Subjects of this chapter - Propagation harmonic waves - Harmonic waves on an interface (Boundary condition) - Introduction of phasor

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1 CHAPTR 8. LCTROMAGNTIC HARMONIC WAV Maxwell s equatns Dffeental wave equatn Tavelng wave A hamnc wave An abtay wave A lnea cmbnatn f hamnc waves. Supepstn pncple. Lnea meda, ( ε, μ ae cnstant) Hamnc waves A cnstant ampltude Tme-nvaant Snusdal steady-state Tme-vayng and H Hamnc waves n 3D-space. cs ( k ± ωt ), cs ( ω t ± k ) Maxwell s eqs. Subjects f ths chapte - Ppagatn hamnc waves - Hamnc waves n an nteface (Bunday cndtn) - Intductn f phas Tme-nvaant pat f the cmplex fm f a hamnc wave. + phase change, phase lead - phase change, phase lag 8- TH PHASOR In a medum f ρ = v 0, J = 0 and cnstant ε and μ Maxwell s equatns H = μ t (8-a) H =ε t (8-b) = 0 (8-c) H = 0 (8-d) Cmbnng tw cul equatns = με (8-) t Its slutn j j t (, t) Re e k = + ω Unfm plane wave (8-3) Hamnc wave M Hamnc Wave 8- Ppetay f Pf. Lee, Yen H

2 whee, vect cmplex ampltude k =ω μεa, wavevect (8-4) k q. (8-) k, but n a k Fm suce bunday cndtn In a medum f fnte extent The same dffeental wave eq. Unfm plane waves Bunday cndtn Lnea cmbnatn (Dffeent k 's ) A wave n a bunday A patal eflectn Ttal wave = ncdent wave + eflected wave, Re j j j t t e k e k = + e ω (8-5) ( ) ( ) Slutn f wave equatn f k = k =ω με. In geneal fm j t (, t) Re ( x, y, ) e ω (8-6) = Phas, tme nvaant. It assumes snusdal steady-state cndtn. It shws elatve phases, tme delays, n space. It s useful n cmbnng multple hamnc waves. It s vald n lnea meda. Dffeental wave eq. f H has the same fm as j t H (, t) Re H( x, y, ) e ω (8-7) = Phas f H xample 8- jk3 Reslve a hamnc wave = a y cs ( kx) e nt tw unfm plane waves and sketch the wavefnts. Slutn Usng ule fmula, we ewte jkx jkx jk3 = a y e + e e = ay e + a + j ( kx + k 3 ) j ( kx + k 3 ) y e M Hamnc Wave 8- Ppetay f Pf. Lee, Yen H

3 The ght sde s the sum f tw unfm plane waves f wavevects = k x + k3 and k = kax + k3 a, espectvely. Bth waves have the same wave numbe k = k = k + k 3 k a a The black( blue) paallel lnes epesent plane wavefnts, vewed fm the tp, f the wave f k ( k ). They cespnd t phases π n, whee n ae nteges. The dts n x axs ae the pnts at whch wavefnts f the same phase meet, and theefe the ttal electc feld ntensty has the maxmum ampltude. The ampltude vaes snusdally alng x axs as s ndcated by the tem cs ( kx ). 8- MAXWLL S QUATIONS IN PHASOR FORM Insetng qs. (8-6) and (8-7) n q. (8-a) Re (,, ) j ω xy e t Re ( xy,, ) e j ω = μ t t H (8-8) Re j ω t (,, ) Re j ω e xy = μ e t jωh ( xy,, ) Cul, tme devatve and eal pat ae mutually exclusve Igne Re and exp ( jω t ) Faaday s law n phas fm = jωμh Maxwell s equatns n phas fm wth cnstant ε and μ = jωμh (8-9a) H = jωε (8-9b) = 0 (8-9c) H = 0 (8-9d) and H ae tme-nvaant cmplex vects M Hamnc Wave 8-3 Ppetay f Pf. Lee, Yen H

4 Cmbne tw cul equatns = jωμ H =ωμε (8-0) Usng a vect dentty, =, and q. (8-9c), + k = 0 : Vect Helmhlt s equatn (8-) ω π k =ω με= = : Wavenumbe (8-) v λ whee ω, angula fequency μ, pemeablty ε, pemttvty v, velcty λ, wavelength. 8-3 PLAN WAV IN LOSSLSS DILCTRIC In Catesan cdnates, vect Helmhlt s equatn a + a + a + k a + a + a = 0 (8-3) ( x x y y ) ( x x y y ) Mutually exclusve x + k x = 0 (8-4a) y + k y = 0 (8-4b) + k = 0 (8-4c) Rewtng q. (8-4a) x x x k 0 x = x y (8-5) Its slutn jkx ˆ ( x ky y k ) = e = ˆ e ± + + ± jk x x x (8-6) k k = k kx + ky + k Scala cmplex ampltude Smlaly ˆ y = e y = ˆ e jk jk k a a, wavevect (8-7) (8-8a) (8-8b) Helmhlt s equatns n q. (8-4) The same k n qs. (8-6), (8-8a) and (8-8b) But n estctn n a k M Hamnc Wave 8-4 Ppetay f Pf. Lee, Yen H

