Spring Semester, 2012

Transcription

1 清大物理系 電動力學 ( 二 任課老師 : 張存續 loamis (II (PHYS 5 Spig Sms, Dpam of Phsis, Naioal Tsig Hua Uisi, Taiwa Tl. 4978, -mail: hshag@phs.hu.u.w Offi hou: :-4: 47 助教 : 趙賢文 : 95865, s9887@m98.hu.u.w 張家銓 : , 45966@homail.om

2 一百學年度第二學期 PHYS5 電動力學二 ( 學分 教師 : 張存續 教學目標 : 鼓勵學生思考 整合 創造 課本 : J. D. Jakso, Classial loamis, Thi iio 參考書 : D. J. Giffihs, Iouio o loamis. D. K. Chg, Fil a Wa lomagis o D. M. Poa, Miowa giig.. P. Fma,. B. Ligho, a M. Sas, Th Fma Lus o Phsis II. 講義下載 : D: hp:// 全部講義 : hp:// 上課時間與地點 : ( 表訂 每星期二 (: : am 與星期四 (: : am 在物理館 9 上課進度 : 如附表 學期成績 : 期中考 4%, 期末考 4%,4 次小考 % 給用心且認真同學加分 : 點名 6%+ 課程參與度 4% 學期成績會視考試狀況稍作調整, 期望能正確的反應大家的學習效果 習題 : 有 6-75% 題目會類似勾選的習題 ( 確保有努力就有收穫 另外 5-4% 組合觀念題 ( 讓程度好的同學可以發揮 考試 : 以英文命題 考試於原上課時間舉行, 考試地點為原上課教室, 除非另有宣佈 考試時, 除非學生有充分理由, 不得請假 未事先請假者, 該次考試以 分計 演習課 : 每週 次, 每星期二第四節 (: : am 暫定 Shul (pig o sus' laig oiio 週次 時間 上課內容 一 / ( 二 導論 ( 課程簡介 評分規定 複習 Mawll quaios / ( 四 Chap.8 Waguis, soa Caiis, a Opial Fibs 二 /8 ( 二 和平紀念日 / ( 四 Chap.8+ Qui # 三 /6 ( 二 Chap.8 /8 ( 四 參加國際會議 四 / ( 二 Chap.8 /5 ( 四 Chap.9 aiaig Ssm, Mulipl Fils a aiaio 五 / ( 二 Chap.9 / ( 四 Chap.9 六 /7 ( 二 Chap.9 /9 ( 四 Chap.9+ Qui # 七 4/ ( 二 校際活動週 4/5 ( 四 Chap. Saig a Diffaio 八 4/ ( 二 Chap. 4/ ( 四 Chap. 九 4/7 ( 二 Chap. 4/9 ( 四 Chap. 十 4/4 ( 二 Chap. Spial Tho of laii 4/6 ( 四 Mim Chs. 8, 9, a 十一 5/ ( 二 Chap. 5/ ( 四 Chap. 十二 5/8 ( 二 Chap.+ Qui # 5/ ( 四 Chap. 十三 5/5 ( 二 Chap. 5/7 ( 四 Chap.4 aiaio b Moig Chags 十四 5/ ( 二 Chap.4 5/4 ( 四 Chap.4 十五 5/9 ( 二 Chap.4+ Qui #4 5/ ( 四 Chap.4 十六 6/5 ( 二 Chap.4 6/7 ( 四 Chap.4 十七 6/ ( 二 Chap.4 6/4 ( 四 Fial Chs., a 4 十八 6/9 ( 二 6/ ( 四 Makup am if. * 此進度表僅供參考, 實際情形視學習狀況調整

3 清大物理系 電動力學 ( 二 任課老師 : 張存續 loamis (II (PHYS 5 Spig Sms, Dpam of Phsis, Naioal Tsig Hua Uisi, Taiwa Tl. 4978, -mail: hshag@phs.hu.u.w Offi hou: :-4: 47 助教 : 趙賢文 : 95865, s9887@m98.hu.u.w 張家銓 : , 45966@homail.omom.Tbook a Cos of h Cous: J. D. Jakso, Classial loamis, iio, Chaps 8-, 4. Oh books will b f i h lu os wh.. Cou of Class : Lu os will b poj squiall o h s uig h lass. Phsial ops will b mphasi, whil algbai ails i h lu os will of b skipp. Qusios a ouag. I is assum ha sus ha a las go hough h algba i h lu os bfo aig lasss (impoa!.. Gaig Poli: Mim (4%; Fial (4%; Qui 4 (% a a pois (%. Th oall so will b omali o fl a aag osis wih oh ouss. 4. Lu N Nos: Saigi fom basi quaios, h lu os follow Jakso losl wih algbai ails fill i. quaios umb i h foma of (8.7, ( f o Jakso. Supplma quaios i i lu os, whih will la b f, a umb (, (... [saig fom ( i ah hap.] quaios i Appis A, B of ah hap a umb (A., (A. a (B., (B. Pag umbs i i h (.g. p. 95 f o Jakso. Sio umbs (.g. S. 8. f o Jakso (p fo sios i Ch.. Mai opis wihi ah sio a highligh b bolfa haas. Som wos a p i ialii haas fo aio. Thial ms whih a iou fo h fis im a uli. Chap 8: Waguis, soa Caiis, a Opial lfibs 8. Fils a h Sufa of a Wihi a Goo Couo* Noaios: H, : fils ousi h ouo; HC, C: fils isi h ouo; : a ui o o ouo sufa; :a omal ooia io h ouo.,, i Assum: (i fils (ii goo bu o pf ouo, i.. H, H,, bu [S Ch. 7 of lu os, q. (4]. b (iii H ( is kow. Fi: (, H (, a pow loss,. i ms of ( H *Th mai suls i S. 8. [ (8.9, (8., (8., (8.4, a (8.5] ha b i wih a muh simpl mho i Ch. 7 of lu os. [S os followig q. (6]. So, w will o o his sio i lasss.

4 8. Fils a h Sufa of a Wihi a Goo Couo (oiu Calulaio of C (, H C( : I h ouo, w ha B ( i H goo ouo assumpio H J D ( i b I a goo ouo, fils a apil alog h ( omal o h sufa, s Ch. 7 of lu os. H (4 (, (, ( H i (5 C ski ph H i H Sub. (4 io (5: (8.7 i H( H( b.. a : H( H ( i i ( ( ( H H H (6 H J= (5 H( H ( H( H (7 i Sub. (7 io (6 H ( ( H o ( Fils a h Sufa of a Wihi a Goo Couo (oiu i H H H i C i H C i Sub. ( ( io ( ( ( ( ( (8. ( ( ( ( H ( (8. b.. a (4 P loss a Pow Loss P Ui Aa: C C C im aag pow io ouo p ui aa ( ( H * * ( ( H C H( H ( (8. 4 ( C H 8. Fils a h Sufa of a Wihi a Goo Couo (oiu Alai mho o i (8.: ( i (8. J( ( ( i ( H (8. im aag pow loss i ouo p J ( ( J ( ui olum (8. P loss ( ( a J H H (, sam as (8. ffi sufa u K ff f K ( i ( f f ( i ( J H H ( (8. (8.4 P (8. & (8.4 loss Kff (8. 5 a : Mos i a Wagui Cosi a hollow ouo of ifi i lgh a uifom oss sio of abia shap (s figu. W assum ha h fillig mium is uifom, lia, a isoopi ( B H ; D, wh a a i gal ompl umbs. This is a suu ommol us o gui M was as wll as a a as wh a soluios a possibl (fo som simpl oss sios. Mawll quaios a b wi B (8, B (9 ompl a ( B ( (8 B ( ( Similal, ( 9 B B (

5 Mos i Waguis (oiu ik (, ( i : ooias ass o, L,.g. (, o (, ik i B (, B ( k h k i Jakso wh, i gal, a k a ompl osas. To b spifi, w assum ha h al pas of a k a boh posii. Th, ik i ik a i ha fowa a bakwa phas loiis, ik spil. As will b s i (, i ik a i also ha fowa a bakwa goup loiis, spil. H, w all iki ik i a fowa wa a a bakwa wa. Wih h assum a ps, w ha k + k, Casia,, liial Mos i Waguis (oiu Thus, ( k (8.9 ( B B B ( k (4 B ( I is i gal o possibl o obai fom (8.9. So ou sag h is o sol (4 fo ( a B(, a h pss h oh ompos of h fils [ ( a B( ] i ms of ( a B ( hough qs. (7 a (8. is : Wiig ( a usig h liial ooia ssm, i h quaios fo a fom (8.9. (hi: =, = Giffihs 9.5 Gui Was 95W 9.5. Wa Guis Ca h lomagi was popaga i a hollow mal pip? Ys, wa gui. Waguis gall ma of goo ouo, so ha = a B= isi h maial. Th bouaoiiosa a hi wall a: // a B Th gi fom of h moohomai was: (,,, ˆ ˆ ˆ (, ( B(,,, B ˆ ˆ ˆ (, ( B B B ik ( ik ( i( k i( k Giffihs Gal Popis of Wa Guis I h iio of h wa gui, h was saisf Mawll s quaios: B B Wh f a J f? B B wh W obai B B i (i i B (i B B i (ii i B ( B B i (iii i B (i

6 Giffihs T, TM, a TM Was Dmiig h logiuial ompos a B, w oul quikl alula all hohs ohs. i ( B k ( / k W obai B B k k B i B ( k ( / k i B ( k ( / k i B ( k ( / k T o i hs laios b ouslf. If T (ass li was; If B TM (ass magi was; If a B TM was. L B B B ik Gal Appoah, Casia ik i B (5 B B, liial ik B i B B ( Usig h laios: ( ompos of (5 a (6: ik ib (6, w obai fom h ass (7 B ik B i (8 upp fowa low bakwa I (5-(8, h sig applis o h wa. wi (7 a ( Mos i Waguis (oiu ik ib (7 B ikb i (8 Si a B ha ala b sol fom (4, (7 a (8 a algbai (ah ha iffial quaios. W ow maipula (7 a (8 o limia i B a hus pss i ms of a B. (7 ( ik ib a a a If, a a boh ip of, h a aa i B ik (9 Sub. (9 io ( Mos i Waguis (oiu B ik ik i i B k i B k ( Mulipl ( b i : ib ik k i k B (8.6a k Simi lal, B i k B k (8.6b Thus, o a B ha b sol fom (4, h soluios fo a B a gi b (8.6a a (8.6b.

7 Mos i Waguis (oiu Disussio: i (i, B,, B i (8.6a a (8.6b a fuios of ol. (ii a a b ompl. Im( o Im( implis issipaio. (iii B lig B =, w ma obai a s of soluios fo,, a B fom (4, 8.6a, a (8.6b, spil. I a b show ha if h boua oiio o is saisfi, h boua oiios o a B a also saisfi. H, his gis a s of ali soluios all h TM (ass magi mos. Similal, b lig =, w ma obai a s of ali soluios all h T (ass li mos. (i is h gaig fuio fo h TM mo a B is h gaig fuio fo h Tmo. Th gaig fuio is o b Ψ i Jakso Mos i Waguis (oiu TM Mo of a Wagui (B =: (s pp s ( wih boua oiio ( ik (a Assum pfl ouig wall. H (b k Z Z wa impa k, k of TM mos ( T Mo of a Wagui ( =: (s pp H H s ik Z h k, wa s H impa of T mos b b.. H s Zh k H s (a H s k H s ( wih boua oiio ( H (a H H (b ( Mos i Waguis (oiu Disussio: (i ih ( o ( osius a igalu poblm (s lu os, Ch., Appi A. Th igalu will b a ifii s of is alus fi b h boua oiio, ah psig a igmo of h wagui (A ampl will b poi blow. (ii (b a (b show ha is ppiula o B (also u i a ai. (iii (b a (b show ha a B a i phas if,,, k a all al (o u i a ai. (i ( [o (] is h ispsio laio, whih las a k fo a gi mo. ( Th wa impa, Z o Z h, gis h aio of o H i h wagui. Fil Pas of Ciula Wagui Mos -fil (a Isio o Loss ( B (b Is sio Loss (B sufa u (mm L (mm T T T T TM TM T T T T TM TM

8 Fil Pas of Ciula Wagui Mos -fil (a Isio o Loss ( B (b Is sio Loss (B sufa u (mm L (mm T T T T TM TM T T T T TM TM Chaaiaio of Ciulal Smmi T Mo (a simulaio S S (B - -4 = o = 6 o = 9 o = 45 o = 8 o Fq (GH (b masum (B S GH = o = 45 o = 6 o 8.5 GH = 9 o = 8 o = 8 o Th Fq (GH T. H. Chag a B.. Yu, Yu High-Pow Millim-Wa oa Joi,. Si. Isum. 8, 47 ( Mos i Waguis (oiu TM Mo of Coaial a Paalll-Wi Tasmissio Lis ( = B = : (s Jakso p. 4 i k B k wi i B k B k (8.6a (8.6b Ths quaios fail fo a iff lass of mos, all h TM (ass lomagi i mo, fo whih hih B. How, h gi h oiio fo h is of his mo: quaios i agula bos a k. (8. basi quaios fo h TM mo. 7 (8.7 is also h ispsio laio i ifii spa. This maks h TM mo usful baus i a popaga a a fqu. To alula a B, w o go bak o Mawll quaios Mos i Waguis (oiu TM( B B ( L B a, TM ik i h, baus B, h -ompo of B gis TM TM TM (, A ( a, baus, gis A ( ( (, TM TM wh TM is h gaig fuio fo h TM mos. Baus a o h sufa of a pf ouo, TM is subj o h boua oiio TM os. o h ouo. This gi s TM os. o TM wh, if h is ol o ouo. So, TM mos is ol i -ouo ofiguaios, suh as oaial a paalll-wi asmissio lis. Fiall, B is gi b h TM ass ompos of B: BTM TM. k

9 Wh sigl ouo ao suppo TM was? (I L s osi h pop of D Lapla quaio. Suppos Φ TM ps o wo aiabls. TM a paial iffial quaio (PD; TM o a oia iffial quaio (OD. Hamoi fuios i wo imsios ha h sam popis as w o i o imsio: Φ TM Φ TM has o loal maima o miima. All ma ou a h bouais. (Th sufa ma o b a qualpoial. If Φ TM=os., TM= & B TM= (No a flawlss agum Wh sigl ouo ao suppo TM was? (II Dai Chg s plaaio. Chap., p.55.. Th magi flu lis alwas los upo hmsls. Fo a TM wa, h magi fil li woul fom los loops i a ass pla.. Th gali Amp s law quis ha h li igal of h magi fil aou a los loop i a ass pla mus qual h sum of h logiuial ouio a ispla ouio u isi h wagui.. Th is o logiuial ouio u isi h wagui a o logiuial ispla u ( =. 4. Th a b o los loops of magi fil lis i a ass pla. (wak olusio Th TM wa ao is i a sigl-ouo hollow wagui of a shap. (Agai, o a pf agum Mos i Waguis (oiu I summa, h TM mos a go b h followig s of quios: TM ( ( TM TM ( (a k B TM TM (b k (o H TM TM TM Y TM k ( wh Y ( / is h (iisi amia of h fillig mium fi i Ch. 7 of lu os (h las pag of S. II. Disussio: : Mos i Waguis (oiu (i Fo h TM mos, w sol a -D quaio TM( fo TM (. Bu his is o a -D poblm baus TM is o h (, full soluio Th full soluio is TM ( ik i (, B ( full soluio. Th full soluio is B TM k TM TM ( TM TM wih ( a B. Fo a aual -D losai poblm [ ( ( ], w ha (, whih h gis h full soluio ( (. (i i No h iff bw h sala poials isuss h a i Ch. a Ch. 6. TM TM ( ga TM as a mahmaial ool. ( ( ga as a phsial quai. (, (, A (, ga a Aas mahmaial ools.

