Giáo trình vật lý lượng tử

The space that was at rest this privilcged obser\'cr was called “ absolute spacc.” A ny other observer moving wilh 3 absolute space would find the speed o f light to be differenl from c

ĐẠI HỌC QUỐC GIA HÀ NỘI KHOA CỎMG NGHỀ

HỆ ĐÀO TẠO CHẤT LƯỢNG CAG

NGÀNH VẬT LÝ KỸ THUẬT

GIÁO lld ữ m VẬT IvÝ I.ƯỢIVG TỂ

Tóm tắt lý thuyết và bài tập

2 2 T he M ic h e lso n -M o rley Experiment 2.3 Length and T im e M easurem ents— A Q uestion o f Principle

2 4 T he Postulates o f E instein

TH E LO R E N T Z C O O R D IN A T E TRA N SFO R M A T IO N S 3.1 T he C onstancy o f the S peeđ o f Light

3.2 T he Invariancc o f M axw eU ’s Equations 3.3 G eneral C onsiderations in S o lv in g Problem s Involving Lorentz Transform ations

3 4 Sim ultaneity

RELATIVISnC LEN G TH C O N T R A Q IO N 4.1 T he D eíìn itio n o f L ength

R E L A H V I S n C H M E D ILA T IO N 5.1 Proper Tim e

5 2 T im e D ilation

RELATIVISnC S P A C E - n M E M EA SU R EM EN TS

R E L A n V I S n C V E L O C IT Y TR A N S P O R M A n O N S 7.1 T he L orentz V elocity Transform ations and the Speed o f Light 7.2 G eneral C onsiderations in S o lv in g V elocity Problems

10

1010

10

11

1 5 15 16

20

20

2 3 23 23

2 7

3 7 37

38

8 1 T he N e e d to R edefìne C lassical M om entum

8.2 The Variation o f M ass with V elocity

8.3 N e w to n ’s Second Law in Rclativity

8.4 M ass and Energy R elationship: E = mcP'

8.5 M om entum and Energy Relationship

8.6 U n its for Energy and M om entum

8.7 G eneral Considerations in S o lv in g M a ss-E n erg y Problem s

PART I I THE QU ANTUM T H E O R Y 0 F E L E C T R O M A G N E T IC R A D IA T IO N

9.3 The C om pton Effect

9-4 Paứ Production and A iưiihilation

9.5 A bsorption o f Photons

M ATTER W AVES

10.1 D e B roglie W aves

10.2 Experim ental V'erification o f D e B ro g lìe’s H ypothesis

10.3 The Probability Interpretation o f D e B roglie W aves

10.4 T he H eisenberg U n cen a in ty Principle

PART I I I H Y D R O G E N L IK E ATOMS

CHAPTER 11 TH E B O H R ATOM

11.1 The H ydrogen Spectrum

11.2 The B ohr T heory o f the H ydrogcn Atom

11.3 E m ission o f Radiation in B o h r s Theory

11.4 E nergy Level Diagram s

11.5 H ydrogenic A tom s

CHAPTER 12 E L E Q R O N O R B IT A L M OTION

12.1 Orbital A ngular M om entum from a C lassical Vievvpoint

12.2 C lassical M agnetic D ip o le M om ent

12.3 C lassical Energy o f a M agnetic D ip o le M om ent in

an Extem al M agnetic Field 12.4 T he Z eem an Experim ent

12.5 Q uantization o f the M agnitude o f the Orbital Angular

45 45

60

61

62 63

8 0 80 81 82 83

1 0 3 103 103

104

107 107

1 1 7 117 118 119 119

C H A P TER 1 3

M om entum 12.7 Explanation o f the Z eem an E ữ ect

ELEC T R O N S P IN 13.1 T he S tem -G erla ch Experim ent 13.2 Electron Spin

13.3 S pin -O rb it C ou plin g 13.4 F m e Structure 13.5 Total A ngular M om en tu m (The Vector M od el)

PART I V M A N Y-EL EC T R O N ATOM S

120

1 2 6 126 127

128

128 129

1 3 5 135 135 135 136

M A N Y -EL EC T R O N ATOM S AN D T H E P E R IO D IC T A B L E 1 4 0 15.1 S pectroscop ic N otation for E lecừ on C oníìgurations in A tom s 15.2 T he Periodic Table and an A tom ic S hell M odel

15.3 S pectroscop ic N otation for A tom ic States 15.4 A to m ic E xcited States and L S C oupling

15.5 T he A n o m a lo u s Z eem an Efifect

X -R A Y S 16.1 X -R a y Apparatus 16.2 Production o f B rem sstrahlung 16.3 Production o f Characteristic X -R ay Spectra 16.4 T he M o se le y R elation

16.5 X -R a y A bsorption E đges

1 6.6 A u g er EflFect 16.7 X -R a y Pluorescence

140 141 142 142 143

1 5 7 157 157

158

160

160

161 161

1 7 3 173

174 174 174

175

2 0 2 C lassitìcation o f Nuclear R eactions

2 0 3 Laboratory and Center-of-M ass S ystem s

2 0 4 Energetics o f Nuclear R eactions

2 0 5 N u clear Cross Sections

2 1 7 Conservation o f Isotopic Spin and Parity

2 1 8 Short-L ived Particles and the R esonan ces

2 1 9 The E ightíold Way

22.1 M olecu lar Bonding

2 2 2 E xcitations o f Diatom ic M o lecu les

K IN E T IC TH EO R Y

23.1 Average Values in a Gas

181 182

1 9 3 193 193 194 194 195

2 0 7

2 07

2 0 7

2 0 8 209

24 2 C ontinuous D istribution punctions

24 3 Pundamental D istribution Punctìons and D ensity o f States

C L A S S IC A L S T A n S n C S : T H E M A X V V ELL- BỌLTZM ANN D IS T R IB U n O N

QUANTUM S T A H S n C S : P E R M I- D IR A C AND

B O S E - E IN S T E IN D IS T R IB U n O N S 26.1 F erm i-D ừ a c S taóstics 26.2 B o se-E in stein Statistics 26.3 High-Temperature L im it

