Electromagnetic Engineering MAPTele

lcomagc gg MAPTl 8-9 Faculdad d ghaa Pogam lcomagc wavs Popagao Icdc Wavguds Tasmsso ls Radao Iês Cavalho Asssa Pofsso Faculdad d ghaa, Uvsdad do Poo www.f.up.p/~ms/ ms@f.up.p

Popagao Faculdad d ghaa Sudg maal a good book o lcomagsm: Mcowav gg, Davd Poa, Addso-Wsl Fld ad Wav lcomagcs, Davd K. Chg, Addso-Wsl aspacs of lcus poblms o cdc, wavguds ad asmsso ls hp://pagas.f.up.p/~ms// 89 Icdc

fohcomg lcus Faculdad d ghaa lcomagc Wavs Pla wav cdc oda Wavguds wk Tasmsso Ls 6/ ad / Radao 89 Icdc

Pla wav cdc Faculdad d ghaa Cos omal cdc flco ad asmsso coffcs saoa wav fac wh pfc coduco mulpl facs oblqu cdc ppdcula ad paalll polaaos flco ad asmsso coffcs fac wh pfc coduco oal al flco 89 Icdc 4

Mall s quaos losslss mda Faculdad d ghaa H µ H ε H H ε µ lcc fld magc fld dlcc pmv magc pmabl ( ε ε ε ) ( µ ) µ µ 8 µ ε c m/s ε µ H ε µ H wav quaos losslss mda 89 Icdc 5

Pla lcomagc wavs mpoal pssos Faculdad d ghaa Pla wav losslss mda lal polad alog popagag alog No: h a soluos of wav quaos losslss mda cos( ω k)ˆ v f s h amplud ( ω πf ) ω s h agula fquc k ω µε s h wavumb k s h phas vloc µ ε π λ cosω ( ) f cos( k) λ cosω ( k) T / dco of popagao 89 Icdc 6

Hlmhol s quaos Faculdad d ghaa wav quaos ε µ H H ε µ phaso oao: ( ) X cos( ω φ) phaso s X X jφ Hlmhol s quaos k H k ( k ω µε ) H No: { } j ( ) ( ω φ R X ) R { X } jω phaso of d( ) d s jωx mpoa θ j ( θ ) ( θ ) cos j s 89 Icdc 7

Pla lcomagc wavs phaso oao Faculdad d ghaa Pla wavs losslss mda jkaˆ jkaˆ H pˆ H pˆ h wav popagag alog h â k ω µε s h wavumb dco pˆ ad pˆ h dca h polaao dco No: h polaao dco of h lcomagc wav s dfd b h polaao dco of h lcc fld 89 Icdc 8

Pla lcomagc wavs phasos Faculdad d ghaa No: H ˆ aˆ ( a ) ( H ) ε µ Ω s h sc mpdac of h mdum lcc ad magc flds a ppdcula o ach oh ad o h dco of popagao asvs lcomagc (TM) wav 89 Icdc 9

Pla lcomagc wavs phasos Faculdad d ghaa ampl jkaˆ pˆ jkaˆ H ph H ˆ H ˆ ˆ ( a ) ( a H ) W h lcc ad magc fld phasos of a GH TM wav popagag alog h dco f spac. Th wav s lal polad alog h dco ad h lcc fld amplud s V/m V/m ˆ ˆ p aˆ ˆ ˆ ˆ ˆ π f k ω µ ε π ad/m c jπ ˆ V/m µ π ε H Ω j π ( aˆ ) ˆ A/m π 89 Icdc

Pla lcomagc wavs phasos Faculdad d ghaa ampl jkaˆ pˆ jkaˆ H ph H ˆ H ˆ ˆ ( a ) ( a H ) Th lcc fld phaso of a TM wav popagag a omagc mdum wh dlcc j 4π ( 4 ) cosa 4 s gv b ˆ V/m Fd h fqucof h wav ad oba h magc fld phaso. kaˆ 4π ( 4) ˆ ˆ ˆ kaˆ 4π ( ˆ 4ˆ) ˆ a k 4π 4 π m 4π (ˆ 4ˆ) aˆ.6 ˆ.8ˆ π 5.º 89 Icdc