10 Mos i Waguis (oiu ampl : T mo of a agula wagui wi h basi quaios fo h T mo: H H s ( wih boua oiio ( ik H (a H H ZhH (b k k ( agula gom Casia ssm H, h wa quaio i ( boms: k H ( Mos i Waguis (oiu wi (4: H (4 k ik ik H Assumig p fo k k k H I o fo, w mus ha H k k k,, w obai ik ik whih his saisfi fo k, k, k. Si (,, ik ik ik ik (,, a (, a all lial ip pais, h ompl soluio fo is H i ik ik ik ik H A A B B ik ik C C ( Mos i Waguis (oiu Applig boua oiios [s (] o (5: H ik H ik ik ik ik ik ik i H A A B B C C B B ik AikA A A b B B ikbik B B B o a i ik iik H os k os kc C B B si ka k m a, m,,, a B B si kb k b,,,, b m ik os os i iki H C C a b (6 fowa wa bakwa wa Sub. k m, k a io, w obai k k k b k m, m,,,, (7 a b Giffihs 9.5. T Was i a agula Wa Gui, a B (, X( Y( spaaio of aiabls X Y X Y ( k X Y k a k X Y wih k k k *Giffihs iaio uss iff boua oiio --- =. X ( A si k B os k Y( Csik Dosk

11 Giffihs T Was i a agula Wa Gui (II B Cos k Dsi k i (@ C (@ b si kb, k (,,,... b B B Aos k Bsi k (@ A H m (@ a sik a,, k ( m,,,...,,, a B (, B os( m / a os( / b ht mo m k m a b ( / [( / ( / ] Mos i Waguis (oiu wi (7 as k, (8 m / wh m (, m,,,, (9 a b m k ah pai of ( m, gis a omal mo (T m mo of h wagui. m a ao boh b, baus ha will a a siuaio o (8.6 o (a, makig H a imiabl. m is h uoff fqu (h fqu a whih k of h waui fo h T m mo. Was wih m ao popaga as a T mo baus boms pul imagia. m (8 is h T m mo ispsio s laio of awagu i fill wih a ili mium wih osa (i gal ompl a. Fohusualasofaufillwagui of a ufill wagui, w ha a (, a (8 (9 a b wi k / fo ufill k wih ( m m m ( a b wagui m k m, m Mos i Waguis (oiu m a b g gui walgh k uoff walgh m p mo & wa o gom fq. f f spa walgh g s s m k al popagaig was s o m k g f o s m k imagia as fils T ( ( =.9a a T, T ( ( = aa b a T ( ( =a a usabl bawih ( a f a Qusio : A pial wagui has k a = b. Wh? Qusio : Ca w us a wagui o aspo was a 6 H? f g Mos i Waguis (oiu Oh quaiis of is: ( Diffiaig k m wih sp o k k k m k k k g goup loi i ufill wagui k g g as m ( Th maiig i fil ompos (,, H, a H a b obai fom H hough ik H (a H k m s ( a (. H (b k upp fowa wh h sig applis o h wa. low bakwa

12 Mos i Waguis (oiu T mo fil pas of agula wagui Mos i Waguis (oiu TM mo fil pas of agula wagui fom. L. Gio, "Miowa masums". : uoff fqu soli u: -fil lis; ash us: B-fil lis fom. L. Gio, "Miowa masums". : uoff fqu soli u: -fil lis; ash us: B-fil lis Mos i Waguis (oiu Disussio : Wagui a miowas A pial wagui has a b o maimi h usabl bawih ( a f a o whih ol h T mo a popaga p a h mo pui is maiai. Was a omall aspo b h T mo o his fqu ag. Waguis om i iff sis. Usabl bawihs of waguis of paial imsios (. m a m o h i miowa ba ( MH o GH. Compa wih oaial asmissio lis, h wagui is apabl of halig muh high pow. H, i is ommol us i high- pow miowa ssms. I a aa ssm, fo ampl, i is us o aspo miowas fom T ( ( = 9.9.9a a h gao o h aa. T, T ( ( = aa T ( ( = a a a usabl bawih ( a f a a k Mos i Waguis (oiu ampl : : TM mos of a oaial asmissio li TM mos a go b h followig s of quios: ( TM ( TM TM( (a H TM Y TM (b k ( TM TM ( gis (. TM Ngl h mos ( TM CI( C. ( TM a V C V / I( a/ b I( / b Appl b.. TM V. / I( a b TM( b C CI( b V TM (, / ik i I b a (a, b h gi ( YV TM ( / ik ( (, i H I b a

13 M j P m iig a Spifi Mo j ˆ A oal 4 4 P 4 p 4 ( p J ( p oal 4 4 P4 p4 ( p4 4 J4 ( p4 8 B (a T (b T sio (B Tasmiss ( T 4 (B Tasmissio B smissio ( Ta Th. u 8. GH Th. u Th. u p Fqu (GH Th. u 4. GH Th. u Th. u p Fqu (GH Th. u Th. u. GH Th. u p Fqu (GH T. H. Chag, C. S. L, C. N. Wu, a C. F. Yu, iig iula Tm mos a low ah gio, Appl. Phs. L. 9, 5 (8. Diffiulis of iig a High-O Mo: Tak T as a ampl Dsi mo T Couplig suu oa-f Wagui aius.86 mm Paasii mos T,A, T,B T,A, T,B T T,A, T,B T,A, T,B TM TM,A, TM,B T M,A, TM,B T 4,A, T 4,B T,A, T,B TM TM,A, TM,B T 5,A, T 5,B T,A, T,B TM,A, TM,B Appliaios of Wagui Mos (I Appliaios of Wagui Mos (II Plasma hambs Maial possig Mo os oa jois Appl. Phs. L. 94, 4 (9 TH wagui, iulao, isolao, pow ii, aa

14 8.7 Mos i Caiis W osi h ampl of a agula ai (i.. a agula wagui wih wo s los b ouos, fo whih w ha wo aiioal boua oiios a hs. wi (7: H m ik os os i ik i a C C b oss -sioal b.. (i: H ( C C iw b H i H m os a os si b k o a b.. (ii: H si iw ( b si k k l, l,, o ( i m l m,,,, H H os a os si, ( b l,, Sub. ( io k, wh ( m m m ml ( m l ml a b / :soafq u (4 of h T ml mo 8.5 g Flow a Auaio i Waguis Pow i a Losslss Wagui : Cosi a TM mo (, H H i a mium wih al, (h al, k. * * TM S H [ H H ] ompl Poig o * * [ ( ( ] fo TM mos k * (b, k * [ ik ] 4 al a k (a k i * [ ] 4 k P TM im aag pow i h -iio k A [ STM] a [ A: ossioal aa] k * ( (5 4 A a a b 8.7 Mos i Caiis (oiu C+ m ik Fom (6 : os os i iki H C C, a b w s ha a ai mo is fom of a fowa wa a a bakwa wa of qual ampliu. Th fowa wa is fl a h igh o bom a bakwa wa, a us io a fowa wa agai a h h lf. Th fowa a bakwa was suppos io a saig wa [s (]. Thus, w ma obai si iw h oh ompos of h ai fil b b supposig h oh ompos of fowa wa bakwa wa h wo alig was, as i (6. Compaiso wih ibaioal mos of a sig: p aiabl(s ip aiabls mo i sig (osillaio i amp., l ai,, B, B, (o B,,, m,, l C 8.5 g Flow a Auaio i Waguis (oiu ( a G's fis ii: s (.4 L a b ip of a appl (.4 o a slab of sufa aa A (o h - pla a ifiissimal hikss i, sufa igals o A( a l wo s of h C slab, whih aish. A ( a l C * L a, h * * a A ( a [ * l ] C A a = b boua oiio i l b (4 A a. (6 k * Sub. (6 io (5: PTM (, w obai 4 A a k PTM, [wh ] (7 A a k

15 8.5 g Flow a Auaio i Waguis (oiu (i.. a k is h k (8 uoff fq. of h mo. ( ( k (9 Sub. (8 a (9 io (7 TM ( ( P a [f. (8.5] (4 A Similal, fo h T mo a al,,, a k, w obai fom (, (a, a (b, k i * T 4 [ H H H ] k k T [ T] A S A ( ( A H a S (4 P a H a [f. (8.5] (4 No: P a P a pss i ms of h gaig fuio. TM T 8.5 g Flow a Auaio i Waguis (oiu : + m g i a Losslss Wagui : s S a J i w w (6.4 w D if, a al, w a wm a 4 4 also al a ps im (6. w m BH B 4 4 aag fil g siis. Appl (6.4 o a sio of a losslss S a, H wagui [i.., a al a h wall H = ouii ]. * S H ( (isi olum J o boh s J o h si wall S o h si wall, (h, k a al a H a i phas [b (b&(b] H is al S is al o boh s s Sa is al [(6.4] s S a (o pow io o ou of olum Im[(6.4] w wm (B-fil g -fil g ( 8.5 g Flow a Auaio i Waguis (oiu Fo h TM mo ( H : UTM fil g p ui lgh (b A( w wm a A wma A a H k A a k 4 ( (4 k A a A a A a (a A a b (6 Similal, fo h T mo ( : UT A Wa A a ( (44 A H a Fom (4, (4, (4, a (44 Us (a,b a P TM PT k G s s ii ( g (8.5 UTM U T k (9 (. p pg k ( g Flow a Auaio i Waguis (oiu Auaio i Waguis Du o Ohmi Loss o h Wall: W pss k fo a losslss ( a loss ( wagui ( ( k, as k (8.55 k i, ( k wh k is h soluio of h ispsio laio fo, i.. [i i (8] (45 Th pssio fo i (8.55 assums ha h wall loss ( k moifis b a small al pa a a small imagia pa, wh a a o b mi. Phsial aso fo α : ffi wagui aius iass b b a amou ski ph. A lag wagui has a small. H,. Phsial aso fo β : Pow issipaio o h wall.

16 ( 8.5 g Flow a Auaio i Waguis (oiu I k k i, is o of pima is baus i moifis h gui walgh slighl. How, suls i auaio, whih a b sigifia o a log isa. W ouli blow how a b alua. pow issipaio/ui lgh * ik ik k P pow flow [ H ] i P P P P (8.56 fil auaio osa (8.57 (8.5 ff l K (46 (8.4 K ff H l (47 (46(47 P (8.58 l H Si h wall loss a b ga as a small pubaio, w ma us h o-o H i fo i S.8. o alula P. 8.5 g Flow a Auaio i Waguis (oiu Spifiall, w alula h o-oo a H, a us h o-o a H o alula P fom (8.5 a P/ fom (8.58. is h fou fom (8.57. Fomula fo fo agula a liial waguis a bl abula i ma miowa books, b.g... Colli, Fouaio of Miowa giig (. p. 89 & p.97 (wh h auaio osa is o b isa of. No: (i has b alula b a pubaio mho. Th mho is iali a h uoff fqu, a whih h is a lag pubaio. S 8.6 gis a mho whih alulas boh a (u o wall loss ali fo all fquis. (ii Oh ps of losss (.g. loss fillig mium o ompl a also oibu o a. 8.5 g Flow a Auaio i Waguis (oiu (iii No h a wo fiiios of h auaio osa. I Ch. 8 of Jakso, h auaio osa fo h wagui is o b a i is fi as P P, (8.57 This is h fil auaio osa, i.. B,. I Ch. 7 of Jakso, h auaio osa fo a uifom mium is o b [s (7.5] a i is fi as P P This is h pow auaio osa, i.. P Obiousl, h pow auaio osa is wi h alu of h fil auaio osa. Tah Wagui (I K. Wag a D. M. Milma, Mal wis fo ah wa guiig, Nau, ol.4, No. 8, p.76, 4.

17 ( B/m.... Tah Wagui (II: Usig Th Lows Loss T Mo T 5 T 4 T T T Fq(GH fs. Poa, p.6.. Colli, p.97. Q: How o i h T mo a fabia i a h ah gio? A possibl soluio: X-a mio-fabiaio (LIGA. 8.8 Cai Pow Loss a Q Dfi Dfiiio ii of fq : W ha so fa assum a al fo M was i ifii spa o a wagui. Si fils a so i a ai, i amps i im if h a losss, ps b a ompl. Thus, fils a a poi i h ai ha h im p gi b i, ( ( i i i( (8.88 Q Q, wh is h soa fqu [.g. (4] wihou h wall loss. (8.88 assums ha h wall loss moifis b a small al pa a a small imagia a pa, wh Q a a Q a aob mi. Phsial aso fo Δ ω : ffi ai si iass b a amou ski ph. A lag ai has a low fqu. H,. Ph sial aso fo Q : pow issipaio o h wall 8.8 Cai Pow Loss a Q (oiu * i i Q so g i h ai [ i ] Q U i ( Q ( U U (pow loss (8.87 Q i Q U so g Q (im-spa fiiio of Q (8.86 pow loss (8.88 pss a amp osillaio whih os o ha a sigl fqu. To am h fqu of (. w wi wh i ( (, i Us (8.88, assum ( = fo < i ( Q ( ( i i( Q 8.8 Cai Pow Loss a Q (oiu Th fqu spum is bs s fom h fil g isibuio i -spa ma, ( (8.9 ma, ( Q Q full wih a half-maimum pois Q Q (fqu-spa fiiio of Q (8.9 No: Fig. 8.8 is h soa fqu of h ai i h abs of Q a loss. is h soa fqu i h ps of losss. I mos ass, h iff is isigifia.

18 8.8 Cai Pow Loss a Q (oiu Phsial Ipaio of Q : so g (i Us h im-spa fiiio: Q pow loss f wa pio so g a im of pow loss so g so g Q (48 pow loss (48 shows ha Q, whih h suls fom h pow loss, is appoimal ims h umb of osillaios uig h a im. A lag Q alu implis ha h fil g a b so i h ai fo a log im. H, Q is of f o as h quali fao. 8.8 Cai Pow Loss a Q (oiu (ii Us h fqu-spa fiiio: Q (s Fig. 8.8 Fo a loss ai, a soa mo a b i o jus a o fqu (as is h as wih a losslss ai bu a a ag of fquis (. Th soa fqu ( + s Fig of a loss ai is h fqu a whih h ai a b i wih h lags isi-fil ampliu, gi h sam sou pow. Th soa wih of a mo is qual o h soa fqu ii Fig. 8.8 b h Q alu of ha mo (s Fig No ha ah mo has a iff Q alu. Q Figu 8.8 a basil ga i pim o masu h Q alu. 8.8 Cai Pow Loss a Q (oiu so g Q pow loss Usig h suls of S. 8., w a alula Q (bu o u o h ohmi loss. W fis alula h o o a H of a spifi ai assumig, h us h o o a H o alula U a pow loss, H w so g ( w wm wm (8.5 pow loss ff s K a (6. s H a ( Cai Pow Loss a Q (oiu Fomula fo Q (u o ohmi loss fo agula a liial aiis a b fou i, fo ampl,.. Colli, "Fouaio of Miowa giig", i p. 5 a p. 56. Q u o oh ps of losss : If h a sal ps of pow losss i a ai (.g. u o Im a ouplig losss, Q a b pss as follows: so g Q (49 (pow loss Q Q -h p of pow loss (5 wh Q (Q u o h -h p of pow loss is gi b Q so g (pow loss

19 8.8 Cai Pow Loss a Q (oiu A Compaiso bw Waguis a Caiis Wagui Cai. Fuio aspo M g so M g High-Q Miowa/Maial Appliao Chaai- ispsio laio a soa fqu aio auaio osa a Q ampls of aspo of high ( pail alaio appliaios pow miowas ( fqu masum (mosl fo (suh as muli-kw miowas, was fo log-ag.- GH aas a ommuiaios so g Q (pow loss Couo loss, ili loss, aiaio loss, iffaio loss Homwok of Chap. 8 Poblms:,, 4, 5, 6, 8, 9,

20 Chap 9: aiaig Ssms, Mulipol Fils a aiaio A Oiw of Chaps o M Was :(o i his ous sou m i wa quaio boua Ch. 7 o pla wa i spa o i wo smi- spas spaa b h - pla Ch. 8 o ouig walls - Ch. 9 J, ~ i ougoig g wa o psib, as i a aa -i Ch. J, ~ ougoig wa o iu b ii M was, as i h as of saig of a pla wa b a ili obj. Ch. 4 moig hags, ougoig wa o suh as los i a shoo 9.6 Sphial Wa Soluios of h Sala Wa quaio Sphial Bssl Fuios a Hakl fuios : Alhough his hap als wih aiaig ssms, h w fis sol h sala sou-f wa quaio i h sphial ooia sam. Th pupos is o obai a ompl s of sphial Bssl fuios a Hakl fuios, wih whih w will pa h fils pou b h sous. Th sala sou-f f wa quaio is [s (6.] (, (, (9.77 i L (, (, (9.78 ah hfoui ompo saisfis h H lmhol l wa q. ( k (,, (9.79 wh k 9.6 Sphial Wa Soluios (oiu I sphial ooias, ( k is wi k si si si L UP ( ( Q(, w obai U P Q si si PQ UQ si UP k UPQ si Mulipl b Th ol m wih -p, so his UPQ m mus b a osa. L i b -m U P Q [ k ] U Psi Q ll ( m si ( (si Diiig all ms b si, w s ha his is h ol m wih -p. l ( l m! m im P (os So i mus b a osa. L i b ll (. 4 ( l m! l Thus, as i S.. of lu os, m m im im P P (os, Q (os ; Q, PQ Y lm (, j baus of ig a 9.6 Sphial Wa Soluios (oiu U U( ( is go b ( (. wi k U ll U U ll ( (Th as f l(. Th, k f l( (9.8 ( l / L f l ( / ul ( k u l ( (9.8 u ( J ( k, N ( k [Bssl fuios of faioal o] l l l l l l l f ( J ( k, N ( k / / k j ( l ( k J ( l k l ( l ( h k j k il ( k Dfi a l ( k ( k N l k h ( l ( k jl( k il( k sphial Bssl fuios Hakl fuios ( ( ( ( A lm h l k A lm h l k Y lm k lm (, ( ( (, [ ] (