2 6 4 Two U se íu l Integrals 26.5 B lackb od y Radiation

2 6 6 Free Electron Theory o f M etals

2 6 7 S p eciíic H eats o f Crystalline Solids

2 6.8 T he Q uantum -M echanical Ideal Gas

2 6 9 Derivation o f the Q uantum Distribution Punctions

S O L ID S 27.1 T he B and T heory o f S olid s

28 9

29 2

29 6

301 305

3 0 9

30 9

31 8

3 2 5 325 325 325

32 6

3 3 3

Galllean Transformations

I ì

Ể 1 E V E ÌST S A N D C O O R D IN A T E S

\Ve b e ^ n by considering the co n cep t o f a physical cvent T he cvcnt niisiht be the striking o f a tree by a liglu iiin g bó)t or the co llisio n o f tw o particles, and happens at a point in space and at an instant in time The particular ẹvent is sp eciíìed by an ob ser\’cr by a ssicn in g to it four coordinales: the three position

coorđinates >r, y , z that measure the distance from the oricin o f a coordinatc sỵstcm \vhere the observer

is located, à i d the tim e coordinate í that the obser\'cr rccords wiih his clock.

C onsiđer n o w tu ’0 observers, oand 0 \ where o 'travels with a constant velocit\' V \vith respect to o

along th eừ cồ m m o n ĩ — x ' axis (Fig 1-1), Both o b s e n ers are equipped \vlih m elcrsticks and clo ck s so that

they can m ẹ ^ u r e coordinates o f events Purther, su pp ose that both observers adjiist their clo c k s so that

w h en they'*p^s each other at J = y = 0 , the clo ck s rcad' I = i' = 0 Aiiy gÌN cn eveni p will have eight

num bers assợciated with it, the four coordinates assiuncd by o and tlic four coordinates

ự , / / , i') a isig n ed (to the sam e evem ) by O '.

(X /.Í I)

Fig 1-1

G a LILE AN c o o r d i n a t e t r \ n s f o r m a t i o n s

The relationship betwecn the measurements (,v.y /) o f o and the measurements (.r ',y , t') o f O '

a palicular event is obtaincd by exaniining Pic 1- 1:

t ’ = -V - V! y ' = V r ' = z additon, in classical phvsics it is im plicitly assumed that

í' = /

:se fcur equations are called the G a lilea n coord in a te tra n sfo rm a tĩo n s.

G AL ILE AN V E L O C IT Y T R \N S F 0 R M A T 1 0 N S

In aidition lo the coorđinates o f an event, the velocity o f a particle is o f interest Observers oand O'

1 describe the particles velocity by assigning three com ponents to it, with (u ,, u ) being the velocity

nponents as measured by o.and ịú^, U y, uỊ) being the velocit>' com ponents as meãsured by O '.

The relationship benveen (Uj, u ) and (u^, iiỊ) is obtained from the lim e diữerentiation o f the

lilear, coordinate transformations Thus, from v' = X — 17,

ogether, the Gaiiìean velocit}- lra n sfo n n a tio n s are

G A L ILE A N A C C E L E R A T IO N T R A iN S P O R M A T IO N S

The acceleration o f a particle is the time derivative o f its vclocity, i.e., a = d u / d i , etc To find the

lilean a ccelem tion :ransform arions we diíTcrentiate the velocity transformations and use the facts that

= l and L' = constant to obtain

a', = a.

JS the measurcd acceleration com poncm s are thc sam e for all observers m oving with uniform rclalive

Dcity.

INVARL-VNCE O F A N E Q U A T IO N

B y invariance o f an equation it is meant that the equation vvill have the sam e form vvhen determined by

I observers In classical theory It is assum ed ihai space and time measurements o f two observers a r é

Ited by the Galilean transformations Thus, when a particular fom i o f an equation is determined by One

er\'er, the Galilean transformations can be applled to this fon n to determine the form for the other

erver l f both fom is are thc sam e, the equation is invariant under thc Galilean transformations S ee

blems 1.11 and 1 12.

S o lv e d P ro b lem s

A passenger in a train m oving at 30 m /s passes a man standing on a station platform at í = í" = 0

Twenty seconds afler the train passes him, the man on the platform determincs that a bird Aying

along the tracks in Ihe sam e direction as the train is 8 0 0 m away What are Ihe coordinates o f ứie

Ans.' The coordinates assigned to the bird by the man on the station platform are

Ans The coordinates assigned to the bird at the second position by ứie man on the platíorm are

f c y 2 - 2 ' 2) = (8 5 0 m, 0 ,0 , 25 s) Hence, the velocity u, o f the bird as measured by the man on the platform is

u', = u , - v = - 0 7 c - 0.6c = - 1 3c

This problem demonstrates ihai vclocities greater than Ihe specd o f lighl are possible with t Galilcan transformations, a result that is inconsistent with Special Relali\ ity.

A train m ovin g with a v elocity o f 6 0 m i/h r passes through a railroad station at 12:00, Twcn

second s later a bolt o f lightning strikes the railroad tracks one m ile from the station in the san direction that the train is m oving Find the coordinatcs o f the lightning flash as m easured by i observer at the station and by Ihe engineer o f the train.

Ans Both observers measure the limc coordinatc as

/ = /' = (2 0 s)

The-observer at the station measures the spatial coordinate to be

determined by the engineer o f Ihe Irain is The spatial coorđinatc as

y = ,x - VI = 1 mi - (60 mi/hr)

1.5 A hunter on the ground íìres a bullet in the nonheast đirection which sưikes a deer 0.25 m iles from

airplane is directiy over the hunter at an allitude h o f one m ile and is traveling due east with a

an observer in the airplane?

Ans Using the Galilean transformations,

: I0 -' hr 1800/hr

x' = x - v t = (O :5m ĩ)cos45“ - (600 mi/hr)( 1.39 X 10”* hr) = 0.094 mi

>■' = y = (0.25 mi) sin 45° = 0 177 mi

z' = 2 — / i = 0 — lm i = —Imi

1.6 An observer, at rest with respect to the ground, observes the follow ing colHsion A particle o f mass

m, = 3 kg moving with v elocity Uị = 4 m / s along the x-axis approaches a second particle o f mass

m, = 1 kg moving with velocity u, = - 3 m /s along the v-axis After a head-on collision the

ground observer finds that has velocity = 3 m /s along the A:-axis Find the velocity o f m,

alter the collision.