5 The same a k Thee slutns nt a sngle wave ˆ ˆ ˆ jk ( ) = a + a + a = a + a + a e = whee x x y y x x y y e j k = ˆ a + ˆ a + ˆ a ˆ j = a e ϕ a x x y y Scala cmplex ampltude Vect cmplex ampltude (8-9) (8-0), ampltude ϕ, phase angle = e jk, a unfm plane wave n 3D space. k, wavevect ppagatn dectn and wavelength, vect cmplex ampltude dectn and ampltude f, pstn vect k, tme-phase f the feld ˆ a ˆx, ˆy and ˆ have the same ϕ phase angle A sngle wave n an nfnte medum ϕ= 0 Tansvese wave Insetng q. (8-9) n q. (8-9c) j j ( e = k k ) = j( k ) e = 0 k = 0 (8-) Ppagatn dectn Oscllatn dectn s n the wavefnt M Hamnc Wave 8-5 Ppetay f Pf. Lee, Yen H

6 H fm q. (8-9a) x y x y = = jkx jky jk = j k e x y ˆ jk ˆ jk ˆ jk ˆ jk ˆ jk ˆ jk e e e e e e x y x y = jωμh a a a a a a jk (8-) H = ( a ) (8-3) k = μ ε : Intnsc mpedance (8-4) Rewtng (8-3) ( ) e j k j k k e H = a H Insetng ˆ = ˆ a (8-5) Same fm as H = ( a a ) Hˆ a (8-6) k H = ˆ / ˆ H, n hms [ Ω ] = μ / ε = 0π 377[ Ω ] Lssless delectcs s eal and H ae n phase Lssy delectcs s cmplex and H ae ut f phase H ( and k ), fm q. (8-5) k, fm q. (8-), H and k ae mutually thgnal Tansvese wave! Real nstantaneus expessn (, t) = a cs( ωt k + ϕ) (8-7a) H (, t) = a cs( ωt k + ϕ) (8-7b) H Unt vects a = a a (8-8) k H M Hamnc Wave 8-6 Ppetay f Pf. Lee, Yen H

7 xample 8- jk A plane wave, = a x ( + j) e, has a fequency f GH and ppagates n a delectc f ε = 4 and μ =. Fnd (a) and H (b) wavelength Slutn (a) Rewtng the phas /4 e jπ = a e jk (8-9) x The eal nstantaneus expessn = = ω + 4 jπ/4 jk jωt Re π ax e e e ax cs t k Insetng q. (8-9) n q. (8-3) jπ/4 H = ( a ) = ( a a ) whee ak = a. k x = a Intnsc mpedance μ = = 88.5[ Ω] εε e e jπ/4 jk y e e. jk Real nstantaneus expessn jπ/4 jk jωt π H = Re ay e e e = ay cs ωt k + 4 M Hamnc Wave 8-7 Ppetay f Pf. Lee, Yen H

8 (b) Fm q. (8-) ω ε π k =ω εμ= = / με 3 0 π = λ λ= m = cm POYNTING VCTOR AND POWR FLOW negy denstes n statc felds D and B H negy s sted n and H f an electmagnetc wave. Same velcty and dectn as the wave. Pwe delvey Tw cul equatns B = t D = + t (8-30a) H J (8-30b) A vect dentty ( ) = ( ) ( ) H H H (8-3) Insetng q. (8-30) n q. (8-3) ( ) = D B H J H + H (8-3) t t Assumng ε, μ and σ ae cnstant D ε = ( ) = ( D ) (8-33a) t t t B μ t t t H = ( H H ) = ( B H ) (8-33b) Fm qs. (8-3) and (8-33) = + t t ( H ) J H ( D ) ( B H ) : Pyntng s theem (8-34) Integatng bth sdes ve a vlume and applyng dvegence theem = + t t ( H ) d s J d v H ( D ) d v ( B H ) d v (8-35) S V V V Ohmc pwe-lss n V Inceased enegy n V pe unt tme. Net pwe flwng nt V. Ttal nstantaneus pwe flwng nt V. (Integand = pwe densty) M Hamnc Wave 8-8 Ppetay f Pf. Lee, Yen H

9 Pyntng vect s defned by = S and H. S H : W / m S k Tme-aveage pwe densty S ~, a nnlnea functn f and H cannt be expessed by phass n S (8-36) Usng cmplex expnental and ts cmplex cnjugate. jϕe k jϕe = a cs ( ωt k + ϕ e) = ( ) ( ) a e e + a e e * ( ˆ jk jω t ) ( ˆ + jk jωt = a e e + a) e e jk jωt * + jk jωt = e e e e + jωt * jωt e + e jωt j jω t+ jk (8-37a) jωt * jωt H = a H cs ( ωt k +ϕ ) = He + H e (8-37b) H h whee, phas, vect cmplex ampltude ˆ, scala cmplex ampltude, ampltude ϕ e, phase angle *, cmplex cnjugate. Insetng q. (8-37) n q. (8-36) jωt * jωt jωt * jωt S = e + e e e H + H j ωt * * j ωt * * = e + e H H H H (8-38) Tme aveage f S s defned by T / S = dt T S : T, tempal ped f the wave T / Tme-aveage Pyntng vect S Re * = H (8-39) M Hamnc Wave 8-9 Ppetay f Pf. Lee, Yen H