21 iw.7 Lapla quaio i Cliial Cooias; Bssl Fuios ( L ( ( Q( Z( P Z k Z Z k Q Q Q i k J ( k, N ( k wh J a N a Bssl fuios of h fis a so ki, spil (s followig pags. J k i k ( ( N ( k i k 5 iw Bssl Fuios : If w l k, h quaio fo aks h saa fom of h Bssl quaio,.7 Lapla quaio i Cliial Cooias; Bssl Fuios (oiu (.77 wih soluios J ( a N (, fom whih w fi h Hakl fuios: H ( J in ( H ( J ( in ( (.86 H ( J ( in ( a h moifi Bssl fuios (Bssl fuios of imagia agum I( i J( i (. ( K ( i H ( i (. S Jakso pp. -6, 6 Gash & hik, a Abamowi & Sgu fo popis of hs spial fuios Sphial Wa Soluios (oiu j ( ( j ( j ( ( ( j ( Fom G. Afk, "Mahmaial Mhos fo Phsiiss" l j l l ( si, l l j ( l (l!! (l l ( os l l, l (l!! ( l l ( l ( l l h ( ( i [ spaial l i p of sphi al was.] ( ( S Jakso pp fo fuh popis of jl, l, hl, a hl Sphial Wa Soluios (oiu pasio of h G fuio : Soluio of h G quaio ( k G(, 4 ( (6.6 is gi b (i i S ik G(, i ifii spa a fo ougoig- (6.4 wa boua oiio. W ma sol (6.6 i h sam wa as i S..9, i.. wi * G(, g (, Y (, Y (,, lm l lm lm sol fo gl (, f o a [wh ( ], a h appl boua oiios a,, a. Th sul is ( l * l l lm lm l ml G (, 4 ik j ( k h ( k Y (, Y (, quaig h wo pssios abo fo G (,, w obai ik ( l * ik jl k h lm k Y Y l lm l ml 4 ( ( (, (,, (9.98 wh a a, spil, h small a lag of a. 8

22 Pa I 9.6 Sphial Wa Soluios (oiu Summa of Diffial quaios a Soluios : Sou-f D.. Lapla q. Hlmhol q. k Soluios ik ik ik Casia i i,,,.,,,. (S..9 (S. 8.4 im k im J (,,,. ik,,,. liial m k J m k (S..7 (S. 8.7 l (,,,. Ylm Ylm(,, jl ( k, l ( k,. sphial (Ss..,. (S. 9.6 D.. wih a G(, 4 ( ( k G(, 4 ( poi sou b..: G ( b..: ougoig wa Soluios (G fuios Sis pasi of G fuio G ik G [q. (6.4] qs. (.7, (.48, (.68 q. ( Pa II 9.6 Sphial Wa Soluios (oiu Summa of Diffial quaios a Soluios : k Sou-f D.. Hlmhol q. Wa q. ik ik ik,,,. A (, Soluios (S. 8.4 Casia (, im ik Jm k,,,. ( liial J, ( (S. 8.7 sphial 4 (, / Ylm (,, jl ( k, l ( k,. (S. 9.6 ( G (,,, D.. wih a ( k G(, 4 ( 4 ( ( poi sou b..: ougoig gwa b..: ougoig wa Soluios ( ik G [q. (6.4] G (,,, [q. (6.44] (G fuios Sis pasi of G fuio q. ( aiaio of a Loali Osillaig Sou iw of Ihomogous Wa quaios a Soluios : (6.5 i f spa, a A A A J saisf Lo gaug. (6.6 Basi suu of h ihomogous wa quaio: 4 f (, (6. Soluio of (6. wih ougoig-wa b..: (, (, G (,,, f(, (6.45 i homogous soluio ( f(, i (6.45 wh G (,,, is alua a (6.44 h a im. is h soluio of ( G G (,,, 4 ( ( (6.4 wih ougoig wa b.. 9. aiaio of a Loali Osillaig Sou (oiu Usig (6.45 (assum i o (6.5 & (6.6, w obai h gal soluios fo A a, whih a ali fo abia J a. ( (, J, / A (, (6.48,(9. (, 4 ( I gal, h sous, J (, a (,, oai a sai pa a a im p pa. Fo sai i J( a (, (9. gis h sai A a i Ch. 5 a Ch., spil. ( A ( A ( J (5. 4 ( sou ( ( (.7 4 Qusio: I is sa o p. 48 ha (9. is ali poi o boua sufas a ps. Wh? [S isussio blow (6.47 i Ch. 6 of lus os.]

23 9. aiaio of a Loali Osillaig Sou (oiu Fils b Hamoi Sous : Ol im-p sous a aia. aiaio fom moig hags a a i Ch. a Ch. 4. H, spiali o sous of h fom (as i a aa: i (, ( (9. i J (, J ( Sub. (9. i io (9. a a ou h -igaio, i w obai ik i A(, A( wih A (, (9. 4 J ( wh k. W shall assum ha J( is ip of A(, i.. h sou will o b aff b h fils fil h aia. i Ohwis, (9. is a igal quaio fo A(. 9. aiaio of a Loali Osillaig Sou (oiu A simpl iaio of (9.: W spiali o hamoi sous fom h ous. Th, ol (6.6 is qui. A (, A (, J (, (6.6 i L J (, J ( a A (, A ( i k A ( J ( [ihomogous Hlmhol wa q.] Th G quaio fo h abo quaio is k G k (, 4 ( (6.6 Soluio of (6.6 wih ougoig wa b.. k ik G (, (6.4 ik ( G (, ( (, A J J whih is (9.. k aiaio of a Loali Osillaig Sou (oiu ik wi (9., A ( J (, (9. 4 H A (wh (9.4 Mawll qs. gi iz (ousi h sou (9.5 k H wh Z 77 (impa of f spa, p. 97. Thus, gi h sou fuio J (, w ma i piipl alua A ( fom (9. a h obai h fils Ha fom (9.4 a (9.5. i No ha p has b assum fo J, h fo all oh quaiis whih a pss i ms of J. No: Th hag isibuio a sala poial a o qui fo h miaio i of H a? ( w h? 5 9. aiaio of a Loali Osillaig Sou (oiu ik Na- Fil pasio of A( ( (9. 4 J Bfo goig io algbai ails, w ma ail obs som gal popis of A ( a h sou (. ik Fo ousi h sou a (o k, w l l l * a us 4 Y lm (, Ylm (,. (.7 l l m l l Si, w ha a. sou l l * A ( Y (, l lm ( Y (, (9.6 m l lm l l J k Th igal i (9.6 ils mulipol offiis as i (4.. Thus, (9.6 shows ha, fo k, A ( a b ompos io mulipol ( l fils, whih fall off as jus as h sai mulipol fils, bu wih i h p. How, w will show la ha, fa fom h sou ( k, A ( bhas as a ougoig sphial wa. 6

24 9. aiaio of a Loali Osillaig Sou (oiu Full llpasio of A( : W ma i fa pa A (, wihou appoimaios, b usig (9.98. Fo ousi h sou, w ha,. H, ( a b wi ik ( ( l * l l lm l ml 4 ik j ( k h ( k Y (, Y (, ik Sub. his quaio io A( (, w obai 4 J ( * l lm J l lm lm, ik ( ( l!! l hl k l a ( ( ik ik sou ( ( l! (l!!(l!!! a a A ( ik h ( k Y (, ( j ( k Y (,, (9. wh ( wih a (,, (S Abamowi & Sgu, "Habook of Mahmaial Fuios, " p. 49. lm 7 9. aiaio of a Loali Osillaig Sou (oiu (9. is a a pssio fo A (. W ow assum k (i.. sou imsio walgh. Th, k a jl ( k us o l ( k j k l ( k k (l!! (9.88 ( ik ( l!! l Sub. h l ( k ( i( k l a ik (,, a (9.88 io (9., w obai J, (,, (, ik l Y lm l [ ( ( l ( ] l a ik a ik a ik A ( ( lm, l * J ( Ylm (, ( is h ombiaio of (9.6 a (9. i Jakso. I is ali fo k a a ousi h sou. Th gio ousi h sou is ommol ii io os (b hi iff phsial haas: Th a (sai o: ( k Th imia (iuio o: ( k Th fa (aiaio o: ( k 8 Giffihs.. li Dipol aiaio Giffihs li Dipol aiaio: Appoimaios Cosi wo poi hags of +q a qq spaaig b a isa (. Assum ( a b pss i siusoial fom. Th sul is a osillaig li ipol: p( q( ˆ qos( ˆ p os( ˆ, wh p q. Th a poial is (, V(, 4 qos[ ( / ] qos[ ( / ] 4 9 Appoimaio #: Mak his phsial ipol io a pf ipol. sima h spaaio isas b h law of osis. os ( ( os ( os os[ ( / ] os[ ( os ] os[ ( ]os( os si[ ( ]si( os Appoimaio #: Th walgh is muh log ha h ipol si.

25 Giffihs Th a Sala Poial os[ ( / ] os[ ( ]os( os si[ ( ]si( os os[ ( ] si[ ( ] os Th a sala poial is: os os[ ( ] si[ ( ] os ( os (, V 4 os[ ( ] si[ ( ] os ( os p os si[ ( os[ ( ] 4 Giffihs Th a Sala Poial Appoimaio #: a h aiaio o. Th a sala poial is: p os V (, si[ ( 4 Th appoimaios ( Giffihs Th a Vo Poial Giffihs Th lomagi Fils a Poig Vo Th a o poial is mi b h u si. q I ( ˆ q siˆ / i[ /] ˆ J(, q si[ ( A (, 4 4 / p si[ ( ] 4 a poials: p os V (, si[ ( 4 p A (, si[ ( ] ˆ 4 VV A BA p A si V ( os[ ( ] θˆ 4 p si B A ( os[ ( ]ˆ φ 4 p si S ( B ( os[ ( ] ˆ 4 Th oal pow aia is 4 p si P S a ( si 4 p 4

26 9. li Dipol Fils a aiaio wi (: ik l Y (, lm [ a( ik a( ik al ( ik ] l ( l A ( lm, l * J ( Y lm (, Tak h l m [ Y J ] J p 4 a o i b A ( J J p ik ( ( l 4 A A ( J J J i ik J ( p, (9.6 4 gi o oibuio baus J wh p ( (4.8 is loali: J J (9.6 gis h li ipol oibuio o h soluio. I is J ali fo k a a ousi J J h sou. i( ip Qusio: Wh is h o moopol J p m (s p. 4? 5 9. li Dipol Fils a aiaio (oiu p i ik wi (9.6: A ( p (9.6 4 p p p iz p Fom (9.4, a fom (9.5, H A H k ik p k H p 4 ik p ik ik ik k p 4 pp I h fa o ( k, (9.8 us o a sphial wa p k H H p ik 4 p ompo p p of sou ZH p p I (9.9, w s ha a H k p p a i phas, a, H,a a muuall ppiula. This is a gal pop of M was i ubou, uifom spa. Gi a wo of hs quaiis, w a fi h hi. (9.8 ( li Dipol Fils a aiaio (oiu p ik H k p 4 ik (9.8 p ik ik k ik p p p 4 I h a o ( k, (9.8 us o p i p ompo H 4 p of sou p (9. p p 4 k p p o (i ( a H a 9 ou of phas aag pow. p (ii has h sam spaial pa as ha of h sai li i ipol i (4., bu wih p. (iii H ( k -fil g B-fil g. p Qusios : (i Wh os ha h sai i fil pa? (ii To obai (9., w ha gl a fw ms i (9.8. Bu som of h gl ms a sill impoa i h a o? Wha a h a i wha ss a h impoa? 7 P 9. li Dipol Fils a aiaio (oiu im-aag pow i h fa o/ui soliagl p p * (9. H Z 4 k p (9. This o gis h iio of p, i.. h polaiaio of h aiaio (s figu blow. Zk p i i i P oal pow aia I gal, p p p p. If i, h p has a fi iio, p p wih p p p p, a P Z 4 k si. (9. p Ohwis, h iio of p (h P a wih im, bu P is sill gi b ( p (9.4 p p H ipol aiaio pa 8

27 9. Magi Dipol a li Quaupol Fil wi (: ik l Y (, [ a ( ik a ( ik a ( ik ] l lm l l A ( ( lm, l * J ( Y lm (, Tak h l ms [ a ] l * A ( ik ( ik Y J Y (, ( (, m m m,, p. 9 i Ym(, Ym(, si si 8 m,, os 4 4 * ( osos sisi i( si si os( os os s l i (.68 9 Thus, 9. Magi Dipol a li Quaupol Fils (oiu 4 l ik ( ( ik ( ( (9. { J J J } A ik J ik ( ik ( ( ( 4 A Q A m, li quaupol aiaio magi ipol aiaio m ik wh ( ( m ik fo k a a (9. 4 ik ousi i h sou m wih m ( J [magi ipol mom]. A gis h A magi ipol oibuio hough (9.4 a (9.5 (s p.5: ik m ik H k 4 m m m ( ( (9.5 Z ik m ( k (9.6 4 m ik ik 9. Magi Dipol a li Quaupol Fils (oiu I h fa o ( k, w ha h sphial wa slouio: ik Z 4 m P H k m k m 4 4 m ZH m P Z k m iio of I h a o ( k, (i a H a 9 ou of phas m aag pow. H 4 m m m (ii H has h sam spaial pa m Z k m 4 i as ha of h sai magi ipol i i (5.56, bu wih p. (iii B -fil g -f il g. Th li quaupol aiaio, isuss i (9.7-(9.5, is mo quaupol aiaio omplia. H, w ol illusa is pa aiaio i pa b h figu o h igh. m m o m Compaiso bw Sai a Tim-p Cass laios -p of mulipol fiiio of mulipol bw, a B ( : imsio pasio moms J,, a B of h sou l sphial ( o / q B hamois Fo, all mulipol sai ( ( pasio p ( fils a b sig ifia. [(.7] o Fo, mulipol as J ( B ( Talo sis Q ( ( ij ij ij fils a omia b [(4.] of h lows-o m ( J oaishig m. Th is o im-p (a a o o B i / l moopol fo a isola sou (s p. 4. Appo. h sam fil p, Q ij, a m ha h pa a -p sam pssios as hos as fo h ospoig ( ( sphial i l b im- hamois of hi sai oupas, sai mulipol, bu wih p pasio ( ( bu wih h i im i p (h as J B all quasi-sai fils. [(9.98] of p. (b fa o M was ik I im-p ass, li mulipols a B, ik i / ga B-fils a (sphial M was magi mulipols a All mulipol fils /, ga -fils. lai pow lls uhag wih isa.

28 Iu li a Magi Dipols 9.4 C-F Lia Aa A Qualiai Look a h C - F Lia Aa : C Talig wa g sou i L Tasmissio i li LC osillao C-f lia aa i + i a o fa o I h a o, a B a piipall ga b a J, spil ( lagl sai fil pas. I h fa o, a B a gai hough B a ( M was C-f Lia Aa (oiu Dail Aalsis: Th -f lia aa is a as of spial is, baus i allows h soluio of (9. i los fom fo a alu of k, whas i Ss. 9. a 9., w assum k<< ik A ( J (, (9. 4 wh J ( Isi k k ( ( (9.5 ( / si k I / ik k A 4 No: (i J is smmi abou J. J( J( J J (ii I is h pak u ol wh k. Qusio: : Th aa appas o b a op iui. How a h b u flowig o i? C-f Lia Aa (oiu ( os [ ( ] [ [ ( ( ] 8 [ ( ] = ( if = H, if, w a wi os. ik os / si k ik I k A ( ( / os ik k k I os os os ( k 9.55 si No: os i a b gl if. Bu os os ik ( os i maks a impoa oibuio o h phas agl a. 6

29 I h fa o, 9.4 C-f Lia Aa (oiu H Z (9.6 & 9.9 ik k si A H A A H * Z Z os( k os os( k fo si a a os ( os /si, k 4 4os ( os /si, k P k H H k si A ( ZI, ( k ZI 8 half-wa aa ( k J P J J (9.57 full-wa aa ( k supposiio of half-wa aas i i phas P mos oh lss oh aow bam wih 7 wi ( C-f Lia Aa (oiu k k os os os, fo ( si P ZI a a k Limiig as ( ipol appoima io: k (i.. os ( k k 8 k k os 8 os os os P ZI k k 8 8 ZI os 8 si 5 4 ( k si [ali fo k ] (4 This has h sam k a p as i (9., li ipol, whih was i b assumig k C-f Lia Aa (oiu aiaio sisa a quial Ciui: : k k J ( Isi k ( ( I ( ( k I (pak u, ZI k ZI 4 k 5 8 ZI o P k 48 Thus, fom (4, P ( k si ( k si (9.8 P P I Z 4 a wh ( k a o s ( (9.9 a : aiaio sisa., is pa of fh fil lfiiio ii of a impa, s m i (6.7. 5( k ohms [S pp. 4-.] i i I I quial i V iui fo i V a a -f iui i aa imsios C-f Lia Aa (oiu Poblms:. Th full-wa aa aiaio i (9.57 a b hough of as h supposiio p of wo half-wa aas, o abo h oh, i i phas. Dmosa his b iig P fo h full-wa aa k = b supposig p h fils of wo half-wa aas (ah of lgh /, s figu blow.. If h wo half-wa aas i poblm a i 8 ou of phas, i P agai b h mho of supposiio.. Plo h appoima agula isibuio of P i poblms P a. plai h iff qualiail. aa of lgh Ths li a al aa of paalll wh poi P is lgh sigl aa fa fom h aa, as of lgh 4 os is assum h. 4

