10.7 Power and the Poynting Vector Electromagnetic Wave Propagation Power and the Poynting Vector
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1 L 333 lecmgnec II Chpe 0 lecmgnec W Ppgn Pf. l J. l Khnd Islmc Unves f G leccl ngneeng Depmen Pwe nd he Pnng Vec neg cn be sped fm ne pn (whee nsme s lced) nhe pn (wh eceve) b mens f M ws. The e f eneg nspn cn be bned fm Mwell's equns: Usng Mwell equn: Dng bh sdes wh : Bu fm vec denes:... () Usng Mwell equn :, Dng bh sdes wh : Subsue n equn (): Tke vlume negl f bh sdes: dv dv dv v v v Applng he dvegence heem he lef hnd sde: Pwe nd he Pnng Vec Pnng s heem ds dv dv S v v Tl pwe levng he vlume Re f decese n eneg sed n elecc nd mgnec felds Ohmc pwe dssped he PnngVec (Ws/m ) s defned s: = I epesens he nsnneus pwe dens vec ds dv dv ssced wh he M feld gven pn. S v v 3 4
2 Pwe nd he Pnng Vec Pnng heem: ses h he ne pwe flwng u f gven vlume v s equl he me e f decese n he eneg sed wh v mnus he hmc lsses. Illusn f pwe blnce f M felds. Ne h = s nml bh nd nd s heefe lng he decn f ppgn k Assume h (, ) e cs - 0 Pwe nd he Pnng Vec snce csacsb= cs cs 0 hen (, ) e cs - 0 nd (, )= = e cs - cs - 0 (, )= e cs cs - A B A B 5 6 The me-ge Pnng vec ( ) ve he ped T= / s: ( ) (, ) d T I cn ls be fund b: T 0 * ( ) Re s s e 0 F (, ) cs cs - 0 ( ) e cs The l me-ge pwe cssng gven sufce S s gven b: P S ds Pwe nd he Pnng Vec 7 0 Pwe nd he Pnng Vec (,,, ) Pnng vec (,,, ) (,, ) me-ge f Pnng vec (me-nvn vec) (w T ( d,,, ) T P P me-vng vec (ws/m ) s/m ) * Re s s 0 ( ) e cs f 0e cs- l me-ge pwe hugh sufce (scl) ws ds S
3 mple 0.7 In nnmgnec medum 7 = 4 sn ( 0 0. ) V/m Fnd (), (b) The me-ge pwe ced b he w (c) The l pwe cssng 00 cm f plne + = 5 () Snce =0 nd /c, he medum s n fee spce bu lssless medum. ence mple 0.7 Slun 7 0., 0, (nnmgnec), c c (b) sn ( ) mw/m T 6 d T 0 0 (c) On plne + = 5 (see mple 3.5.5), n 5 ence he l pwe s P mple 0.7 Slun. ds. S W n 0. Reflecn f plne w nml ncdence When plne w fm ne medum mees dffeen medum, s pl efleced nd pl nsmed. The ppn f he ncden w h s efleced nsmed depends n he pmees (ε,μ,σ) f he w med nvlved. Nml ncdence (plne w s nml he bund) nd blque ncdence wll be suded. 3
4 Reflecn f plne w nml ncdence Suppse plne w ppgng lng he + decn s ncden nmll n he bund =0 beween medum (<0) chcesed b ε,μ,σ nd medum (>0) chcesed b ε,μ,σ. Reflecn f plne w nml ncdence Incden W (, ) s lng lng + n medum. Assume he elecc nd megnec fled (n phs fm) s fllws: s( ) 0e, hen 0 s mgnude f he ncden elecc 0 s( ) 0e e feld =0 3 4 Reflecn f plne w nml ncdence Refleced W (, ) s lng lng n medum. If s( ) 0, e hen e e 0 s( ) 0 0 s mgnude f he efleced elecc feld =0 Reflecn f plne w nml ncdence Tnsmed W (, ) s lng lng + n medum. If s ( ) 0, e hen 0 s( ) 0e e 0 s mgnude f he nsmed elecc feld =
