UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 10.5 Prof. Steven Errede LECTURE NOTES 10.5

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1 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede LECTURE NOTES.5 EM Standng Waves n Resnant Cavtes One can ceate a esnant cavt f EM waves b takng a wavegude (f abta shape) and clsng/cappng ff the tw pen ends f the wavegude. Standng EM waves est n (ected) esnant cavt (= lnea supepstn f tw cunteppagatng tavelng EM waves f same fequenc). Analgus t standng acustcal/sund waves n an acustcal enclsue. Rectangula esnant cavt use Catesan cdnates Clndcal esnant cavt use clndcal cdnates t slve the EM wave eqn. Sphecal esnant cavt use sphecal cdnates A.) Rectangula Resnant Cavt: ( LWH ab d) wth pefectl cnductng walls (.e. n dsspatn/eneg lss mechansms pesent), wth a, b, d. n.b. Agan, b cnventn: a > b > d. Snce we have ectangula smmet, we use Catesan cdnates - seek mnchmatc EM plane wave tpe slutns f the geneal fm: E t E e B t B e,,,,,,,,,, t t Mawell s Equatns (nsde the ectangula esnant cavt awa fm the walls): () Gauss Law: () N Mnples: E B E Subject t the bunda cndtns E and B at all nne sufaces. (3) Faada s Law: (4) Ampee s Law: B E E B E t B c t B B E c Take the cul f (3): = {Gauss Law} E B E E B E {usng (4) Ampee s Law} c E c E c E c E E E E E,, E E,, e.. each s a fcn,, E E,, Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved.

2 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede F each cmpnent,, f E,, sepaatn f vaables technque:,, f,, E X Y Z we t pduct slutns and then use the E E whee subscpt,,. c X Y Z c Y Z X Z X Y X Y Z c Dvde bth sdes b X Y Z : The wave equatn becmes: fcn nl fcn nl fcn nl X Y Z X Y Z c Ths equatn must hld/be tue f abta (,, ) pts. nte t esnant cavt Ths can nl be tue f: X X Y Y Z Z k k k cnstant cnstant cnstant X Y k X ky Z kz a b d n.b. We seek scllat (nt damped) slutns!!! wth: k k k k c Geneal slutn(s) ae f the fm:,, : chaactestc equatn,, cs sn cs sn cs sn E A k B k C k D k E k F k n.b. In geneal, k, k and k shuld each have subscpt,,, but we wll shtl fnd ut that k same f all,,, k same f all,,, and k same f all,,. Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved.

3 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede Bunda Cndtns: bundaes and bundaes: E E E, b at, d ceffcents E C, a at, d ceffcents A E, a at, b ceffcents A C and and and k n b, n,,3,... k d,,,3,... k m b, m,,3,... k d,,,3,... k m a, m,,3,... k n d, n,,3,... n.b. m =, and/ n = and/ ae nt allwed, thewse E,, (tval slutn). Thus we have (absbng cnstants/ceffcents, & dppng,, subscpts n ceffcents):,, cs sn sn sn,, sn cs sn sn,, sn sn cs sn E A k B k k k E k C k D k k E k k E k F k E E E But () Gauss Law: E Thus: sn cs sn sn k A k B k k k k sn k Csn k Dcs k sn k ksn k sn k Esn k Fcs k Ths equatn must be satsfed f an/all pnts nsde ectangula cavt esnat. In patcula, t has t be satsfed at,,,,. We see that f the lcus f pnts asscated wth ( =,,) and (, =,) and (,, = ), we must have B D F n the abve equatn. Nte als that f the lcus f pnts asscated wth m k,,, n k, and,, k whee mn,, dd nteges (, 3, 5, 7, etc. ) we must have: Ak Ck Ek. Nte futhe that ths elatn s autmatcall satsfed f mn,, and even nteges (, 4, 6, 8, etc. ). Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved. 3