10 xample 8-3 Gven an electmagnetc wave, (, t) = a cs( ωt k) and H (, t) = a cs( ωt k), fnd S usng (a) b and H (b) and H Slutn (a) Fm q. (8-36) y ε μ ε S = H = a cs ωt k μ Usng tgnmety ( ) ε S = a + cs ( ωt k) μ Afte tme-aveage x ε S = a (8-40) μ (b) Phass f and H jk = a e x H = a y ε e μ jk Fm q. (8-39) * * jk ε jk S = Re = Re ( xe ) y e H a a μ Rewtng t ε = Re μ jk jk ( axe ) ( aye ) ε S = a (8-4) μ The esult s the same as q. (8-40). M Hamnc Wave 8-0 Ppetay f Pf. Lee, Yen H

11 8-5 POLARIZATON OF PLAN WAV Phas f a unfm plane wave = exp ( jk ) k fm Maxwell N lmts n and a ˆx, ˆy and ˆ f In phase scllates wth tme alng a staght lne Out f phase changes magntude and dectn wth tme Oentatn f s called the plaatn H s just gven by q. (8-3) Lnea Plaatn A plane wave n + -dectn shuld be n xy -plane jk = e = ˆ a + ˆ a jk e (8-4) ( ) x x y y Scala cmplex ampltudes. If ˆx = x and ˆy y =, eal values jk jωt Re ( x x y y ) e e a a ( xax y ay ) cs ( t k) = + = + ω (8-43) It scllates wth tme alng a lne paallel t xax + ya y Lnealy plaed n the dectn xax + ya y. Lnealy plaed alng x -axs + Lnealy plaed alng y -axs Tw cmpnent waves ae n phase. M Hamnc Wave 8- Ppetay f Pf. Lee, Yen H

12 xample 8-4 Gven a lnealy plaed wave, = ( x + 3 y) jk a a e, ppagatng n fee space, fnd (a) tatn angle f the plaatn vect wth espect t x -axs (b) tme-aveage pwe densty Slutn (a) The tatn angle θ= tan = 60 3 (b) Fm q. (8-3) ε ( 3 ) ( 3 ) jk H ε a a a e a a e = x + y = y x μ μ Fm q. (8-39) S Re ( 3 jk ε = ax + ay) e μ ( ay 3 ax) e ε = ( ax + 3ay) ( ay 3ax) μ ε = + μ ( 3) a The tme-aveage pwe densty s the sum f thse f the waves that ae plaed n x and y -dectns, espectvely. jk jk * 8-5. Ccula Plaatn y -cmpnent lags behnd x -cmpnent by 90 n tme dmensn / ( x y) ( x y) jk j jk = a j a e = a + e π a e (8-44) whee ˆ = ˆ =. x y Instantaneus expessn = a cs ( ωt k) + a cs ( ωt k π/ ) (8-45) x y At = 0 ( ) a ( ) ( ) a sn ( ) = a cs ω t + cs ωt π/ x y = a cs ω t + ωt x y (8-46) At ω t = 0, = ax At ω t = π /, = ay Rght-hand cculaly plaed wave A ccula lcus n xy -plane M Hamnc Wave 8- Ppetay f Pf. Lee, Yen H

13 Fg. 8-4, ght-hand ccula plaatn y - cmp. lags behnd x -cmp. by 90 n tme phase y -cmp. aves at p at a late tme than x -cmp. by a quate f the tempal ped. y -cmp. s dsplaced backwad by a quate wavelength wth espect t the x -cmp. The phase elatnshp s mantaned thughut the space at all tmes. y -cmp. leads x -cmp. by 90 n tme phase Left-hand cculaly plaed wave When the left fnges fllw the tatn f the electc feld vect, the left thumb pnts t the ppagatn dectn f the wave. / ( x y) ( x y) jk j jk = a + j a e = a + e π a e (8-47) Instantaneus expessn = a cs ( ωt k) + a cs ( ωt k + π/ ) (8-48) x y At = 0 ( ) a ( ) ( ) a sn ( ) = a cs ω t + cs ω t + π/ x y = a cs ωt ωt x y (8-49) At ω t = 0, = ax At ω t = π /, = ay Clckwse tatn A ccula lcus n xy plane M Hamnc Wave 8-3 Ppetay f Pf. Lee, Yen H