5 Feld n medum :, Feld n medum :, Snce he ws e nsvese, nd felds e enel ngenl he nefce. Applng he bund cndns he nefce 0: ( = nd = ) hen : Reflecn f plne w nml ncdence (0) (0) (0) (0) (0) (0) Reflecn f plne w nml ncdence Fm he ls w equns: Re flecn Ceffcen =, nd Tnsmssn Ceffcen =, 0 0 Ne h:. +. Bh nd e dmensnless nd m be cmple. ( nd e el f lssless med, nd cmple f lss med) 3. 0 Reflecn f plne w nml ncdence When medum s pefec delecc (lssless, σ=0), nd medum s pefec cnduc (σ= ): F cnduc = 45 η= 0 Γ=- τ=0 0 The w s ll efleced nd hee s n nsmed w ( =0). The ll efleced w cmbnes wh he ncden w fm sndng w. A sndng w "snds" nd des n l; cnsss f w llng ws ( nd ) f equl mpludes bu n ppse decns. 9 Reflecn f plne w nml ncdence The sndng w n medum s: = + = e + e s s s Bu = =, =0, =0, = 0 e e j j s 0 s ja ja e e s j0sn (snce sn A= ) j j Thus =Re e, j = sn sn 0 Smll, cn be shwn h: = cs cs 0 0 5
6 Reflecn f plne w nml ncdence Sndng Ws mples Sndng w n sng Sndng ws sn sn. The cuves 0,,, 3, 4,..., e, especvel, mes 0, T/, T/4, 3T/, T/,... ; /. hp:// fend.de/ph4e/swefl.hm Reflecn f plne w nml ncdence Medum : pefec delecc =0 Medum : pefec delecc =0 η nd η e el nd s e Γ nd τ. Thee s sndng w n medum bu hee s ls nsmedwnmedum.(ncden w s pl efleced nd pl nsmed). weve, he ncden nd efleced ws h mpludes h e n equl n mgnude. Tw cses: cse : when η>η cse : when η < η 3 CAS Medum :pefec delecc =0, Medum : pefec delecc =0 If,, 0, e j0 0 nd e el j j s s s ( e e ) e j j s ( e ) e j m m m m j s mmum when e 0,,4,6... n n n 0,,,3 0,,,3,... s mnmum when e j mn 3 5 mn,3,5... mn,,... (n) (n) 0,,,3 mn n 4 4 6
7 CAS Medum :pefec delecc =0, Medum : pefec delecc =0 If,, 0, j e 0 nd e el j j s s s ( e e ) e j j s ( e ) e j m m n 4 j mn mn mn mn j s mmum when e 3 5 m,3,5... m,,... (n ) (n ) 0,,,3 s mnmum when e 0,,4,6... n n n0,,,3 0,,,3,... 5 Sndng ws due eflecn n nefce beween w lssless med; /. 6 Mesues he mun f eflecns, he me eflecns, he lge he sndng w h s fmed. The f m mn s Sndng W R, SWR m mn m mn s s Snce 0 Γ, fllws h s. * When Γ=0, s=, n eflecn, l nsmssn. * When Γ =, s=, n nsmssn, l eflecn. s s dmensnless, epessed n decbels (db) s: s db=0lg0 s 7 mple 0. In feespce ( 0), plne w wh Slun : 0 c (4) 4 c 3 =0 cs(0 ) ma/m s ncden nmll n lssless medum ( =, = ) n egn 0. Deemne he efleced w, nd h e nsmed w, 7
8 mple 0. slun cnnued Gven h =0 cs(0 ) we epec h = cs(0 ) whee nd = =0 k ence, = 0 cs( 0 ) mv/m Nw = =, Thus cs 0 + mv/m fm whch we esl bn s cs0 + ma/m 9 Smll, Thus whee mple 0. slun cnnued cs 0 +.ence, 40 4 cs0 mv/m 3 3 fm whch we bn 0 4 cs0 ma/m mple 0.9 Gven unfm plne w n s () Fnd =40 cs( ) + 30 sn( ) V/m (b) If he w encunes pefecl cnducng ple nml he s = 0, fnd he efleced w nd. (c) Wh e he l nd felds f 0? (d) Clcule he me-ge Pnng vecs f 0 nd 0. 3 mple 0.9 slun Slun () Ths s sml he pblem n mple 0.3. We m e he w s cnssng f w ws nd whee =40 cs( ), = 30 sn( ) A msphec pessue, hs = Thus m be egded s fee spce. Le cs( ) ence = cs( ) k 3
9 mple 0.9 slun Smll, whee ence nd = sn( ) k = sn( ) 4 sn( ) + cs( ) ma/m 4 3 Ths pblem cn ls be slved usng Mehd f mple mple 0.9 slun (b) Snce medum s pefecl cnducng, << h s, = 0 shwng h he ncden nd felds e ll efleced. = = ence, = 40 cs( ) 30 sn( ) V/m = cs( ) sn( ) A/m 3 4 (c) The l felds n nd cn be shwn be sndng w. The l felds n he cnduc e 0, 340. mple 0.9 slun (d) F 0, s k [ ] = =0 40 F 0, 0 s k becuse he whle ncden pwe s efleced. 35 Oblque ncdence W ves n ngle. Assume lssless med. Unfm plne w n genel fm e j( k ) (,) cs( k ) Re[ ] ˆ ˆ ˆ psn vec k kˆ ˆ ˆ k k w numbe ppgn vec k k k k F lssless unbunded med, k = 36 9