30 9.4 C-f Lia Aa (oiu Soluio o poblm : : Piipl of supposiio quis ha w a h fils (o h pows of h aas, ah of oal lgh. wi (9.55 ik os k I os os k A ( (9.55 k si (9.55 applis o a sigl aa of oal lgh (s fig. abo. So h fil of ah of h aas i his poblm a b obai fom (9.55 b plaig i (9.55 wih a pssig wih sp o h of ah aa (i.. b a aa. ik, os k os os k I 4 4 A,, (5 k, si aa wh os a 4 os. 4 4 os W ma appoima, i h omiao of (5 b, bu mus us h o fo h phas agls i h poial ms., C-f Lia Aa (oiu I is assum ha ah aa i his poblm is i i h half-wa pa, h w s k i (5 a h suppos fil of h aas (i i phas is gi b os( os AA A I ik i os i os [ ] ( (6 k si os ( os I ik k aa si Fom (, P Z k si A aa ZI 4 sam as h full wa os os si soluio i (9.57 Sl Soluio o poblm : o If h wo half-wa aas i poblm a i 8 ou of phas, w simpl pla h " " sig i (6 wih a " " sig C-f Lia Aa (oiu Thus, I i os i os os( ik os AA A [ ] k si si( os os( os I ik i k si aa Z Fom (, ( P k si A aa ZI 8 si ( os os ( os si os ZI Soluio o poblm : si si Homwok of Chap. 9 Poblms:, 6, 8, 9, 4, 6, 7,, i phas ipol aiaio ou of phas quaupol aiaio Qusio: How os a phas aa aa wok? 4 44

31 Chap : Saig a Diffaio. Saig a Log Walgh Diffial Saig Coss Sio : Cosi a pla wa ik i Assum f spa. ε (. i H s i i Z Z / H i Hs ii oo a obj of imsio, wh ε a b al (lial polai o ompl [.g. p, m sa fo iulal polai wa, ε ( ε ]. i ε i a Hi will iu mulipols o h obj, whih i u ga sa aiaio ( s, Hs. Fo, ol h iu p a m a impoa. Fom (9.9 a (9.6, w ha ik k s 4 pm [i fa o] (. ik ik p k m k H 4 4 H s s Z H p + H m p p m m ZH ZH H, o fi a H, w o fi h iu p a m. s s. Saig a Log Walgh (oiu Fo saig poblms, a usful figu of mi is h sa pow alai o ii pow. Fuhmo, i is of impoa o kow h polaiaio sa of h sa aiaio. i Thus w fi a iffial saig oss sio (wih imsio m as aia pow i -iio wih ε-polaiaio ui soli agl, ε;, ε ii pow i -iio wih ε -polaiaio ui aa ε* Z s Th maig of will (. ε * bom la i (.. Z i No: (i Fo a iulal polai sa, ε a b wi ε ( ε iε, wh ε ε. (ii ε a ε* ; ε a ε* ; εε* ; εε* (iii ε is o ssail h iio of s. ε * s gis h ε-ompo of. s. Saig a Log Walgh (oiu ε* Z s ε ε * Z i ε wi (.:, ;, (. ik i ε Sub. ik io (. k s 4 ( p m 4 k * * m, ε ;, ε ε 4 [( p ] ε ε* pp ε* p ( ε* ( p ε* p 4 k ( * ε* p ε (.4 m 4. Saig a Log Walgh (oiu ampl : Saig b a small (a <<, uifom ili sph wih μ=μ a abia ε m a i / / (lai pmiii oal li fil Fom (4.56, w obai h li ipol mom p iu o h sa b i a ε ik p 4 b assumpio (4.56 & (.5 i 4 Sub. (.5 io k * * ε p ε m ( ε k a, ε ;, ε* ε (.6 Qusio: is i fo a ili sph i a sai fil Qusio: is i fo a ili sph i a sai fil. Wh is i also ali fo h im-p fil h? 4

32 mi 4.4 Boua-Valu Poblms wih Dilis (oiu ampl: A ili sph is pla i a uifom li fil. Fi wh. a W hoos h sphial ooias a ii h spa io wo gios: a a a. I boh gios, w ha wih h l m im P (os l soluio: l m im Q (os [S.. of lu os] l is ip of. l i A lpl(os b.. is fii a os. l l l is fii a. i ou B l Cl P l (os l Qusio : If l, ou as. Wh h kp h l ms i? mi ou 4.4 Boua-Valu Poblms wih Dilis (oiu (7, ( A B os. (l i b. / (9, ( A ; C a / / (, ( Al Cl fo l This is h ol wa ( & i os (6 a boh b saisfi. / / a (4.54 ou os os / appli fil / ipol fil wih p 4 a [f. (4.] / polaiaio hag [s (4.58] u o polaiaio hag oal li fil 7 5 mi 4.4 Boua-Valu Poblms wih Dilis (oiu l l l i Al Pl ou B l Cl P l l l wi: (os, (os b.. (i: ou ( os. os os. B os.; B ; Bl ( l P (os os i ou b.. (ii: ( a ( a [ ( a ( a] i ou A B C a (8 l l Cl Aa l Ba l A C a (9 l a l A l C l a, l ( i ou a a i a ou a l l l l l ( l / b.. (iii: ( ( la a lb a l C a C/ a, l ( A [ C / a ], l ( l la ( l C / a, l ( l l 6. Saig a Log Walgh (oiu W fi h - pla as h saig p la. L b alog h -ais a li o h - pla. Th oiaios (, of ui ( ( os ε, ε, a ε a spifi aoigl as fol lows ε ε ε (, polaiaio of ii i wa polaiaio sa (, of sa wa o saig pla polaiaio sa (, of sa wa o saig pla ( ( ( ( ε ( ( ε iio of fiii wa ( ε ε wh ε is o h - pla makig a agl wih h -ais, ε is o h - (saig pla, ( is o h saig pla, a, ε, a ε a muuall ohogoal. Polaiaio o ( ε ( ( of h ii wa a polaiaio sas [ ε, ε ] of h sa wa a all assum o b al, psig lia polaiaio. ( 8

33 . Saig a Log Walgh (oiu Applig q. ( i Ch. of luos: os sisiososos iio [ : agl bw (, a (, ] of ii ( wa o ε (,, ε (,, a ( ( ( ε ( ε (,, w obai ε ε ( ε ε si si os os os os os ( si si os os os ε ε si 4 6 wi (.6: (, ;, ε ε k a * ε ε 4 6 ( 4 6 k a k a ε ε os os 4 6 ( 4 6 k a k a si ε ε 9. Saig a Log Walgh (oiu Assum ha h ii aiaio has a fi iio, bu is upolai (i.. is aom. W ak h aag o : 4 6 k a os ( k a si % lial (.9 os polai a wh gi s hgofpolaia io ofhsa aiaio. 4 6 k a k a a ka os (. [ ] (. Qusio : I (., wh a pows isa of aig fils?. Saig a Log Walgh (oiu (. gis aa, implig ha ol a small faio of h aiaio ii o h ili sph is sa. This is u if h sa is a pfl ouig sph (wih aius. S ampl. ampl : Saig b a small pfl ouig sph Th ii aiaio will iu boh li a magi ipol moms ( p a m o h ouo. p a m a gi b p 4 a [S S.. of lu os.] (. i m = a H [S poblm.] i ik i ε Fom Hi i Z [Z / ] 4 k pm (. (. (.4 w obai k a (.4. Saig a Log Walgh (oiu As i ampl, fo uploai ii aiaio, (.4 4 ils 4 6 k a (os ( k a ( os 4 6 k a [ 5 ( os os ] (.6 8 si [pak a 6 ] (.7 5( os 8os 4 6 k a a [ ka ] Agai, w fi a. B gomi opis, h sa (a ouo woul b opaqu o h ii aiaio, a h ii aiaio woul ha b oall blok [ a ]. This ampl mosas ha gomi i opis ompll l baks ow fo a, wh w phsial opis, as i saig/iffaio ho.

34 . Saig a Log Walgh (oiu Poblm : Di h ipol mom i (.: : m= a H i. Soluio: Si a, w ma assum H i o b uifom. H a i Fo a pf ouo, w B ha B isi i h sph. H i I S. 9., w ha show ha i h a o (, h magi ipol aiaio has gligibl ibl -fil. H, w assum B ousi h sph a wi B. Th, B = wih h soluio: [S.. of lu os] l m im P (os l This mol is ali fo, l m im whih is suffii fo us o Ql (os fi h ipol mom of a subj o boua oiios: sph wih aius. B( Hi ( Hi Hios B ( a a. Saig a Log Walgh (oiu l m im Pl (os l m im l (os wi P Q (os os is ip of. l l b.. A l C l Pl(os is fii a os. l b.. ( H os A H & A if i i l l Hios Cl Pl (os l l Hi C a a l l l l a C P i & l if H os ios a Hi a ( u o h sph ( m H i b b.. ( os (os C a H C os si B Compaig wih (5.4, w fi ha his is a magi ipol fil pou b a (iu ipol mom of m= a H. i 4 Opioal. Pubaio Tho of Saig Gal Tho: Cosi a slighl o-uifom mium wih ( I S.., of h sa a b ( of a alu, bu h soluio is mo ( ( si b h sa gom. wh a a ip of a ( a a o ssail h f spa alus. B B ( H D H D ( ( ( B H D ( D D D D (4 Th pupos of h abo maipulaio f p is o obai his small quai, whih ( (4 a b a as a pubaio. D DD (. B H 5 Opioal. Pubaio Tho of Saig (oiu i Assum D,, B, H, (. ( DDi BH (. k ik ( k G(, 4 ( G(,. H, ik D i B H ( DD 4 (.4 No: (i D( is a ii pla wa whih saisfis h homo- gous Hlmhol q. [i.. h HS of (. = ] (ii (.4 is a igal laio, o a soluio. L h iga i (.4 b of imsio a, h a w a wi D as ik ik ( ( ik fo D D As wih gl ik D As 4 (.6 i B H 6

35 Opioal igaioi b pas. Pubaio Tho of Saig (oiu ik a [ a is a o fuio of.] ik [ ( ( ( ] a a a a a a ik ik ik [ i ( k a k a ( ( ] k a ik a i Th sul is o pla " " wih " ik " k ik D (.6 As (.7 4 k B H * : polaiaio Fom (., w obai ε As ε o of h (.8 ( D sa wa No: (i A s gis h sa fil D s = A s ik / [h H s hough (.]. s is NOT a o poial. (ii (.7 is a igal quaio fo A s, NOT a soluio. 7 Opioal. Pubaio Tho of Saig (oiu Bo Appoimaio: : wi (.7 ik A k s D B H 4 k Fo a lia mium, (.7 D ( ( ( D ( (.9 B ( ( H ( B H ( H W s fom (.9 ha h iga of (.7 is ompos of small quaiis a H. To fis o i a, w ol ( ( o us h o o (o upub a H fo a H i a H. Thus, w wi ( ( This appo., all h ( D D Bo appo., us h igal q. (.7 io (. ( ( B H ( H B a soluio fo. As 8 Opioal. Pubaio Tho of Saig (oiu L h upub fils b hos of a pla wa, ( ik ( ( D ε B D ( D, ( ( ( ( Sub. D ( a B ( io (., h sub. (. io (.7, a fiall mulipl up h sul b ε* D ( ( * * ε As k i ε ε q ( D 4 ( ε* ( ε (. wh q k(. Th absolu squa of (. gis h iffial saig oss sio hough (.8. ε* D As ( (.8 9 Opioal. Pubaio Tho of Saig (oiu ampl: Saig b a uifom ili sph wih = + a q[ k( ] iq a iqos os auum a iq (, iq 4 a aosqa siqa si( 4 q q q q Thus, fom (. (l ε* A s siqa qa osqa k ( * D q ε ε q qa k a ( ε* ε ( * + 6 si, os,

36 Opioal. Pubaio Tho of Saig (oiu * * s s k a ( * D qa ε ε ( Sub. ε A io ε A (.8 D qa 4 6 k a Bo lim ε * ε (. 4 6 i agm wih k a * (.6 i h limi ε ε. Qusio: (.6 a (. boh gi h iffial saig oss sio ( of a ili sph wih aius muh small ha h walgh. (.6 is ali fo abia alus of ( /. I us o (. i h limi. A phsial ff i ilu i (.6 [bu o i (.] ha kps a a fii alu i h limi? Wha is i? plai wh i kps fii.. Pubaio Tho of Saig (oiu Blu Sk a Sus: Saig b gass D P (4.4 DNp= Nmol Maosopiall, w ha p: ipol mom p molul Nmol p mol Miosopiall, w ma wi mol : molula polaiabili [s (4.7 & (4.7], wh spa o h si of h molul N: o of moluls/ui olum ( ( ( ( (. mol j mol j j j Si ( fluuas miosopiall wih a wak aiaio (, w ma appl h pubaio ho jus lop. Sub. ( io (., h sub. (. io (.8, w obai 4 k mol * F(, [assum ] 6 ε ε q fo aoml isibu moluls iq j iq jj oal o of moluls l F q j j j (ioh aiaio wh ( (.9. Pubaio Tho of Saig (oiu W ow la mol o h maosopi quaiis ii,, a NN. ( N mol mol N N N i of 4 faio k mol ε* ε F( q 6 4 k * ( 4 N ε ε F q Toal saig oss sio p molul is gi b F ( q [ F( q : oal umb of sas] 4 os * ε ε k 4 N k4 ε ε N 4 N * os si si os (.4 is o h - pla fo ipol sa (p Pubaio Tho of Saig (oiu L I b h isi (pow/ui aa of h ii wa, h I (.4 a (.5 sib wha IN I, is kow as aligh saig. 4 k wh N [auaio offii] (.5 N Disussio : 4 Viol ligh ( 4 m is sa mo ha (i k 65 4 ligh ( 65 m b a fao of ( (ii I (.5, Nmol (s las pag. H, N if aoms (o moluls of h sam p a a o ak ou. (iii Th aoms i a gas aia iohl, bu h hags wihi a aom aia ohl. Suppos h a lo-io pais i ah aom a w w abl o spli all h aoms io a gas of sigl lo-io pais, ah wih h sam p. Th, h maosopi mais h sam, bu h spli pais o log aia ohl, suligiasaisiims sa isi ims wak. This plais h fao i (.5 (S p N 4

37 I I /. Pubaio Tho of Saig (oiu iliaio o I h ah amosph, is a fuio of. Th, I( I( ( ( I( I high oo iol 4 i Amosph agl:.5 fom "Alas of h Sola Ssm", oal Aso. So. suis, sus su Qusios: (i Wh is h sk blu isa of iol? (ii Wh is i mo likl o g a subu i h summ? (iii Ho summ/ol wi suls mosl fom a iff aus ha i (ii. Wha is i? 5 pla wa pla wa.5 Sala Diffaio Tho lows-o appoimaio a aig (gomial opis -o appoimaio iffaio ho (phsial opis Assum suh ha hs lis a almos paalll Fis miimum (ompl allaio ous a si if. Nau of h iffaio poblm: Phsiall, h iffaio poblm h is o spaabl fom h saig poblm. How, h ams a iff Th saig poblm a i his hap assums. Th sala iffaio ho is mos ali wh >>, fo whih i gis h -o oio o h gomial opis (s p Sala Diffaio Tho (oiu Jusifiaio of h Sala Diffaio Tho: Phsiall, loi sposs (J, of h apu maial o h ii wa ga lomagi fils i aiio o issipaig som of h ii wa. Fa fom h gs of h apu, J a piipall sul i flio of h ii wa, whil J a a h gs pou fils ha pass o h igh of h apu ogh wih h ii wa. Th suppos fils fom h iffaio pa. I h fa o of h iffaio gio ( >a fw λ fom h apu, h fils ak h fom of a M wa, whih obs Z H [s (9.9] wh Z ( / = ( / / is h impa of auum, a is h iio of wa popagaio. 7.5 Sala Diffaio Tho (oiu Thus,,, H,, a a muuall ohogoal, a h amplius of a H ha a kow aio Z. Thfo, o ompo of h fils gis mos of h ifomaio (phas a isi, bu o h polaiaio abou h fa fils. This jusifis a sala ho fo h iffaio phomo a plais wh i has b h basis of mos of h wok o iffaio. Th Kihhoff Igal Fomula: I h saig poblm, w alula h sa fils u o J a assoia wih h ipol moms iu b h ii fils I h iffaio poblm, h fils a pou i pa b h iu J a o h apu maial, bu J a o o appa pliil i fil quaios. Th a implii i h boua oiios. Th Kihhoff igal fomula psss h iffa fils i ms of h boua fils. Dmiaio of h a fils quis aua halig of h b.. s ( fw ass a b sol ompll. How, h fa fils a b fail aual mi wih u b.. s. 8

38 .5 Sala Diffaio Tho (oiu f o h figus o h igh. S is a opaqu sufa wih apu(s o i. Th iffaio gio (gio II is h olum los b S a S. i L (, ( b a sala fil (a ompo of o B, h k (, k (.7 No: gis h phas a isi, bu o h polaiaio, of h fils. Blow, w will pss i gio II i ms of a o h boua sufas b makig us of G's hm. ( s ( a ( Sala Diffaio Tho (oiu wi ( s( a (.5 Iou a G's fuio G(, saisfig ( k G(, ( (.74 Appl (.5 o h olum los b S a S (gio II a l a G. k ( [ (, ( ( (, G G ] a G, G, s [ ( ( ( ( ] s kg(, ( Fo a obsaio poi isi gio II, ( s (.75 s a [ ( G (, G(, ( ] No : is iwal i io h olum isa of ouwal i as i (.5. Is his a goo hoi?.5 Sala Diffaio Tho (oiu Soluio of ik (.74: G (, wih. (.76 4 G fuio wih ougoig wa b.. (o: ik G(, G ik 4 i k H, ik ik ik 4 4 ik ( 4 ( i s ( (.77 s a ik k W assum ha o S is asmi hough S. Th, S a h oibuio o h igal i (.77 fom S aishs as h is of h aius of h sph. Assum fuh ha h aius gos o ifii a h gl h oibuio fom S. (.77 h gis h Kihhoff igal fomula ik ( ( i ( (.79 a ik 4 s k i gio II is ow pss i ms of a o S..5 Sala Diffaio Tho (oiu Kihhoff Appoimaio: o: wi (.79, ik ( ( i ( (.79 4 s a ik k (.79 is a igal quaio fo. I boms a soluio fo u h Kihhoff appoimaio, whih osiss of. a aish wh o S p i h opigs.. a i h opigs a hos of h ii wa i h abs of a obsals. Th a, how, mahmaial iosisis wih h Kihhoff appoimaio:. If a aish o a fii sufa, h wh (u fo boh Lapla a Hlmhol quaios.. (.79 os o il o S h assum alus of a. Appoimaios ma h wok bs fo, a fail bal fo ~ o > ( : si of h ap u o obsal. S p.478.

