Notes follow and parts taken from sources in Bibliography

Transcription

1 PHYS 33 Noes follow and pars aken from soures in ibliograph leromoie Fore To begin suding elerodnamis, we firs look a he onneion beween fields and urrens. We an wrie Ohm s law, whih ou hae seen in inroduor phsis as V=IR, in he form f where is he usual urren densi, f is he fore per harge, and is known as he onduii of he maerial. The onduii is er losel relaed o he resisii ha ou saw in, as he wo are reiproals of one anoher. The mos general elerodnami form of f is gien b he Loren fore law. Diiding b he harge q gies f In he ase of a harge (like an eleron) moing hrough a wire, we know he eloi of he harge, known as is drif eloi, is in general er small less han m/s. Unless here is a huge eernal magnei field, he erm will probabl be er small ompared o he erm. We ould hen wrie whih is a more familiar form of Ohm s law, desribing he response of he harges in a maerial o an applied eleri field. If we eamine his relaionship for a wire wih a onsan ross-seional area A and a onsan onduii, we will quikl see (as shown in our book) ha he urren I in he wire is proporional o he poenial differene V beween he ends, and ha he onsan of proporionali is /R. If is uniform and he urren is sead (no hanging in ime), aking he diergene of he equaion aboe leads o where he lef hand side is rue beause he onduii is uniform and herefore here is no wa for o hange in he maerial (meaning di() = ). The resul on he righ hand side, ha he diergene of is also ero, means he ne harge densi is ero. This is reasonable, sine he oal harge in he posiie laie is eal balaned b he harge of he negaie elerons (boh bound and free). Noie ha he means we are alking abou wha happens inside he olume of he maerial; here an sill be eess harge on is

2 PHYS 33 surfae, sine we haen said anhing abou ha. The harge would hae o be added b some oher means, of ourse jus puing a wire in an elerial irui doesn hange is oal harge densi. Your book breaks he fore per uni harge f down ino he sum of wo erms: fs, he fore (per uni harge) proided b he baer or whaeer is aing o power he irui, and, he elerosai fore (per uni harge) beween he harges in he irui. If we look a he line inegral of f oer he omplee irui, we ge d f f s d The wo inegrals are equal beause he inegral of oer a losed loop is ero in elerosais (a onsequene of he url of being ero). These inegrals are equal o he eleromoie fore, or MF, whih ou hae known sine is no a fore, bu raher an eleri poenial or olage. If a baer is ideal (no inernal resisane), hen f = and = - fs. The inegral on a pah ending in he wo erminals a and b is V ab b a d b a f s d bu beause fs is ero ouside of he baer, we an hange ha inegral ino one oer a losed loop b a f s d f s d The purpose of esablishing hese formulas is o eend he problem from baeries o generaors. Here, harges are phsiall moed when a wire loop is moed inside a magnei field. In a resul illusraed in he Griffihs book, imagine pulling he loop shown below ou of a magnei field poining ino he page. The free harges in he lef side of he loop suddenl hae a eloi o he righ. The Loren fore on (posiie) harges moing o he righ in an in-he-page magnei field is direed upwards, giing a lokwise urren. F In he lef hand segmen of he loop, he harges now hae omponens of eloi boh o he righ (aused b whaeer pulled he loop o he righ) and upward (aused b he Loren

3 PHYS 33 fore). If he upward eloi is u and he righ-going eloi is, we an all heir resulan eor w. The magnei fore is auall perpendiular o w, meaning i has a omponen parallel o u and a omponen parallel o. Now work an be done! The eor diagram shown in he book is reprodued below: u w u fmag fpull Inegraing he do produ of he fore per harge fpull (equal in magniude o u ) and he elemen of pah lengh dl aken b a harge moing up he lef side of he loop a disane h/os (where h is he lengh of he lef side of he loop) gies f pull h d u sin h os whih mahes he resul found in for he MF produed moing a loop hrough a field using =-d/d. To onfirm his, we alulae he magnei flu in he loop o be =h where is he disane from he lef edge of he loop o he poin where he field is ero. eause and h don hange oer ime as he loop is remoed from he field, he MF is =-d/d = -h d/d = -h. This is he same magniude as wha we found in he preious paragraph for work done per uni harge. As for he direion, we know ha Len s law means ha he flu produed b he urren generaed in he loop as o ouner he hange in flu ha reaed i. Remoing he loop dereases he ino he page flu, so we need a urren ha generaes a field ino he page. he righ hand rule, ha urren has o be lokwise (as we found earlier). Induion Farada demonsraed ha hanging he magnei flu hrough a wire loop would produe an MF ha would drie a urren in he loop. One of relaii s ke feaures is ha i doesn maer if a magne is held fied while ou moe he loop or if he loop is fied and he magne is moed he onl imporan ariable is he relaie eloi of he wo. We an wrie his resul mahemaiall as 3

4 PHYS 33 d d d whih we an use Sokes heorem o wrie as Noie ha we had a differen epression for he line inegral of around a losed loop in elerosais i was ero. Tha s he basis of Kirhoff s olage rule ha he sum of poenial jumps and drops around a irui is ero. Of ourse, one wa o epress he fa ha ou are limiing ourself o suding elerosais is o sa ha he url of is ero. We an now see ha here are wo differen soures for eleri fields: harges, and hanging magnei fields. The parallel is eplored in he book b noiing ha, if here are no harges around, we hae and Tha allows us o use Sokes heorem in he oher direion and find ha d d d d Induane Sine we now know hanging magnei flues will indue eleri fields ha drie urrens, i shouldn be er surprising ha, gien wo loops of wire near one anoher, hanging he urren in one loop will indue a urren in he oher loop. This ineraion beween he loops is alled induane. eause he magnei field a loop due o loop (all i ) depends on he urren in loop, we an wrie he following formula for he flu hrough loop (): da We an more direl onne he flu hrough loop wih he urren in loop b I en 4