14 Fg. 8-5, left-hand ccula plaatn y - cmp. leads x -cmp. by 90 n tme phase y -cmp. aves at p at an eale tme than x -cmp. by a quate f the tempal ped. y -cmp. s dsplaced fwad by a quate wavelength wth espect t the x -cmp. xample 8-5 A lnealy plaed wave can be expessed by the sum f tw cculaly plaed waves. Fnd tw cculaly plaed waves cntaned n jk (a) = a xe jk = a + a e (b) ( x x y y ) Slutn (a) By addng and subtactng ( /) j a y jk jk = a xe = ( x j y) + ( x + j y) e a a a a (8-50) On the ght sde, the fst tem s a ght-hand cculaly plaed wave and the secnd s a left-hand cculaly plaed wave. Thus, the answe s jk ( ax j a y) e jk and ( ax + j a y) e The lnealy plaed wave n x -dectn s the sum f tw cculaly plaed waves, wth a half the ampltude f the lnea plaatn, whch tate n the ppste dectns and meet each the at x -axs nce n evey cycle f the tatns. (b) Magntude f the ampltude = + x y Phase angle f the ampltude θ= tan y x M Hamnc Wave 8-4 Ppetay f Pf. Lee, Yen H

15 The plaatn vect s paallel t a efeence lne that s tated by θ wth espect t x -axs. Refeng t the esult f (a), we can expess the lnealy plaed wave by the sum f tw cculaly plaed waves, whch tate n the ppste dectns and meet at the efeence lne nce n evey cycle f the tatns. We add a phase θ t the ght-hand cculaly plaed wave n q. (8-44), and add a phase θ t the left-hand cculaly plaed wave n q. (8-47), s that the plaatn vects becme paallel t the efeence lne at = 0 and t = 0, a j a e θ e, (8-5a) j jk ( x y) ( j j jk x y) e θ e + a a, (8-5b) Reducng the ampltude f the waves n q. (8-5) by half and addng tw waves j jk j jk = ( ax j ay) e θ e + ( ax + j a y) e θ e (8-5) whch s cetanly a sum f tw cculaly plaed waves Let us check the esult n q. (8-5). Reaangng t = ( e e ) x j( e e ) y + a a e jk = cs θ ax + sn θa ye jθ jθ jθ jθ jk Snce cs θ= x and sn θ = tated by θ wth espect t x -axs. y, t s a lnealy plaed wave n the dectn that s llptcal Plaatn jϕ jk = a + e a e : llptcally plaed wave (8-53) ( x x y y ) Lnealy and cculaly plaed waves ae specal cases f an ellptcally plaed wave Instantaneus expessn = a cs ( ωt k ) + a cs ( ωt k + ϕ) (8-54) x x y y M Hamnc Wave 8-5 Ppetay f Pf. Lee, Yen H

16 At = 0 plane x = x cs ( ω t) (8-55a) = cs ( ω t + ϕ ) (8-55b) y y F 0 <ϕ<π / At ω t = 0, x = x and y y cs = ϕ ( n fst quadant) ω = π, = 0 and = sn ϕ ( paallel t y -axs) At t / x llptcal lcus by n q. (8-54) New cdnate system, x ' and y ' Paallel t the maj and mn axes y y s tatng clckwse n xy -plane Left-hand ellptcally plaed. llptcal lcus n geneal fm x = x cs ( ω t + ϕ ') = x cs ( ω t + ϕ' ) (8-56a) y = y sn ( ω t + ϕ ') = y sn ( ω t +ϕ' ) (8-56b) ccw tatn n x ' y ' -plane Cd. tansfmatn f q. (8-55) t pmed cd. x x cs ( t) cs y cs ( t ) sn cs ( t) sn cs ( t ) cs = ω θ+ ω + ϕ θ (8-57a) = ω θ+ ω +ϕ θ (8-57b) y x y By equatng q. (8-57) wth q. (8-56) xy csϕ tan ( θ ) = ( ) ( ) x y (8-58) tan θ= / f ϕ= 0, lnealy plaed wave. y x M Hamnc Wave 8-6 Ppetay f Pf. Lee, Yen H

17 xample 8-6 jϕ Fnd tme-aveage pwe densty f a unfm plane wave, = ( x x + y y ) ppagatng n fee space. Slutn Vect cmplex ampltude = a + e ϕ a j x x y y Fm q. (8-3) ( ) ( j ϕ H = a ) jk k = a xax + ye ay e jk H e Vect cmplex ampltude j = x y y x ( e ϕ ) H a a Tme-aveage Pyntng vect S * * = Re Re H = H = Re = ( x + y ) + jϕ jϕ ( xax ye ay ) ( xay ye a x ) jk a e a e, The pwe delveed by an ellptcally-plaed wave s equal t the sum f the ndvdual pwes f tw lnealy-plaed waves. 8-6 PLAN WAV IN LOSSY MDIA An electmagnetc wave n a lssy delectc. Induced electc dples Pwe lss (Same feq. ω) Dampng fce (Cllsn) Cnductn cuent als causes pwe lss. In a lssy medum, ˆ ε=ε jε Pwe lss Magnetc nteactn s neglgble μ s eal M Hamnc Wave 8-7 Ppetay f Pf. Lee, Yen H