39 .5 Sala Diffaio Tho (oiu mo h mahmaial iosisis s i h Kihhoff Appoimaio b h hoi of a pop G fuio. If is kow o h sufa S, a Diih G G fuio G D (,, saisfig GD (, = fo o S is qui. A gali Kihhoff igal: ( a [ ( G (, ] (.8 s Cosi a pla s wih apau (s. Th mho of imags a b us o gi h Diihl G fuios plii fom: ik ik (.84 GD (, 4 (,, wh (,, ik ( k i ( a (.85 i s k.5 Sala Diffaio Tho (oiu A Spial Cas*: Diffaio of sphial was oigiaig fom a poi sou a P s. ik s ( ( b Kihhoff appoimaio (5 s ik 4 ik s s i ( ik G (, (6 k s s s Sub. (5, (6 io (.79, assum k & k a h gl k k O a O ms, w obai ik s ik s s ( P 4 s a (7 s s obsaio P s (poi sou poi s P P s * Mo ass a b fou i Maio & O (oigi of ooias Hal, Classial lomagi aiaio, followig q. ( Sala Diffaio Tho (oiu As w will s fom h followig ampl, h sala iffaio ho ags wih obsaios, alhough i is highl aifiial. ampl: : Diffaio b a iula isk. Fo simplii, w assum (i P s a P a o h ais of h isk. (ii P s a P a a qual isa fom h isk. Ps s P b b s b a H, a s os b, os b s (8 5.5 Sala Diffaio Tho (oiu ik 4 ik ik s s Sub. (8 io ( P s a (7 ( P ikb (9 b Igaig b pas [ a a u u a a ik, a u u, ] a s ik ik ik ik b b ( P ikb (igaig b pas agai ik ik ik b b ikb ( ik 4k b b b gligibl, si k 6 s

40 .5 Sala Diffaio Tho (oiu Qusios: (i Isi a P: I( P ( P b /[4 b ] ( Si I ( P fo all b, h is alwas a bigh spo ( Fsl bigh spo a a poi o h ais. Wha is h phsial aso? ik b (ii lim ( P ( I h limi of o obsil (, ( P us o h a soluio fo a poi sou a Ps, i.. h appoima soluio i ( boms h a soluio i (. Wha is h mahmaial aso? Th iffaio pa of a isk (fom Hallia, sik, a Walk. No h Fsl bigh spo a h of h pa. Th oi iffaio igs a o piabl b (, whih applis ol o fils o h ais. 7.5 Sala Diffaio Tho (oiu A hisoial ao abou h Fsl bigh spo: (Th followig paagaphs a ak fom Hallia,sik,aWalk. Diffaio fis a a plaaio i h wa ho of ligh. How, his ho, oigiall aa b Hugs a us as la b Youg o plai oubl-sli if, was slow i big aop, lagl baus i a ou o Nwo s ho ha ligh was a sam of pails. Nwo s iw was h pailig iw i Fh siifi ils of h al ih u, wh Augusi Fsl was a oug milia gi. Fsl, who bli i h wa ho of ligh, submi a pap o h Fh Aam of Sis sibig his pims a his wa-ho plaaios of hm. I 89, h Aam, omia b suppos of Nwo a hikig o hallg h wa poi of iw, ogai a pi ompiio fo a ssa o h subj of iffaio. Fsl wo. Th Nwoias, how, w ih o o sil. O of hm, S. D. Poisso, poi ou h sag sul ha if Fsl s hois w o, h ligh was shoul fla io h shaow gio of a sph as h pass h g of h sph, pouig a bigh spo a h of h shaow. Th pi ommi aag a s of h famous mahmaiia s piio a iso ha h pi Fsl bigh spo,as wallioa, wasi h! Nohig buils ofi i a ho so muh as haig o of is up a ouiuii piios ifi b pim. 8 Bso Nwo s ig Wh a ls wih a lag aius of uau is pla o a fla pla, as i Fig. 7.9, a hi film of ai is fom. Wh Nwo is illumia wih moo-hoomai ligh, iula figs, all Nwo s igs, a b s..8 Babi s Piipl ik wi ( ( i ( (.79 4 s a ik k P s o iffaio s, imagima sufa 4 ash sufa ( P ( Wh h spo is ak ulik Fsl bigh spo? This is h wa au. 9 iffaio s P s a P 4 ash sufa ( ( omplma iffaio s P s b P 4 ash sufa ( ( o h obaal : a B Kihoff's appo.:, lswh : a hos of h sou w ha ( P ( P ( P [Babi's piipl] 4 a b

41 .8 Babi s Piipl (oiu ampl: a ligh bam of fii wih o s ( P fii wih a ( b ( omplma s s Babi's piipl ( P a( P b( P ( P ( P a b 4 Fsl a Fauhof Diffaio: (s p.49 Th is a la iffaio pa ol wh. So, I igals suh as (.77, ( a b appoima b ( P ik wh p i, wh h phas agl k mus b alua mo aual. l Cosi h lgh sals:,, a. / ( os / ( ( ( 8 ( k O ( k O ( k O ( k If h a high ms a gl, w ha h Fauhof iffaio (fa fil. If h m is kp, bu high o ms a gl, w ha h Fsl iffaio (a fil. 4 Homwok of Chap. Poblms:,, 7,, 4 4

42 Chap : Spial Tho of laii (f.: Maio & Hal, Classial lomagi aiaio,., Ch. 4 isi s spial ho of laii is bas o wo posulas:. Laws of phsis a iaia i i fom i all lllo fams (I laii, w of all h iial fam a Lo fam.. Th sp of ligh i auum has h sam alu i all Lo fams, ip of h moio of h sou. Th basis of h ho a o i Appi A o a lma ll wih a mphasis o h Lo asfomaio a laiisi momum/g. H, w ami laii i h fou-imsioal spa of a, whih pois h famwok fo us o ami h laws of mhais a lomagism. Th os of h lu os pa osiabl fom Ch. of Jakso. Isa, w follow Ch. 4 of Maio. I h lu os, sio umbs o o follow Jakso. Sio : Dfiiios a Opaio uls of Tsos of Diff aks i h 4-Dimsioal i Spa Th Lo Tasfomaio : Cosi wo Lo fams, K a K. Fam K mos alog h ommo -ais wih osa sp lai o fam K. Assum ha a, ooia as of fams K a K olap. Posula las o h followig Lo asfomaio fo spa a im ooias. [i i Appi A, q. (A.5, wh h lai moio is assum o b alog h -ais.] (,,, (,,, K K, ( ( Fams K a KK oii a. ( wh is h Lo fao fo h asfomaio.. Dfiiios a Opaio uls of (oiu A o abou oaio : I ma books, h lai sp bw wo fams is o b a h pail loi i a gi fam is o b u. This uall las o wo fiiios ii fo h sam oaio : Lo fao fo h asfomaio, ( Jakso (.7 Lo fao of a pail i a gi fam, ( u. Jakso (.46 a (.5 To aoi ofusio wih h oaio (.g. wh w pfom a Lo asfomaio of h Lo fao of a pail, w will o h lai sp bw wo fams b ah h pail loi b houghou his hap, a hus fi ( [Lo fao fo h asfomaio] ( [Lo fao of a pail i a gi fam].. Dfiiios a Opaio uls of (oiu Fou - Dimsio Spa Quaiis a Opaio uls : Dfi a posiio o i h 4-imsioal spa of a as (,,, i (, i 4-o spaial o a a 4-D mai as a, i / 4, ow umb i 4, olum umb h, h Lo asfomaio i ( a b wi 4 o a ( i i i i 4 a h is Lo asfomaio is: a. ( 4

43 . Dfiiios a Opaio uls of (oiu Th a mai i ( shows ha h Lo asfomaio is a ohogoal asfomaio baus i saisfis fiiio of ohogoal a a asfomaio* (4 *S H. Golsi, "Classial Mhais," iio, p.4. b ( b ( b (4, Thus, a a a a, whih is a sam of posula [s qs. (B. a (B. i Appi B.] Jus as h -imsioal os (a sos i gal a fi b hi asfomaio popis i h -spa, w ma fi 4-os (a 4-sos i gal b hi asfomaio popis i h (, spa a fi uls fo hi opaio. 5. Dfiiios a Opaio uls of (oiu. A s of 4 quaiis A ( 4 o A ( A, A, A, A4, whih asfom i h sam wa as, i.. A a A, (5 is all a 4-o 4o(o 4-so of h fis ak. Th posiio o [ (,,, i] of a poi i h 4-D spa is obiousl a 4-o. As aoh ampl, h momum o of a pail i h 4-D spa, fi as i i p ( p, p, p, ( p,, is a 4-o baus i asfoms as [s q. (A.8, Appi A.] p p p p 4 o p a p p ip i i i 6. Dfiiios a Opaio uls of (oiu. If a quai is uhag u h Lo asfomaio, i is all a Lo sala (o 4-o of h oh ak. Th Lo sala is also all a Lo iaia. Th Lo sala is i gal a fuio of h ompos of a 4-o. Fo ampl, w ha jus show ha H, is a Lo sala. 7. Dfiiios a Opaio uls of (oiu ( i. Dfi h 4-D opao, [,,, ], as h oupa of h opao i h -spa. Th, h 4-gai of a Lo sala, [,,, ( i ], is a 4- o. Poof : (6 a a a b ( Tasfoms as a 4-o. A 4. Th 4-ig of a 4-o, A, is a Lo sala. Poof : A : 4-o a A b (5 A : ompo of A A A A A A a a (7 A a b ( b (4 A is uhag u h Lo asfomaio 8

44 . Dfiiios a Opaio uls of (oiu 5Th 5. Th 4-Laplaia opao,, (8 is a Lo sala opao, i.. [ : a Lo sala]. Poof : W ha show i im 4 ha h ig of a 4-o is a Lo sala, i.. A A. L b a Lo sala, h A a A a boh 4-os (s im. H, A A. 6. Th o pou of wo 4-os, ABAB, is a Lo sala. Poof : A B b (4 A B A B a A a B a a A B AB AB A A = A A a usful pop of h A A ohogoal asfomaio (9 9. Dfiiios a Opaio uls of (oiu ampl : I fam K, a pail's posiiohags b i a im ial. Th, (,,, i is a 4-o. H, ( is a Lo iaia, i.. i fam K, ( is gi b. Spial as : Th pail is a s i fam K (h s fam of h pail. H, = a (,,, i, wh w ha o h iffial im i fam K b isa of, baus fam K is a uiqu fam. is all h pop im of h pail. [a Lo iaia] ( wh is h loi of h pail i fam K. Disussio : (i Fo h spial as ha K is h s fam of h pail, is also h lai loi of h fams. H,.. Dfiiios a Opaio uls of (oiu (ii Th Lo asfomaio applis ol o iial fams. If h pail has a alaio, ( i ( is h iffial im i h isaaous s fam of h pail, i whih h pail has o loi bu a fii alaio. I gal, h sp of h s fam (h is a fuio of im, i.. ( [Jakso, (.6]. (iii Cosi a spial as of osa pail loi. Th muo has a lifim of. s i is s fam bw bih a a. If h lifim is masu i a Lo fam i whih h muo has a osa, h b (, h s-fam lifim i ( a h masu lifim a la b a a a. bih bih bih This psss h phomo of im ilaio; aml, wh h im ial of a lok's s im (.g. abo is obs i a moig fam, i is ga b a fao of. Th iaia of (= mas ha will ha h sam alu i all Lo fams.. Dfiiios a Opaio uls of (oiu 7A so of fh so ak ( T i is a s of f6 quaiis, T (, 4, whih asfom aoig o T a a T (, 8. Th o pou of a 4-so of h so ak a a 4-o, ( T A T A, is a 4-o. aat a A : ( T A Poof T A a a a T A a T A a ( TA ( Tasfom as a 4-o.

45 . Dfiiios a Opaio uls of (oiu T T 9. Th 4-ig of a 4-so of h so ak, ( T, is a 4-o. Poof : T ( T a a T aat, a T T a aa a a ( ( T,, Tasfom as a 4-o.. Dfiiios a Opaio uls of (oiu. A 4-so of h hi ak is a s of 64 quaiis, G ( - 4, whih asfom aoig o G a a a G i j k ijk ijk Pobl m : If is a 4-so of h so ak, show ha F F Soluio : F a a F ( - 4 is a 4-so of h h i ak. j k jk jk (4 ai F Fjk i ajak i jk i i a a Fjk ja k (5 ijk i Tasfom as a 4-so of h hi ak. 4. Dfiiios a Opaio uls of (oiu Poblm : Show ha h s of quaios, F F F ( -4 (6 is iaia i fom u h Lo asfomaio. F F F jk i j k ijk i Soluio : wi (5: a a a,, Chag iis i (5 as follows: ik, k j, j i F Fij a i a j a k (7 ijk k,, Chag iis i (7 as follows: k j, i k, j i F Fki a a a (8 ijk i j k Combi (5, (7, a (8, b (6 F F F F jk Fij Fki aiajak ijk j i k j 5. Dfiiios a Opaio uls of (oiu. If a phsial law a b pss as a laio bw 4-sos of h sam ak, h i's fom is iaia i all Lo fams. ampl : If h phsial law i fam K is of h fom A B, h, A a A ab B, i.. ABAB. (9 B ampl : If h phsial law i fam K is of h fom T F, h, T aa T a a F F, i.. F T FT F [i.. iaia i fom] ( I h followig sio, w ami laiisi liii mhais i 4-o fomalism. I S., w will mosa ha laws of lomagism a i aia u h Lo asfomaio b pssig hm as laios bw sos of h sam ak. Fom h Lo asfomaio of hs sos, w also obai h asfomaio quaios fo aious lomagi quaiis. 6

46 Sio : laiisi Mhais W bgi wih a o o h ms "osaio", "iaia", a "oaia". Th osaio of a quai mas ha i mais uhag i im i a gi Lo fam. Fo ampl, h laiisi momum a g of a isola ssm of pails a boh os af a ollisio. This is a fuamal law o b isuss i his Sio. Th i aia of a quai mas ha i is iaia i alu u a Lo asfomaio. Suh a quai is all a Lo iaia o Lo sala. Fo ampl, h o pou of wo 4-osisaLoiaia iaia. How, i ma o ma o b a os quai. A ampl will b poi i his sio. Th m oaia fs o phsial laws. A phsial law is "oaia" if i is " iaia i fom u h Lo asfomaio." As will b show, h w laws of laiisi mhais a isig laws of lomagism a all oaia. 7. laiisi Mhais (oiu Th 4 - Momum ( p of a Sigl Pail : As show i (A.8, if w fi h momum of a pail as p m a g as m ( m is all h s mass*, h h 4-momum, p ( p, p, p, i, is a 4-o, whih asfoms as p p P, P, P, (. p p K (. p ( p P, P, P, (. K ( p (.4 *Thoughou his hap, m a M o h s mass. Disussio: I Appi A, w fis fi pm a m, h show ha h law of osaio of momum a g is oaia. Cosl, fom h quim of h oaia of his osaio law, w a u h fiiios of p m a m (s Jakso S laiisi Mhais (oiu Th o pou of wo 4-os is a Lo sala, h pp pp p p ( i.. p is a Lo sala (iaia. If fam K is h s fam of h pail (i.. p, m h p(,,, im a pppp gis p m, o 4 p m ( Si p is a Lo iaia,,( shows ha h s mass m is a Lo iaia. This has i fa b assum i S. of Appi A, wh w i h Lo asfomaio quaios fo p ( m a ( m. ( is a usful fomula fo i las s h pail's oal g ( o is momum ( p. (Momum i pail phsis is of pss i ui of GV/. Fo a laiisi pail, w a sill spak of is (maosopi kii g K, fi as: K m ( m. (4 9. laiisi Mhais (oiu Th 4 - Momum ( P of a Ssm of Pails Cosi a ssm of pails, ah wih h 4-momum p ( p, p, p, i / ( p, i /, j,,, j j j j j j Si h Lo asfomaio is a lia asfomaio, h sum of a umb of 4-os also obs h Lo asfomaio. Thus, P p is a 4-o a is ompos asfom as j j pj pj j j pj pj j j pj pj j j j j j j p j j j j j (5. (5. ( (5. ( (5.4 a PP ( p ( p ( p ( p ( (6 is a Lo iaia. j j j j j j j j j j