4 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede Thus: m k m,,3, 4, E,, A csksnk snk a n E,, C sn kcsk sn kk n,,3,4, b E,, Esnk snk csk k,,3, 4, d n.b.: mn smultaneusl s nt allwed! Wth: E,, E ˆE ˆE ˆ Nw use Faada s Law t detemne B : B E E E B Ek sn k csk csk Ck sn k csk csk E E B Ak csk sn k csk Ek csk sn k csk E E B Ck cskcsksn k Ak csk csk sn k : B,, B ˆB ˆB ˆ Ek ˆ Ck sn k cs k cs k Ak cs sn cs ˆ Ek k k k Ck cs cs sn ˆ Ak k k k Ths epessn f B,, (alead) autmatcall satsfes bunda cndtn () B : B at, a B at, b B at, d m n wth k wth k wth k a b d m,,, n,,,,,, 4 Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved.

5 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede,,??? Des B B B B B,, cs cs cs cs cs cs k Ek Ck k k k k k E kkc k Ak Ek k k k kka kke kck Ak cskcskcs k kkc kka B,, YES!!! k k k cs cs cs F TE Mdes: E ceffcent E. Then Ak Ck Ek tells us that: Ak Ck : Thus: k C A k The lwest TEmn,, mde abd s: TE,,, cs sn sn t,,,3, k m E t A k k k e m a k t E,,, ta sn kcsk sn k e n k,,,3, k n b E,,, t k,,,3, d (n = s NOT allwed f TE mdes!!!) k t B t,,, A k sn k csk cske k t B,,, t ka csk sn k cske k t B t,,, A k k csk csk snke k The angula cutff fequenc f th mn,, mde f TE mdes n a ectangula cavt s: mn m n c a b d and: vg c v n dspesn. k Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved. 5

6 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede F TM Mdes: B Ck Ak : Thus: The lwest TM mn,, mde abd s: TM k C A k t E,,, t A csksn k sn k e m k,,,3, m a k,,, t E t A snkcsk csk e n k, n,,3, k b (m = s NOT allwed f TM mdes!!!) k,,,3, d k k k t E,,, ta snksnk csk e k k k k k k B t,,, Ak k A k snkcskcske k,,, k t B t Ak Ak k csk sn k cske k k B,,, t t k k k F ethe TE TM mdes: k k k k c wth: k m n, m,, k, n,, k,,, a b d The angula cutff fequenc f th mn,, mde s the same f TE/TM mdes n a ectangula cavt: mn m n c a b d and: vg c v n dspesn. k 6 Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved.

7 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede B.) The Sphecal Resnant Cavt: The geneal pblem f EM mdes n a sphecal cavt s mathematcall cnsdeabl me nvlved (e.g. than f the ectangula cavt) due t the vectal natue f the E and B -felds. F smplct s sake, t s cnceptuall ease t cnsde the scala wave equatn, wth a scala feld t t,, satsfng the fee-suce wave equatn t, c t t whch can be Fue-analed n the cmple tme-dman t,, e d wth each Fue cmpnent, satsfng the Helmhlt wave equatn: k, wth: k c.e. n dspesn. In sphecal cdnates, the Laplacan peat s:,,,, sn sn sn T slve ths scala wave equatn we agan t a pduct slutn f the fm: Plug ths, R, PQe t m, m f Y, m sphecal hamncs nt the abve scala wave equatn, use the sepaatn f vaables technque: d d Get adal equatn: k f whee: =,,, 3,... d d Let: f u. Then we btan Bessel s equatn wth nde v : d d k u d d m m Slutns f the (adal) Bessel s equatn ae f the fm: f J k N k A m B Bessel fcn f st Bessel fcn f nd knd f de knd f de It s custma t defne s-called sphecal Bessel functns and sphecal Hankel functns: j J whee: k n N The Y, m satsf the angula ptn f scala wave equatn Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved. 7

8 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede and: h J N j n, n.b. If = k s eal, then h h * sn d j d d cs n d j n j n sn cs F, : j n sn cs cs sn...!! 3!!... F, : j sn n cs h h e e e e h h!! whee: The geneal slutn t Helmhlt s equatn n sphecal cdnates can be wtten as: t, Am h k Am h k Ym,, m Ceffcents ae detemned b bunda cndtns. F the case f EM waves n a sphecal esnant cavt we wll (hee) nl cnsde TM mdes, whch f sphecal gemet means that the adal cmpnent f B, B. We futhe assume (f smplct s sake) that the E and B -felds d nt have an eplct -dependence. 8 Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved.