5 PHYS 33 M I where M is known as he muual induane. I should be lear from wha we know abou he direionali of magnei fields and heir rae of derease wih disane ha M is a funion of he geomer of he siuaion where are he loops, wha are heir shapes, how are he oriened relaie o one anoher, and how far are he from eah oher? A formula for alulaing M relies on wriing in erms of he eor poenial A and hen I d d d da A d d M 4 4 Noie ha his las epression (for M) is smmeri in erms of pahs around he loops. For ha reason, M = M so we an drop he subsrips. If he urren in loop hanges, a urren will be indued in loop. We an wrie his as d d di M d I migh now seem obious ha he hanging urren in loop will indue a urren in loop, bu here will also be an effe on loop iself from he hange in I. This is known as selfinduane and is usuall represened as L raher han M. The flu in a loop due o is own urren is and he MF produed is L I di L d and is also known as bak MF. The work done agains his bak MF o esablish a urren is sored as is magnei field. Traeling agains he MF requires ou o do work equal o - per harge. If we use he preious formula o define and onsider moing harges a a rae I, we ge dw d I L I di d W L I 5

6 PHYS 33 whih should be familiar from. We an wrie his sored energ in a more general form using he fa ha he flu is equal o he line inegral of he eor poenial around a losed loop, as well as being equal o LI. We an hen wrie L I A d W I A d A I d A d for a general olume urren disribuion. Using Ampere s law and inegraion b pars, we an find W d A d d A da and he surfae S bounds he olume V. The ineresing par abou his form of he energ is ha he surfae an be hosen o be er large; his makes he olume inegral larger, bu beause he ross produ of A and is inegraed oer his (now er large) surfae, and beause he drop off wih disane, he beome less imporan. In he limi of V he surfae inegral disappears and we ge V S V W all spae d Mawell s quaions Gahering ogeher all of our fundamenal equaions of elerodnamis in one plae, we an wrie hem as The indiaion ha hese equaions are no omplee is ha we an look a he boom wo equaions and alulae he diergene of eah. For he url of, we ge di(url ) = ideniall and he righ side is similarl ero sine he ime deriaie of di is also ero. The problem ours when we ake he diergene of he url of : 6

7 PHYS 33 The lef side is ero b he ideniies of eor alulus, bu he righ side is no neessaril ero. A sead urren has a ero diergene, bu a general urren migh no. The pial wa o illusrae he inonsisen is o realie ha in Ampere s law d da he line inegral of is aken around a losed loop ha bounds some surfae, and he enlosed urren = urren moing hrough ha surfae. We end o pik he surfae as jus being he irle bounded b he Amperian loop, bu ha s no a requiremen. We ould look a a balloon he lip of he balloon is he Amperian loop, and he surfae ould be he smalles irle bounded b he loop, bu i ould also be he enire inflaed balloon s surfae. In he image below, he red irle is he Amperian loop, he purple disk is he surfae we ordinaril use o inegrae (i.e., we look a he urren flowing aross he purple disk) and he green surfae is an equall-alid surfae bounded b he same Amperian loop. Now imagine a simple irui onsising of a baer and a parallel-plae apaior. If we sreh he balloon around one of he plaes, so i separaes he wo pars of he apaior, we ge er differen answers o he inegral of oer he area depending on whih surfae (green or purple) we hoose. Choosing he purple surfae, our inegral for he do produ of and da would be jus he urren flowing in he irui (all i I). On he oher hand, if we pik he green surfae, here is no urren rossing i, so he inegral for he do produ of and da would anish. This is a problem, sine from our er firs eperiene wih Gauss law or Ampere s law, we hae known ha our pariular hoie for he arbirar surfae of inegraion an hae phsial signifiane. This means we hae o modif Ampere s law so ha we ge he same answer regardless of whih surfae we use. The use of he law of urren onseraion larifies he hange we need: We an fi Ampere s law (whih is wha Mawell did) b adding he erm on he righ o our urren, like his: 7

8 PHYS 33 8 I s lear from he form of he righ side ha he new erm has o hae he dimensions of muliplied b a olume urren densi. For his reason, / is alled he displaemen urren and represened b d. Our soluion o he preious problem is now lear. Inside he plaes, he field saisfies a d I d A I A Q en For he purple surfae, he righmos erm in he final epression disappears sine disappears, bu Ien suries. For he larger green surfae, Ien is ero sine no harge passes hrough i bu he harging apaior plae means / is no ero on ha surfae. The new (omplee) Mawell s equaions an be wrien in differenial form as Griffihs groups hese equaions wih he Loren fore law F= q ( + ) and rewries hem o pu he fields on one side of eah equaion and he harges and urrens ha generae hem on he oher side. If here are no harges or urrens around (as in a auum), we would ge

9 PHYS 33 The and equaions would be more ompleel smmeri if magnei harges (i.e., monopoles) eised. We ould inlude boh magnei harges and magnei urrens (he moemen of hose harges) in Mawell s equaions b wriing hem as e m m e Taking he diergene of he lower wo (url) equaions, we would ge m m e e Mawell s quaions in Maer From our preious work, we know ha eernal eleri and magnei fields an be epeed o indue polariaion and magneiaion of maer. The polariaion as like a olume or surfae harge densi (b or b) and he magneiaion as like a olume or surfae urren densi (b or Kb). Speifiall, we an wrie P b b M I urns ou ha, o presere he onseraion of harge, a hange in polariaion will require a polariaion urren p o eis. This an be shown b eamining he effe of a hange in polariaion. If P inreases, he surfae harge densiies a eah end of he maerial are epeed o inrease (wih opposie signs, of ourse). Assume a polariaion P auses a surfae harge densi b a one end and -b a he oher end. When i inreases, he surfae harges will also inrease: b di da P da p P 9

10 PHYS 33 This polariaion urren is ompleel unrelaed o he bound urren produed b he maerial s response o an eernal magnei field, and represens onl he hange in he (mirosopi) posiions of he posiie and negaie harge eners. To hek ha harge is onsered, we look a he diergene of his new urren: P P b p as epeed. So, he harge densi sill has wo separae pars, bu he urren densi now has hree. P f b f P M f p f b We an rewrie some of our earlier epressions o inlude he new erm: P D for D P f f and M H where D H P M f f for a final form of f f D H D These are made slighl more ompa if we assume linear media so ha

11 PHYS 33 P e M m H We an also reognie ha we hae preiousl alled D/ he displaemen urren d. oundar Condiions The boundar ondiions on he eleromagnei fields are more eas o derie using he inegral forms of Mawell s equaions: S P D da d Q free d d, en da S da S P H d I free, en d d S D da Using he firs equaion and hoosing our surfae o be a linder wih is ais of smmer perpendiular o he boundar beween wo regions and haing infiniesimal heigh, we find D A D A f A In he same irumsane, he equaion for gies D D For he wo line inegrals, we reangular loops whose long sides are parallel o he boundar and on opposie sides of i, and whose shor sides are infiniesimal (see below). f Inegraing oer his loop, we ge da d S sine a loop of infiniesimal area will onain no flu. We an hen wrie d

