT DEFINED. 6.2.MICROSTRIP TRANSMISSION LINES ................................................................................... 50 6.3.PROPAGATION IN OPTICAL FIBERS .................................................... ERROR! BOOKMARK NOT DEFINED. REFERENCES: .................................................................................... ERROR! BOOKMARK NOT DEFINED. APPENDICES ...................................................................................... ERROR! BOOKMARK NOT DEFINED. APPENDIX A: GRAPHICAL SOLUTION TO DOUBLE STUB MATCHING ....... ERROR! BOOKMARK NOT DEFINED. CHAPTER 1: RELATIONSHIP BETWEEN FIELD AND CIRCUIT THEORY 1.1.INTRODUCTION Conventionalcircuittheory,wherewedealwithVoltage,VandCurrent,I,and fieldtheory,whereweusethefieldvectorsE,D,B,H,andJareinter-related. Considerationofcircuitsfromeitherpointofviewgivesthesameresults. However,therearecertaininherentassumptionsinthecircuittheoryapproach, which become invalid as circuit dimensions and the impressed signal wavelength become comparable. This necessitates either the use of field theory, which is the more general approach or a modification of the circuit theory approach. In this chapter, we shall see how the two are related, and why circuit theory has limitations.Itwillbeshownthatthenormalexpressionscanbeobtainedusing fieldtheory,andthatMaxwellsequations,thefourcommandmentsof electromagneticfieldpropagation,canbeobtainedasgeneralizationsofcircuit expressions. 1.2.CIRCUIT LAWS OBTAINED USING FIELD QUANTITIES (1)Ohms law: Consider the conducting rod in figure 1.1 with parameters as shown Figure 1.1: Conducting Rod of Uniform cross-section and current density Ifis the electric field at a point, thenJ =and q qp pJ Jl ldl dl J = = = 1.1 Note that we have assumed a uniform rod with a uniform current density, J. Since: qpdl V =(potential difference between p and q) J I = (Current through the rod) p q A l J,4 lR=(Resistance of the rod) Equation 1.1 states that V=IR, which is Ohms law derived from field theory. (ii)The series R-L-C circuit Figure 1.2 shows a simple series R-L-C circuit Figure 1.2: Simple R-L-C Circuit Recall Faradays law in integral form: dl dst = [Surface not changing]1.2 ConsidertheRHS.Sincethecircuitinfigure1.2istimeinvariant,thepartial derivativecanbereplacedbyanordinaryone:furthermore,ds = ,thetotal flux (we assume it links all turns). The RHS can therefore be written as: ( )d d dIds LI Lt dt dt dt = = = 1.3 The right hand side can be broken into five parts:The integral from 0 to 1 - V01, is the applied voltage. Note that V01=-V10; The integral from 2 to 3; 3 32 2Jdl dl IR = = 1.4 The integral from 4 to 5, wheredl =voltagedropacrossanelement. Voltagedropacrosstheresistorisnotthesameasthatacrossthe capacitor.Acrossresistor,energyisactuallylost.Acrossthecapacitor, energy is stored as5 54 4D Qdl dlC = = 1.5 Note:D=Q/A,andtheintegralgivesthecapacitorplatespacingd multiplied by Q/A. we then use C =/ d With no charge on the capacitor at t = , the charge Q will be given by0 1 7 6 5432 V C R L 5 541,t tQ Idt dl Idtc = = 1.6 Theintegralfrom6to7;Byvirtueofthefactthatweassumeaperfectly conducting filament, which must have zero tangential electric field; this part of the integral is identically zero. Combining equation 1.3 to 1.7 then gives us the following result: 101tdIV IR L Idtdt C= + +1.7 Equation1.7isthe familiarexpressionfortheseriesR-L-Ccircuit,but thistime derived from field theory. Several assumptions were used: (A) Afilamentaryconductordefinestheclosedpathorcircuit.Thisconductor haszerotangentialelectricfield(E)everywhere.Forperfectconductor, tan0 = and0 dl = . No voltage drop along conductor.(B) Maximum circuit dimensions are small compared to the wavelength. (C) Circuitelementsareideal,i.e.,displacementcurrent,magneticfluxand imperfectconductivityareconfinedtocapacitors,inductorsandresistors respectively. Theabovetwoexampleshavedemonstratedthatordinarycircuits,canbe analyzed using field theory. 1.3.MAXWELLS EQUATIONS AS GENERALISATIONS OF CIRCUIT EQUATIONS Maxwells equations can be obtained as generalizations of Amperes, Faradays, and Gausss laws, which are circuit equations. (i)Amperes law:d l I =