Pla lcomagc wavs phasos Faculdad d ghaa ampl jkaˆ pˆ jkaˆ H ph H ˆ H ˆ ˆ ( a ) ( a H ) Th lcc fld phaso of a TM wav popagag a omagc mdum wh dlcc j 4π ( 4 ) cosa 4 s gv b ˆ V/m Fd h fqucof h wav ad oba h magc fld phaso. k ω µ ε ω µ ε k π m µ ε π f c f.5 GH ε 4, µ ˆ.6ˆ.8ˆ a µ µ µ 6π ε ε ε Ω H π j4π ( 4) (.8ˆ.6ˆ) A/m 89 Icdc

Icdc of a TM wav a a pla bouda Faculdad d ghaa flcd â agl of cdc θ asmd pla of cdc pla â θ θ θ pla coag h omal o h bouda sufac ad h vco dcag h dco of popagao of h cd wav â cd mdum ( ε, µ σ ), mdum ( ε, µ σ ), dcos of popagao: aˆ aˆ aˆ s θ ˆ s θ s θ ˆ ˆ ˆ ˆ ˆ 89 Icdc

Sll s laws law of flco Faculdad d ghaa wavfo go wh cosa phas â / O pla wavs wavfos a ppdcula o â / A pos O ad A hav h sam phas A B â / / pos O ad A hav h sam phas θ θ O θ phas k ds. / / AO koa k â mdum ε, µ σ ( ), mdum ε, µ σ ( ), / / OO sθ OO sθ θ θ ω kω µε v f 89 Icdc 4

Sll s laws law of faco Faculdad d ghaa pla wavs wavfos a ppdcula o â â / A / O pos O ad A hav h sam phas / pos O ad B hav h sam phas A B â phas k ds. θ θ / k AO k OB θ O / OO k sθ k OO sθ / â mdum ( ε, µ σ ), mdum ( ε, µ σ ), sθ sθ k k v v f f ω kω µε v f 89 Icdc 5

Id of faco Faculdad d ghaa Id of faco ao of h spd of lgh f spac o ha h mdum lag low spd c v f : losslss mda µ ε vf µ ε µ ε µ ε sθ s θ v v f f sθ sθ Sll s law of faco 89 Icdc 6

Bouda codos Faculdad d ghaa ( ) ( H H ) J S ( D D ) ρs ( B B ) aˆ aˆ aˆ aˆ,a,a B,om B,om â l b h omal o h fac pog fom mdum o mdum â mdum ( ε, µ σ ), mdum ( ε, µ σ ), a B om couos Ha couos f J S Dom couos f ρ S couos No: J S ad ρ ol dal coducos S 89 Icdc 7

Bouda codos dal coducos Faculdad d ghaa Idal coducos σ û mdum ( ε, µ σ ), sk dph δ π f µσ mdum ( ε, µ σ ), D cod cod H B cod cod J S ρ S ampl σ J ρ S S aˆ ˆ a ( H H ) ( D D ) aˆ H H ˆ D D,aaˆ a,omaˆ 89 Icdc 8

Nomal cdc Faculdad d ghaa omal cdc θ θ θ s s θ θ θ θ mdum mdum â H H â cd flcd H H jk ˆ jk jk ˆ ˆ jk ˆ mdum ( jk jk )ˆ H H H jk jk ˆ H â asmd H jk ˆ jk ˆ mdum 89 Icdc 9

89 Icdc Faculdad d ghaa Nomal cdc flco ad asmsso coffcs bouda codos mdum mdum â H H â â H couous a couous a H ( ) f J S ( ) jk jk ˆ H jk jk ˆ mdum mdum jk ˆ H jk ˆ a H H

Nomal cdc flco ad asmsso coffcs Faculdad d ghaa Γ τ flco coffc asmsso coffc â H H â H â No mdum mdum. Γ τ. Γ. τ 4. jk jk ( Γ )ˆ τ jk ˆ mdum ( jk jk )ˆ H jk jk ˆ mdum H jk ˆ jk ˆ 89 Icdc

Nomal cdc saoa wav Faculdad d ghaa jk jk ( Γ )ˆ Γ τ τ jk jk [( Γ) Γ ]ˆ â H H â τ jk Γ jk ( jk )ˆ ˆ H â s ( ) j j mdum Γ mdum τ τ jk ˆ j s ( k )ˆ Γ popagag wav saoa wav 89 Icdc