12 PHYS 33 Repea he proedure for he line inegral of H o find H H I free, en Onl a surfae urren densi ould flow hrough his loop, so we are lef wih H H K free nˆ If here are no free surfae urrens or harges, his an be summaried as Conseraion of Charge Alhough we hae known abou global onseraion of harge for some ime (i.e., he oal harge of all he pariles in he unierse is onsan), here is a more resriie form of his idea. The global onseraion of harge would be saisfied if + C of harge appeared here and C of harge appeared on he Moon, bu i isn allowed phsiall. Charge has o rael from one plae o he ne, and ha inoles a urren moing hrough a boundar surfae S. We an wrie his as d Q d S da V d V d whih we an simplif o showing ha harge won disappear from one plae and appear a anoher; i will onl rael beween he wo loaions. We an deelop a similar equaion for energ. The energies sored in eleri and magnei fields are

13 PHYS 33 3 d W d W m e whih we an ombine o ge he oal energ. Relaed o his is he work done b he Loren fore: d q d q d F Using q = d and = we an find he work done on all harges in he olume V V d d d W This means is he power per olume. We an also wrie his as whih we an rewrie using and see ha We hen use so we an wrie

14 PHYS 33 and hen dw d d d V d S da The olume inegral gies he oal energ sored in he fields while he surfae inegral shows he energ flowing hrough he surfae. The Poning eor S is his power per uni area, and i is equal o S Idenifing Uem as he oal energ sored in he and fields, we an wrie he power as dw d du d em S Wriing in erms of energ densi, we an break i ino mehanial and eleromagnei pars umeh and uem, and hen use he diergene heorem on he inegral of S oer he area o ge or d d V S da u u d S da S meh em S u u S meh Noie he resemblane beween his and he oninui equaion for harge i s he same idea, onl for energ. em Momenum An imporan resul from elerodnamis is he fa ha he fields hemseles arr momenum. We an see ha his mus be he ase beause we an arrange a siuaion where i appears ha he Newon s hird law is iolaed. The eample in our book is if a poin harge moes in he direion and a nearb poin harge is moing in he direion. As he approah he origin, we see ha he magnei field eah reaes in he V d 4

15 PHYS 33 neighborhood of he oher leads o Loren fores ha are no along he same line. Reproduing he diagram in our book, we see hese fores and fields below: q X Fm Fm q The wo magnei fores are no ani-parallel, meaning Newon s hird law is no working here. Tha s wh he fields hae o arr momenum. A beer wa o epress he eleromagnei fields and make he disribuion of energ and momenum learer is o moe from he eor epressions o he Mawell sress ensor, also alled he energmomenum ensor. To begin, he eleromagnei fores on he harges in a olume V are F V d d We hen use Mawell s equaions o wrie he fore per olume in erms of he fields as f V The ime deriaie of S and Farada s law ( = -/) are ombined o le us wrie making our fore densi equal o f This ompliaed epression will eenuall be simpler if we wrie i as smmeriall as we an. We an do his b adding a erm () whih will be ero anwa. The eor ideniies in he fron of he book le us rewrie he gradien of boh and o ge 5

16 PHYS 33 6 f Now we make i look muh simpler wih he sress ensor. us as a eor has one inde (usuall ranging oer he direions,, and in hree dimensions), a ensor (of rank wo, like his one) will hae wo. I is defined here as T j i j i j i j i j i Here, we hae nine omponens sine eah inde (i or j) an range oer hree oordinaes. Taking he diergene of his ensor will urn i ino a eor (noie ha his is no oo unepeed, sine aking he diergene of a eor will drop i b one rank and produe a salar, so reduing our rank-wo ensor o a eor is reasonable). The omponens of his eor are found as shown below: T j j j j j j j This is wha will finall simplif our fore densi formula: S T f or, if we wan o inegrae o ge he oal fore, we hae V S S d d d da T F To ge bak o he momenum quesion ha sared his, we use he fa ha he basi definiion of fore is he ime deriaie of momenum. We an all his he mehanial momenum of he pariles inoled. Using our preious epression for fore, we ge V S meh d S d d da T d p d The wo erms on he righ hae differen inerpreaions: he olume inegral gies he momenum sored in he and fields, and he surfae inegral desribes he momenum

17 PHYS 33 flowing ino he surfae S per uni ime. We an hen wrie he eleromagnei momenum (he olume inegral) as p em V S d The formula aboe an quikl be adaped o a momenum densi for he fields: em S The ineresing hing abou his appears when we rewrie he fore epression T em meh Noie he similari beween his epression and boh he oninui epression for onseraion of harge and he epression for he diergene of S. This means ha jus as S is boh he power per area and momenum per olume, he ensor T is boh fore per area and momenum urren densi (momenum urren = flow of momenum). We ould use his eleromagnei momenum o alulae he eleromagnei angular momenum. We ge r em em r The Theor of Relaii To begin he sud of relaii, we will use he wo basi ideas proposed b Alber insein in he earl par of he h enur. These posulaes are gien wihou proof; relaii is esseniall wha ou ge if ou assume hese wo hings are rue and hen ask wha he mahemaial and phsial onsequenes of hem would be. ) The laws of phsis are he same for all inerial obserers Inerial obserers are hose who are no aeleraed in an wa. While relaii ells us ha here is no suh hing as an absolue eloi (onl he relaie eloi beween wo objes makes sense), aeleraion an be deermined easil. If ou are in a small, windowless room, ou ould deermine our aeleraion b holding on o a wall while releasing a mass. If he room aeleraes, ou will moe wih i (sine ou re holding he wall) and he mass will no. You ll see i fall o he floor, or eiling, or a wall. If our room is rul inerial, he mass will jus hang here in he air foreer. Someone else in a similar room ha is also inerial will 7

