UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 3 Prof. Steven Errede LECTURE NOTES 3

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1 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E ECTURE NOTES Cnsatin aws (cntinu): Angula Mntu Assciat with Fils W ha lan that th acscpic fils ha assciat with th: Engy: Engy Dnsity: u, t E, t B, t U t u t E t B t Engy:,,, Pynting s Vct: S, t E, tb, t Juls/ Juls Watts Juls -sc ina Mntu: ina Mntu Dnsity: t, St, S t, Et, Bt, c ina Mntu: p t, t S, t E, tb, t c kg -sc kg- sc Th acscpic fils can aitinally ha assciat with th: Angula Mntu: Angula Mntu Dnsity:, t, t S, t S, t c kg E, tb, t -sc Angula Mntu: t, t, t S, t c kg - E, tb, t sc Nt that n STATIC E& B fils can cay nt lina ntu p fcn t an nt angula ntu fcn t as lng as E B is nn-z! Again, at th icscpic ll, itual phtns assciat with th acscpic fils cay angula ntu as wll as lina ntu p an (kintic) ngy E! Only whn th fil cntibutins a inclu f th ttal lina ntu p Tt an th ttal angula ntu Tt, i.. p Tt p ch p an Tt ch is cnsatin f lina ntu an cnsatin f angula ntu spaatly, inpnntly satisfi. Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

2 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Giffiths Exapl 8.4 Angula Mntu Assciat with a ng Slni & Caxial Cylinical Capacit. Cnsi a lng slni f aius R an lngth R, with n tuns p unit lngth n N caying a stay/dc cunt f I Aps. Tt Caxial with th lng slni is a cylinical (i.. caxial) capacit cnsisting f tw lng cylinical cnucting tubs, n insi th slni, f aius a < R an n utsi th slni, f aius b > R. Th cylins a f t tat abut th ẑ -axis. Bth cylinical cnucting tubs ha sa lngth with a & b. Th inn (ut) cnucting cylinical tub cais lctic chag +Q ( Q) unifly istibut thi sufacs, spctily. Whn th cunt I in th lng slni is slwly/gaually uc (s.g. Giffiths Exapl 7.8, p. 06-7), th cylinical cnucting tubs bgin t tat - th inn (ut) cnucting cylinical tub tating cunt-clckwis (clckwis), spctily as iw f ab!!! ẑ ng Slni z B sl in R z a Rb R Q a b Q cap E in z ab, ; z z I Cylinical Capacit QUESTION: F wh/hw s th chanical angula ntu iginat? ANSWER: Th chanical angula ntu ipat/tansf t th cylinical cnucting tubs was initially st in th fils assciat with this syst: sl B niz R, z Th B -fil assciat with th lng slni: insi Th E -fil assciat with th cylinical capacit: E cap insi an Q a b, z Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

3 n.b. UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E cap E insi is nn-z f { a b.an. z }, z sl B insi is nn-z f { R R b.an. }. Hnc, in th gin {a R & z } nly: Pynting s Vct (Engy Flux Dnsity): Q SEB nqi niz z Watts Vy usful tabl: z z z z z z S nqi Watts ngy/ pw ciculats in th ictin in th gin a R & z : fil lina ntu nsity: S EB nqi nqi kg -sc angula ntu nsity: E B n.b. in cylinical cinats, thus: Nt that: nqi kg -sc nqi nqi nqi z z cnstant!!! pints in ẑ ictin. W thn cput th angula ntu cspning t th gin a R & z : R z nqi z z a 0 z cnstant ct! R z nqi z 0 z a z ẑ by intgating = lu f gin { a R.an. z } nqi ẑ R a nqi R a z ẑ ẑ Cylinical Cinats S b a R kg -sc th lu Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