1.8 Note:ACapacitorstoresenergypredominantlyintheelectricfieldwhilean Inductor stores energy predominantly in the magnetic field. Stokes theorem coverts the line integral in equation 1.8 around a closed path to an integral over the surface enclosed by the path. Consequently, a more general relationisobtainedbysubstitutingforIusingtheconductioncurrentdensity,J. Anevenmoregeneralexpressionisobtainedbyincludingthedisplacement current density,D t to give: s s sD Ddl J ds ds J dst dt| | = + = + |\

1.9 6ThisistheloopormeshformofoneofMaxwellsequationsderivedfrom Amperes law. Using Stokes theorem, LHS of the integral in equation 1.9 can be converted to an open surface integral. We thus get the point form of the equation: DJt = +1.10 (ii)Faradays Law (for constant flux): ddt= 1.11 Where; V is the induced emf in a circuit andis the total magnetic flux linking the circuit.Sincevoltageistheintegralaroundthecircuitof, dl and istheintegralof ds over the surface enclosed by the circuit, the more general form of equation 1.11 is: sdl dst =

1.12 The surface may be changing so the time derivative should be inside the integral sign. This is another one of Maxwells equations. The point relation is obtained by applying Stokes theorem to get: t = 1.13 (iii)Gausss law (electric field)Dds Q =

1.14 Generally, total charge is the integral, over the volume of interest, of the charge density, p. Equation 1.14 becomes: Dds dv =

1.15 The relation is obtained by applying the divergence theorem (which converts an integral over a closed surface to a volume integral within the volume enclosed) to the LHS of equation 1.15 to give: D =1.16 (iv)Gauss law (magnetic field) 0 ds =

1.17 The magnetic field does not have source points. Thus, there is no such things as amagneticcharge,implyingthatmagneticcharge=0asinequation1.7. Applying the divergence theorem gives 0 =1.18 To summarize these results: 7 0J DD = += ==&&

( )0dl D J dsd l dsDds dvds = + = = = & &

IIIIIIIV Theabovefieldequationshavebeenobtainedasgeneralizationsofcircuit equations. These four equations contain the continuity equation, J = orJ ds dv =

1.19 1.3.1.Free space relationships Infreespace,andformostpracticalpurposesinair,theconductioncurrent densityandthechargedensityarezero,permittingsimplificationofMaxwells equation: 00DD== ==&&

1.3.2.Harmonic fields Forharmonictimevariationofafield, jwte = ;jwt= .Inotherwords, taking a partial derivative with respect to time for harmonic fields is equivalent to multiplyingthefieldbyj.Similarly,adoublepartialderivativewithrespectto time is equivalent to multiplying by -2w . For harmonic time variations, Maxwells equations therefore are: I. ( ) jw = + Circulationofthemagneticfieldgenerates an orthogonal electric field. II.JW = Circulationoftheelectricfieldgeneratesan orthogonal magnetic field. III. , , D = = = SourcepointofanElectric Field is a charge. Charge enclosed by the surface determines the flux out of the surface. IV.0 = Magnetic field has no source points Note that the constitutive relations, ; D = = andJ = have been used, and that a homogeneous isotropic medium has been assumed. 81.4.BREAK DOWN OF SIMPLE CIRCUIT THEORY IN PROBLEM ANALYSIS Simplecircuittheoryassumesacurrent(conductionordisplacement)whichis constantthroughoutacircuitelement,i.e.,evenifthecurrentisalternating,the same current in the same direction exists at all similarly aligned cross-sections of the circuit element at any instant in time. This is because at low frequencies the wavelengthismuchgreaterthanthedimensionsofthecircuitelement,sothe field strength can be assumed constant. This is illustrated in Figure 1.3A. Athigherfrequencies,wavelengthapproachescircuitdimensionssothatthe assumptionsofconstantelectricfieldandcurrentarenolongervalid(Figure. 1.3B). These vary from point to point in circuit element at any instant in time.