18 PHYS 33 ge he same resuls from phsis eperimens ha ou do, regardless of our relaie eloiies. We frequenl alk abou an obserer s frame, whih is shor for inerial frame. Two unaeleraed people moing relaie o one anoher are in wo differen inerial frames. ) All inerial obserers measure he same alue for he speed of ligh in a auum This is wha ould be hough of as he ra par of relaii: he speed we measure for anhing we hae eperiene wih depends on our eloi relaie o i. A ar moing a 7 mph (relaie o he road) on he opposie side of he inersae seems o be moing er quikl o us if we re moing W a 7 mph. If here s a ar in he ne lane, whih is also moing 7 mph W, i seems o hae no eloi. If ou an hrow a baseball a 6 mph and ou re hanging ou of a ar window while he ar is moing a 7 mph when ou hrow i, ou know ha is speed will be 3 mph relaie o he ground. Wha he seond posulae ells us is ha we an measure he speed of he ligh leaing a flashligh parked on he road, and we ll ge eal he same speed for ha ligh as we would ge if we measured i from headlighs on a ar moing a 3 mph. Ligh is er differen from eerhing else! This migh be eas enough o aep, bu he onsequenes of hese wo posulaes lead o some hard-o-beliee resuls. ens In Newonian phsis, we an assign a posiion (3 numbers) and a ime ( number) o eerhing ha happens. Newon belieed ha ime was absolue: here ould be one maser lok in he unierse and all oher loks ould be snhronied wih i sine all people would eperiene ime he same wa. In ha sense, ime was seen as ompleel differen from spae, jus as we would sa ha he ehange rae beween US and Canadian dollars has nohing o do wih spae. I s he same eerwhere and posiion does no ener ino i a all. Relaii fores us o see he four numbers for spaial oordinaes and ime as poins in a four dimensional spaeime. When somehing happens (a flashligh is urned on, a ball is dropped, wo ars rash, e.) in his 4-D spaeime, we all i an een and gie is four oordinaes. eause i s er hard (or impossible) for humans o imagine 4 dimensions, we ll someimes drop one or wo of he spae dimensions and jus use he remaining spae oordinae(s) and he ime oordinae. This will presere he essenial pars of relaii wihou ausing headahes. Wha ma sound like nipiking is imporan when we alk abou eens: we alk abou measuring hings raher han seeing hings beause we know ligh akes a finie amoun of ime o ge beween wo plaes. You e probabl noied a similar effe wih sound ou an wah a flash of lighning and oun he seonds unil ou hear hunder, and ha ells ou how far awa he flash was. The ime when ou heard he sound is differen from he ime he sound was auall reaed. We don usuall noie for ordinar arh disanes, bu he ime beween somehing happening and he ime ou see i happen is no ero. To fi his problem, we ll assume ha ou hae an infinie number of helpers saioned a all poins in spae and no moing relaie o ou (i.e., in our inerial frame). There will alwas 8

19 PHYS 33 be one of hem posiioned so ha his/her disane o an een will be esseniall ero, so he/she will see hings as he happen, wihou dela, and reord he oordinaes whih are hen sen o ou. This idea allows us o measure phsial reali insead of human perepion. The Ligh Clok Imagine a simple lok whih keeps ime b leing a ligh pulse boune bak and forh beween wo mirrors whih are a disane D awa from one anoher. If he spae beween he mirrors is a auum, i s eas o see ha eah round rip for he ligh pulse will ake a ime D = D/. Now, imagine a lok like his on a roke moing pas ou a (high) speed. You ll see he piure below (noe ha he roke is moing perpendiular o he long ais of he lok) Noie ha ou will see he ligh pulse moing boh up and down and lef o righ. Wha we would hae said before relaii is ha he ligh pulse speeds up i has a omponen of eloi = in he up-down direion, and a omponen = in he lef-righ direion. Then, we d use he Phagorean formula o find he magniude of he eloi eor. This is how hings work (or seem o) in he low-speed phsis we re used o. insein sas ha he hing ha doesn hange is reall he speed of ligh. Therefore, he ligh moing along he diagonal line (as seen b ou) mus sill hae he speed ha ou d 9

20 PHYS 33 measure for i if he roke were saionar. This will ause some weird effes. Firs, ou ll sill see he disane beween he mirrors o be D. You ll also measure he lower leg of he riangle (whih has been drawn longer han he updown leg for lari) as haing a lengh ( )/, meaning he lengh of he pah ligh akes is s D The rael ime for ligh will be as far as ou re onerned, so ou ge D or D We re going o ge a disagreemen on he ime aken for ligh o make his rip! A person on he roke (who onl sees he ligh going up and down) will sa ha he oal disane raeled b he ligh was D and i had o be moing a, so he ime aken was = D/ ( means he ime as measured b someone who is no moing relaie o he lok, also known as being in he lok s res frame. This ime is alled he proper ime beause i s ime measured in he res frame of he lok). We an rewrie he ime ou measure () in erms of he ime measured b he roke s passenger ( ) as

21 PHYS 33 If ou re moing relaie o he roke lok, ou will hink i s running slow while an oupan of he roke will be onined i s running a he righ rae. For eample, if =.9 (abou 7,, m/s), seonds on he roke s lok will seem o ake se.9.94 se on our lok! A er imporan poin here is ha his does no represen a problem wih our lok s design or manufaure. Tha s wh we inrodued suh a simple lok in he firs plae. This means ha ime goes slower on he roke as measured b someone on arh. This is alled ime dilaion. The person on he roke will be onined eerhing is OK and ime is moing normall for him/her. In oher words, ou an use his effe o eend our life in he sense of doing more wih i. If ou like o read books, ou won ge an more of hem read b hopping on a high-speed roke han ou would b saing a home. The differene is ha ou ould ride a roke awa from arh for 6 monhs (measured on board), urn around and ome bak, and housands of ears would hae passed on arh if our roke moed quikl enough! Wh don we eer noie his effe? Calulae i for somehing moing a abou 8 km/seond (abou arh s esape eloi, and muh faser han non-asronaus rael). We haen speified a ime, bu we an see ha he imporan hing is he raio of he wo imes he ime measured on he saionar lok diided b he ime measured on he roke. This faor will appear so ofen in relaii ha we frequenl jus refer o i as gamma or, and he formula for i is For he 8 km/s roke, we ge a of.36. In oher words, we ould rael a ha speed for a ear as measured on arh and our lok would onl be. seonds behind one ha saed on arh!! We don ommonl noie his effe beause we rarel enouner speeds whih are een lose o he speed of ligh. In fa, if we ould ge o % of he speed of ligh (around he equaor in abou a seond!), would sill be onl.5. Tha s suh a in hange i migh sill be hard o noie.