ínitial morĩìenmm = final momentum

m|U, + m,Uj = n t,u ’Ị + »12«;

(3 kg)(4 m /s) + (1 k g)(-3 m/s) = (3 kg)«; + (1 kg)(3m /s)

9 kg • m /s = (3 kg)«; + 3 kg • m /s Solving u ’ ị = 2 m /s,

1.7 A second observcr, O ' w ho is walking wiih a velocity o f 2 r a /s rclative to thc ground along the ;t-

axis observcs ihe collision described in Problem 1.6 What are the system momenta before and after

the collision as determined by him?

Ans Using ihc Galỉlcan vclocity transformations,

Thus, as a result o f the Galilcan lransformations, o ’ also dctermines that momentum is conserved (but

at a difĩcrent vaíuc from ihal found by O).

1-8 An open car traveỉing at 100 ft /s has a boy in ìt \vho throws a baịi u p v ^ d with a veỉocity o f 2 0 f t /s

Write thc equation o f motion (giving posiiion as a fiinction o f time) for the ball as w e n bỹ (õ ) the

boy, (Ế) an observer stationary on the road.

(a) For the boy in the car the ball ơavels straight up and dovra, so

y = Vot' + ị a í ^ = (2 0 f t /s ỵ + ị ( - 3 2 = 20?" - 16f'2

y = V = o (Ề) For the stationaiy observer, One obtains (rom the Galilean transformations

t = í

I = x ' + p /= 0 + lOOí y = ý = 2 0 t— l 6 r 2 = r* = 0

1.9 Consider a mass anached to a spring and m ovm g on a horizontal, frictionless suríace Show, frorr the classical transíormation laws, that the equations o f motion o f the mass are ứie same ai constant velocity along the direction o f the spring.

Ans The equation o f motion o f the mass, as determined by an observer at rest with respect to the surface, i:

1.10 Show that the electromagnetic wave equation,

is not invaríant under the Galỉlean tiansfonnations.

Áns Tlie equation will be invariaat i f it retains the same fonn when expiessed in teiim o f the ftew yariable

y , y , r' We tìrst find from the Galilean transíormations thai

Thercforc the wavc equalion is m l invariant unđer the Galllean tiansformations, for the form o f the

equation has changed because of the extra terni on thc left-hand side.

The electromagnetic wave cquation follows from Ma.\weH’s equations o f electromagnetic

thcory By applying the procedure described here to Maxwell's equations, One finds that Maxwell's

shows that the electrornagnetic wave cquation is invariant unđer a Lorcntz transformation.

S u p p lem en ta ry P roblem s1.11 A man (O') in ihe back o f a 20-ft Hatcar moving at 30 ft/s records that a Aashbulb is fìrcd in the fiont o f the

Aatcar tv,'0 seconds after he has passed a man (O) on the ground Find the coorđinates of the evenl as

d c te r m in e d b y e a c h o b s e r v c r A n s ụ f ) = ( 2 0 f t, 2 s)- ( i í) = ( 8 0 ft 2 s)

1.12 A boy sees a deer run dlrectly away from him The deer is running wilh a speed of 20 mi/hr The boy gives

Chase and nins with a speed of 8 mi/hr \VhaI is the speed of the decr relativéto the boy? Ans 12 mi/hr

1.13 A boy in a train throws a ball in the fo™-ard direction with a speed o f 20mi/hr If the train is moving with a

speed of 80 mi/hr, whai is the speed o f the ball as mcasuređ by a person on the ground? Ans 100 mi/hr

1.14 A passenger walks backvvarđ along ihe aisle o f a train wiih a speed of 2m i/hr as the train moves slong a

^ i g h t track at a constant speed of 60m i/hr with respect to the ground Wiat is the passengcr’s speed as

measured by an obscrver siandmg on the grounđ? Arts 58 nií/hr

1.15 A conductor standing ọn a railroad platíbmi synchronizes his vvatch with the engineer in the front o f a train

ữaveling at 60 mi/hr The traín is I /4 milc long Two minutes after the train leaves the plalfomi a bĩakeman m

and by the conductor, w h en the coíTee is poưred? ^„ s ự r") = ( - 1 m i, 2 m in ); (x 0 = (I 2 mĩ 2 m in)

1.16 sitting in a tTdin pours two cups o f coAee One 10 m inutes a íte r Ihe oiher T he train is m oving in a

straight line with a velocìty of 2 0 m/s UTial is the distance separation berween the two pourings as m eíured

by a person on the g ro u n d ? A ns 1 2 000m

1.17 A one-kilogram ball is constrained to move to ihe north at 3 m/s It makes a perfectly elastic collision with an

Ịđentical second ball which is al rest, and both balls move on a north-souứi áxis aflcr the coUision Compute

in the laboratory system the toul momcntum bcfore and aftcr the collision /íns 3 kg - 111/5

1.Ỉ8 For Problem 1.17, calculatc the total encrgy before and aíter the collision /Ins 4 5 J

1.19 Refcr to Problcm 1.17 Caỉculaic the toial momcntum bcforc and after the collision as measurcd by an

1.20 For the observer in Problem 1.19, calculale ihe tota! energy before and aỉìer ihe collision, Ans 2.25J

1.21 Repeat Problems 1.19 and 1.20 for an observer moving eastuard at 2 m/s Í/Ỉ5 5 kg • m /s 37^ north ol west; 8.5 J

1.22 A person is in a boai moving eastwarđ vvith a speed o f 15 f\/s At the instant ihat ứie boat passes a dock, a person on the dock throws a rock northward The rock strikes ihe water 6 s later at a đistance o f 150 ft from thc đock Finđ ứie coordinates o f the splash as measured by ihe person in ứie boat.

Ans (x ,y 0 = (- 9 0 ft I 5 0 ft,6 s)

1.23 Consiđer a one-dimensional, elastic collision Ihat takes place along ihe x-axis o f o.Show, from ứie classicai

ưansformation equations, ihal kinetic energy will also be conserv'ed as determined by a second observcr, O ' who moves with constant velocity u along the x-axis o f o.