Figure .1.3: circuit component relative size at low frequencies. When simple circuit theory breaks down, it is necessary to use distributed circuit analysis. Circuit quantities (V and I) are permitted to change incrementally along thecircuit.Definingrelationshipsareintheformofdifferentialequations.The physicalcircuitisthendescribedintheformofequivalentimpedance,towhich simple circuit theory can be applied. This approach will be used when analyzing transmission lines. 1.5.WAVES & WAVEGUIDING SYSTEMS In all applications, electromagnetic energy must be guided either for transmission from a point (telephone wires, component interconnections, etc), or for feeding antennas before radiation and consequent unguided (unbounded) transmission can occur. Figure 1.4: Examples of wave guiding structures Wave guiding systems are classified into two broad categories: Component Half a Wavelength A wavelength B 9(i)Transmission Lines:These are characterized by having at least two conductors, and supporting the TEM mode in normal operation (see examples in Figure 1.4). (ii)Wave guides: These are guiding systems, which support the transverse electric (TE) or transverse magnetic (TM) modes in normal operation. They are incapable of supporting the TEM mode and are characterized by having a cut- off frequency for each mode below which propagation cannot occur. Examples include rectangular and circular wavegides (Figure 1.4) . We shall study these guiding systems in their normal mode of operation and derive the important relationships and parameters pertaining to them, starting with Transmission lines in this chapter. 1.6.ASSIGNMENT ONE: 1.6.1.1.1. Starting with Maxwells equations derive the continuity equation1.2. Show that for harmonic time variation of a field, given as jwte = 222wt= 1.3.Showthatthepartialdifferentialequation 2 22 2A Ax t = hasageneral solution of the form: ( ) ( )0 2 0 IA f Vt f vt = + + ; with Vo appropriately defined 10CHAPTER TWO: UNBOUNDED WAVE PROPAGATION 2.1.THE WAVE EQUATION IN A PERFECT DIELECTRIC Definition: Wave motion: A group of phenomena constitute a wave if a physical phenomenon occurring at oneplaceatagiventimeisreproducedatotherlocationslater,thetimedelay being proportional to the space separation from the first location. Consider, e.g., ( )1 0f Vt at times 1tand 2t(Figure 2.1). At any fixed time (e.g. t=t1,t=t2 etc) the function only depends on X. Evidently the phenomenon travels in thepositivexdirectionwithavelocity 0V .Similarly, ( )2 0f X t + representsa phenomenon traveling in the negative x direction.

Figure 2.1: Illustration of a propagating phenomenon. Weshallnowdeveloptheequationgoverningthepropagationoffieldsina perfectdielectric(nocharges,noconductioncurrent),startingwithMaxwells equations. ( ) jw = + I JW = II D = III 0 =IV WedifferentiateIw.r.ttimeandsincethecurloperationisw.r.tspacewecan reverse the order of differentiation: LHS:( )t = & 2 1( )oV t t 1( )of x vt 2( )of x v t + 11RHS: ( )Dt t = = & & && Where andhave been assumed time- independent.i.e. 22t| | = = |\ & 2.1 Taking the curl of LHS and RHS of II, and use = & & for time invariant : = &2.2 Use 2.1: = &&2.3 Use identity: 2 = i.e. 2 = && 2.4 Therefore:2 = &&2.5 Similarly, 2 = && 2.6 Equation 2.5 and 2.6 are the wave equations in a perfect dielectric and must be satisfiedbyand forelectromagneticwavepropagation.Forfreespace, 0 =and 0 =and, assuming harmonic time dependence, we get Helmholtz equation (a similar equation can be derived for ):