9 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede m! m m Hence: Ym, P cse 4 m! Wll have sme estctns mpsed n t Asscated Legendé Plnmal If B and B eplct functn f, then: B B {necessal} B But: E eques: E t TM mdes wth n eplct -dependence nvlve nl E, E and B B E Cmbnng E and B wth hamnc tme dependence t c t t e f slutns, We btan: c BB The -cmpnent f ths equatn s: B B B sn c sn B B sn sn sn sn But: snb T pduct slutns f the fm: ~Legendé equatn wth m u B P, cs Substtutng ths nt the abve equatn gves a dffeental equatn f u f the fm f: du Bessel s equatn: u wth =,,, 3,... defnng the d c angula dependence f the TM mdes. Let us cnsde a esnant sphecal cavt as tw cncentc, pefectl cnductng sphees f nne adus a and ute adus b. If u B, P cs, the adal and tangental electc felds (usng Ampee s Law) ae: c c u E, snb P cs sn c c u E, B P cs Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved. 9

10 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede But E E whch must vansh at = a and = b u u a b The slutns f the adal Bessel equatn ae sphecal Bessel functns ( sphecal Hankel functns). u The abve adal bunda cndtns n a lead t tanscendental equatns f b the chaactestc fequences, {eeeeek}!!! Hweve {dn t panc!}, f: (b a) = h s such that h a then: And thus n ths stuatn, the slutns f Bessel s equatn: a cnstant!!! u d c a du du ku d whee: ae smpl sn (k) and cs (k)!!!.e. u AcskBsn k k c a u Then: kasn kakb cska a F b a h a u and ka kb kb kb an appmate slutn s: u Acsk ka wth: kh k b a n, n =,,,... b sn cs Thus: k n n c a h, n =,,, 3,... and =,,, 3,... The cespndng angula cutff fequenc s: n n c kn c f h a, n =,,, 3,... and =,,, 3,... a h a Because h a, we see that the mdes wth n =,, 3,... tun ut t have elatvel hgh n fequences n c h f n. Hweve, the n = mdes have elatvel lw fequences: c c f h a. a a An eact slutn (cect t fst de n (h/a) epansn) f n = s: c a h These egen-mde fequences ae knwn as Schumann esnance fequences. =,, 3,... (W.O. Schumann Z. Natufsch. 7, 49, 5 (95)) Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved.

11 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede F n =, the EM felds ae: Ve Useful Table: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ E, E P cs and B P cs b a ẑ ˆ, B S ˆ ˆ ˆ E ˆ, ŷ ˆ Pntng s vect: S E B P P P P ˆ ˆ ˆ cs cs cs cs 3 3 Ccumpla N-S waves! The Eath s suface and the Eath s nsphee behave as a sphecal esnant cavt (!!!) wth the Eath s suface {appmatel} as the nne sphecal suface: a 6378 km m (= Eath s mean equatal adus), the heght h (abve the suface f the Eath) 5 f the nsphee s: h km m( a ) b = a + h m. F the n = Schumann esnances: c a h f h a. : : 3: 4: 5: c h a c 6 h a c h 3 a c h 4 a c 3 h 5 a f f f f f H H H H H (... etc.) n.b. F the n = Schumann esnances: f.5 KH The n = Schumann esnances n the Eath-nsphee cavt manfest themselves as peaks n the nse pwe spectum n the VLF (Ve Lw Fequenc) ptn f the EM spectum VLF EM standng waves n the sphecal cavt f the Eath-nsphee sstem!!! Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved.