The Postulates of

Einstein

ỈL U T E SPACE A N D THE E TH ER

iquence o f the G alilean velocity tt-ansrormations is that if a certain observer nieasurcs a light

vel with the v elo city c = 3 X 10* m /s , Ihen any other observer m oving relative to him will

gsam e light signal to travel with a velocitv different from c What determines the particular

ne such that if an o b scn er is at rest rclative to this frame, this privileged observer will

^alue c for the velocity o f light signals?

nstein it was gcnerally believed that this privĩleged observer was the same observer for whom

^ations were valid, MaxweH’s equations describe electrom agnetic thcory and predict that

Bc vvavcs will travel with the speed c = 1 / y r ^ = 3 X 10® m /s The space that was at rest

this privilcged obser\'cr was called “ absolute spacc.” A ny other observer moving wilh

3 absolute space would find the speed o f light to be differenl from c Sincc light is an

; wave, it was felt by 19th ceniury physicists Ihat a m edium must exist through which the

Thus it \vas postulatcd thai the "ether" permeated all o f absolute space.

p lE L S O N -M O R L E Y E X P E R IM E N T

cxists, then an observer on ihe earth m oving through the eiher should notice an “ether

ptus with the sensitiviiy to measure the earth’s motion through thc hypothesized ether was

y chelson in 1881, and reíìncd by M ichelson and M orley in 1887 The outcome o f the

pat no m o tio n through th e ethcr w as d eiecled S ee Problcms 2.5, 2 6 and 2.7

^ND T IM E M E A S U R E M E N T S — A Q U E S T IO N O F P R IN C IP L E

5ent cotnm on to both the null result o f the M ichelso n -M o rley experiment and the fact that

Jfons hold only for a pnvileged obsereer is the G alìlean transíomiations These "obvious”

fw ere rc-exam ined by Einstcin from what might be termed an "opcrational" point o f view

a well-defined proceđure by which it is measured lf such a procedure cannot be formuIated, then the quantity should not be employed in physics.

Einstein could find no way to justify operationally the Galilean transíormation t' = I, i.e., the statement that two observers can measure the time o f an event to be the same Consequently, the transformation

t' = I, and with it the rest o f the Galilean transformations, was rẹjected by Einstein.

2.4 T H E PO ST U L A T E S O F E IN ST E IN

Einstein’s guiding idea, which he called the Principle o f Reìativịty, was that a ll nonaccelerating

observers should be treated equally in all respects, even if they are movừig (at constant velocity) relative to each oứier This principle can be formalized as follows:

Posruỉate 1: The Iaws o f physics are ứie same (invarianl) for aỉl inerĩial (nonacceleratìng) observers Newton's lavvs o f motion are in aecord with the Principle o f Relativity, but M axw ell’s equations together with the Galilean transformations are in conAict with it Einstein could see no reason for a basic difference between dynamical and elecưomagnetic laws Hence his

Postuỉate 2: In vacuum the speed o f light as measured by all ỉnerlỉal obsen'ers is

A Aashbulb is located 30 km from an observer The bulb is fưeđ and the observer sees the f1ash at

1:00 P.M What is the actual time that the bulb is fired?

The timc for Ihe iỉght signal to travel 30 km ỉs

Ai 30 X

I X lO -S

c 3 x lO * m /s Thcreforc, the Aashbulb was fired 1 X 10“* s bcfore 1:00 P.M.

A rod is moving from left to right W hen the left end o f Ihc rod passes a camera, a picture is taken

o f the rod together with a stationary calibrated meterstick In the developed picturc the left enđ ol the rod coincides wiưi the zcro mark and the right end coincides \vith the 0,90-m mark on ứiE

mcterstick If the rod is m oving at 0 8c with respect to the camera, dctermine the a cíu a l length ol

Ans In order th.ll thc light sitỉnaỉ from thé right end o f the rod be recorded by the canicra it must have

siancd from the 0.<50-m mark at an earlicr time HÌvcn by

(a) Signal starts from rìpht

end; ciimcra shutler closcd.

C l ] (ố) SÌRnal arrivcs ừom rlght end and is recorded hy op«n caniera loịỊcther wiih signal from li*n cnd.

Fìg 2-1 Thcrefore the acmal length o f the rod IS I = 0.90 m + 0.72 m = 1.62m This resull illusirales that

pholographing a moving rod will no! give Ils correct leneth.

2.4 Let tu'0 events occur ai equal distances from an observer Suppose the obsen'er adopts the

follow ing statement as a deíìnition o f simultaneity o f equidistaiit evems: “The two events are

simullaneoiis i f the light signals em itted from each event reach me at the same time.” Show that

according 10 this dclìnition, if the observ'er detcrmines Ihat two evcnts are simultaneous then

another ob sen x r, moving relative to him, will in general determine that the two events are nol

simultaneous.

Ans From Fig 2-2 ii is secn that if the r»'0 lighl signals reach thc firet observer (O) at the same lime they

will necessarily rcach the second obscrvcr (O') at diíTcrcnl times Since the two sịgnais slanẽd oul

cquiđistant from 0 \ he will, according lo the above definition, determine thai Ihe Mo events did not

occur simullaneously, bui ihal event fl happcned bcforc cvcnt A.

(a) SÌ^naLv A and B start (b» Slpnol B rcaches 0\

(c) SỉgnaLs.4 and B rcach 0 simulUineously (đ) Signal A nrachcs o.

2.5 Pigure 2-3 diagrams a M ichelson-M orley interíerometer oriented with one arm (/^) parallel to the

“ether wind," Show that if the apparatus is rotated through 90“, the number o f íringes, A N , that

move past the telescope crosshairs is, to first order in (d/c) ,

To travel along the other arm a light ray musl be aimeđ such that its resultanl velocity vecior (velocitỊ

to arm A This gives a speed o f for both directions along path /j, so the time for the rount trip ỉs

Now if the interierometcr is rotated 90°, and /j

time difference Thus

: ìnterchanged, and there is also a reversal o f the

and the imerfcrence pattem observed would show a ữinge shift o f ầ N fringes, where

_ ẵ - & ‘ _ c í ò - ẵ ') ụ , +

Here r and X are the period and wavelength o f the light.