22 PHYS 33 This effe has been measured man imes. One of he mos dire was o do i inoles he lifeime of a subaomi parile alled a muon. This parile is unsable and deas ino an eleron or posiron and some oher hings wih a half-life of.56 miroseonds. In oher words, if ou hae a jar full of, muons, ou ll hae abou 5, afer.56 miroseonds,,5 afer 3. miroseonds, e. These pariles are being produed high in arh s amosphere all he ime. We an measure heir speeds as he approah he ground, and he are piall moing a >99% of he speed of ligh, as measured b someone on arh. Sine somehing moing a.998 will rael abou 467 m in.56 miroseonds, we would epe o see half as man muons making i o he surfae of he arh as we would see on a small hill 467 m aboe sea leel. Wha we auall find, when we measure, is ha far fewer han 5% of he muons sill around a 467 m disappear b he ime he reah he ground. Wh are he siking around? I s beause for hese muons is in he neighborhood of 6! A lok raeling wih he muons would onl moe abou nanoseonds in he rip from 467 m o he ground, so i s no surprise ha more han half of hem make i down o sea leel. An ineresing quesion here is wha ou would see if ou ould rael wih he muons raher han siing here on he arh. How would ou eplain he fa ha he lie long enough o rael hrough so muh of he arh s amosphere? The rik here is ha someone in he res frame of he muons would laim ha he arh is rushing owards hem a.998! The will beliee heir loks are working fine, and he ll agree wih people on arh abou he relaie speed beween he arh and he muons. Sine we re basiall reling on he equaion = here, he onl hing ha an hange from he muons perspeie is. In oher words, someone in he muon frame would laim ha he arh s amosphere is onl abou /6 h as hik as we sa i is. Therefore, ha person would no be surprised a he number of muons whih made i hrough he (muh hinner, in heir opinion) amosphere. This brings us o he ne odd prediion of relaii, lengh onraion. The fa ha wo obserers will disagree on he rae of he passage of ime will also mean ha he will disagree on he ime ordering of some hings! For eample, imagine ou re sanding on he raks while a (er fas) rain goes b a.8. If someone on he rain in he middle of a ar srikes a mah, he people on he rain will laim ha he ligh reahed he opposie ends of he ar a he same ime (ha s wha being in he middle of he ar would mean). Wha will ou obsere on he raks? Lengh Conraion As we saw in he muon ase, moing obserers disagree on boh elapsed imes and disanes. If we see he lok in he roke frame as moing slowl, passengers on he roke an onl eplain hings if he beliee he unierse rushing pas hem is onraed in he direion of moion. The lengh onraion faor is he same one ha gies us ime dilaion. The onl differene is ha, while he proper ime beween wo eens is alwas he shores ime measured b an obserer, he proper lengh of an obje is alwas he longes disane measured b an obserer. Again, he proper ime beween wo eens is he ime as

23 PHYS 33 measured b someone who sees he wo eens happening in he same plae, while he proper lengh of he disane beween wo objes is he lengh as measured b an obserer who sees hem happen a he same ime. For he onneion beween he lengh measured b a moing obserer L and he proper lengh L, our formula is L L or L L The Ineral The ke onep in relaii is he inariane of he ineral. To undersand his, we need o sop hinking of ime and spae as ompleel separae hings and realie ha he are linked. If spae and ime were ompleel disin (Newonian phsis), eerone would measure he same alue for he ime inerals beween wo eens, so he separaion in ime of wo eens would be an inarian (eerone agrees on i). Also, while differen obserers migh disagree abou he omponens of a lengh due o he differen orienaions of heir oordinae ssems, he would agree on he oal lengh iself. In oher words, he d disagree abou he omponens of a eor bu would agree abou is magniude. The big differene now is ha he hing ha we all hae o agree on is he speed of ligh. Imagine a pulse of ligh as a sphere ha epands in ime (he piures ou migh hae seen of H-bomb eplosions are a good model here). The radius of he sphere r will grow in ime aording o he simple formula r =. Going bak o Caresian oordinaes, wha is he alue of r? We jus need he Phagorean heorem in 3-D, whih ells us r We an square boh sides of r = and subsiue our epression for r aboe and ge These oordinaes are reall oordinae differenes (i.e., we wrie bu we undersand ha o mean ( b a )). Le s pu some smbols in here o make i a lile more obious: We will define a new quani alled he ineral b puing all of his suff on one side of he equaion ( ligh) 3

24 PHYS 33 (Auall, his is he square of he ineral, so is square roo would be he ineral iself.) This four-omponen obje is alled he ineral, and we know ha i has o equal ero for a ligh pulse. For ha reason, we all a ero ineral a ligh-like ineral. No all inerals are ligh-like. Wha if we wan o measure a sik whih is a res along he ais? We jus look a eah end of he sik, and find he hange in oordinaes beween lef and righ ends. If he sik is along he ais, righ lef =. Similarl, righ lef =. The hing isn moing, so we an measure he separaion in ime of he wo ends we ge anoher ero: righ lef =. The onl hing ha s no ero is he differene beween he lef and righ ends. Tha should jus be he lengh (proper lengh, sine i s a res in our frame) of he sik. righ lef = L. Now, our ineral (we ll use he leer s as he smbol for he ineral) beomes L * s or s s L Seems like an awful lo of work jus o find ha he ineral = proper lengh of he sik. We an find a similar resul if we measure he ineral beween wo hings happening a he same plae a differen imes. We all i he proper ime hen (imagine wo amera flashes happening a he origin of our oordinae ssem seond apar in ime - righ lef =, righ lef =, righ lef =, bu = se. so we ge s = - ( se) ). A big differene beween spae-ime inerals and 3-D lenghs is ha lenghs are eiher ero or posiie. We neer measure a disane beween wo hings as 3 meers. The squares of inerals an be negaie, posiie, or ero. For ero, we all he ineral ligh-like, bu for a posiie number, we hae a spaelike ineral. A spaelike ineral beween wo eens (like he lengh of our rod when boh ends are measured a he same ime) means he an be ausall relaed. Le s sa a lok in our house is snhronied wih a lok on Mars (when i s 5 ligh-minues awa from arh, for eample) and ou press a buon on our remoe onrol a : PM. If a bomb eplodes on Mars a :3 PM, here s no wa ou ould hae aused i b pressing he buon beause he wo eens (buon press on arh, bomb eplosion on Mars) are separaed b a spaelike ineral. Our would be minues and he disane beween arh and Mars is (5 minues)*, so we ge our ineral from s 5min min min As an eample, le s look a he muon eperimen from earlier. Do he people on he ground agree wih he muons abou he ineral? We need more ea numbers for his. We ll keep our speed as.998 and he muon half-life as.56 miroseonds. Those of us on he arh will see he muon rael (.998 )*(.56 miroseonds) = m in.56 miroseonds. The square of he ineral will hen be 875. m. For he muon, he spaial separaion is ero (sine he muon sees boh spaial poins going b is origin of oordinaes) and he separaion in ime is.56 miroseonds / or.986 miroseonds. The square of he 4