47 wi (5:. laiisi Mhais (oiu p p j j j j pj pj j j pj pj j j j j j j pj j j j ( ( W s fom (5 ha ol wh all h ompos of P (i.. h h ompos of oal momum plus h oal g a ah os i fam K will all h ompos of P b os. If o ompo of P is o os, a oaio of h spaial ooia ssm a mak a ompo of P (momum o g uos i h w spaial ooia ssm. Thus, h laiisi law of osaio mus ak h fom as sib blow i o fo i o b a oaia law.. laiisi Mhais (oiu Law of Cosaio of Momum a g : Fo asos jus isuss, i laii, h osaio of momum a g oms i o law ah ha sp a a laws fo h momum a g as i olaiisi mhais. Th law sas ha, fo a isola ssm of pails, P bfo ollisio Paf ollisio, (7 whih implis ha p, p, p, a a ah os, i.. j j p j j j j j j j j p bfo ollisio af ollisio (8 j j bfo ollisio af ollisio (9 j Si h law i (7 is pss as a 4-o laio, i has h sam fom i all Lo fams [s (9]. Thus, i fam K, w ha P bfo ollisio P af ollisio. j j j. laiisi Mhais (oiu If P is os, h o pou P P mus also b os. Thus, j j pj pj pj pj j j j j j j ( ( ( ( ( ( ( bfo ollisio af ollisio Disussio : (i PP fo a isola ssm is boh a Lo iaia [s (6] a a os quai [s (]. If h ssm is o isola, i is sill a Lo iaia, bu o log a os quai. (ii P bfo ollisio i P af ollisio i i (7 is a fuamal law (ah ha a i laio, i whih h olaiisi law of osaio of momum has b o ilu h g, m. A impoa asp of his law is ha i applis o all posss i a isola ssm, suh as lasi a ilasi ollisios, i ula aios, a pail as. As a sul, h oal s mass of h ssm ma o b os, as is illusa i h followig wo poblms.. laiisi Mhais (oiu Poblm :Two iial pails of s mass m a qual a opposi loiis olli ha-o ilasiall o fom a sigl pail. Fi h mass a loi of h w pail. Soluio : m, m, (bfo M (af m Th oal momum bfo h ollisio is m m. So h ollisio ous i h -of-momum (CM fam, i.. h fam i whih h sum of h momum of all pails aishs. Fo la ompaiso wih h sul i poblm, w o h mass of h w pail b o iia ha i is a i h CM fam. M m Cosaio of momum Th w pail is sa ioa. Cosaio of g mmm M m m m 4

48 . laiisi Mhais (oiu Disussio:Ihispoblm, w m, m, (bfo fi M m = m >m, i.. s mass Mm (af has b a fom h kii g [(γ-m ] of h olliig pails. Th is o o kow wha s isi h w pail. W ol o kow is s mass a h h g assoia wih i. A ho obj has a s mass ga ha wh i s ol. Th iff i s mass u o a ias i mpau a i piipl b masu b is alaio u a kow fo, a w kow ha a las som of h a mass is i h fom of hmal g. I ma oh ass, i s o possibl o kow wha s isi. Nula fusio a fissio aios a ampls of oosaio of s mass. Th oal s mass is u af h aio a h mass fii appas as kii gis a aiaio. I fa, all aios (hmial o ula i whih g is absob (.g. phooshsis o las (.g. igsio of foo iol a ospoig hag of h aas oal s mass. 5. laiisi Mhais (oiu Poblm : A pail of s mass m a loi ollis wih a saioa pail of h sam s mass a is absob b i. Fi h s mass a loi of h w pail. Solu io : Th ollisio ous i h saioa-ag (ST fam. So, w o h w pail mass b M s, loi b Vs, a Lo / fao b s [ ( Vs / ]. ( m,, a also ST fam quaiis. Cosaio of momum m s Ms Vs ( Cosaio of g ( msms ( ( ( V s m, m (bfo ( M m M s, s VV s (af s s V s Ms m m ( ( m ( [ ] s m ( m [ ( ] m ( M ( m 6 s. laiisi Mhais (oiu Disussio : I poblm (CM fam, h w pail's massis Mm m. ( I poblm (ST fam, h w pail's mass is M m. (4 s m, m, (bfo M (af M m m M VV, m (bfo, (af s s s No ha is h Lo fao of h pail(s bfo ollisio. I pail phsis pims, Mm o Ms is h g aailabl fo h aio of w pails (wh o sms?. Th s g of h lo o posio is m.5 MV. If TV of g is fo pail aio (i.. M m TV o M TV, h h qui of h olliig pail(s is s b (, Mm m T V V.957 [CM fam]. b (4, Ms m TV 7.66 [ST fam] Th g assoia wih is o b obai i a alao laiisi Mhais (oiu Thus, kii g i CM fam ( i 5 kii g i ST fam 7.66 This shows ha fa lss kii g is i h CM fam ha i h ST fam. I fa, all h kii g of h 6 wo olliig pails [ ( MV TV] is pu i us i h CM fam, whil i h ST fam, % of h kii g of h ii pail is was! This is wh h Iaioal Lia Colli (ILC poj plas o ala boh los a posiios o gis upotvsohah ha h h ollisio ous i h CM fam. Qusio: : Wh us a log lia alao isa of a mo ompa iula alao? 8

49 Sio : Coaia of loamis I h spial ho of laii, Nwo's law has b aiall moifi. Th lomagi laws o o a moifiaio baus h a ala oaia. How, h oaia of hs laws (suh as Mawll quaios is o immial la fom h quaios b whih h a usuall ps. Ou pupos i his sio is o po ha h M laws a i oaia b asig hm io laios bw 4-sos of h sam ak [s (9 a (]. W will o his b fis fiig 4-sos i ms of kow M quaiis a fomig quaios wih 4-sos of h sam ak, h show ha o o mo isig M laws a implii i ah quaio. This will po ha h laws a oaia a jusif h fi quaiis o b lgiima 4-sos. Fuhmo, Lo asfomaios of hs sos will il h afomaio quaios fo aious M quaiis. No: : Jakso sw ihs o h Gausia ui ssm saig f o m Ch.. Fom h o, w also aop h Gaussia ui ssm. 9. Coaia of loamis (oiu. Dfi a 4-u as (, J, J, J, Giffihs J ( J, J, J, i ( J, i (5 a us i o fom a laio J (6 Th, (6 gis h law of osaio of hag ( i J J J (5. J ( i Thus, h fiiio of J i (5 as a 4-o las o h oaia psaio [(6] of h M law i (5.. This i u jusifis h fiiio of J as a 4-o. Th Lo asfomaio of J h gis J J J, J, J, J K J J ( J (7 J, J, J, ( K J. Coaia of loamis (oiu V (, A, A, A, Giffihs Dfi. Dfi a 4-poial i as A ( A, A, A, i (8 4 A J (9 a wi h oaia laios: A (4 A A J (6.5 (9 4 V (6.6 A Giffihs V (4 A [Lo oiio] (6.4 4 This agai shows h osis of A big a 4-o a (6.4-(6.6 (6.6 big oaia laws. Th Lo asfomaio A A A, A, A, A A K of A h gis A (4 ( A A, A, A, ( A K. Coaia of loamis (oiu No: Th sou-f wa quaio a b il pu io h oaia fom:. (4. Dfi a 4-waumb as k k, k, k, i k, i (4 ( ( Th, kk kk Iaia of h phas B h sam agum, w fi ha k fi i (4 is a lgiima 4-o. Thus, is Lo asfomaio gis k k k, k, k, k k K (44 k ( k k, k, k, ( k K laiisi Doppl shif

50 . Coaia of loamis (oiu 4. Dfi a fil sgh so of h so ak F [Maio, (4.6]: B B i B B i F (45 B B i D i i i D Th, F 4 4 SI H J J 4 B B J B I h oaia s of quaios [s (6] F F F F 's a lms ( - 4, of F i (45. s ( (,, B s ( (,, 4, (,,4, a (,,4 B.. Coaia of loamis (oiu 4 Th oaia quaios, F F F F J a, gi h s of Mawll quaios i f spa. This shows ha Mawll quais a oaia as wll as jusifis h fiiio of F as a so of h so ak. Thus, F a a F gis, h asfomaio quaios fo a B (s Maio, S. 4.6.,, B, B K B B B ( 46 B B,,, K B B I (46, is h loi of fam K lai o fam K, a " " a " " f o h iio of. S Appi C fo a summa of asfomaio quaios. 4. Coaia of loamis (oiu 5. Th oaia quaio*, P FP ( is a Lo sala, m gis (Maio, p.49 P FP, wh p ( p i, p, p, a p m m B B i B B i F B B i i i i p ( m( B B ( B m. Coaia of loamis (oiu 6. I a simila ma, w a mosa h oaia of h osaio laws fo fil/mhaial g a fil/ mhaial momum, as gi b Jakso (6. a (6.: m h fil s a S p h pfi l s T a (6. mh fil (6. p ( ( B laiisi quaio (47 of moio m This quaio is implii i (47. *I o fo his quaio o b oaia, h hag mus b a Lo iaia. This has b pimall sablish (s Jakso, p

51 . Coaia of loamis (oiu Cosi h gal fom of h laiisi quaio of moio i (47, p F, wh F is a fo, suh as h gaiaioal fo. Spial as : F ( o-imsioal poblm F ( m m m m ( m (48 / ( / / / ( ( / / / / F macosa fo os o aus osa alai o. Spial as : F ( os., as i uifom iula moio F p ( m m (Uulao & Wiggl (49 Qusios: (i I is somims sai ha a pail has wo masss, m a m. Wh? (ii Th alaio is o ssail paalll o h fo. Gi a ampl. (iii la (48 o (A.. 7. Coaia of loamis (oiu Poblm : A poli aa opas o a fqu of. Wha is h fqu i b h poli af h sigal is fl fom a a moig a h loi? Soluio : W o i i sps., k k poli a Sp. I h poli fam, h aa ss a wa (, k owa show i poli fam h a, whih h is i moig i a loi (iio i show h i i hfigu. Tasfommig o h a fam b (44, w obai ( k, wh k is h ompo of k alog, i.. k kos (s figu. Thus, / k / ( os ( k kos( This is h Dopp-shif fqu b h a. I is also h fqu of h wa fl b h a as s i h a fam. 8. Coaia of loamis (oiu Sp. I h a fam (s figu, h a ss h fl wa (, k bak o h a a h fqu os ( poli a k, k I h a fam, h poli is moig a loi (iio show i a fam show i h fugu lai o h a. Tafommig o h poli fam b (44, w obai h fqu obs b h poli ( Doppl shif agai ( k ( k os os ( os os ( ( si. 9 If h aa fqu is f ( / H a h a mos awa fom h poli ( a 5 km/h, h poli woul a fqu f ( / shif b f f 78 H. 9. Coaia of loamis (oiu Poblm : A obs i h laboao ss a ifii lo bam of aius a a uifom hag si moig aiall a loi. Wha fo os h s o a lo a a isa ( a fom h ais? Assum h lo mos aiall a h loi. Soluio: : Th poblm a b ail sol i h lab fam. H, w will ak a log ou fo a is o som of h asfomaio quaios jus i. a Th u si J i, J, ( h lab fam is J. [ has a gai alu.] K B (7, w ha, i h bam fam a J / J (, J, K ( J (. W s ha h lab fam is ga ha h bam fam b h fao. This is baus ui lgh of h bam i is s fam is oa b his fao wh iw i h lab fam. 4

52 . Coaia of loamis (oiu I h bam fam, J, ; h, h is ol a aial li fil. Gauss law, 4,, B ( B a s K h gis 4 (, fo a ( B B. [, ] K W ow asfom ( io lab-fam a B b usig h s asfomaio quaios i (46, i whih w s. ( B B B ( ( Thus, h fo f o a lo (i h lab fam is f B ( ( is posii. Fo ( a lo bam, is gai. 4 Homwok of Chap. Poblms:4569, 4, 5, 6, 6, 9,, 4 4 Appi A: laii i Collg Phsis (f. Hallia, sik,a Wlk Walk, Fuamalsof Phsis Sio : Th Lo Tasfomaio Th Galila Tasfomaio: Cosi iial fams S a S. Fam S mos alog h ommo -ais* wih osa sp lai o fam S. A, h ooias oii a, a im, h posiio of poi P is (,, i S a (,, i S. Th h Galila asfomaio gis is uhag i,,, (A. h asfomaio. * I h mai, h -ais is h iio of lai moio. To S b osis wih h fs S i i his appi, h w P assum ha h lai moio is alog h -ais. Qus io: How o ou mi a f fam is iial? 44

53 .A. Th Lo Tasfomaio (oiu isi's s Posulas: : Th laws of lassial mhais o o a i fom u h Galila asfomaio. Fo ampl, (A. shows F = m a i fam S asfoms o F = m a i fam S. How, w h h sam asfomaio is appli o h wa quaio i auum,, is fom hags ompll (s Jaksop56 Jakso, p. 56. So, wh isi bga his wok o laii, h w wo appoahs o mak all h laws of phsis iaia i fom i all iial fams: ( Moif h ho of lomagism so ha i is iaia i fom u h Galila asfomaio; o ( Moif h Galila asfomaio a h laws of mhais so ha h laws of bh boh mhais a lomagism i a iaia i i fom u h w asfomaio. isi ook h so appoah. His spial ho of laii i is bas o posulas: l. Laws of phsis a iaia i fom i all iial fams.. Th sp of ligh i auum has h sam alu i all iial fams, ip of h moio of h sou. 45.A. Th Lo Tasfomaio (oiu a Simulai: A is somhig (suh as h missio of a ligh puls b a sou whih happs a posiio (,, a im. A [sib ollil as (,,,, i a gi fam] will ha iff ooias i iff fams. Th fams mio h a la a all iial fams. Th im of a a b masu b mhos w omall hik of. Bu, i laii, im masum of quis high pisio (whih a a las b o i a hough pim a w mus ba i mi h fam i whih h im is masu. Th simpls wa o masu im is o a h lok a h posiio of h. If h lok is awa fom h, h im of h is h im show o h lok (a h isa h ligh sigal of h ahs h lok mius h im la u o h al of h sigal (a sp fom h s posiio ii o h lok s posiio. ii Th posiio of h a h masu im of h all f o h fam i whih h h obs a h lok a bh boh a s (bu h sou whih gas h is o ssail a s. 46.A. Th Lo Tasfomaio (oiu Two s a simulaous i a f fam if h ha h sam im ooia i ha fam, whh o o h ha h sam spaial ooias. Simulai a b pimall s as follows. If wo s a a h sam isa b a obs loa miwa, h a simulaous i h obs s s fam. Wihi a gi fam, h op of spa a im i h spial ho of laii is o iff fom ou usual op of spa a im. How, aial iffs ais wh spa a im ooias of a masu i o fam a ompa wih hos masu i aoh fam. I makig h ompaiso, w fi ha spa a im a agl wih ah oh i laii. Fo ampl, wo simulaous s ouig a iff posiios i fam S will o log b simulaous i fam S S, a hi im iff i S ps upo hi spaial spaaio i S. I laii, spa a im ooias asfom aoig o h Lo asfomaio, whih is i blow fom posula. 47.A. Th Lo Tasfomaio (oiu Tim Dilaio: : Cosi a puls of ligh mi b a sou o a ai (. I als iall upwa fo a isa D, h is fl bak b a mio, a la a h sou (. I h ai fam (Fig., h im ial bw h s is D mio. (A. I h lab fam (Fig., h ai, mio, a sou aallmoigasp a buh D Fig., h ligh sill als a sp (b posula. So, h im ial of h s is sou L, (A. mio / wh L [( D ] ( A.4 Fig. limiaig D a L fom L D L (A.-(A.4, w obai, (A.5 / sou (a h sam loaio as sou wh [ / ]. (A.6 48

54 .A. Th Lo Tasfomaio (oiu Qusio: : Wh is D h sam i boh fams? Lghs ppiula o h iio of moio a h sam i boh fams, i.. h a ooias asfom as:, (A.7 Th poof of his is b oaiio. Suppos ha w ha wo iiall maufau pis of pip (s figu. Th ao fi isi ah oh baus h ha iial aius. Imagi ha o fom H. C. Ohaia, "Phsis" pip is a s o h gou a h oh is a s o h ai. If h moio of h ai lai o h gou w o big abou a ass oaio of h ai pip, h b smm, h moio of h gou pip lai o h ai woul ha o big abou a oaio of h gou pip. Bu hs wo ffs aoaio, si i o as h ai pip woul fi isi h gou pip, a i h oh as i woul fi ousi. 49.A. Th Lo Tasfomaio (oiu Goig bak o (A.5: Δ = Δ. I his quaio, Δ is h im ial of s masu i a spial fam i whih h s ou a h sam posiio. I is all h pop im. Viw i a oh fam, hs s will ou a iff posiios a, b (A.5, hi im ial (Δ will b ga ha h pop im b a fao of. This is kow as h ff of im ilaio. Th muo has a aag lifim of. s (bw bih a a i is s fam. I a 977 pim a CN, muos w ala o a sp of.9994, ospoig o a alu of Wihi pimal o, h masu aag lifim of hs muos was i = 6.5 s. I aoh pim, wo shoi loks wih a pf pisio show slighl iff aigs af o ha b flow aou h wol. Th iff was agai i agm wih (A.5. Tim ilaio us ou o ou iuiio, baus i is oo i a posula whih also us ou o ou iuiio. 5.A. Th Lo Tasfomaio (oiu Th Twi Paao: Suppos somo als o a spaship wih a Lo fao of = (i h ah fam a his wi boh sas o ah. Th, b im ilaio, a masu b h alig wi i h spaship fam (his is his pop im will b as whmasu b h ah wi i h ah fam. So h ah wi ags fas a his alig boh will b 9 as oug wh h us o ah af a -a jou (gl h spaship s alaio/laio pios. Th paao is: if h alig wi masus h ag of his ah wi, will h olu ha h himslf ags fas b h sam agum of im ilaio? Th is o paao a all. Ol h ah wi s masum is o baus h is alwas i a iial fam. Th alig wi will ha o b ala a la i h spaship. Duig hs pios, h ao us h spial ho of laii (isi s posulas f o iial fams. I fa, h will ofim h masum of his ah wi if hussisi s gal ho of laii, whih als wih alaig f fams. 5.A. Th Lo Tasfomaio (oiu Lgh Coaio: Assum ha pla pu is saioa i h ah fam a a a isa L fom ah (Fig.. A spaship is alig a sp o pu. Th uaio of h ip, masu o ah, is L /. (A.8 I h spaship'ss fam (Fig., boh ah a pu Fig. mo a sp. Th uaio ah spaship pu of h ip,, is h ial Fig. bw h pau of h fom Giaoli, "Phsis fo Siiss a gis" ah a h aial of h pu. This is h "pop im" of h spaship baus boh s ou a h sam posiio. Thus, b (A. 5 /, (A.9 a b us o alula h ah-apu isa as iw o h spaship L. (A. limiaig a fom (A.8-(A., w obai L L (A. 5