4 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Thus, th angula ntu is nqi R a z kg- sc Whn th cunt I in th lng slni is slwly/gaually uc, th changing agntic fil inucs a changing cicufntial lctic fil, by Faaay s aw: E Ba C t S ẑ sl Sinc: B insi niz R thn R f cntu C R: B in niz IN I I E R n n t t S S C C I f cntu C R: OUT I I R E R n R n t t Th {instantanus} chanical tqu xt n th inn cnucting cylin IN E R is: (f aius a) by th tangntial E -fil ch IN I t N,,, a t FE t a QE t a a nq a t ch I t I t N kg-, a a t nq a nq a z N-= t t z sc t But tqu (by its finitin) = ti at f chang f angula ntu, i.. N t t Th cspning {incas} in th chanical angula ntu th inn cylin acquis in th ti it taks th cunt in th slni t cas f I t 0 I t ch ch t ch ch kg- It t final 0 is gin by: N t t N t t t an: sc ch final a t t ch final ch ch ch a 0 ch final a t t final init a t t t ch = t t final N ch final a t t final a init a t00 = t 0 tt final tt ch ch ch final I t a final a t tfinal N t 0 a t t 0 nq a z t t tt final I t I t 0 nqa t 0 nqa I t nqa z I t0 t I t I ch ch kg- Thus: final a t tfinal a nqa Iz n.b. pints in ẑ ictin. sc th inn cnucting cylin {iw f ab} tats cunt-clckwis@ t t! 0 final 4 Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

5 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Siilaly, th {instantanus} chanical tqu xt n th ut cnucting cylinical tub (f aius b) by th tangntial E OUT -fil E R is: ch OUT I t R N,,, b t FE t b QE t bb 0nQ t ch I t I t kg- N, b b t nq R nq R z t N-= t sc z Th cspning {incas} in th chanical angula ntu th ut cylin acqui in th ti it taks th cunt in th slni t cas f I t 0 I t It t final 0 is gin by: ch final b t t ch final ch ch ch b 0 ch b final b t t final init b t t t ch = t t final N ch final b t t final b init b t00 t 0 tt ch ch final I t tt final I t b final b t t final nq R zt nqr z t t0 t t 0 t It t 0 final 0 nqr z I t It 0 I nqr z I ch ch kg- Thus: final b t tfinal b nqir z n.b. pints in ẑ ictin. sc ut cnucting tub tats clckwis t t final iw f ab! ch ch ch ch Nw nt that, f t tfinal : final Tt final a final b final i i ch final Tt nqia z nqir z nqi R a z But this is pcisly th fil angula ntu, f t 0 : nqi R a z ch i.. 0 t final Tt t tfinal nqi R a z kg- sc Thus, w xplicitly s that angula ntu is cns angula ntu that was iginally st in th fils f this ic is cnt t chanical angula ntu as th cunt in th lng slni is slwly/staily cas! Again, icscpically, angula ntu is cai by itual phtns assciat with th acscpic E& B fils in this gin f spac. Th angula ntu, as initially cai by th fils in th gin a R an z is tansf t th tw chag cnucting inn/ut cylinical tubs as th cunt flwing cnucting in th slni is slwly cas f I 0, th cylins acquiing nn-z chanical angula ntu, th ttal f which = initial angula ntu! Nt als th ti-s situatin (I incasing, i.. f 0 I ) als s th ti-s thing bcaus th fc/intactin (icscpically & acscpically) bys ti-sal inaianc!!! 0 Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs. 5

6 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Th Fil Engy Dnsity u, Pynting s Vct S, ina Mntu Dnsity an Angula Mntu Dnsity Assciat with a Pint Elctic Chag q an a Pint Magntic Mnpl g n.b. This is a static pbl i.. it has n ti pnnc! q q Pint lctic chag at igin: E 4 4 Pint agntic npl.g. lcat at z : g 0 g B 4 40 B x ẑ g z igin g g z 4 4 Vctially: z z cs ŷ P bsatin/fil pint aw f Csins: z & cs cs fil ngy nsity: u E B EE B B q g u 4 4 cs q g u 4 4 cs q u 4 g 6 6 cs but: c 6 Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