18 8-6. Lssy Delectc ( σ= 0) In a smple medum wth n fee chages and cnstant ε and μ Maxwell s equatns = jωμh (8-59a) H = jω( ε jε ) (8-59b) = 0 jωεˆ (8-59c) (8-59d) H = 0 Helmhlt s equatn ˆ + k = 0 (8-60) Cmplex wavenumbe kˆ ˆ ( j ) =ω με=ω μ ε ε (8-6) Plane wave slutn = e jk ˆ a k A wave n + -dectn = e e jk ˆ γ Ppagatn cnstant ( ) γ = jkˆ = jω μ ε jε α+ jβ : a k by the means (8-6) (8-63) (8-64) whee α, attenuatn cnstant β, phase cnstant. με ε α=ω + ε με ε β=ω + + ε / / : Nepes pe mete [Np/m] (8-65a) : Radans pe mete [ad/m] (8-65b) Lss tangent s defned by ε σ tan ξ= = ε ωε : ξ, lss angle (8-66) Insetng q. (8-64) n q. (8-63) j j = e α e β = ( a ) e α e β (8-67a) M Hamnc Wave 8-8 Ppetay f Pf. Lee, Yen H

19 Instantaneus expessn α = ( a ) e cs ( ωt β) (8-67b) Attenuatn. Phase vaatn Phase dffeence s π between tw pnts sepaated by λ βλ = π Rewtng t π β= λ : β and λ depend n ε (8-68) Phase velcty v p ω = β (8-69) F με > με β n medum > k n fee space λ n medum < λ n fee space v p s slwe n medum H fm qs. (8-63) and q. (8-59a) γ γ ( ae ) a ( ae ) = γ = jωμh Rewtng t, wth q. (8-64) εˆ μ ˆ H γ α jβ = ( a a ) e = ( a a ) e e (8-70) Cmplex ntnsc mpedance, = ˆ ˆ / Hˆ μ μ = ˆ = εˆ ε jε (8-7) Cmplex ˆ H and ae ut f phase xample j/ Fm a wave, = a xe e, gven n a lssy delectc f detemne ntnsc mpedance f the medum. μ =μ and ε = ε, Slutn Attenuatn and phase cnstants α= 0.3 β= 0.5 Fm q. (8-65) ε + α ε = β ε + + ε (8-7) M Hamnc Wave 8-9 Ppetay f Pf. Lee, Yen H

20 Insetng α and β n q. (8-7), lss tangent ε =.88 ε Fm q. (8-7), wth μ = ˆ ε j ( ε / ε ) ε = ε and the lss tangent 0π = = 83e j.88 j 0.54 (8-73) (8-74) H lags behnd by 0.54 adan. Nte that α, β and ˆ ae clsely elated t each the Lssy Delectc ( σ 0) Cnductn cuent s the dmnant suce f pwe lss. In a delectc wth σ 0 and n dampng = jωμh (8-75a) H = J + jωε = ( σ+ jωε) (8-75b) = 0 (8-75c) H = 0 (8-75d) and H, geneated utsde D =ρ v = 0, J s nduced N suce chage cuent Rewtng q. (8-75b) σ H = jω ε j ω jω( ε jε ) jωεˆ ˆε s newly defned ε = ε σ ε = ω (8-76) (8-77a) (8-77b) Plane wave slutn = (8-78) e γ Substtutng q. (8-77) n qs. (8-64) and (8-65) σ γ α+ jβ= jω με j ωε (8-79) με σ α=ω + ωε με σ β=ω + + ωε / / (8-80a) (8-80b) M Hamnc Wave 8-0 Ppetay f Pf. Lee, Yen H

Smlaly μ = ˆ ε σ ωε ( j / ) (8-8) Lw-lss delectc, ε << ε σ / ωε

(8-8) σ μ α ε σ β ω με + 8 ωε (8-83a) (8-83b) whee α ~ σ β ω με,

21 Smlaly μ = ˆ ε σ ωε ( j / ) (8-8) Lw-lss delectc, ε << ε σ / ωε << Applyng bnmal expansn σ σ γ jω με j + ωε 8 ωε α+ jβ (8-8) Bnmal expansn n geneal fm n n( n ) ( + a) = + na + a +...! whee a <<. Fm q. (8-8) σ μ α ε σ β ω με + 8 ωε (8-83a) (8-83b) whee α ~ σ β ω με, pefect delectc Smlaly μ σ ˆ + j ε ωε Pstve phase angle leads H n tme phase. (8-84) M Hamnc Wave 8- Ppetay f Pf. Lee, Yen H