10 Medum :, Medum :, k k k Oblque ncdence k =0 n k k =β k k Oblque ncdence cs( k k k ) cs( k k k ) cs( k k k ) whee k k k k cs sn k θ s ngle f ncdence. The plne defned b ppgn vec k nd un nml vec n he bund s clled plne f ncdence Pllel Pln I's defned s s ncdence plne ( feld les n he plne) (cs sn ) e (cs sn ) e s j sn cs s e s s e j sn cs Pllel Pln j sncs j sn cs
11 (cs sn ) e Pllel Pln s j sn cs s e j sn cs Pllel Pln Tngenl cmpnens f nd shuld be cnnuus he bund =0, e e e jsn jsn jsn (cs ) (cs ) (cs ) jsn jsn e e e jsn The epnenl ems mus be equl f he pevus equns be vld: sn sn sn (Incdence ngle = eflecn ngle) 4 sn sn n n (snell's lw) 4 ence, cs cs cs (-cmpnens f ) Reflecn ceffcen (-cmpnen f ) cs cs, cs cs cs Tnsmssn ceffcen, cs cs cs whee ( ) cs Pllel Pln Pllel Pln Bewse ngle, B defned s he ncdence ngle whch he eflecn ceffcen s 0 (ll nsmssn). B seng B : cs cs B cs csb cs csb 0 sn sn B sn Snce, nd sn ( / ) sn B ( / ) B 43 44
12 Pependcul Pln In hs cse, he feld s pependcul he plne f ncdence (he plne) j sncs s e ( cs sn ) e Pependcul Pln s jsncs s e (cs sn ) e s jsncs j sn cs Pependcul Pln j sncs s e ( cs sn ) e s (cs sn ) j sn cs Tngenl cmpnens f nd shuld be cnnuus he bund =0, nd b seng = : (-cmpnen f ) cs cs (-cmpnen f ) cs cs Reflecn ceffcen, cs cs Tnsmssn ceffcen whee Pependcul Pln cs, cs cs 47 4
13 Pependcul Pln Bewse ngle F n eflecn (l nsmssn): B seng : cs cs B csb cs cs cs B sn B sn Snce sn sn sn B 0, nd B ( / ) B ( / ) Ppe Reflecn ceffcen Tnsmssn ceffcen Reln Nml Incdence Summ Pependcul cs cs cs cs cs cs cs Pllel cs cs cs cs cs cs cs cs cs mple 0.0 mple 0.0 slun An M w ls n fee spce wh he elecc feld cmpnen j( ) s = 00e V/m Deemne () nd (b) The mgnec feld cmpnen ( c) The me ge pwe n he w 5 () Cmpng he gven wh s cle h jk. j( k k k ) s = e = e k 0, k 0.66, k 0.5 k k k k Bu n fee spce, k= c ence, kc 3 0 d/s 6.3 m k (0.6 6) (0.5) 5 3
14 (b) he cespndng mgnec feld s gven b s k j( ) j 00 e ; = 0 j( ) s ( ) e A/m (c) The me ge pwe s mple 0.0 slun s k * 00 s s k vg = Re ( ) 0 = W/m s 53 mple 0. A unfm plne w n wh = cs ( 4 3) V/m s ncden n delecc slb ( 0) wh,.5, =0. Fnd () The pln f he w (b) The ngle f ncdence (c) The efleced feld (d) The nsmed feld 54 mple 0. slun () Fm he ncden feld, s evden h he ppgn vec s k 4 3 k 5 c ence, =5c=50 d/s A un vec nml he nefce ( = 0) s. The plne cnnng k nd s = cnsn, whch s he -plne, he plne f ncdence. Snce s nml hs plne, we h pependcul pln (sml Fgue 0.7). k 4 (b) fm he fgue, n 53.3 k 3 Alenvel, we cn bn fm he fc h s he ngle beween k nd, h s, cs. k n mple 0. slun n
15 (c) Le cs( k. ) whch s sml fm he gven. The un vec s chsen n vew f he fc h he ngenl cmpnen f mus be cnnuus he nefce. Fm he Fgue: k k k k k sn, k k cs Bu = nd k = k = 5 becuse bh k nd k e n he sme medum. ence k 4 3 mple 0. slun 57 T fnd, we need. Fm Snell's lw 0 n c sn sn sn sn n c cs cs cs cs whee 0 377, cs cs cs cs30.39 ence, 0.39() 3. mple 0. slun 3.cs( ) V/m mple 0. slun (d) Smll, le he nsmed elecc feld be cs( k. ) 50 whee k c 30 Fm he Fgue, k ksn =4 k kcs 6.9 k Nce h k = k = k cs cs cs 3.4cs cs cs mple 0. slun The sme esul culd be bned fm he eln =+. ence, Fm, s esl bned s 4.cs( ) k cs( k. ) 7.906(3.4) ( ) cs( ) ma/m 60 5
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