25 PHYS 33 5 ineral as seen b he muon is also 875. m. The hing ha s ruial for us is ha we an mi measuremens from wo differen referene frames: if we ombine he.986 miroseonds measured b he muon wih he 467 m arhbound obserers see i rael, we ge nonsense. This isn surprising from our firs work wih eors, ou know ou ll ge nonsense if ou le 3 people wih differenl-oriened oordinae ssems eah gie ou one omponen for an ordinar eor. We an use he ineral o ge he Loren ransformaion equaions relaing disanes and imes measured in one inerial frame o hose measured in anoher inerial frame. If we assume he relaie moion beween he wo frames is along he ais (whih is parallel o he ais, he and aes are parallel, he and aes are parallel, and we sa ha he origins of eah frame oinide a = = ), we ge / / ' / ' ' ' ' ' where Noie ha he ime oordinae in one frame is found b miing he spae and ime oordinaes in he oher frame, and he same for he spae oordinaes. Time and spae are wrapped ogeher in a wa ha is no eas o separae, jus as he 3 perpendiular direions in Newonian spae are ogeher. us as we ould wrie a mari represening he wa and were relaed o and in a roaed oordinae ssem, we an wrie a mari onneing,,, and o,,, and. If we keep he arrangemen aboe (all relaie moion is along he ais = ais, e.), we will hae ha = and =. Finall, we add one more noaional hange: we will wrie / as (meaning is range is from ero o ). Doing ha, we an wrie our 4-D mari (all i L) as or L ' ' ' ' ' ' ' ' '

26 PHYS 33 Noie ha hanging o primed oordinaes from he unprimed ones jus inoles hanging he sign of b. In oher words, he eloi is relaie. There s one final hange we had o make ha ou ma hae noied. Time and disane are no measured in he same unis, so we hae o ome up wih a ombinaion ha makes sense dimensionall. From he firs da or wo of our firs phsis lass, ou should remember ha ime and disane are relaed b a eloi, and here s one obious eloi we an use here: he speed of ligh. This mehod allows us o wrie he er ompa form aboe for he ransformaion mari and i gies us a ommon uni of measure for all oordinaes. If we hoose he meer as ha ommon uni, i will orrespond o / seonds, or abou 3.33 nanoseonds. We ould hen alk abou a ommerial being m long (= 3 seonds), alhough i migh seem like an odd wa o alk abou a ime. If ou hink bak o he las ime ou ook a long drie, hough, ou ma er well hae used his same idea. If one of our passengers said How far are we from, ou ould hae said Abou an hour een hough ha is a ime and he quesion asked for a disane. erone in he ar would hae undersood ha he onersion faor in ha ase is he ar s speed. This is he same idea, bu here is less ambigui sine our ar an rael a a wide arie of speeds bu ligh in a auum has onl one speed. We ould find he meri of speial relaii b aking he ranspose of he mari (all i L) aboe, mulipling i b some 4 4 mari of ariables (a a 4, b b 4, 4, d d 4 if we wan), mulipling he resul b L again, and demanding ha our resul equal our mari of as, bs, s, and ds sine we know we should ge L T g L = g. A firs his looks like a problem of 4 4 = 6 unknowns, bu here are arious mahemaial and phsial properies ha he meri ensor will hae in a reasonable heor, and ha makes he work muh easier. Wha we ge a he end (whih ou an more easil erif han proe from srah) is ha he meri ensor g is g Noie how lose ha is o wha we would ge in 4-D Caresian oordinaes (whih would be he 4 4 ideni mari). The onl differene is in he sign of he erm assoiaed wih ime. Tha is he mos imporan negaie sign in all of relaiisi phsis! Remember ha he hing ha is independen of oordinae hoie in regular -D or 3-D geomer is he lengh of a eor (or he square of is lengh, whih is of ourse also inarian), whih is found using he Phagorean heorem and is in -D and ds 6

27 PHYS 33 ds in 3-D. The eension o four spaial dimensions would be obious, sine we d jus add he square of he differene in he new oordinae under he radial. We ould oninue on o do mah in 47 spaial dimensions if we waned o do so i migh be impossible o imagine or onsru, bu i wouldn be signifianl harder o alulae. Spae like his is known as ulidean spae, or fla spae. I s wha ou almos erainl worked wih elusiel in highshool geomer. Our formula for an inarian obje (hing ha is independen of oordinae hoie or independen of relaie eloi beween inerial obserers) is slighl differen in speial relaii: ds One of he hings done er ofen in phsis is o rewrie a new resul so ha is form mahes ha of an old resul. eause he minus sign aboe is so ruial and so ou of plae as ompared o ordinar ulidean spae, some (mosl older) books would wrie he ime oordinae in he form i. If wrien his wa, we an ake he ulidean approah and sa we re going o wrie he inarian hing as he square roo of he sum of he squares of all he oordinaes, sine squaring i will gie us he () we need anwa. In an een, he hange of sign means his is a fundamenall differen kind of onsruion han ulidean spae, known as Minkowski spae afer one of insein s mah professors who worked wih relaii shorl afer insein formulaed i. One of he ruial differenes in he wo kinds of spae is ha ulidean spae is posiie-definie. Tha means ha he disane beween wo poins (or he lengh of a eor) an onl be ero if eah of he omponens of lengh is indiiduall ero. A quik look a he formula for he ineral shows us ha ha s no he ase in Minkowski spae. We an hae posiie alues for all four of he oordinae hanges and sill ge an ineral of ero or een one less han ero. oh spaes are fla, howeer, meaning he re abou as similar as he an be oherwise. If ou deide o sud General Relaii a some laer poin, he big deal wih i will be ha he meri is no jus he Minkowski spae meri. You ll see ha i depends on he disribuion of maer and energ in he area. Relaiisi Addiion of Veloiies We are going o see ha one of he odd feaures of relaii is ha eloiies are no longer addiie. Le s sa we hae a person riding on a fla railroad ar moing a 3 m/s relaie o he ground who an hrow a baseball a m/s. If he person hrows he ball in he direion he railar is moing, Newonian phsis would ell us ha someone on he ground would measure a eloi of 5 m/s for he baseball (i.e., eloiies are addiie). 7