7 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E g c q g c u 4 4 q cs cs Juls n.b. f (als tu f = 0): u 4 q g c Watts q S g z 4 4 cs In sphical cinats: 0, z cs sin, z sin ẑ z n.b. z sin ŷ is t an t z Pynting s Vct: S EB cs cs,. an x qg z qg S z 6 6 sin sin qg qg S z sin 6 6 cs cs Nt that Pynting s ct S i.. ngy is ciculating in th (aziuthal) ictin in a static pbl! Nt als that S anishs whn = 0 (i.. npl g is n tp f lctic chag q ) an als anishs whn anti- t z (thn z sin 0 )! kg sin qg qg sin z 6 6 cs cs Fil ina ntu nsity: S S c EB -s H again, nt that i.. lina ntu nsity is ciculating in th (aziuthal) ictin in a static pbl! Nt als that anishs whn = 0 an als anishs whn anti- t z (thn z sin anishs)! Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs. 7

8 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Fil Angula ntu nsity: kg -s qg z 6 cs but: qg z 6 cs Nw: z z z z z cs z cs cs sin sin wh: z cs sin cs an: z cs sin q g 6 cs sin kg -s Fil Engy Dnsity: Pynting s Vct: u g c q cs S 6 qg cs sin Juls Watts ina Mntu Dnsity: Angula Mntu Dnsity: 6 q g cs q g 6 cs sin sin kg -s kg -s Th Ttal Fil Engy: U u Juls bcaus E 0 B bth ig/a bth singula (at 0 an spctily) s this is nt a supis!!! Hw, th Pw flwing thugh/cssing th nclsing sufac S is z (!!!): P S a q g a an sin 0 S 6 S cs But: S 0!!! Th Pw 0 Watts!!! P bcaus a an a an 0, i.. is always t!!! fil ngy assciat with lctic chag agntic npl q g syst ciculats! (i.. is fully cntain within nclsing sufac S!!!) 8 Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

9 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Ttal Fil ina Mntu: qg p sin 6 cs Nt that: a 0 bcaus a kg- sc Fil ina Mntu S ciculats (i.. fil lina ntu is als fully cntain within th nclsing sufac S)! p 6 6 q g sin cs sin qg cs t s th -intgal fist (tiial gt ): sin p qg cs Nxt, lt s th -intgal: cs sin p qg cs cs qg 8 8 qg 0 qg 0 4 cs sin 4 cs cs cs sin sin cs qg 8 0 cs 8 cs 0 cs Finally, w cay ut th -intgal: sin p qg q 0 g 8 8 qg 8 qg : p kg- 8 sc Nt that th fil lina ntu p is finit as lng as th lctic chag-agntic npl spaatin istanc 0. Whn th lctic chag q is n tp f th npl g, thn 0 an p bcs infinit Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs. 9

10 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Ttal Fil Angula Mntu: kg- sc sin cs cs sin qg z qg z cs 6 cs t s wk this ut in Catsian cinats: sin csxsinsiny csz 6 6 x y z z sin cs sin sin cs cs sin qg cs x y z sin cs cs sin cs sin cs sin qg Nw nt that: cs sin cs sin 0 sin cs cs Thus, th intgals th -aiabl f bth th x an y ts xplicitly anish, x - an ŷ - cpnnts f bth anish u t anifst axial/aziuthal syty (tatinal inaianc) f this pbl abut th ẑ -axis; nly th z -t ains: Nw: cs qg 6 sin cs sin cs z cs sin qg sin z cs sin t s th -intgal fist - (tiial), sinc intgan has n xplicit -pnnc, gt: qg sin z sin cs Nxt, lt s th -intgal nting that: cs cs 0 0 cs cs cs cs cs cs cs cs cs cs cs cs 0 0 Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

11 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E qg 8 sin qg sin sin z z cs 8 cs 0 0 Finally, lt s th -intgal: t u cs an u cs sin, an sin cs u F 0u an f u Thn: qg u u u z 8 but: u u u u u qg u u u u qg 8 uz u u 8 u u qg qg qg u u z z z u qg z kg- sc 4 u uz Nt that th fil angula ntu assciat with th lctic chag-agntic npl syst is inpnnt f th q gspaatin istanc,!!! Quantu chanically is quantiz in intg ( n half-intg) units f h, wh h = Planck s cnstant, i.. h qg qg h!!! 4 Hw, call th Diac Quantizatin Cnitin (P45 ctu Nts 8) which as f insisting n th singl-alu natu f th lctn s wafunctin cicling/biting a {psu} hay agntic npl: g g nh (SI units) Diac Quantizatin Cnitin c Ths tw fula ag if n n, thus if n =,,, thn,,,... an h 4,,,... qg kg- sc Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