Assume that the earứi’s velocity thiough the ether is the same as its orbital velocity, so that

V = 10“''c Considcr a M ichelson-M orley experiment where the arms o f the mterferometer are each

lOm long and on e arm is in the dừection o f motion o f the earth through the ether, Calculate the

diíĩerence in từne for the two light waves to ữavel along each o f the arms.

Ans Refer to Problem 2.5.

2

(3 X 10* m/s)c^ (5m ) = 3.33 X

IQ-2.7 The original M ichelson-M orleỵ experiment used an interíerometer with arms o f 11 m and sodium

light o f 5900 A The experiment would reveal a ÍTÌnge shift o f 0.005 fringes What upper limit does

a null result place on the speed o f the earth through ửie ether?

Ans From Problem 2.5, the number o f tringes, AN, seen to pass the telescope crosshairs is

0.005 = (5900 X I0-I0m)(3 X 10«m/s)^

-Solving, u = 3.47 X 10’ m/s.

The earth's orbital velocity is 3 X 10* m/s, so the interíerometer w:

this motion No fringe shiít was observed.

ỉ sensitive enough to dctect

Su p p lem en tary P rob lem s2-8 Repeat Problcm 2.3 for ứìC case whcre the picture is taken whcn ihe righi cnđ o f the rod passcs ưic

caniera Ans 0.18m

2-9 At thc instant that the midpoini o f a moving lĩietersĩick passcs a camcra, the camera shuner opcns and a

relative to the camcra Is 0.8c, vvhat will be ihc length o f ihc moving meterslick as recordcd on the

fi1m? Ans 2.778 m

2 10 Rcfcr to Problem 2.4 If Uic two signals reach O ' simuỉtancousỉy, what is iheir limc sequence as determined

by o ? Ans A occurs beforc B

2 11 Assume that thc orbital specd o f the canh, 3 x 1 0 ^ m/s, is cqual to the speed o f thc carth through the eứier If

light takes seconds to travel ihrough an equaỉ-ann Michclson-Morỉcy apparatus in a direction parallel to

this moiion, calculate how long it U'ill take light to travel pcrpcnđicular to this motion.

The Lorentz Coordìnate Transformations

2 o f Section 2.4 requires that the Galilean coordĩnate transfomiations be replaced by the

dinate trans/orm alions For the two observers o f Fig 1-1 these are

Ị(5.í), V is the velocity o f O'with respect to oalong their common axis; V is positive i f O'

jitiv e jr-dircction and negative i f O ' m ovcs in the negative í-direction It has also been

1 origins coincide when the clocks are started, so that f' = f = 0 when y = X = 0 Note 5formations can be obtained fìom the first set o f transformations by interchanging

Ịmed variables and letting t! - > —V This is to be expected from Postulate 1, since both

Ịipletely equivalent and o b server om oves wiứi velocity - i ! w ith respect to o '

ÍS T A N C Y O F T H E S P E E D O F L IG H T

pt at the instant when oand O'pass each other (at í = í" = 0), a light signal is sent from

“ ‘ h in the positive x - x ' direction I f oíìnds that the signaPs spatial and time coordinates

are relateđ by í = ct, then, according to {3.1), O ' will find ứiat

X - 17 c t - VI

V 1 - V l - V[I - (i;/c))[l + (i'/c )] V 1 + (i>/c)

Thus o ' will find thai y = ư , in agreement with the second postulate o f Einstein Note also that for this

event on ưie light signal, í' 5^ (, in deíìnite disagreement with the Galilean assumption.

3.2 T H E INVARIAiNCE O F M Á X W E LL 'S EQUATIONS

As discussed in Chapters 1 and 2, MaxweH’s equations by electromagnetic theory are not mvariant

under Galilean transformations However, as showTi by H A Lorentz (before Einstem), they are invariant

under Lorentz fransformations See Problems 6.21 through 6.23.

3 J G E N E R A L C O N S ID E R A T IO N S IN SOLVING P R O B L E M S INVO LVING L O R E N T Z

T R A N SPO R M A T IO N S

When attacking any space-tim e problem, the key concept to keep in múid is that o f “ event." Most

problems are concem ed with rwo obser%’ers measuring the space and time coordinatcs o f an event (or

events) Thus, each event has eight numbers associated with it: ( x , y , z , t ) , as assigned by o , and

( y , y , / , / ) , as assigned by o The Loreniz coordinate tiansformations express the relationships between

these assignments.

Many times, problems are conceraed with the determination o f the spatial interval an d/or the time

interval between two events In this case a useílil technique is to subtract from each other the appropriate

Lorenti transformations describing each event For example, suppose observer o ' measures the time and

spatiãl intervals between two events, A and B, and its is đesired to obtain the time interval between these

same two events as measured by o.From {3.2) one obtains upon subtracting from tg

ỉữ — lj — - ■ '

Since all the quantities on the right-hand side o f this equation are known, one can determine te - 1^.

3 4 SIM ULTANEITY

Two events are sim u ìta n eo u s to an observer if the observer measures that the two cvents occur at the

sam e time, With classical physics, when one observer determines that two events are simultaneous then

since r' = í from the Galilean lransformations, every other observer also fìnds that the two events ăre

are, in general, not simultaneous to another observer.

Suppose, for exam ple, that evenls 4 and s are simultaneous as detennined by O ', so that = 4 -

According to (J.J), observer omeasures the time separation o f these same two events as

I f the two events occur at the same spatial location, so that xỉg = then the two cvents are also

simultaneous as detcrmined by o. But i f x 'g T Ì x '^ , then o determines ứiat ứie two events are not

simultaneous.

THE LORENT2 COORDINATE TR.W SF0RM AT10NS

Note tha! i f the two events occur at the same spatial location, only o n e clock is needed by each observer to determine i f the events are simultaneous On the other hand, if the two evenls are separated

spatially, then each observer needs rn '0 clocks, properly synchronized, to determine vvheứier or not the two events are simultaneous.