Nomal cdc mama ad mma Faculdad d ghaa jk jk ( Γ )ˆ jk j k ( Γ )ˆ Γ Γ jθγ â H H â [ Γ cos( θ k ) ] ( Γ s ( k ) ) Γ θ Γ ( θ k ) Γ Γ cos Γ H â mdum mdum Γ τ jk ˆ j s ( k )ˆ τ Γ mama: mma: cos θγ k m θγ ( ) π k Γ ( θ Γ k ) ( ) ( θ π ) [ ] cos MAX Γ k ( Γ ) ( ) m MAX 89 Icdc

Nomal cdc cdc o dal coduco Faculdad d ghaa f mdum s losslss ( σ ) jk jk ( Γ )ˆ f mdum s dal coduco ( σ ) Γ τ â H H â o popagag wav, ol saoa wav jk jk ( )ˆ j s ( k )ˆ H â mdum Γ τ mdum µ ε σ j ωε mama: MAX ( )π k ad MAX mma: m π k ad m 89 Icdc 4

Avag pow cad b lcomagc wav Faculdad d ghaa avag pow cad b h wav acoss sufac A: P av S da A av S av R { H } * avag Pog s vco da daˆ ppdcula o sufac A ad wh fsmal absolu valu da A 89 Icdc 5

Avag Pog s vco fo TM wavs Faculdad d ghaa TM wavs H aˆ H ˆ a A ( B C) ( A C) B C( A B) ( aˆ ) * H * * {( ) aˆ ( ˆ )} * a * aˆ * S av R * aˆ S av R { H } * avag Pog s vco pos alog h popagao dco 89 Icdc 6

Avag pow fo TM wavs losslss mda Faculdad d ghaa TM wavs losslss mda H aˆ ˆ H a jkaˆ pˆ s al S md R * aˆ S av aˆ ( W/m ) P av S da A av L A b h sufac ppdcula o h popagag dco da da aˆ P av S av A ( W) P av o A ( W) S av s cosa 89 Icdc 7

Avag cd, flcd ad asmd pow Faculdad d ghaa P P av, av, o o A A P P av, av, Γ â H H â P av, o A P P P av, av, av, P P P P av, av, av, av, τ Γ mpoa H P av mdum S da A av â Γ mdum τ S av A Γ τ o A 89 Icdc 8

Nomal cdc mulpl facs Faculdad d ghaa mdum mdum mdum o M M M. mda ad f. jk ˆ all wavs a lal polad alog d flco ad asmsso coffcs: fac : fac : fac : Γ Γ Γ Γ ad ad ad τ τ τ 89 Icdc 9

Nomal cdc mulpl facs Faculdad d ghaa mdum mdum d mdum jk ( ) ( τ ) jk d τ ( jk τ ) τ τ jk d jk ( d ) Γ jk ( ) Γ τ Γ τ jk d jk jkd ( Γτ ) jkd jk ( d ) Γ τ Γ Γ τ jk d jk jkd jk( d ) τ Γ Γ τ τ ΓΓτ j4k d jk jkd jk( d ) Γ Γ τ Γ Γ τ j4k d jk j5kd jk( d ) τ Γ Γ τ j5kd jk( d ) Γ Γ τ M M M 89 Icdc

Nomal cdc mulpl facs Faculdad d ghaa mdum mdum mdum mdum mdum mdum jk jk jk M M M jk jk d d jkd j4k d j6k d τ Γ τ Γ τ τ Γ Γ τ τ Γ Γ L Γ jk d jkd j k d ( Γ Γ Γ Γ L) τ τ Γ 4 Γ τ τ Γ jk d jkd ( Γ Γ ) jk d τ τγ Γ jkd ΓΓ Γ Γ Γ Γ jk d jk d 89 Icdc

Nomal cdc mulpl facs Faculdad d ghaa mdum mdum mdum mdum mdum mdum jk jk jk M M M jk jk d d jkd jk d jk d jk d j5k d jk d j7k d jk d τ τ τ τ Γ Γ τ τ Γ Γ τ τ Γ Γ L τ j τ ( k k) d jkd j4kd j6k d ( Γ Γ Γ Γ Γ Γ L) τ τ ( kk) d jkd ( Γ Γ ) j τ ( k k ) j d τ jkd Γ Γ 89 Icdc

89 Icdc Faculdad d ghaa Nomal cdc a mulpl facs alav appoach mdum mdum mdum d M M M mdum mdum mdum d ( ) H jk jk jk jk ˆ ˆ mdum jk jk jk jk jk ( ) H jk jk jk jk ˆ ˆ mdum H jk jk ˆ ˆ mdum