12 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Th Fil Engy Dnsity u, Pynting s Vct S, ina Mntu Dnsity an Angula Mntu Dnsity Assciat with a Pint Elctic Chag q an a Pint/Pu Magntic Dipl Mnt z n.b. This is {again} a static pbl has n ti pnnc! q q ) This ti, w lcat th pint chag q at z : E 4 4 ) W lcat th pu/pint agntic ipl nt z at th igin: 8 B 4 in cinat-f f 8 B cs sin 4 in sphical cinats q ẑ z z (igin) P = bsatin/fil pint ŷ Vctially:, an z z z Engy Dnsity: x aw f csins: cs cs u E B EE B B u q 4cs sin 6 4 cs 4 cs 8 8 cs sin cs sin Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

13 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E u u u q 6 4 cs cs sin cs q 6 4 cs cs q cs cs cs cs cs n.b. If ( 0) thn f 0 Pynting s Vct: S EB but: an: q 4 : u cs q S 4 8 cs sin z 4 q 8 cs sin z 6 z z an z cs sin thn: cssin z cssin cs sin cszsinz sin sin cs sin cs sin sin cs z 0 sin cs z z z z zz cs sin cs sin sin Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

14 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E x But: In sphical cinats: an z cs sin cs sin z cs sin sin 0 z cs sin sin cs cs 0 q 8 4 S sin sincs sin 6 q 8 4 cs sin 6 cs 8 4 cs sin 6 cs Nt that:.) S pints in th -ictin!!! anishs (f > 0) whn: cs 0!!! S q.) S Watts i.. whn: cs quatin f a lin-cu (cspns t a sufac in!!).) S als anishs (f > 0) whn: sin 0 i.. at 0 an N/S pls! 4.) Nt als that {h} S s nt anish whn = 0 (i.. whn pint lctic chag q an pint agntic ipl nt z a n tp f/cincint with ach th!! q 5.) F ( 0, with > 0): S sin Watts 5 6 Fil ina Mntu Dnsity: S 6 S q Sa cnts a ab f ẑ ŷ 8 cs apply h f. S 4 cs sin kg -sc 4 Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

15 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E wh an Fil Angula Mntu Dnsity: 6 q E 4 cs 8 sin cs Vy Usful Tabl: Nt that f 0 cs Engy in Fil: that pints in: ictin f 0 ictin f q U u cs cs cs U E igs at =, B igs at =0 sin Pw in Fil cssing/passing thugh nclsing sufac S: P S a 0 S bcaus a an a but S pints in th -ictin. ngy ciculats within lu, ncls by sufac S!!! Ttal ina Mntu in Fil: 8 4 cs p sin q 6 cs Caying ut th -D lu intgal is ~ tius. W nt xplicitly wa thugh this h. Th cntibutins f ach f th ts assciat with th nuat in th intgan a a.) finit, b.) lgaithically-ignt, an c.) z spctily. Thus p h, an als nt that ach f ths ts is pptinal t q, which is stngly ignt as th lctic chag q pint/pu agntic ipl spaatin istanc 0. Nt again that a 0 i.. fil lina ntu ciculats in -ictin. S Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs. 5

16 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Ttal Angula Mntu in Fil: 8 4 cs sin q 6 cs 8 4 cs q 6 sin cs 8 4 cs sin q 6 sin cs W again chs t wk this ut in Catsian cinats, s sin cs z 8 4 cs cs q zsin 6 cs Thn sin csxsinsiny csz, thus: cs z sin sincscsx sincssinycs zz sin sin cscsxsin cssinysincs zsinz sin cscsx sin cssiny sin cs z sin cscsxsin cssinysin z Again, th intgals f th x an ŷ cpnnts f will cntibut nthing whn th intgals f... cs... sin a cai ut nly th ẑ t suis th an 0 -intgatin: cs q sin z 0 0 cs Caying ut th ain f th intgatin is ~ swhat tius, s w n t xplicitly wa thugh this h, but intstingly nugh, it yils a finit sult (f 0 ): q 4 z, which igs as th lctic chag q pint agntic ipl 8 nt spaatin istanc 0, which cincis with that f a al/physical lctn i.. a pint lctic chag with pint agntic ipl nt f agnitu =. 6 Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs.