12 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede Schumann esnances n the Eath-nsphee cavt ae ected b the adal E -feld cmpnent f lghtnng dschages (the fequenc cmpnent f EM waves pduced b lghtnng at these Schumann esnance fequences). Lghtnng dschages (anwhee n Eath) cntan a wde spectum f fequences f EM adatn the fequenc cmpnents f, f, f3, f4,.. ecte these esnant mdes the Eath lteall ngs lke a bell at these fequences!!! The n = Schumann esnances ae the lwest-lng nmal mdes f the Eath-nsphee cavt. Schumann esnances wee fst defntvel bseved n 96. (M. Balse and C.A. Wagne, Natue 88, 638 (96)). Nkla Tesla ma have bseved them befe 9!!! (Befe the nsphee was knwn t even est!!!) He als estmated the lwest mdal fequenc t be f ~ 6 H!!! n = Schumann Resnances: Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved.

13 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede The bseved Schumann esnance fequences ae sstematcall lwe than pedcted, (pmal) due t dampng effects: Q whee Q = Qualt fact = Q f esnance, and = wdth at half mamum f pwe spectum: The Eath s suface s als nt pefectl cnductng. Seawate cnductvt C. Semens!! Nethe s the nsphee! Insphee s cnductvt C 4 7 Semens On Jul 9, 96, a nuclea eplsn (EMP) detnated at hgh alttude (4 km) ve Jhnsn Island n the Pacfc {Test Sht: Stafsh Pme, Opeatn Dmnc I}. - Measuabl affected the Eath s nsphee and adatn belts n a wld-wde scale! - Sudden decease f ~ 3 5% n Schumann fequences ncease n heght f nsphee! - Change n heght f nsphee: hh h.3.5r 4 6 km!!! - Heght changes decaed awa afte ~ seveal hus. - Atfcal adatn belts lasted seveal eas! Nte that # f lghtnng stkes, (e.g. n tpcs) s stngl celated t aveage tempeatue. Scentsts have used Schumann esnances & mnthl mean magnetc feld stengths t mnt lghtnng ates and thus mnt mnthl tempeatues the all celate ve well!!! Mntng Schumann Resnances Glbal Themmete useful f Glbal Wamng studes!! Eath Cdnate Sstem: c f h a E (nth suth) E P cs (up dwn) B P cs (east west) S P 3 cs P cs ˆ (nth suth) F the n = mdes f Schumann Resnances: E ˆ (up dwn) B ˆ (east west) S ˆ (nth suth) Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved. 3

14 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede We can bseve Schumann esnances ght hee n twn UIUC!! Use e.g. Gbsn P-9 sngle-cl electc guta pckup LP 9 Hens, ~K tuns #4AWG cppe we f detect f Schumann waves and a spectum anale (e.g. HP 356A Dnamc Sgnal Anale) ead ut the HP 356A nt PC va GPIB. Oentatn/algnment f Gbsn P-9 electc guta pckup s mptant want as f pckup algned B ˆ (.e. ented east west) as shwn n fgue belw. n.b. nl ths entatn elded Schumann-tpe esnance sgnals {als ted the 9 entatns {up-dwn} and {nth-suth} but bseved n sgnal(s) f Schumann esnances f these.} Electc guta PU s ae ve senstve e.g. the can easl detect ca / bus taffc n steet belw fm 65 ESB (6 th Fl Lab) can easl see ca/bus sgnal fm PU n a scpe!!! n.b. PU hused n 4 clsed, gunded alumnum sheet-metal b t suppess electc nse. 4 Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved.

15 UIUC Phscs 436 EM Felds & Suces II Fall Semeste, 5 Lect. Ntes.5 Pf. Steven Eede Pfess Steven Eede, Depatment f Phscs, Unvest f Illns at Ubana-Champagn, Illns 5-5. All Rghts Reseved. 5