22 xample j (. ) Gven an electmagnetc wave, = a x 300e [ V / m ], ppagatng n a medum f = ˆ 0 + j [ Ω ], fnd (a) S at = 0.5m (b) tme-aveage hmc pwe-lss pe unt vlume at = 0.5m at steady state Slutn (a) Magnetc feld ntensty 300 H = ( a ) = a ˆ 0 + j 0. j (. ) y e Tme aveage Pyntng vect 0.4 * 300 e S = Re Re H = a 0 j 0.4 = a ( 0.7) e (8-85) At = 0.5m ( 0.7) e 64.3 W / m = = S a a (b) Tme aveagng Pyntng s theem n q. (8-34) at steady state H = J S * = J (8-86) whee J s the eal nstantaneus value and J s the phas. valuatng the left sde f q. (8-86) usng q. (8-85) ( 0.7)( 0.4) S = e = 85.e At = 0.5m, tme-aveage hmc pwe-lss pe unt vlume s S = 85.e = 69.0 W / m (8-87) In the the methd, the pwe lss can be calculated by the use f J Fst, fm qs. (8-79) and (8-8) γ σ = jωε j ˆ ωε Fm the equatn we btan γ 0. + j. 3 σ= Re Re.89 0 [ S / m] ˆ = = 0 + j At = 0.5m = a j (. 0.5) x 300e (. 0.5) ( j ) J = a x e M Hamnc Wave 8- Ppetay f Pf. Lee, Yen H

23 Tme-aveage hmc pwe-lss * ( ) J = e = 69.0 W / m 3 (8-88) Tw esults n qs. (8-87) and (8-88) ae dentcal Gd Cnduct Gd cnduct σ/ ωε >> σ + j γ jω με j = ωμσ j / / j /4 j ωε : j = e e π = π = (8-89) α+ jβ α=β= πf μσ (8-90) Plane wave n a gd cnduct α jβ = e e = ( a ) e e πf μσ j πf μσ (8-9a) f ( ) e π μσ a cs ( t f ) = ω π μσ (8-9b) Attenuatn Depth f penetatn skn depth δ= = = πf μσ α β : [m] (8-9) Fm β= π/ λ λ δ= π (8-93) Phase velcty n a gd cnduct v p = f λ = ωδ = πf μσ 4 : p v =.3 0 m/ s at GH n Cu (8-94) M Hamnc Wave 8-3 Ppetay f Pf. Lee, Yen H

xample 8-9 7 Fnd the skn depth n cppe f σ= 5.8 0 [ S / m] as a functn f fequency Slutn Fm q. (8-93), wth μ μ δ= = πf μ σ π f 4π 0 5.8 0 The answe s 0.

(8-8) ˆ = ˆ jωμ σ ( + j ) σδ : leads H by 45 n tme phase (8-95) The eal pat f ˆ R = : Suface esstance pe unt aea (8-96) σδ s Resstance f a gd cnduct f css sectn w δ and length l l R = w δσ (8-97)

24 xample Fnd the skn depth n cppe f σ= [ S / m] as a functn f fequency Slutn Fm q. (8-93), wth μ μ δ= = πf μ σ π f 4π The answe s δ= [m] f 7 7 At a fequency f GH, skn depth n cppe s nly. μ m. Fm q. (8-8) ˆ = ˆ jωμ σ ( + j ) σδ : leads H by 45 n tme phase (8-95) The eal pat f ˆ R = : Suface esstance pe unt aea (8-96) σδ s Resstance f a gd cnduct f css sectn w δ and length l l R = w δσ (8-97) xample 8-0 ( ) / A plane wave = ( ) j a x e + δ ppagates nt a gd cnductn f cnductvty σ n the egn 0. 3 (a) Fnd tme-aveage pwe lss n a vlume l w [ m ] (b) Fnd ttal cuent, n phas fm, cssng the aea w [ m ] n y -plane (c) Assumng the cuent n (b) flws unfmly n a vlume f css sectn w δ and length l, fnd ts esstance, and tme-aveage pwe lss n the vlume M Hamnc Wave 8-4 Ppetay f Pf. Lee, Yen H

25 Slutn (a) Tme-aveage pwe lss * = * P = Re dv w Re d J = σ l V = 0 = / = σw l e δ d = 0 = σwδl 4 (8-98) (b) Ttal cuent σwδ I = σ d = σ w e d = w = ( + j) / δ y= 0 s = 0 = 0 (8-99) + j (c) The cnduct has a css sectn wδ, length l and cnductvty σ. The esstance l R = w δσ (8-00) Tme-aveage pwe lss, wth qs. (8-99) and (8-00) * P = R Re I I w = σ δl (8-0) 4 The esults n qs. (8-98) and (8-0) ae dentcal, fm whch we can assume that the cuent flws unfmly nly n the egn 0 δ, as fa as the ttal pwe lss s cncened. We can als btan the same pwe lss as q. (8-98) (8-0) fm the vltage l that s appled t the esstance R f q. (8-00). Pwe densty n a cnduct lectc and magnetc feld ntenstes Z a ( j ) / xe + δ σδ HZ ( ak ) Z ay e ˆ + j ( + j) / δ Tme-aveage Pyntng vect * / / / / ( ) / Re δ j δ σδ δ j δ j ( xe e ) y e e Re σδ + δ S = a a = a e + j S σδ / = a e δ 4 : Attenuated (8-0) M Hamnc Wave 8-5 Ppetay f Pf. Lee, Yen H