S olved P ro b lem s3.1 Evaluate y i - for (o) V = ì O ~ - c ; ( b ) V = 0.999Sc.

Ans In ihe following we make use o f the binomial expansion.

(1 + x)" = 1 + M +

(a) Sening X = - 1 0 ■* and n = ỉ in thc binomial expansion, and, because X is so small, keeping only the fiisl two terms o f the expansion, we obtain

(1 - lO-")''- 1 + l ( - 1 0 - ' ) = 1 - 0.00005 = 0,99995

(6) y i -(i'2 /c^ ) = y/l - (0.9998)- = y/\ - ( 1 -O.OOOIÝ

To evaluale (1 — 0.0002) we cmploy the binomial expansion to obtain

(1 - 0.0002)- % 1 - 2(0.0002) = 1 - 0.0004 Using this in the above expression we obiain

v/1 - ( i , “ /c2) y i - ( 1 - 0 0 0 0 4 ) = V0.0004 = 0.02

3.2 A s measured by o,a Aashbulb go es o f ĩ at X = 1 0 0 km, y = lO km, r = I km at / = 5 X 10“'' s,

What are the coordinates y , y , z ' , and /' o f this event as đetermined by a second observer, O ',

m oving relative to oat - 0 8c along the com m on x - x ' axis?

Ảns From the Lorentz transformations.

0 6 c along the x - x ' axis, find the equations o f motion o f the panicle as determined by o.

T 1 Be 'ẽuuauuiis u f HIUIÌUII« tm eiiii

đ ạ i h ọ c Q U Ò C g i a H Ầ N O Ĩ j

T R Ư N G T Ầ M T H Ô N G T IN ĩ h ỉ í r # N

ncd by O' i (cos 60°)/' y = Uyt' = ^ ( s i n 6 0 '" )r'

Substituting from (i.ý ) ìn the íìrst expression, we obtain

3.4 A train i mile long (ạs measured by an obscrver on the ữain) is traveling at a speed o f lOOmi/hr

Two lightning bolts strike the ends o f ừie ffain simultaneously as determined by an observer 011 the

ground UTiat is the time separation as measured bv an obsérver on the train?

(100 mi

Let events ^ and B be đetỉncd by the strikuig o f each lightning bolt Wiih o as the ground obser\-er

we have from (i.J) : striking o f each lightni

(^' b - O + - <<)

, + ^ ị Z L i Ị £ l z : ] i Z i ( o , 5 n , 0

Q _ (1.86 X 10’ mi/s)~

Soiving = -4 0 2 X 1 0 '’ s The minus sign denoies that cvent A occurred after event B

3.S Obsei^er 0 notes ihat nvo cvents are scparaled in space and time by 600 m and 8 X IQ-'' s How

fast must an observer O ' be moving relativc to o in order that the évents be simũltaneous to 0 ' ?

Ans Subtractỉng two Lorentz transfomiaĩions, we obtain

THE LOR£NTZ COORDÍNATE TRANSPORMATIONS

Ans Subtracting Uvo Lorent2 transformations:

(I X 0 - s - 2 X 1 0 - s) - í

y i - a -/c ^ ) Solving, v/c = - 1 / 2 Therefore V is in the negative jr-direction.

3.7 Refer to Problem 3.6 What is the spatial separation o f the two events as measured by ớ ' ?

Ans Subtracting two Lorentz transformalỉons:

r ; - r ' =

From Problcm 3.6, v /c = —ịo r t ; = —1 5 x 10®m/s.

, _ (12 X lO^m- 6 X lO^m)- { - 1 , 5 X 10*m/s)(l X 10-'‘ s - 2 X lO-^s)

~ - - p— = 5.20 X 10 m

- ( - 0 5 ) '

S u p p le m e n ta ry P rob lem s3.8 Obtain 0 2 ) from (Ì.I).

3.9 As detennined by O', a lightning bolt strikes at = 60m, >/ = y = 0, ;' = 8 X 10“* s O' has a velocity of

along the jT-axis o f o What are the space-time coordinates o f the strike as delennined by

ơ? Ans ( x y z t) = (93 m 0 0, 2 X 1Q-’ s)

3.10 Observcr O ' has a velocity o f 0.8c relative to o , and clocks are adjusted such ứial / = r" = 0 whenx = y = 0

If o determines thai a Aashbulb goes off at J = 50m and 1 — 2 X 10“’ s, whal is the time o f this event as measured by O'? Ans 1.11 X IO-’ s

3.11 Refer to Problcm 3.10 l f a second Aashbulb Aashes at y = lOm an d / ' = 2 X 10"’ s as determineđ by O'

w hat is the tim e in ler\ al b«tw een the two events as m easurcd by ơ ? Ans 1.78 X 10“ ’ s

3.12 Refer to Problem 3.11 Whaĩ is the spatial sepaiation o f the two c\'enls as mcasured by (a) O ' (b) ơ ĩ

Ans (fl) 6.67 m;(í>) 46.7 m

Relativistic Length

Contraction

ID E P IN IT IO N O F L EN G T H

py is at rest with respect to an observer, its length is determined by measuring the diíĩerence

spatial coordinates o f the endpoints o f the body Since the body is not moving, ứiese

Jts may be made at any time, and ứie length so deteưnined is called the rest length or p ro p er

Ị body.

ving body, hovvever, the procedure is more complicated, since the spatial coordinates o f ứie

he body m u st b e m easured a t the s a m e tim e The diíTerence berween these coordinates is then

I ứie leiĩgth o f the body.

Inovv a ruler, oriented along ứie x - x ' dircction, that is at rest with respect to observer O' We

nine how the length measurements o f oand O'are related to each other when O'is moving

bvith a velocity I' in ứie x - x ' direction Let the ends o f ữie niler be designated by A and B

|n t z transformation {3.1) we obtain

v/1

-= Lq is the (proper) length o f the ruler as measured by O ' If X g and are

at the same time, so that Ig — 1 ^ = 0, then the diíTcrence Xg — Xjf — L w ill be the length o f

'iured by 0 Thus we have

< 1 we have L < io , so Ihat the length o f ữie m oving ruler is measurcd by o to be

esult is called the L on'nl:~F ir:g era ld conlraciion.