2 20 k E + = 2.7 Where, 2 2w = 2.8 And 2 2ww fv v = = = = 2.9 ItcanbeshownthatifEandHareindependentoftheyandzdirections(a common case) 2.5 and 2.6 reduce to2 22 2t = 2.10 2 22 2x t = 2.11 Considerequation2.10whichisequivalenttothreescalarequationsin ,& .x y z Itwillbeshownlaterthat0x = forawavepropagatinginthex direction. Taking say the y component (the z component behaves similarly) gives equation 2.10 as: 122 22 2y yx t = 2.12 This partial differential equation has a general solution of the form (HW 1.3): ( ) ( )0 2 0 y If Vt f vt = + +2.13 Withreferencetothedefinitiongivenearlier,itisevidentthatequation2.13 describes wave motion.2.2.UNIFORM PLANE WAVESDefinition: A uniform plane wave is an electromagnetic wave in which electric and magnetic fieldsareorthogonal,bothlayinginaplanetransversetothedirectionof propagation,eachbeinguniforminanysuchplane(Figure.2.2).Notethat,the fields in the illustration are functions of x and t only.

Figure 2.2: UPW propagating in positive x directionWriting the wave equation 2.10 in terms of its components; 2 22 2x xx t = 2.14 (a) 2 22 2y yx t = 2.14(b) 2 22 2ZE Ex t = 2.14(c) In free space, the divergence of the electric field E is zero, so that: H E y z x yE zHDirection of motion 130yx zEEx y z + + = 2.15 ThelasttwotermsontheLHSarezerobecauseEisindependentofyandz. Thereforeeventhefirstcomponentmustbezero.Thismeansthateither is constantorequaltozero.However,aconstantcannotbepartofwavemotion, therefore 0 = .Asimilarargumentforthemagneticfieldshowsthat 0 = . We can therefore conclude that uniform plane waves are transverse. 2.2.1.Intrinsic impedance ForE,Hindependentofyandzandhavingnoxcomponents,thecurl expressions can be written as:yzy za ax x = + 2.16(a) yzy za ax x = + 2.16(b) Substitute into I and II:y yz zy z y zEEa a a ax x t t | | + = + | \ 2.17(a) y yz zy z y za a a ax x t t | | + = + | \ 2.17(b) Equating components in the y and z directions gives: yzx t = 2.18(a) yzx t= 2.18(b) yzx t= 2.19(a) yzx t= 2.19(b) With( )1 20v =,( )1 0 yf V t = for propagation in positive x direction. Then: ( )( )( )010 1 00yx vt fv f x vtt x vt t = =

; where ( )110ffx vt= 14Using 2.18(a), 0 1, 1zzv f f dxx= =;But ( )011 1x vt ff fx x = = So 11 zfx c f cx = + = +