28 PHYS 33 8 How does i work in relaii? If i were he same, a person riding in a roke a.9 relaie o anoher obserer ould urn on he roke s headlighs, and he obserer would measure he ligh from hem o be moing a.9, bu ha iolaes one of our iniial assumpions abou relaii. We an find he eloi omposiion law b appling wo onseuie Loren ransformaions. Le s sa we hae our same roke aboe moing a.9 relaie o he same obserer, bu now he roke s pilo fires a missile (whih has a speed relaie o he roke of.95 ) in he direion he roke is moing. We an find he speed of he missile as measured b he obserer b performing a mari mulipliaion of he Loren mari wih =.9 b he Loren mari wih =.95. Puing in he aual numbers, and using he fa ha he wo faors would be.94 and 3.6, we ge Sine he firs row of he Loren mari is ( ), we an find he new from our produ mari b jus diiding he seond number in he firs row b he firs one, giing us =.9979 insead of.8. We ould find a general formula his wa wihou haing o use maries eer ime. To see wha he formula would be, we ll use for he speed (diided b ) in our firs mari and for he speed in he seond one. We would ge he produ mari below: ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ) ( ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ Again diiding he firs elemen in he seond olumn b he firs elemen in he firs olumn, we ge I seems lear now ha eloiies don jus ombine he same wa in relaiisi phsis ha he seem o in Newonian phsis. If ou re sanding on a flaar on a rain and ou hrow a baseball in he direion of he rain, he speed of he ball as measured b someone on he ground is simple enough o find. We would alulae i as

29 PHYS 33 all Ground all Train Train Ground (assuming ou re hrowing owards he fron of he rain). This an be righ in relaiisi phsis. We ould replae he ball wih a flashligh, and his formula would ell us ha he person on he rain migh measure for he speed of ligh, bu someone on he ground would ge +. This would iolae one of our wo basi priniples. The aual formula is A AC AC C C In his ase, if one of he eloiies is, we ll ge A AC AC AC AC We an epress all eloiies as fraions of he speed of ligh. The denominaor is alread fied, so we jus need o diide boh piees of he numeraor b. Our ne eloi A will also hae o be diided b, and we ge A AC AC Now he person on he ground will measure he eloi of ligh as jus as he person on he rain. In anoher eample, if he ball-rain eloi is.6 and he rain-ground eloi is.7, Newon would predi a ne eloi of.3. Wha relaii ells us is *.7.95 This is also reassuring wheneer we add wo eloiies whih are hemseles less han, our final eloi, while larger, will sill be less han. We an ge an eleron o he speed of ligh b puing a parile aeleraor on a rain. 9

30 PHYS 33 Momenum Anoher area where phsis deiaes from Newonian mehanis a high speeds is he momenum of an obje. Careful measuremens of ollisions inside parile aeleraors (where we are auall able o ge maer (small piees of i) up o er high speeds) show us ha he law of onseraion of momenum will be iolaed in hese ollisions if we mainain ha p = m. If we adjus our definiion of momenum, i will be onsered a high speeds as well as in he low-speed Newonian limi. We now wrie he relaiisi momenum as p m m This gies us esseniall he same formula we e alwas had for ordinar eloiies, bu he relaiisi momenum will be larger han he Newonian momenum b he faor. In he ase of an eleron in a parile aeleraor, we an obsere alues for of >,. Using he Newonian formula for momenum will gie wrong answers. nerg The epression for he energ of a parile is also differen in relaiisi phsis. The larges hange is ha we now reognie ha maer and energ are inerhangeable, and a small amoun of maer represens an enormous amoun of energ. This energ ha he maer has wihou een moing (known, herefore, as res energ whih is dependen on he res mass of he obje) is added o he kinei energ o gie he oal energ of a moing parile: m m represens he res energ, and m he res mass. There has been a grea deal of onfusion abou res mass and he hange of mass wih speed in he ears sine relaii was proposed. The general opinion now is ha i s a bad idea o alk abou an kind of hange of mass wih speed, and we ll see wh shorl. Anwa, wha does his formula predi for he res energ of a kg mass? We ge kg * (3,, m/s) = 9 6! Tha s an unbelieable amoun of energ. I s abou wha would be released b a megaon H-bomb! (Tha s abou imes he energ released b he bomb ha 3

31 PHYS 33 desroed Nagasaki). In oher words, if we ould ge % of he res energ ou of a paper lip ( g), ha would be equialen o he bomb ha ended World War II. If kinei energ is he differene beween oal energ and res energ, our new formula for ha should be K m This is going o be an awkward wa o find K for hings whih aren moing relaiisiall, sine will be so lose o anwa. eause of is small sie, we an use an approimaion. We ll use he binomial epansion o rewrie Remember ha his ompliaion onl omes abou beause <<, so here s no poin in keeping he higher-order erms like he one wih 3/8 in fron of i (if / <<, (/) 4 mus be er lose o ero). Now use our epansion aboe o alulae - in our K formula: K m m whih should erainl look familiar. I s imporan ha our relaiisi resuls alwas redue o he Newonian ones in he low-speed limi. For ha reason, we probabl shouldn sa ha Newon s phsis is wrong, bu raher ha i is inomplee. For he kinds of speeds mos people are familiar wih, he differene beween Newonian phsis and relaiisi phsis isn measurable. The roo of hese epressions for momenum and energ an be found in anoher ineral whih we haen seen e. us as we had one kind for hanges in displaemen and ime, we hae one for momenum and energ. I is p m We hae he same odd kind of formula we had for lenghs and imes. In going from he - D Phagorean heorem ( + = ) o he 3-D ersion ( + + = r ), we add he square of he hange in he new oordinae, in he same wa (and for he same reason) ha we find eor magniudes in -D wih V V V and in 3-D wih V V V V. Here, we are subraing he square of one omponen from he 4 3