Mulpl facs bouda codos Faculdad d ghaa dlcc mda J S a H a couous couous mdum mdum mdum d d jk d jk d jk d jk d jk d jk d 4 quaos 4 ukows (,, ), assumg 89 Icdc 4

89 Icdc 5 Faculdad d ghaa Mulpl facs bouda codos mdum mdum mdum d d jk d jk d jk d jk d jk d jk d k j d k j Γ Γ Γ Γ ( ) d k j d k k j Γ Γ τ τ pvous pssos, as pcd

Mulpl facs ampls Faculdad d ghaa mdum mdum mdum lmao of flcos a fac b sg mdum mdum s a machg dvc d ampls lmaos of fls lss Auao of ada chos (vsbl aplas) 89 Icdc 6

Mulpl facs l/4 asfom Faculdad d ghaa lma flcos jkd Γ Γ mdum mdum mdum jkd jkd jkd ( ) ( )( ) ( ) d l/4 asfom : jk d k d mπ, m odd g Γ Γ Γ Γ τ Γ jk d jk d ( k k ) j d τ jkd Γ wavlgh mdum d m λ, odd 4 m g 89 Icdc 7

l/4 asfom dff wavlgh Faculdad d ghaa Γ Γ Γ Γ Γ mdum mdum mdum f p dsg fquc λ, p v f p vloc mdum d v d 4 jkd ( ) fquc f Γ Γ jk d f p Γ Γ Γ Γ τ Γ jk d jk d ( k k ) j d τ jkd Γ * P P f c Γ Γ 4 ( cos( kd )) Γ cos( k d) F cos F cos δ δ wh Γ F Γ π δ k d f f p 89 Icdc 8

l/4 asfom dff wavlgh Faculdad d ghaa P P f c F cos δ F cos δ wh F Γ Γ ad π δ f f p mdum mdum mdum Γ F f δ Γ F f δ d P P f c.9.8.7.6 F F.5.4... F F. 4 5 6 7 8 9 π π 5π δ 89 Icdc 9

Poblm Faculdad d ghaa fomula No: coduco 89 Icdc 4

Poblm Faculdad d ghaa A pla lcomagc wav wh h followg lcc fld phaso popagas a ad s cd omall o a fac a wh a omagc mdum wh facv d ha occups h go >. Fd j4π ˆ ( V/m) a) h flco ad asmsso coffcs; b) h lcc fld phasos of h flcd ad asmd wavs; c) h avag Pogvcos of h cd, flcd ad asmd wavs; d) h faco of cd pow ha s asmd o mdum. fomula 89 Icdc 4

Poblm Faculdad d ghaa fomula P P f c F cos δ F cos δ F Γ Γ π δ f f p 89 Icdc 4

Oblqu cdc of a TM wav a a pla bouda Faculdad d ghaa TM wav H aˆ. omal o pla of cdc ad H a pla â flcd θ â asmd â θ jk aˆ ˆ ppdcula polaao θ â. paalllo pla of cdc jk aˆ ( ˆ s ˆ ) θ paalll polaao cd mdum ε, µ σ ( ), mdum ε, µ σ ( ), Gal cas: jk aˆ a ( ) jk ˆ ˆ s ˆ ˆ θ,, paalll polaao ppdcula polaao 89 Icdc 4

Ppdcula ad paalll polaaos covo ppdcula polaao paalll polaao Faculdad d ghaa â H â â H â θ θ θ H θ θ θ H â â H H mdum mdum mdum mdum Compos of, paalll o h fac kp h dco 89 Icdc 44

Ppdcula polaao lcc ad magc flds Faculdad d ghaa cd aˆ s θ ˆ ˆ jkaˆ ˆ H aˆ flcd aˆ s θ ˆ jkaˆ ˆ H aˆ jkaˆ θ ( s θ ˆ cos ) ˆ ˆ s θ ˆ ˆ jkaˆ θ ( s θ ˆ cos ) ˆ â â H θ θ H θ H â asmd mdum mdum aˆ s θ ˆ ˆ jkaˆ ˆ H aˆ jkaˆ θ ( s θ ˆ cos ) ˆ laoshps bw ad obad fom bouda codos, 89 Icdc 45

Ppdcula polaao lcc ad magc flds Faculdad d ghaa bouda codos a couous H a couous ( f J S ) â H â a θ θ θ H H H H â jksθ jk sθ jk sθ jk sθ jk sθ cos θ jk sθ H mdum mdum k sθ k sθ ( ) 89 Icdc 46