We can ignore the constant C since it is not part of wave motion, giving: ;yz yzEE = =2.20(a) Similarly, zy= 2.20(b) Since, 2 2 2 2;y z z y = + = +=2.21 E is volts/m, and H is in amps/m, so that E/H has dimensions of impedance. This ratio,whichdependsonlyonthedielectric,iscalledtheintrinsicimpedanceof the medium. In free space the intrinsic impedance is ( )0 0/ 377 =ohm 2.3.FREQUENCY DEPENDENCE OF THE CLASSIFICATION MATERIALS Beforeobtainingthewaveequationinconductingmedia,itisinstructiveto establishguidelinesbywhichdielectricsandconductorscanbedistinguished. Consider equation I ( ) jw E = +We see that the term on the RHS has two components: conduction current ( ) and a displacement current( ) jw . While the conduction current is independent of frequency, the displacement current increases with frequency. This means that as frequency increases, a material can change from a conductor to a dielectric. It thereforemakessensetoclassifymaterialsdependingontherelative magnitudes of conduction and displacement currents: w >1100 w< Dielectrics w 1100100 w< < Quasi conductors w < 100w< Conductors 15Itisthereforepossibleforthesamematerialtobehaveasadielectric,aquasi conductor, or conductor depending on frequency (See example in HW 2.1). 2.4.WAVE PROPAGATION IN A CONDUCTIVE MEDIUM 2.4.1.Propagation Constant for a Conductive medium Maxwells equations for a conductive medium will retain both the conduction and displacecurrentcomponents,buttherewillbenostoredcharge.Asbefore,we differentiate I with respect to time; take the curl of II, and carry out the necessary substitutions to get the wave equation for the electric field E. A similar derivation canbeusedtogetthewaveequationforthemagneticfieldH(seeequation 2.22) 2 = + && &2.22(a) 2 = + && &2.22(b) For harmonic time dependence, Helmholtz equation for a conducting medium is ( )2 20 w jw + = 2.23(a) ( )2 20 w jw + =2.23(b) Rearranging 2 20 =2.24(a) 2 20 = 2.24(b) where, ( )2jw jw = + 2.25 a j = +is a complex number known as the propagation constant. For a UPW propagating in the x direction, 2.24 gives: 222Ex= 2.26 2.26 has a solution of the form: ( )( )00, Rexx jwtx ext e + = ( = ( )0Rej wte e (= 2.27 Evidentlyequation2.27representsawavetravelinginthepositivexdirection, attenuating(decaying)accordingtoe-Xwithasthephaseshiftperunit 16distance. is therefore called the attenuation constant and the phase constant of themedium. Using 2.25 and considering only positive square roots, it can be shown that:22 21 12ww | |= + | |\ and22 21 12ww | |= + +| |\ 2.28 From the definition of , 2=so that v =wf =2.4.2.Good dielectricAgooddielectricwillalwayshavesomelosses(asopposedtoaperfect dielectric). However since( ) / 1 w , it can then be shown that (HW 2.3): 2a 2.29(a)

22 218ww | | + |\ 2.29(b) The wave velocity, v, will be: 122 2118wvw | |= = + |\ 20 2 218vw| | |\ 2.30 ( )120v =is the velocity of propagation in the unbounded lossless dielectric. It canbeseenthattheeffectofsmalllossesisareductioninthevelocityof propagation of the wave.Good conductor For a good conductor, ( ) / 1 w This gives: ( ) 45 jw jw jw w w = + = < 2.31 2wa= 2.32 2w= 2.33 17 2 w wv = = 2.34 2.4.3.Skin Effect From2.32and2.33,itisevidentthat and willbeverylargeforagood conductor, especially at high frequencies. This has several consequences: (i)Velocity of propagation will be very low (see 2.34) (ii)The wave attenuates very rapidly as it propagates through a conductor. Consequently, radio frequencywaves penetrate only to a small depth in a good conductorbeforetheybecomenegligiblysmallcomparedtotheirsurface magnitude. We define the depth of penetration, or skin depth, , as the depth at which the wave is 1/e (approximately 37%) of its surface value. If the electric field strength at the surface is E, then at a depth , the field strength Es, is given by: 11ssas s se ee = = = Or 1sae e ==>1 2sw = = 2.36 Using the result of equation 2.32 for a good conductor Example:Copperwith 7 705.8 10 / ; 4 10 mhos m = = = ,thedepthat 100Hz,1MHz,1GHzand100GHzare6.6m,6.6x10-2mm,2.1x10-3mmand 2.1x10-4mm respectively. Surface Impedance: From the above example, we see that current is confined to a very thin sheet on thesurfaceofagoodconductorathighfrequencies.Itisconvenienttodefine surface impedance, tanssJ =2.37 Where, tan is the tangential electric field at the surface and sJis the resulting linearsurfacecurrentdensity(totalconductioncurrentpermeterwidthofthe surface). Consider a thick flat plate with a current distribution as shown in figure 2.3: 0yJ Je= 2.38 The limit is justified only if the thickness, t>>s so that 18 0 0tJ Jdy Jdy = = 000yJJ e dy= = 2.39 Figure 2.3: Conduction current distribution in a thick plate Since, tan0 tan,sJ J= = , then tanssZJ= =Recall that for a good conductor, 045 w = < (equation 2.33) ( ) ( ) 12s mjw wZ j = = + =