55 .A. Th Lo Tasfomaio (oiu I (A., L = L /, L is h lgh of a obj (o, i h abo ampl, h ah-pu isa masu i h s fam of h obj (i.. h fam i whih h obj is a s. Lgh masu i his spial fam is all h pop lgh. Viw i a oh fam, h obj will b moig a, b (A., is lgh will b lss ha h pop lgh b a fao of. This is kow as h ff of lgh oaio. No ha h oaio ff applis ol o lghs alog h iio of moio. Lgh oaio is a i osqu of im ilaio [s (A.9]. I is hfo o supisig ha im ilaio a b if fom lgh oaio. If, fo ampl, h spaship p has a alu of. Th ah-pu isa, as masu i h spaship, woul b half of ha masu o ah. Bu h sp of ah/pu lai o h spaship is sill. So, o h spaship, h jou s uaio is ol half of ha masu o ah. H, o miu laps i h spaship will b mius laps o ah. 5.A. Th Lo Tasfomaio (oiu Th Lo Tasfomaio: Assum fams S a S oii a a S mos alog h ommo -ais wih sp lai o S (s figu. A poi P has ooias (,,, i S a (,,, i S. Th lgh, wh masu i S, is (lgh oaio. So, o (. (A. B smm o b simila agum, (A. limiaig fom (A. a (A. [usig ], ( (A.4 (A.7, (A., a (A.4 gi h Lo asfomaio: S S (A.5 P ( S Appi B fo a mo fomal iaio. 54.A. Th Lo Tasfomaio (oiu Tasfomaio of Cooia Diff bw s : Si h Lo asfomaio is lia, h ooia iffs bw s: i S:,,, (A.6 i S :,,, (A.7 asfom i h sam ma. Thus, ( (A.8 ( ( (,,,,,, (,,, (,,, S S 55.A. Th Lo Tasfomaio (oiu Disussio o simulai: : Cosi h asfomaio quaio fo h im ial bw wo s ( [fom (A.8] A B sou I iias ha simulaous S s i fam S ( whih A sou B ou a iff posiios ( S will o b simulaous i fam S (. This a b plai o h basis of posula hough h followig ampl. I fam S, a puls of ligh mi miwa bw pois A a B (s figu will ah A a B a h sam im, i.. h wo s (aials of h sigals a A a B a simulaous i fam S. I fam S, h sigal sill als a sp i boh iios, bu B is moig owa h ligh a A awa fom i. So, h sigal will ah fis a h wo s a o log simulaous. B 56

56 .A. Th Lo Tasfomaio (oiu Th ampl isuss abo a b ami quaiail as follows. A B sou Assum ha, i fam S, h S wo s a spaiall spaa b a isa. Obs i S, h isa is sho b a fao A sou B of u o lgh oaio, i.. S. Thus, i fam S, h sigals ah A a B wih a im iff of B A. This is pisl h piio of (A.8, (. [ i fam S] 57.A. Th Lo Tasfomaio (oiu Poblm : I fam S, s A a B ou a iff posiios, a B ous af A. Is i posibl fo B o p A i aoh fam S moig a sp lai o fam S? If so, os his ma ha a ff a p is aus? Soluio : I fam S, l h s ha a spaial ial B A a im ial B A. Th h im ial isgii(a8: i fam S,, i (A.8: (. B A W s ha if /, h, whih mas ha h o of ip s i fam S ma b s i fam S. Suppos, how, ha h s a o, i.. B is aus b A. This woul qui a bo, o a sigal, o al fom A o B. wi (A.8 as (. Si h fass sp fo a sigal o al fom A o B is, w mus ha i o fo. This is o possibl [s(a6]ah (A.6] hus sho of o s (aus a ff ao b s. 58.A. Th Lo Tasfomaio (oiu Poblm : : Show ha h ffs of im ilaio a lgh oaio a implii i h Lo asfomaio. Soluio : Th im ial bw s asfom as ( o (. If is h pop im i S, h h s ou a h sam posiio (. So w us h la quaio a obai (im ilaio. Th iff i h ooias of h s asfom as o (. Agai, h qusio is whih quaio o us. If is h "pop lgh" i S, h h wo pois a a s a hi ooias o o ha o b masu simulaousl (i.. w o o kow. Bu si h o is moig i S, is pois mus b masu simulaousl i S (. So w us h fom quaio a obai (lgh oaio. 59.A. Th Lo Tasfomaio (oiu Tasfomaio of Vloi : Th loi of a pail is gi b lim (i fam ; lim (i fam. (A.9 S S L, S S ( ( ( (A. ( ( ( ( ( Poblm: A spaship mos awa fom h ah a sp. A puls of ligh is mi fom h ah i h iio owa h sapship. Wha is h sp of ligh masu o h spaship? Sl : Soluio 6

57 .A. Th Lo Tasfomaio (oiu Tasfomaio of Alaio : Fo simplii, w fis osi h asfomaio of alaio i h iio of lai moio (i.., [ fom (A.] S ( (, ( ( S ( [fom (A.5] ( H, (A ( ( (A. (.A. Th Lo Tasfomaio (oiu B h sam mho, w ma obai h asfomaio quaios fo alaio i abia iios (s Jakso Poblm.5. a a a S a (A. a a a S wh " " a " " f o h iio of. 6 6.A. Th Lo Tasfomaio (oiu Poblm : A ok is lauh fom h ah io ou spa. I mos o a saigh li wih a osa alaio ( a wih sp o is s fam (Wh is a spifi i h ok's fam? Calula h im qui fo h ok o ala fom o sp o h fial sp f, aoig o ah a ok loks. Soluio : L S b h ah fam, S b h ok s fam, a h o-imsioal moio b alog h -ais. Th is asfomaio of (A. gis (omiig subsip " ": a a., ( / Lo asfomaios appl o wo iial S fams. So, S is h isaaous s fam of, a h ok, bu S os o ala wih h S ok. I S, w ha a / a. This gis a (alaio i S a /. Si S is h s fam of h ok, fo h asfomaio quals of h ok i S. Thus, / a a/ a/, wh ( / [ No: aa] 6.A. Th Lo Tasfomaio (oiu Fom h pssio of h alaio i h ah fam ( S, / a a/ a( /, (A. w ma alua h oal alaio im as masu o ah T f f f f T a a a / /. ( / a ( f / Th ok fam is alaig. So, o fi h oal alaio im as masu o h ok, w mus sill wok i h ah fam b usig h laio /. T T T / f f f. ( a a / a I f / W fi ha, i h limi f /,boh T a T appoah h p alu of f / a. How, T / T iass apil as f f / u o h ff of im ilaio (s figu. I h figu, f ( f / is h im ilaio fao a h fial sp. / f f T / T 64

58 S S p p,, Sio : laiisi Momum a g (f.: H. C. Ohaia, Phsis Phsis,., pp Th law of osaio of momum sas ha, fo a isola ssm of pails, mi i (bfo ollisio mi ui (af ollisio. U h Galila asfomaio, h sam is u i all (iial fams. How, u h Lo asfomaio, i m i i, hough os i o fam, will i gal o b os i aoh fam. Thus, posula is iola if w oiu o fi h momum as m. Th ho of laii aks a majo sp b fiig (o posulaig h momum a g as p m / No: ( / is h Lo fao (A.4 i f of a pail. I is o b isiguish fom h m (A.5 Lo fao fo h asfomaio. Fo simplii, i w will osi ol o-imioal i moio alog h ais. Th momum a g of a pail a h p m a m 65.A. laiisi Momum a g (oiu Fom (A., h loi i fam is S. H, h momum of h pail is (assumig m hashsamalui sam S m m ( p ( [( ( ] m (,, p ( S Si (, p, ( (, p boms S m p m ( p (A.6 Similal, w i h Lo asfomaio quaio fo g: ( p (A.7 66.A. laiisi Momum a g (oiu B h sam mho, w a h moio o imsios a i h Lo asfomaio quaios fo p a. Th sul is p ( p p, p = S p (A.8 p p p, ( p S (A.8 shows ha p a i S is a lia ombiaio of p a i S, wih osa offiis (i.. h offiis a ip of p a of h pail. Th sam quaios will hfo hol u fo h oal momum a g ( p j, j of a ssm of pails, pj ( pj j pj = pj (A.9 pj pj ( p j j j 67.A. laiisi Momum a g (oiu pj ( pj j p j = p j wi (A.9. pj pj j ( j p j Fom his s of quaios, w s ha if (a ol if h oal momum ( p p j a oal g (( j ofassmofpails a boh os i S, h oal momum a oal g will b boh os i S. Disussio: (i This shows ha h posulaio of p m a m will ps h osaio law u h Lo asfomaio. How, h osaio law mus ow b o ilu boh h momum a g. (ii Wiig m ( m m, w ma ii h oal g io h kii ( m (u o moio a a w fom g m (a iisi g all h s-mass g. 68

59 .A. laiisi Momum a g (oiu Poblm: A pail of s mass m mos o h -ais is aa o h oigi b a fo m ( os. I pfoms osillaios of ampliu a. pss h laiisi osillaio pio as a fii igal, a obai h laig ms of his igal fo small a. a Soluio : Th pio is gi b 4, (A. wh h loi a b alula fom h g quaio / m ( / m m m a (A. Subsiuig fom (A. io (A., w obai 4 a ( a / / / ( a ( a /(4 paig h iga i pows of ( a / a usig b a (fo, w obai ( b / b ( b / b b b a 6 69 Appi B: A Fomal Diaio of h Lo Tasfomaio I Appi A, w bgi wih h iaio of "im ilaio" a "lgh oaio" fom posula, follow b a iaio of h Lo asfomaio. H, w ps a mo fomal (bu phsiall lss aspa appoah, whb h Lo asfomaio is i il fom posula. Th followig paagaphs a ak fom Aloso & Fi "Phsis Phsis, " p9 p.9. fig o h figu o h igh, suppos ha a a flash of ligh is mi a h ommo posiio of h wo obss. Af a im, obs O will o ha h ligh has ah poi A a will wi, wh is h sp of ligh. Si,w ma also wi (B. 7.B A Fomal Diaio (oiu Similal, obs O, whos posiio is o log oii wih ha of O, will o ha h ligh ais a h sam poi A i a im, bu also wih loi. Thfo h wis, o (B. Ou ask is o obai a asfomaio laig (B. a (B.. Th smm of h poblm suggss ha a. Also si OO fo obs O, i mus b ha fo (poi O. This suggss makig k(, wh k is a osa o b - mi. Si is iff, w ma also assum ha a( b, wh a a b a osas o b mi (fo h Galila asfomaio, k a a b. Makig all hs subsiuios i (B., w ha ( a ( bb o k ( k b a ( k ba ( a k /. 7.B A Fomal Diaio (oiu This sul mus b iial o (B.. Thfo k b a ba k a k / Solig his s of quaios, fo k, a, a b, w ha / k a a b / Isig hs alus of k, a, a b i k( a ( ab, w obai h Lo asfomaio / ( B. / / 7

60 Appi C: Summa of Lo Tasfomaio quaios ( Fo all quaios,. B smm, quaios fo h is asfomaio iff ol b h sig of (o. (,,, (,,,. K K, Fams K a K ( oii a. ( ( K. ( K 7.B Summa of Lo Tasfomaio quaios a a a. K a a a a K wh " " a " " f o h iio of. Spial as: o imsioal moio, a a ( a K, a p p P K, P, P, p p K 4. p ( p P, P, P, ( p K 74.B Summa of Lo Tasfomaio quaios.b Summa of Lo Tasfomaio quaios 5. J J J, J, J, J J K J ( J J, J, J, ( J K 7. k k k k K k ( k K ( k k, k, k, k, k, k, 6. A A A, A, A, A A K A ( A A, A, A, ( A K B 8. B B B B,, B, B K,, B, B K wh " " a " " f o h iio of

61 Chap 4: aiaio b Moig Chags o o Gaussia ui ssm, iw of Basi quaios : s p.78 fo osio fomula. 4 ( (6.5 f-spa ihomogous wa quaios (SI A 4 A J( J (6.6 gal fom of 4 f(, (6. (6.5 a (6.6 Soluio of (6. wih ougoig-wa b.. : (, i (, G (,,, f(,, (6.45 wh h a G' s fuio [ ( ] G (,,, / (6.44 is h soluio of (wih ougoig-wa b.. G (,,, 4 ( ( (6.4 ( iw of Basi quaios (oiu Appl (6.45 (assumig o (6.5 & (6.6 i (, (, [ ( ] (9. (, (, A J No : W boh a A o spif a B, u lss h sou has hamoi im p (as i Chs. 9 a. A Qualiai Piu of aiaio b a Ala Chag: -fil lis suouig a saioa hag. Fom. M. isbg, "Fuamals of Mo Phsis" A faio of -fil lis showig h ff of hag alaio. 4. Liéa-Wih Poials a Fils fo a Poi Chag Lia-Wih Poials fo a Poi Chag : A (, [ ( ] (, wi (9.: J (, (, (,, J(, u o a poi hag ( ais a sig moig alog h obi ( a h loi ( ( / a b wi (, ( [ (,, A(, ] a poi of obsaio J (, ( ( (a ( [ ( o ( (, ] ( ( ( ( ( ( [ β ] (a (, A ( obi of wh ( ( a β( (. 4. Liéa-Wih Poials (oiu ( ( [ ] [ f ( ] ( ( ( β( β( f ( ( ( (, wi (:, [ ] [ ] A (, wh f ( (. ( g ( Ui Usig g ( f ( a [ ], w obai bi ( i i f is h soluio i (, [ ] ( of f ( i a. f( ( ( β ( A (, [ ] ( f ( wh [ ] implis ha quaiis i h bak a o b alua a h a im [ ( ]. Qusio : Wha ifomaio is i o o fi? 4

62 4. Liéa-Wih Poials (oiu ( ( [ ( ( ] ( is a fi posiio, ip. of im [ (,, A(, ] a poi of obsaio ( ( ( (a [ ( ( ] ( o ( ( [ ( ] ( ( ( (a ( ( obi of (4 ( ( f [ ] β( ( ( (5 Sub. (5 i o ( gis h Lia-Wih poials (, ( β (, β A ( β (4.8 5 Fils fo a Poi Chag : 4. Fils fo a Poi Chag (oiu wi ( a (4.8: ( (, ( ( β [ ] (, (, ( (4.8 β( [ ] (, (, β A A ( ( β To obai (, a B (,, [ (,, A(, ] a w o iffia (, poi of obsaio a A(, wih sp o. (a Th HS of (4.8 ps o ( hough a, bu h HS of o ( ( ( ( ps o hough ol. ( (a H, i is mo oi o obi of us (. 6 (, wi (: A (, L F( b a fuio of, h 4. Fils fo a Poi Chag (oiu [ (,, A(, ] a ( [ poi of obsaio ] ( (a ( ( o ( β( [ ] ( ( ( (a obi of ( F F ( ( F F (6 ( Us (. S S.. of lu os. ( ( ( ( [ ] [ ] (, ( ( & (6 ( ( [ ] (, A β ( 7 4. Fils fo a Poi Chag (oiu Thus, ( f ( f ( f (, f( (, A wh β ( (. Us (5 ( β ( [ ] [ ] β [ f( ] [ f( ] f( [s o blow] β g ( ( a g( a f ( β (7 f f ( ( No: Baus of h [ f ( ] fao i h iga, igaio ( o f( mas f( o. Bu,,, a i h iga a all fuios of [o f ( ]. H,,,, a ao b alua a h a im [o ]. 8

63 4. Fils fo a Poi Chag (oiu ( To pu i a simpl fom, w o alua a (. ( ( ( ( ( ( ( ( β ( β (8 ( ( β β [ (,, B(, ] a ( b (4 poi of obsaio (a ( ( o [ β( ( ] ( ( β ( β ( (a β obi of ( β ( β β β Sub. (8 fo ( β ( β β ( β ( β ( β β β ( β ( β ( Fils fo a Poi Chag (oiu (, β ( fom (7 β Us (8, (9 { β [ ( ] ( ( } β β β β β β β ( β β ( β( β ( β( { [ ] [ ]} β ( β[ ( β ] β ( ( ( ( β β β β ( ( ( β β β β { [( β( ] [ ( β β [ β( β β( β] ] } / ( β ( ββ β ββ 4 β (4.4 β 4. Fils fo a Poi Chag (oiu To i B (,, w wi (7 (, [ β ( ] β ( Us (8 ( (, [ β ( ] β [ ] β [ β β ( ] β β β β ( β β opas o ( ol ( [ol ( ps o ] β( ( / B (, A [ ] ( ( ( / a aa [ ] β ( ( ( ( / ( / [ ] ( β( β β ( followig h sam sps as B (, ( (, i iig (7 (4. 4. Fils fo a Poi Chag (oiu β β β β β loi fil ( alaio fil ( a (, (4.4 wi B (, ( (, (4. Disussio : [ (,, B(, ] a (i Th loi fils a ssiall poi of obsaio sai fils fallig off as /. (a (ii Fo h alaio fils, ( (4. a (4.4 4 show ha o ( ( (,, B (,, a ( a ( (a muuall ohogoal, as is pial obi of of aiaio fils. No : (i Ui o ( pois fom h a posiio o. (ii a a quaiis i h sam f fam.