Ppdcula polaao flco ad asmsso coffcs ( ) â H Faculdad d ghaa â θ θ θ H â H mdum mdum Γ τ flco coffc asmsso coffc 89 Icdc 47

Ppdcula polaao flco ad asmsso coffcs Γ τ flco coffc asmsso coffc â H Faculdad d ghaa â o θ θ θ H. Γ τ (as omal cdc) â. s possbl ha Γ (Bws s agl) θ θ B H s θ s θ mdum mdum s θ B µ ε ( µ µ ) µ ε ol wh µ µ. f mdum s dal coduco, Γ τ 89 Icdc 48

lcc fld mdum ppdcula polaao Faculdad d ghaa jk aˆ ˆ jk aˆ ˆ â H â Γ τ aˆ s θ aˆ s θ θ θ θ H jk jk jk sθ [( τ Γ ) Γ ] ˆ â τ jk jk jks ( ) ˆ jk aˆ θ ˆ Γ H τ jk aˆ θ ˆ jγ s jks ( k ) ˆ mdum mdum wav popagag alog â wav popagag alog, wh dpd amplud 89 Icdc 49

Mama ad mma mdum ppdcula polaao jk aˆ ˆ aˆ aˆ jk aˆ s θ s θ ˆ â H Faculdad d ghaa â j ( ksθ k ) ( jk Γ ) ˆ Γ Γ jθ Γ θ θ θ H [ Γ cos( θ k ) ] Γ s( θ k θ ) [ ] Γ Γ cos Γ Γ cos Γ ( θ k ) â H mdum mdum mama: k ( θ π ) MAX Γ mma: Γ k [ θ ( ) π ] m ( ) ( Γ ) MAX Γ m 89 Icdc 5

Icdc o dal coduco ppdcula polaao Faculdad d ghaa If mdum s dal coduco Γ τ â H jk jk cos ( )ˆ jk sθ θ θ θ jk sθ j s( k )ˆ â wav popagag alog, wh dpd amplud H mdum dal coduco mama: ( ) π MAX k MAX mma: π m k m 89 Icdc 5

Paalll polaao lcc ad magc flds Faculdad d ghaa cd aˆ s θ ˆ ˆ jk aˆ ˆ s jkaˆ H ˆ flcd aˆ H s θ ( ˆ ) θ ˆ jk aˆ ˆ jk aˆ ˆ s θ ˆ ˆ ( ˆ s ˆ ) θ â H H θ θ â θ H â asmd mdum mdum aˆ s θ ˆ ˆ jkaˆ ( ˆ s θ ˆ ) laoshps bw, ad jkaˆ H ˆ obad fom bouda codos 89 Icdc 5

Paalll polaao lcc ad magc flds Faculdad d ghaa bouda codos a couous H a couous ( f J S ) â â H H â â a H H H θ θ θ θ θ θ H H â â jk sθ jk sθ cos θ jk sθ H jk sθ jk sθ jk sθ H mo mdum mo mdum k s θ k sθ ( ) ( ) 89 Icdc 5

Paalll polaao flco ad asmsso coffcs Faculdad d ghaa ( ) ( ) â H â θ θ H θ H â mdum mdum Γ τ flco coffc asmsso coffc 89 Icdc 54

Paalll polaao flco ad asmsso coffcs Faculdad d ghaa Γ τ flco coffc asmsso coffc â H θ θ H â o θ. Γ τ â. s possbl ha Γ H (Bws s agl) θ θ B s θ s θ mdum mdum s θ B µ ε ( ε ε ) µ ε wh µ µ s θ B ( ε ε ). If mdum s dal coduco, Γ τ 89 Icdc 55

lcc fld mdum paalll polaao Faculdad d ghaa jk τ Γ aˆ jkaˆ ( ˆ s θ ˆ ) ( cos ˆ s ˆ θ ) jk aˆ ( ˆ s θ ˆ ) θ jk aˆ a ( ˆ ) jk ˆ s θ ˆ Γ ( ˆ s θ ˆ ) â H θ θ θ H â Γ τ â τ Γ Γ jk sθ jk sθ jk aˆ ( ˆ s θ ˆ ) jk jk ( ) ˆ jk jk ( ) s θ ˆ H mdum mdum τ jγ jk aˆ jk sθ s ( ˆ s θ ˆ ) ( k ) ˆ Γ jk sθ cos ( k ) s θ ˆ 89 Icdc 56

lcc fld mdum paalll polaao Faculdad d ghaa wav popagag alog â â H â jk aˆ τ ( ˆ s ) θ ˆ θ θ θ H jk sθ jγ s ( k ) cos ˆ θ â jk sθ Γ cos ( k ) s ˆ θ H mdum mdum wavs popagag alog, wh dpd ampluds 89 Icdc 57