I to keep clear the distinclion between the concepts o f “spatial coordinate separation” and

non m istake in solving problems is sim ply to multiply or divide a given spatial interval by

^ /c ^ ) This approach will work i f One is con cem ed with íĩnding ửie relations between

he concept o f “ length" is deíìned precisely above However, i f One is interested in the

fb etw een t»'0 evcnts that do not occur simultaneously, Ihen the ansvver is obtaineđ from ứie

subtraction technique o f Section 3.3; the c o ư ect answer w ill n o l be obtained by multiplying or dividing the original spatial separation by y i — { v ^ /c ^ ị

S olved P ro b lem s4.1 How fast does a rocket ship have to go for its lengứi to be contracted to 99% o f its rest length?

Ans From the cxprcssion for length cootraction {4.1),

^ = 0.99 = ự \ - ( i^ /c ^ ) or u = 0.141c

4 2 Calculate the Lorentz contraction o f the earth’s diameter as measured by an observer O ' who is

stationary wiứi respect to the sun.

Ans Taking the orbial velocity o f the earth to be 3 X 10^ m /s and the diameter o f the earth as 7920 mi, the expression for the Lorentz contiaction yields

D = D j ì - (v^/c^) = (7.92 X 10’ m í ) ị i - « (7.92 X lo' mi)(l - 0,5 X IQ-*)

Solving, Do - D = 3.96 X 10"’ mi = 2.51 in It is seen that relativistic eíTecIs are very small at speeds that are normally encountered.

4.3 A meterstick makes an angle o f 30° with respect to the y -a x is o f O ' What must be the value o f V if the meterstick makes an íUigle o f 45° with respect to the x-axis o f ơ ?

4.4 Refer to Problem 4.3 What is the lengUi o f the meterstick as measurcd by ơ ?

Àns Use the Pythagorean thcorcra or, more simply,

sin 45° sin 45°

4.5 A cube has a (proper) volum e o f lOOOcm’ Find the volum e as detcrmined by an obscrver O ' who

moves at a velocity o f 0 8c relative to the cube in a direction parallel to one edge.

' Ans The obsCTver measures an edge o f the cube parallel to the đirection o f moiion to have the contracted

ỉengứi

ỉ = I V l = (lO c m )/! - ( 0.8)= = 6 cm

The lenglhs o f the other edges are unchanged:

fy = ly = í = ì = 10 cm

ThereTore,

= ự x = ( 6 a ĩi)(1 0 c m )(1 0 c in ) = 600 c m ’

S u p p le m e n ta ry P rob lem s4.6 An airplane is movmg with respect to the eanh al a speed o f 600 m/s Its proper length is 50 m By how much

will il appcar to be shonened to an observer on eanh? Ans lO"'” !!! ,

4.7 Compute the contraction in lengđi o f a ttain ị mile long when it is traveling at lOOmi/hr.

A rs 5.58 X 1 0 -'= m i = 3.52 X 1 0 - '“ in,

4.8 At whal speed must an observer inove past ứie earứi so that ứie eanh appears like an ellipse whose major axis

is sLx tũnes its minor axis? Ans 0.985c

4.9 An obscrver o ' holds a 1.00 m stick at an angle of 30° with respect to thc positive y-axis o ' is moving in the

positive x - x direction vvith a vclocity 0.8c vvith respect to observer o What are the length and angle o f the

stick as m easuređ by o ? A n s 0.721 m; 43.9°

4.10 A squarc o f area 100 cm^ is at rest in the reference frame o f o Observer O ' moves relative to o at 0.8c and

parallcl to One side o f ứie squarc What does O ' measure for the area? Ans 60 cm^

4.11 For the square of Problem 4.10, finđ the area mcasured by O' if O' is moving at a velocity O.Sc relative to o

and along a diagonal o f the squarc Ans 60 cm^

4.12 Repcat Problem 4.5 if O ' moves with Ihe same spccd parallel to a diagonal o f a face o f the cubc.

' owith his single clock, is called the p ro p er tim e interval between the events.

sider the same two events A and B as viewed by a second observer, 0 ' , m oving with a

Ih respect to o.The second observer will necessarily detemiine that the two events occur at ũons and will therefore have to use t\vo diíĩerent, properly s^Tichroniĩed clocks to determine

tion t'g — = ầ t ' between A and B To find the relationship benveen the time separations as

' and o ' we subtract two Lorentz time transformations, obtaining

nines that the two events occur at the same location, Xg - = 0 Thus

A/ a

A/' =

-■ < 1, Aí* > A/ q , so that the time intcrval between the two evems as measured by O ' is

bxample ứie single clock was taken to be at rest with respect to 0 The samc result would

Be clock were takcn to be at rest with respect to O ' Thus, in general, suppose a single

"ough a time interval Alg l f this clock is m oving wilh a velocity V with respect to an etermine that his two clocks advance through a time intcrval Aí given by

CH í ! l :

Clocks I t n d 2

â dvance throuRh ủ í > Afo

QSingle clock A dvuìccs through ửíQ < A /

Fig 5-1 Time Dilarion as Vieweđ by Observer o

Time dilation is a very real effect Suppose in FÌ2 5-1 cameras are placed at the location o f clock 2 and

at the location o f the single clock, and a picture is taken by each camera when the single clock passes clock

2 WTièn ứie pictures are developed, each picture will show the same thing— that the single clock has

advanced through AÍQ w hile clock 2 has advanced Ihrough A t > A/q, with Af and A/q related by the time

dilation expression.

A W arning!

It is importanl to keep clear the distinction between the “time separation" o f two events and the

“proper time interval” berween two e\'ents l f observers oand O'measure the time separation between two

events ứiat, for both observcrs, occur at different spatial locations, then these time separations are n o l

related by sim ply multiplying or dividing by

S olved P rob lem sThe average lifetim e o f /j-m esons with a speed o f 0 9 5 c is mcasured to be 6 X 10“‘ s Compute the

average lifetim e o f /j-m esons in a system in which they are at rest.