26 xample 8- Refeng t Fg. 8-0, fnd the ttal pwe tansmtted nt the cnduct thugh an aea w l f the cnduct suface. Slutn Fm q. (8-0), tme-aveage pwe densty at = 0 σδ S = a (8-03) 4 Tme-aveage ttal pwe acss the aea P l w = d = σδ x 0 y 0 4 w = S s l (8-04) = Nte that ds s n a dectn, whch s n the same dectn as S, nt the utwad suface nmal. The pwe acss the suface decays t e at =, as s shwn n q. (8-0), mplyng that the tansmtted pwe s dsspated all n the cnduct. It s vefed by the fact that the esults n qs. (8-04) and (8-98) ae dentcal. 8-7 PLAN WAVS AT BOUNDARY A unfm plane wave Reflectn Tansmssn Inteface (Dffeent ε and μ ) 8-7. Nmal Incdence f Plane Wave M Hamnc Wave 8-6 Ppetay f Pf. Lee, Yen H

27 Incdent wave plaed n x dectn j = a xe (8-05a) j y e β H = a (8-05b) whee β, phase cnstant, ntnsc mpedance, ampltude, assumed eal The bunday cndtn ( k and k t ) a, unfm ( and t ) a x and t at = 0 The eflected plane wave j = a xe (8-06a) j y e β H = a (8-06b) Tavelng n dectn. Same β and as, same medum The tansmtted plane wave t t j = a xe (8-07a) t t j y e β H = a (8-07b) Fm the bunday cndtn at = 0 plane ( 0) + ( 0) = t ( 0) (8-08a) H ( 0) + H ( 0) = H t ( 0) (8-08b) Insetng qs. (8-05)-( 8-07) t + = (8-09a) t = (8-09b) By slvng the equatn = + t = + (8-0a) (8-0b) M Hamnc Wave 8-7 Ppetay f Pf. Lee, Yen H

28 We defne Γ= = + : Reflectn ceffcent (8-a) t τ= = + : Tansmssn ceffcent (8-b) Tue nly f nmal ncdence Γ < 0 f < a = a Γ 0 f Mandaty eflectn Γ= 0 f = N eflectn q. (8-) can be appled even t an nteface between tw lssy meda Cmplex and Cmplex Γ and τ Fm q. (8-) +Γ=τ (8-) Vald nly f nmal ncdence xample 8- F nmal ncdence f a unfm plane wave, pve Γ + ( / ) τ = fm the cnsevatn f enegy Slutn Tme-aveage pwe denstes f the ncdent, eflected and tansmtted waves S * = Re H = a (8-3a) S = a = Γ a (8-3b) S t t = a = τ a (8-3c) Fm the cnsevatn f enegy t S = S + S (8-4) Substtutng q. (8-3) Γ + τ = (8-5) Tw tems n the left sde cespnd t the eflected and the tansmtted eneges elatve t the ncdent enegy. M Hamnc Wave 8-8 Ppetay f Pf. Lee, Yen H

29 A. Standng Wave Rat Ttal electc feld ntensty n medum jβ j β = + = axe + Γe a ˆ x (8-6) Γ s cmplex n geneal j Γ= Γ e φ (8-7) Rewtng q. (8-6) wth q. (8-7) ˆ j j( ) = e β + Γ e β +φ (8-8) The maxmum ampltude and ts lcatn ˆ = + Γ (8-9) max max ( ) = φ+ π β ( n) ( n = 0, ±, ±,...) (8-0) The mnmum ampltude and ts lcatn ˆ = Γ (8-) mn mn ( ) = φ+ π +π β ( n ) ( n = 0, ±, ±,...) (8-) The ntemaxmal dstance s π /β, a half wavelength f the wave n medum. Rewtng q. (8-8) ˆ jβ = e Γ+Γ+Γe j( β +φ) φ/ ( ) cs ( /) jβ j = e Γ + Γ e β + φ (8-3) Real nstantaneus expessn = Γ cs ωt β + Γ cs β +φ/ cs ω t +φ/ (8-4) ( ) ( ) ( ) ( ) Tavelng wave Standng wave M Hamnc Wave 8-9 Ppetay f Pf. Lee, Yen H

30 Tavelng wave, a blue cled lne; standng wave, black cled lne ω t = 0, π / 4 and π f Γ= 0.5 and φ = 0 at ω t = 0, π/ 4, π/, 3 π / 4, and π f Γ= 0.5 and φ = 0. Black cled lne, ; blue cled lne, ampltude f as a functn f pstn The standng wave at s defned by S ˆ = max = : Dmensnless (8-5) Γ ˆ mn +Γ Γ anges fm 0 t, and S anges fm t. M Hamnc Wave 8-30 Ppetay f Pf. Lee, Yen H