17 UIUC Physics 46 Fils & Sucs II Fall Sst, 05 ct. Nts Pf. Stn E Th ain pups f th ab xapl, asi f its instuctinal us as acaic xcis t illustat a sipl static lctagntic syst in which ngy, lina ntu an angula ntu a all inl, is als t phasiz/unsc th iptant pint that al/physical lctns siultanusly ha bth a pint lctic chag an a pint agntic ipl nt bth f which a ncssay ingints in t b abl t tansf {appantly} abitaily lag aunts f ngy, lina an angula ntu t th such paticls ia th lctagntic intactin. Withut th siultanus psnc f bth an lctic chag an a agntic ipl nt, tansf f lina & angula ntu cul nt ccu! It is nt supising that classical acscpic lctynaics fails h t cctly quantitatily xplain th physics pati at th icscpic scal th ain f quantu chanics (an byn i.. th stuctu f spac-ti itslf at th sallst istanc scals). Dspit than 00 yas f cllcti fft, sinc xplicit iscy th lctn by J.J. Thpsn in 897, an th iscy f lctn spin an th lctn s agntic ipl nt by fist bs xpintally by O. Stn & W. Glach in 9 an subsquntly xplain thtically by W. Pauli an S. Gusit an G. Uhlnbck in 95, tay, w still ha gain n funantal insight as t what pcisly lctic chag is, n w unstan th physics igins f intinsic spin angula ntu (assciat with ith spin-½ fins {an th accpanying Pauli xclusin pincipl } intg spin bsns, such as th phtn {an thi accpanying ggaius natu at th quantu ll th ppsit f that f fins!}, n any funantal xplanatin f th xistnc f th intinsic agntic ipl nt(s) assciat with all f th funantal, pint-lik lctically-chag paticls th gnatins f intg-chag pint-lik lptns,, an six pint-lik quaks : uct,, an : sb,,. Nt that th W bsn th spin- lctically-chag iat f th wak intactins als has a agntic ipl nt, as wll as an lctic quaupl nt. Ths sa funantal paticls als intact ia th wak intactin an thus ha wak chags an wak agntic nts {th W bsn als aitinally has a wak quaupl nt}. Th spin-½ quaks aitinally intact ia th stng intactins, an hnc ha stng chlctic chags (, gn & blu ) as wll as stng ch-agntic ipl nts. Thus, pint chag an pint agntic ipl nts, tc. assciat with th all f th funantal paticls w knw an l tanscns ach f th iniiual fcs, an in fact pints t/hints at a singl cn xplanatin. W knw that intinsic spin an th accpanying agntic ipl nts f ths paticls a in anifstly fully-latiistic phnna, an thus hint at an xplanatin pati nly at th sallst cnciabl istanc scal, wh th quantu bhai f spac-ti itslf bcs anifst i.. th s-call 5 Planck istanc scal, als knwn as th Planck lngth: G c.6640 ts, 5 44 with cspning Planck ti tp P c GN c scns! It ay s supising that Nwtn s gaitatinal cnstantgn nts h hw, Einstin s gnal thy f latiity tlls us that th is an intinsic link btwn th gaitatinal fc as w unstan it acscpically in th y-ay wl an th cuatu f spac-ti! P N Pfss Stn E, Dpatnt f Physics, Unisity f Illinis at Ubana-Chapaign, Illinis All Rights Rs. 7

Propagation of Light About Rapidly Rotating Neutron Stars. Sheldon Campbell University of Alberta

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