64 4. Fils fo a Poi Chag (oiu (, β β β (4.4 β β wi loi fil ( alaio fil ( a B (, ( (, (4. (iii a B i gal ha a [ (,, B(, ] a boa fqu spum. Si w poi of obsaio ha i (4. a (4.4 (a fom (9., whih applis o a o- ( o ( ispsi mium (i his as, h ( auum, sigals a all fquis ( (a al a sp. H, a B obi of a p ol o h isaaous moio of h poi hag a a sigl a posiio (. 4. Fils fo a Poi Chag (oiu β β β β β loi fil ( alaio fil ( a (, (4.4 wi B (, ( (, (4.,, B (i Quaiis i h baks a [ ( (, ] a poi of obsaio o b alua a h a im, whih is h soluio of (a ( ( /, o ( ( wh h obi ( is a spifi ( fuio of. Thus, ps o (a a. This maks h fial pssio obi of fo a fuio of a, as show o h LHS of (4.4. Fo h sam aso, h ui o ( i (4., h h fial pssio fo B, also ps o a [s (4.7a blow] Fils fo a Poi Chag (oiu ( Th laio bw obs's s im a h a im, ( /, iias ha a sigal fom h hag als a sp owa h obs, ip of h moio of h hag (isi's posula. A Illusaio i of ftim aaio i a Lgh Coaio: i Compu ga gaphis show h isual appaa of a h- imsioal lai of os a balls moig owa ou a aious sps. (fom Bso, Uisi Phsis A.99, h Th omal A.5, h os A.95, h os lai appas iw a s appa saigh. appa b. sl iso Fils fo a Poi Chag (oiu Chag i Uifom Moio : os. PP isa bw poi P a poi P PQ os β OQ β OQ PQ β PQ b b si b b ( b I h abo pssios, a a a quaiis ( β is a / β b osa. H, ( 6

65 4. Fils fo a Poi Chag (oiu β os. [ ] [loi fil] β b / β β [ ] [ b ] β b / b [sam as (.5] (4.7b (4.7a / ( β ( b, las pag os β os / β β ( b [ ] [ ] b smm. os os. o h lf si of h oigi ( Fils fo a Poi Chag (oiu B ( (, os si os si So, h ol oaishig ompo of B is B B os si /[ b ( b] b b Disussio : (i wi h - a B-fils / b b / b B As p, h fial pssios fo a B a fuios of h obs'ss posiio ( b a im (, alhough h fils a ga b h hag a h a posiio ( P a im ( Fils fo a Poi Chag (oiu 4. Fils fo a Poi Chag (oiu (ii wi h -fil a poi O b / b / b b If, is i fom h hag's ps posiio P (i.. posiio a h im of obsaio o h obsaio poi O, alhough is ga b h hag a h a posiio P. Si b a a b gi abia (posii o gai alus, his iio laio applis o all obsaio pois aou h hag. Thus, -fil lis aou h hag a saigh lis maaig fom (o, if, ogig o h ps posiio P. 9 (iii wi h -fil a poi O b / b / b -fil lis 6

66 4. Fils fo a Poi Chag (oiu (i wi h -fil a poi O b / b / b has a maimum alu a, wh passs hough poi M. b ma ( b ma is ow o a b. ( b ma sam as (.5 Duaio of appiabl : b b 4. Fils fo a Poi Chag (oiu loamis i a Cai : As show i h figu, a lo buh moig uifoml o h ais wih 6 is abou o a ai. Si (6, h -fil lis of lo a oa i a fla isk wih h lo a h (loi fil. As a sul, h los hal "s" ah oh, baus h (aial li fos bw hs los a gligibl*. Th, as h buh s h ai, h alaio fils mg ( pag. -fil ai lis 6 lo buh *Qusio: Th gligibl ibl li fo bw a los implis ha h aial alaio of ih lo is gligibl. How, h alaio will b o-gligibl gligibl wh i is iw i h lab fam. Wh? [S lu os, Ch., q. (A..] 4. Fils fo a Poi Chag (oiu Fils i h ai pou b a = 6 lo buh ( (4 ( (5 ( Combi "loi" a "alaio" fils fom b a sigl lo buh (fom Ch. Wag, NSC. Fils bhi h buh a all h wak fils. Qusio: How o h los g la i h ai? 4. Fils fo a Poi Chag (oiu Th lows o (TM mo -fil lis of sal ai mos (fom L. H. Chag, NSC is i b h ijio of high pow miowas fom a klso. Th aial li fil of his mo is us o ala h los. Wak fils lf i h ai b h lo buh a b iw as h supposiio of h ompl s of ai igmos. O o mo of h high-o mos ma hus b soal ifo b a sussio of lo buhs o gow o sigifia ampliu a if wih h alaio poss. 4

67 4. Toal Pow aia b a Ala Chag β wi (4.4 : (, β β β β loi fil alaio fil S (, B (, ( (, (, ( Lamo's Fomula : Ngl h loi fil a ak h limi ( a,, β, ps,, β,. Th, lim (, ( β β si lim S ( β 4 β lim β P 4 ( 4 β β (4. pow aia si, pak a (4. 4 ui soli agl 5 4. Toal Pow aia b (oiu p Lamo's lim P (4. fomula No ha all quaiis i Ss a al. H, P m p p p. p I Jakso, his is o b ( laiisi Galiaio : Th pssio i (4. a b gali o a laiisi fom i whih P is a Lo iaia a appliabl o all lo gis. Th pou is as follows. i p P ( p, (4-o p P P P (4.4 (Lo sala m I ms of a : p p P (4.5 m Co o lab im b : p P ( m ( ags wih suls i il fom (4.4 (S S p m 4. Toal Pow aia b (oiu P i ( a b pu i iff foms: ( ( p ( m p p ( m m p m m ( m p p m p No: psss boh p p p m m iio a ampliu p ( p m aiaios of p, bu p ol psss h Sub. fo i ( ampliu aiaio of p. p p P ( ( m 7 m 4. Toal Pow aia b (oiu ββ β ββ ββ ββ p ββ m β β ββ m 6 4 m ββ ββ β 4 4 m m ββ β β ββ 4 m β ββ (ab( ( ( a ( b ( a ( b Sub. ( io ( 6 AB A B AB P (4.6 β β β 8

68 4. Toal Pow aia b (oiu ampl : Lia alao ( p alaig fo F wi (: P m p p wi (: P ( P Fo lia alaio, p p p m (4.7 P p F p (4.8 m F F P m m m m 5.7 m/mv P: aia pow. : all suppli pow aiaio losss a ompll Tpiall, 5 MV/m gligibl i lia alaos Toal Pow aia b (oiu ampl : :Ciula alao (.g. shoo p p wi (: P ( m p p Fo iula alaos,. Thus, p P m p p ( p p p p p gligibl p p., p m 4 4 p P (4. m No ha (4. is a a pssio fo P if h pail is i p uifom iula moio, i.. if. 4. Toal Pow aia b (oiu 4 4 wi (4.: P p aiaio loss p oluio P [ (i GV ] 8.85 MV (i ms kv, fo al shoos (alaos 7 k V, fo h. GV NSC shoo soag 8.85 MV, fo h GV Coll shoo igs Toal pow aia i iula lo alaos: 6 P(i was (i MV J (i amp 4. Toal Pow aia b (oiu Poblm: If a hag is i uifom iula moio, (4. is a a pssio fo h oal pow i aias. Show ha h oal pow has h sam alu as iw i h s fam of h pail. Soluio: Cosi a isaaous posiio of h hag loa a h boom of is obi, wh h hag mos hoioall o h igh a loi (upp figu a h alaio a pois iall upwa wih a / ( is h aius of h il. Viw i h s fam of h hag (low figu, w ha [s q. (A. i Ch. of lu os] a a ( a a a ( a ( a ( / Thus, a a a a. [ ] K K a a a

69 4. Toal Pow aia b (oiu Thus, h alaio of h hag is iall upwa i boh fams a h a la b a a Si h hag is a s i fam K, a Lamo's fomula i (4. boms a, whih gis K 4 P = a a a K a This is h sam pow as iw i h lab fam [s (4.]. Th sul h, P P, is osis wih h fa h oal aia pow is alo iaia [s (4.4]. How, h agula isibuio of aiaio will b iff i h wo fams. W will show la i (4.44 ha fo h sam alaio, h agula isibuio ps ssiil o pail's loi. 4. Agula Disibuio of aiaio mi b a Ala Chag β 4 β β wi (4.4 : (, β β pow p ui aa a obsaio poi loi fil alaio fil S (, 4 (, B (, [ (,, B (, ]a 4 (, ( (, poi of obsaio (a 4 (, ( ( o ( S (, ( (, ( 4 ( (Ngl h loi fil (a obi of β β 4 (4.5 β 4 4. Agula Disibuio of aiaio (oiu I his sio (as i S. 4., w a is i h agula isibuio of pow aia b h hag. Bu S (, ( (, i (4.5 gis h pow p ui aa i a 4 h obsaio poi. Pow aia b h hag io a ui soli agl [ P( / ] is i gal iff fom h pow i o h aa subig h soli agl [ P( / ]. Th aso is ha moio of fh hag owa ( (awa fom h obsaio poi will sho (lgh h aia puls, whih suls i ias (as pow a h obsaio poi baus h oal g i mus qual h oal g aia (osaio of g. P( Thus, o pss h pow aia i ms of h pow i, w o mi h aio of (i puls lgh o (aia puls P( lgh Agula Disibuio of aiaio (oiu Obsaio im a aiaio im a la b (. ( Us (4: ( ( ( Thus, β( ( A puls of fuaio i a a is aia b h hag a ( a fo a uaio of β( (. No ha a a quaiis i h sam f fam (lab fam. ( S (, ( ( S (, ( P(/ P( / pow i pow aia b a a hag a ( a ui. soli agl ui soli agl P( P( I boh a, is wih sp o h hag. P( P( 6

70 4. Agula Disibuio of aiaio (oiu wi ( S (, ( ( S (, ( P(/ P( / P( P( ( S (, ( β( ( β( ( β β 4 b (4.5 β P( 5 β β P(, (4.8 4 β wh, β, β a o b alua a h a im. (4.8 gis P( h pow aia io a ui soli agl i h iio of i ms of h hag a isaaous β a β of h pail Agula Disibuio of aiaio (oiu Cas : β β P( β β wi (4.8: 4 β 5,β β β P ( si (4.9 ( si 5 β β 4 os P P( 6 ag wih (4.6 ( (4.4 a (4.7 Fo, (4.9 us o Lamo's sul (4., wih h o aiaio pakig a 9. Bu as, h agula isibuio is ipp fowa mo a mo a iass i magiu, wih h maimum isi a os 5 (4.4 8 ma 4. Agula Disibuio of aiaio (oiu As, w ha. H, os ( P( si os lim (4. 4 [agl of maimum isi] (4.4 ma ( P m oo ma P( squa agl ( Agula Disibuio of aiaio (oiu P ( β β Cas : β β. I, (4.8 4 β 5 β, β l sios sisi os P ( si os 4 [ ] (4.44 os os 4 4 ( 4 [ag wih (4.] P P ( p (4.47 m P( 6 4 os lim (4.45 ( ( ma [agl of maimum isi] aow o lik a sahligh 4

71 4.4 aiaio mi b a Chag i Abia, ml laiisi i Moio I Ss. 4. a 4., w ami h aiaio poblm fom h iwpoi of h hag pail a psss h aia pow i ms of h isaaous β a β of h pail. Fom h o, w will swih ou iwpoi o h obs. Th mphasis will also b swih fom h pow of aiaio o h fqu spum of h sigal i a h obsaio poi. To fi h spum, w o fis kow h im hiso of h obs aiaio. H, w a o log sik o isaaous quaiis as i Ss. 4. a 4.. W mus ow follow h pail's obi. As h pail als alog is obi, i oiuousl aias owa h obs. A Foui asfom of h im-p sigal i h als is spal os. Wwillbisoli is i ppiula alaio ( β β. Th aso is as follows aiaio mi b a Chag wih γ>>(oiu p m p m P (, fo β β (4.7 wi P (, fo β β (4.47 ββ Pβ β whih implis P fo h sam alaig fo. H, fo a hag wih i abia moio, w ma gl P ( u o β β a osi ol P ( u o β β. Th isaaous s aius of uau a b pss pssims of h ppiula ompo of h alaio ( as follows. m F m Fo alaio o, h ffi mass is m. S (4.48 lu os, Ch., q. ( aiaio mi b a Chag wih γ>>(oiu Th Spal S l Wih fo β β : obs Agula isibuio of aiaio:. Th obs is illumia b ligh mi i a a of lgh, ospoig o a (a im ial of missio. I h ial, h fo g of h puls als a isa D, whil h a g of puls is bhi h fo g b a isa ( L D ( Puls uaio (o h obs: T L/ A boa spum agig fom a up o a iial fqu of T L, (4.5 wh is h maimum fqu of appiabl aiaio aiaio mi b a Chag wih γ>>(oiu Shoo aiaio- A Qualiai Disussio : If h hag is i iula moio wih oaio fqu, h a T L / Th pulss ou a h ( L P ( obsaio poi a gula T L/ ials of ( L / T Pulss of shoo ampl : Coll GV shoo aiaio popagaig 4 aiall ouwa 6 /s L 9 L.4 /s (6 kv -as 44

72 4.4 aiaio mi b a Chag wih γ>>(oiu Disussio: I (4.5,, h iial fqu (maimum fqu of appiabl aiaio sals as, whih plais h ml high fqu aiaio fom a shoo. Th fao is u o h sho uaio of h plus s b h obs. Th puls is sho b wo ffs:. Baus h agula wih (/ of h aiaio is aow, ol h aiaio mi b a lo o a a of flgh ( / a ah h obs. obs Thus, o h lo, h missio ial is.. Th lo is "hasig" is aiaio. H, o h obs, h i puls lgh is o. Isa, i is ompss b a fao of ( ( ( β ff is ploi i a i all h f lo las (FL aiaio mi b a Chag wih γ>>(oiu ampl : As a paial ampl of fh puls uaio o h obs, osi agai h Coll GV shoo, fo whih w ha 9.4 / s. Si T, h puls uaio T of a sigl lo is iibl sho, T 4. s. This plais h boa spum. How, h aual plus i a shoo os o om fom a sigl lo, bu fom a lo buh of fii lgh (piall a fw mm. los i h buh aia iohl. So h spum of h buh is h sam as ha of a sigl lo, bu h puls uaio ( quals h passag im of h lo buh ( buh lgh/. Fo ampl, fo a buh lgh of 6 mm, w ha s aiaio mi b a Chag wih γ>>(oiu Th Shoo as a Ligh Sou : Th shoo mis is aiaio wih a boa fqu spum i a bam of ml small agula spa (/. I is a uiqu sah ool a a also b us fo mio-fabiaio a oh appliaios. Th phoo blow shows h ligh sou faili a h Naioal Shoo aiaio sah C (NSC i Taiwa. 4.4 aiaio mi b a Chag wih γ>>(oiu lo buhs a fis ala o a g of f. GV i h boos shoo, a h s o h soag ig (also a shoo, wh h g is maiai ia GV. GV whil h los poi shoo aiaio o uss aou h ig. Th los a pow b miowas fom h F ssms. soag ig soag ig boos shoo F ssms boos shoo F ssm 47 48

73 6 kw klso oaial li 4.4 aiaio mi b a Chag wih γ>>(oiu pow suppl Th F ssm low ll F ssms oolig ssms lo buhs 5 MH miowa ouplig suu -fil lis alaio ai 4.4 aiaio mi b a Chag wih γ>>(oiu Phoo of h NSC boos shoo showig som k ompos of h alao oaial asmissio li alaio ai mag iulao 49 auum pump lo hal (i high auum aiaio mi b a Chag wih γ>>(oiu sah saios aou h NSC soag ig 4.7 Uulaos a Wiggls fo Shoo Ligh Sous Th boa spum of aiaio mi b laiisi los b b h magi fils of shoo soag igs pois a usful sou of gi phoos. As appliaio gw, h fo bigh sous wih h aiaio mo oa i fqu l o h magi "isio is" all wiggls a uulaos o b pla i h shoo ig. Th magi popis of hs is aus h los o ugo spial ilmoio ha suls i h oaio of h aiaio i io a muh mo moohomai spum o sis of spa paks. 5 5