31 xample 8-3 A nmally ncdent wave s eflected ff an nteface at = 0 plane. Measued n medum ae the standng wave at f 4, a dstance f 0.m between the fst maxmum f the electc feld ntensty and the nteface, and a dstance f 0.5m between tw adjacent maxma. Fnd the ntnsc mpedance f medum. Slutn The ntemaxmal dstance s equal t a half wavelength f the wave n medum λ= 0.5m π β = = π. λ The fst maxmum cespnds t n = 0 n q. (8-0) max = φ= 0.m β Cmbnng tw equatns φ= 0.8π (8-6) Fm q. (8-5) S 4 Γ= = = 0.6 S + 4+ (8-7) The eflectn ceffcent, fm qs. (8-6) and (8-7) e j π Γ= (8-8) Rewtng q. (8-a) +Γ = Γ (8-9) Insetng q. (8-8) and = n q. (8-9) j0.8π j e 0.99e = = = 0.8e j0.8π j e.e The answe s 0.83 = e [ Ω ] 305 j j 0.83 B. Inteface Invlvng Pefect Cnduct Inteface between a lssless medum and a pefect cnduct < 0 0 Pefect cnduct σ= = 0 Γ = and τ = 0 M Hamnc Wave 8-3 Ppetay f Pf. Lee, Yen H

32 Incdent plane wave n medum j = a e β : Plaed n x -dectn (8-30a) x e β j H = a y (8-30b) Reflected plane wave ppagates n dectn jβ jβ = a Γ e = a e (8-3a) x x e β j H = a y (8-3b) whee Ampltude f fm =Γ H fm q. (8-3). = 0 at cnduct suface. Ndes wth a ped f λ /. Fst nde f H at λ /4 fm the cnduct suface. The same ped as M Hamnc Wave 8-3 Ppetay f Pf. Lee, Yen H

33 Ttal feld n medum a = + = x e e jβ jβ ( ) = a j sn β H = H + H = a y e + e = ay cs β x ( ) jβ jβ Instantaneus expessn Z ax sn( β) cs ( ωt π/) = a sn( β ) sn( ωt) x (8-3a) (8-3b) : Standng wave (8-33a) H = a cs ( β ) cs ( ωt) (8-33b) x Standng wave cannt tanspt enegy n space, because the fact j n q. (8-3a) * makes H be magnay and esults n S = 0. xample 8-4 A ght-hand cculaly plaed wave, = ( x y) j a j a e β, ppagates n fee space and mpnges n the suface f a pefect cnduct at = 0 plane. Fnd (a) plaatn vect f the eflected wave (b) suface cuent densty n the cnduct suface (c) n fee space Slutn (a) Intnsc mpedance f the pefect cnduct = 0 Reflectn ceffcent Γ= Reflected wave j β j e j e j β =Γ a a = a + a (8-34) ( x y) ( x y) Real nstantaneus expessn f the eflected wave jπ/ jβ jωt = Re a + e a e e ( x y) π = ax cs ( ω t +β ) + ay cs ω t +β + = a cs ω t +β ω t +β ( ) a sn ( ) x y Let us check at = 0 plane At ω t = 0, = a x At ω t = π /, = a (8-35) y When the left fnges fllw n q. (8-35), the left thumb pnts t the ppagatn dectn f, the dectn. Snce the x and y -cmpnents have the same ampltude, the eflected wave s left-hand cculaly plaed. M Hamnc Wave 8-33 Ppetay f Pf. Lee, Yen H

34 If a ght-hand cculaly plaed wave s eflected, t becmes a left-hand cculaly plaed wave, vce vesa. (b) Magnetc feld ntenstes f the ncdent and eflected waves H = ( ak ) = a ( ax jay) e jβ ( ay jax) = + e jβ H = ( ak ) = ( a) ( ax + jay) e jβ ( ay jax) = + Ttal feld at = 0 plane H = H + H = a + a ( y j x ) Suface cuent densty J = a H = a a + a e jβ ( ) ( j ) s n y x = ( ax jay) whee a n s the unt suface nmal. (c) Ttal feld n fee space β ( x y) ( x y) = + = a j a e + a + j a e j jβ ( ) a ( ) = a j sn β sn β x y Instantaneus expessn ( ) ( ) jωt j( ωt π/) jωt x y = Re e = Re sn β e sn β e a a ( ) a ( ) a ( ) = sn β x sn ωt y cs ωt M Hamnc Wave 8-34 Ppetay f Pf. Lee, Yen H

Electric potential energy Electrostatic force does work on a particle : Potential energy (: i initial state f : final state):

Electric potential energy Electrostatic force does work on a particle : Potential energy (: i initial state f : final state): Electc ptental enegy Electstatc fce des wk n a patcle : v v v v W = F s = E s. Ptental enegy (: ntal state f : fnal state): Δ U = U U = W. f ΔU Electc ptental : Δ : ptental enegy pe unt chag e. J ( Jule)