32 PHYS 33 squares of he ohers. Tha s jus he weird rule we hae o follow o ge he righ answer. Noie wha happens o he formula aboe when he obje is a res (and herefore p =): we jus ge m 4 or m Tha famous equaion ha is so ofen quoed is reall onl par of he sor, jus as we saw in Phsis I ha F = m a is reall onl a speial ase of F = dp/d. The full equaion aboe ells us ha he mass of an obje is an inarian, jus like he ineral is for displaemens & imes. Tha s wh i s a bad idea o sa ha an obje s mass depends on is eloi. Where would people ge he idea o do ha in he firs plae? The problem here is ha we an eer aelerae a parile wih mass (een a in mass, like an eleron) o he speed of ligh. We an ge an eleron up o , or faser, bu we ll neer quie be able o ge i up o eal. One wa o undersand wh is o look a wha he kinei energ would be for a parile wih mass m moing a. You d ge m /, whih is undefined (bu whih approahes infini as approahes ). In oher words, we d need o gie his eleron (or oher mass) an infinie amoun of energ, whih we an do. The idea of assoiaing his wih hanging mass probabl omes from he noion ha is wihin 3 m/s of he speed of ligh, so giing i ha las era kik should be eas. If we were allowed o use Newonian phsis, we d sa ha he work we need o do would be equal o he hange in K we wan o ause, or W K m * f m Ki 9.8 ( Wrong) m f i whih is large, bu no infinie. We deide i's worh he os, so we dump he needed energ aboe ino he eleron. Wha we find is ha he eleron s energ wen up, bu is eloi is sill slighl less han. eause we wan o use Newonian quaniies, we guess ha sine energ inreased bu eloi didn jump like we epeed, he mass of he obje mus hae inreased, hereb making i harder o aelerae. This les us undersand wh we an aelerae his in eleron o he speed of ligh i keeps geing more and more massie unil i s like ring o push an oean liner, and ou hae o dump a huge amoun of energ in i o ge he slighes hange in eloi. Again, his is probabl no he bes wa o hink abou i. We should onsider mass o alwas be he inarian mass, whih sas he same regardless of eloi. I s jus our Newonian ideas ha need o be hanged. We ould ge around i b wriing 3

33 PHYS 33 dp F d d d m sine ha redues o our old epression a low speed where and also learl shows ha he fore will be less and less effeie a inreasing he eloi of a mass as inreases. Relaiisi Collisions The wo big hings o keep in mind when eamining ollisions a relaiisi eloiies are ) oal energ and oal momenum (eah omponen) are onsered and ) he onneion beween energ, momenum, and mass is he familiar p m 4 To anale hese ollisions more easil, we will use a mehod from he book Spaeime Phsis, b. F. Talor &. A. Wheeler. We represen he energ, momenum, and resmass of eah parile b drawing a righ riangle. The erial leg of he riangle is he parile s oal energ, he horional leg is he momenum, and wha would be he hpoenuse is he mass. We hae o sa would be beause onneing he erial and horional legs on a piee of paper will gie us a hpoenuse of lengh ( + (p) ), bu he mass is reall ( (p) ). We an draw i eal, so we ll label i. Also, i s er ommon o hange o a ssem of unis where =. This means he formula aboe redues o he simpler one below: p m As an eample of his, le s sa we hae an eleron (m = kg) moing a.997. In MKS unis, is momenum would be kg m/s and is oal energ would be.6 - or abou 658, ev. In our new ssem of unis, we an replae he in he relaiisi momenum formula wih, so we ge p = m where m is he eleron s res mass. This gies us p =.889 m and = m =.996 m. Verif for ourself ha = p + m sill works. We an now see if i is possible for a single eleron o absorb a single phoon (whih seems like i would be eas enough). We don lose anhing b analing his from he frame of he eleron (sine he onl hange if we didn do his would be a Doppler shif of he phoon s energ). We hae wo inoming pariles: an eleron wih momenum p =, mass = m, and energ = m (b he formula aboe) and a phoon wih some energ and momenum p whih are equal sine i mus hae a mass =. 33

34 PHYS 33 We an alk abou he phoon s energ in erms of he eleron s res energ o make hings easier. Le s sa (he phoon s energ) =. me (he eleron s mass). This would make he phoon a low-energ X-ra. We an hen sa ha our inoming energ = + e =. me and our inoming momenum = p = / = =. me in our new unis. If he phoon is o be ompleel absorbed, onl he eleron remains. To onsere energ and momenum, i mus hae =. me and p =. me. u does i sill saisf he ineral equaion? p =. me in his ase, whih does no work, sine i s supposed o be jus me. A free eleron is herefore unable o absorb a phoon sine i would iolae he onseraion of momenum & energ (alled momenerg in he Spaeime Phsis book). I is able o saer a phoon, meaning absorb and re-emi he phoon (no neessaril in he same direion) almos insanl, and his proess is known as Compon saering. Pair Produion & Annihilaion Somehing ha is allowed is he reaion of maer from phoons or he desruion of maer (wih an aompaning produion of phoons). Firs, desruion: wha if we hae an eleron and a posiron (whih is he animaer ersion of an eleron hink of hem as omplee opposies of one anoher)? When he ombine, he will oall annihilae eah oher in a burs of energ (gamma ras, in his ase). Could he wo pariles reae jus one phoon when he are annihilaed? Le s sa our eleron and posiron aren moing relaie o eah oher, and we ll go o heir ommon referene frame o anale his. In ha frame, eah parile has ero momenum and herefore an energ equal o is mass: = m e for eah (he posiron s mass is he same as he eleron s). If we ombine hem and laim ha one phoon will emerge, i will hae o hae an energ of m e. There s no problem wih ha, bu we also require ha momenum be onsered. This is where he problem appears he posiron-eleron pair had no momenum, bu he phoon has o hae a momenum equal o is energ, whih mus be m e in his ase. We an hae hese wo pariles annihilae and produe jus one phoon (and similarl, if we run he lok bakwards, a single phoon b iself an produe an eleron-posiron pair). If, on he oher hand, wo phoons are reaed, eah ould arr m e of energ and herefore m e of momenum. If he momena are direed opposiel, he oal momenum would hen be ero, whih is wha we need. This, in fa, is wha happens in he majori of ases (i is also possible o onsere momenum and energ if hree phoons are produed). The fa ha he phoons leae in opposie direions is used in PT (posiron-emission omograph) sans o dramaiall redue noise and improe image quali. An eleron-posiron pair an be produed when a single phoon ineras wih some oher parile o allow boh energ and momenum o be onsered. The phoon will need more 34