1sj+= 2.40 Surface resistance 12sswR = = 2.41 And Surface reactance1ssX= 2.42 Weseethereforethataconductorhavingathickness>>swithexponential currentdistributionhas thesameresistanceas aconductorof thicknessswith the total current as before uniformly distributed throughout its thickness. Power loss in the conductor is thus 2effs sJ R =2.43 With Jseff as the effective value of the linear current density 2.5.POWER FLOW IN ELECTROMAGNETIC FIELDS Consider I:xH J D J = + = + & & J xH = & 2.44 Dimensions of 2.44 are those of current density (A/2m ). Multiply through by: J x = & 2.45(a) Thick conductor Thickness s t y tan E y yo J J e = J 19Dimensionsof2.44(a)arethoseofpowerperunitvolume (Amps/m2xVolts/mWatts/m3)Applying vector identityxF F x xF = to first term on the right: x x x = Orx x x = Substitute into 2.45(a)J xE x = & From II, = = & And Substituting:J x = & & Since212 t = &and 212 t = & (see below) 2 22 2J xt t = 2.45(b) Consider the integral of 2.45(b) over some volume V( )2 22 2v v vEJdv dv x dvt | |= + |\ 2.45(c) Apply divergence theorem to last term: ,vx dv x ds =

over S the Surface enclosing V, gives 2 22 2v v sJdv dv x dst | | = + |\

2.46 (1) (2) (3) Evidently 1.J ispowerdissipation/unitvolume vJdv isthetotalpower dissipated in a volume v. 2. 212is Stored electric energy/unit volume and 212is Stored magnetic energy/unitvolume.Therefore,thevolumeintegral(2)representstotal stored energy. The negative time derivative represents the rate of decrease of stored energy.3.From the law of conservation of energy, the rate of dissipation of energy (1) mustequaltherateatwhichstoredenergyisdecreasingplustherateat 0jwte a = 0jwtjw e a = & 2 20jwtjw e = &

2 2 202jwtjw et = 212 t = &

20whichenergyentersthevolumeV,i.e.,(3)mustrepresenttheofflowof energy inwards through the surface of V. sx ds

is the rate of energy flow outwards from the volume V. sds

is the rate of energy flow inwards through surface of V Poyntings theorem:P x = andcalled,Poyntingsvector,atanypointisameasureof therate of flowofenergyperunitareaatthatpoint.Thedirectionofflow(directionof Poyntings vector) is perpendicular to both & . Note that is normal to& Perfect Dielectric (UPW): Total energy density due to electric and magnetic fields is ( )2 212 + . Given that wave velocity is 0v , the rate of energy flow per unit area ( )2 2012v = + 0012sin90vv | |= + | |\ | | = = = |\ 2.5.1.Conducting Medium ThenormalcomponentofPoyntingsvectoratthesurfaceofaconductor accountsforpowerlossintheconductor.Assumingaflatmetalplatewith thickness sThetangentialcomponentsofelectricandmagneticfields, tan tan& are related by tan tan s = 2.47 Where45sw = (see equation. 2.40) Since& are no longer in time phase we use the complex Poyntings vector. 12x = 2.48 21

tan tan12x =

2.49 Then ( )tan tan1Re2avx = 2.50 Note: tan tan& areinspacequadraturesothatthecrossproductmaintains both magnitudes. However, tanleads tanby 45 in time (see equation 2.47) so that a factor of cos45 is introduced.i.e, 0tan tan1cos 452av = 2.51

22 tantan2 21 12 2ss= =2.52 Now sJis equal in magnitude to the tangential magnetic field 2 221/2av s sJ watts m = ( )2s sRJ eff = 2.53 i.e., Poyntings vector can be used to account for power loss in the conductor. 2.6.REFLECTION AND REFRACTION OF UNIFORM PLANE WAVES Weshallconsideronlynormalincidence.(seeJordan&Balmain, ElectromagneticWavesandradiatingsystems,forthecaseofincidenceat angles 090