Mama ad mma mdum paalll polaao Faculdad d ghaa jk aˆ jkaˆ ( cos ˆ s ˆ ) ( cos ˆ s ˆ ) θ θ θ θ a ( ) ( ) jk jk j a a a jk βˆ ˆ ˆ ˆ ˆ s θ ˆ â H â jk aˆ jk aˆ jk ( aˆ aˆ ) ( Γ ) jk ( ) Γ θ θ θ H Γ Γ jθ Γ â [ Γ cos( θγ k ) ] Γ s ( θ k θ ) [ ] Γ cos H ( θ k ) Γ Γ cos Γ mdum mdum mama: k mma: ( θγ π ) [ θγ ( ) π ] MAX m k ( Γ ) ( Γ ) MAX m 89 Icdc 58

Icdc o dal coduco paalll polaao Faculdad d ghaa f mdum s dal coduco Γ τ â H j a [( ) ( ) ] jk jk a a jk a jk ˆ ˆ ˆ ˆ ˆ s θ ˆ jk sθ jk sθ cos ( k ) ( k ) s θ ˆ s ˆ â θ θ H wavs popagag alog, wh dpd ampluds mdum dal coduco mama of : MAX ( ) k π mma of : m π k MAX m 89 Icdc 59

Mallc wavguds jksθ : j s ( k )ˆ : j jk sθ jksθ s cos ( k ) ( k ) s θ ˆ ˆ Faculdad d ghaa ppdcula polaao: paalll polaao: a a π k π k θ θ mdum dal coduco fo boh polaaos, a coducg pla paalll π o h pla could b sd a k whou modfg lcc fld mdum π k 89 Icdc 6

Mallc wavguds Faculdad d ghaa lcomagc wav s gudd b h wo coducg sufacs mallc wavgud could b possbl o gud a lcomagc wav wh ol dlcc mda? θ θ mdum dal coduco β π 89 Icdc 6

Dlcc wavguds Faculdad d ghaa gal cas: ach cdc pa of h wav s asmd o dlcc dlcc dlcc dlcc af som dsac, h lcomagc wav mdum s cosdabl auad gall, dlcc mda do gud lcomagc wavs ffcl θ θ h soluo would b o hav o g asmd o mdum s hs possbl? 89 Icdc 6

Toal al flco Faculdad d ghaa Sll s law of faco: s θ sθ â flcd â asmd > θ > θ θ θ Ccal agl: θ c θ such ha θ 9º θ c acs θ â cd θ θ c o asmd wav o mdum mdum mdum Toal al flco s θ s θc s θ ± s θ ± j s θ s θ 89 Icdc 6

Toal al flco Faculdad d ghaa Toal al flco: s θ ± j s θ â flcd â asmd θ Losslss ad omagc mda: µ ε θ θ µ ε ε â Rflco coffcs: cd mdum mdum Γ Γ Γ Γ Γ cos cos θ m j ( ) s θ ± j ( ) s θ Γ θ ± j ( ) s θ ± j ( ) s θ Γ Γ 89 Icdc 64

Toal al flco vasc wavs Faculdad d ghaa spaal vaao of flds mdum : jk aˆ â flcd â asmd aˆ s θ ˆ ˆ θ θ jk ( sθ ) θ ± j s θ â cd mdum mdum k s jk θ sθ wav popagag alog amplud dcasg poall wh vasc flds 89 Icdc 65

Poblm Faculdad d ghaa Th lcc fld of a ufom pla wav popagag a s gv b jπ (4 ) ˆ. Ths wav s cd o a fac wh a dlcc mdum wh facv d ha occups h go >. Fd (a) (b) (c) (d) () h dco of popagao of h cd wav ad h agl of cdc; h magc fld phaso of h cd wav; h popagao dcos of h flcd ad asmd wavs. h lcc fld phasos of h flcd ad asmd wavs h faco of h cd pow ha s asmd o h dlcc fomula 89 Icdc 66

Poblm Faculdad d ghaa fomula 89 Icdc 67

Poblm Faculdad d ghaa fomula Losslss mda: 89 Icdc 68