Ans The time measured syslem in which the ;i-mesons are at rcsi is the proper lime

Afo = (A r)\/l - = (6 X l O - S ) ^ ! -(0 9 5 )2 = I g7 X lQ-‘ s

5 2 An airplanc is moving vviih respect to the eanh with a speed o f 6 0 0 m /s A s determined by earth

clocks, how long will it take for the airplane’s clock to fall behind by two microseconds?

Ans From ửie time dilaiion exprcssion.

= lO^s = ll.ó d a y s

S J Observers o and O' approach each other with a relative velocity o f 0.6c If o measures the initial

distance to ỡ ' to be 20 m, how much tứne will it take, as determined by o , before the two observers

meet?

distance _ 20 m

velocity 0.6 10* m/sec = 11.1 X 10-*s

5.4 In Problem 5.3, how much time will it take, as determined by O', before the two observers meet?

Ans The two events under consideration are: ịA) the position of O ' when omakes his initial measurement,

and (B) the coincidence o f oand O'.Both o f these events occur at the origin o f O'.Thcrefore, the

time lapse measured by O ' is equal to ửie proper time between the two events From the time dilarion

• expression.

Aío = (A /)V l -(!.►*/<-) = (11,1 X 10“*

This problcm can also be solvcd by noting ứiat the inirial distancc

to the đistance measured by othrough ứie Lorenlz contraction:

5.6 Determine the ansvver to Problem 5.5 relativistically.

Am The half-life o f 1.8 X 10”* s is determined by an observerat rest with respecl to the pion beam Froir the poúit o f view o f an observer in the laboratory, ứie half-life has becn increased because o f the timt dilaiion, and is given by

A;o 1.8 x l 0-* s ,

^ t = = 3 X 1 0 ' * s

y i - ( 0,8)2 Therefore, the distance travcled is

d = v M = (O.g ; 3 X 10*m /s)(Jx 10-‘ s) = 7.20m

For an observer at rest wiứi rcspcct to the pion bcam, the distance dp the pions have to ữavcl ìs shorter than ứie laboratory distance d, by the Lorentz contraction:

d, = = í/,v/l - ( 0 8)2 ^ 0 6í/, The time elapseđ when this distance is covered is

0.6d,

10-" s = :

: 3 X 10*m/s

S u p p le m e n ta ry Problem s5.7 An atom dccays in 2 X 10“^ s W"hat is ihc decay lime as measured by an observer in a laboratory vvhen the

atom is moving with a speed o f 0.8c? Ans 3.33 X 10"^ s

5.8 How fast woulđ a rocket ship have to go if an obsen-er on the rocket ship aged at half ihe rate o f an obsen-er

on the eanh? Ans 0.866c

5.9 A n ia n \v ith 6 0 y ears to live w an ts lo visit a distant galaxy which is 1600Ó0 lieht years away \V hat m usl be his

constant sp eed? A n s v / c = 1 — (0.7 0 3 X 10"^)

5.10 A panicle moving at O.Sc in a laboraiory decays after ưaveling 3 m How long did it exist as measured bv an

observer in ihe laboratory? Ans Ỉ 2 5 x l0 ~ ® s

5.11 \Miat does an obser\er moving with the particle ofProblem 5.Ỉ0 measure for the time ihe particle lived beforc

đecaying? Ans 0.7 5 X 10"® s

Relativistic Space-Time Measurements

ous chapters we discussed, more or less separately, relativistic space measurements and Tie mcasurements There are, hoNvever, many types o f problems where space and túĩie

; are intemvined and cannot be treated separately.

S olved P rob lem sick moves with a v elocity o f 0 6 c relative to you along the direction o f its length How tll it take for the meterstick to pass you?

he iengứi o f the meterstick as mcasurcd by you is obiaìncđ from ihe Lorcntz contraciion:

L = io^ /l - (t-2/c2) = (I m)v/l - (0.6)= = 0.8r

: time for the meterstick to pass you is then found from

dìstance = velocity X timc 0.8 m = (0.6 X 3 X 10*m/s) X Ai

A / = 4 4 4 X 1 0 " ’ s

ycars for light to reach us from the most distant pans or our galaxy Coulđ a human

b at a constant speed, in 30 years?

6 dislance traveled by light in 10’ yeare is, according to an obscrver ai resi with respect to the carth,

»here c is cxpressed in, say, mi/yr If this obsener now moves with constant speeđ V wiih respect to

the carth the dístance d that he has to travel is shoncned according to the Lorentz contraction:

d = ư„v/l = (10’c ) / r ^ õ ? 7 ? ) The time inienal available to travel this distance is 50 years, so thai

Solving,

lO'cv/1

50

- = V l - 2.5 X 10'- a 0.999999 S75 Therefore a human traveling at ihis speed will finđ ửiat when he completes ửie ưip he has aged 50

years.

6 3 A ;i-m eson with an average lifetjm e o f 2 X 10"* s is created in the upper acmosphere at an elevation

o f 6 0 0 0 m \Vhen it is created it has a velocity o f 0 9 9 8 c ứi a direction toward the earth What is the

average distance ửiat it will travel before decaying, as determined by an observer on the earth?

{Classically, this distance is

= 1' A f = (0 9 9 8 X 3 X 1 0 * m /s)(2 X 10“S ) = 599 m

so ihat //-m eson s would not, on thc average reach the earth.)

.•1/15 As detennined by an observer on the eanh, the !ifetime is increased because o f time dilalion:

Aí 2 x l 0 - ‘ s , / r r ; - (0.998)-' : = 3 1 6 x 10-‘ s The avcrage distance travelcd, as determineđ by an earth observ'er, is

= (0.998 X 3 X 10* m/s)(31.6 X 10-‘ s) = 9470 m Thus, an obsener on thc carth delermines that, on the average, a ;j-meson will rcach the earth.

6 4 Consider an o b sen er at resi with rcspect to the íi-m eson o f Problem 6.3 How far will he measure

the earth 10 approach him before ữie /j-meson disintegrates? Compare ứiis distance with the

diitance he m easures from the poin l o f creaiion o f the p-m eson to the earth.

Am As deiennined by an obscr\’er at rest with respect to the /j-meson, the distance travcled by the earth is

d = v à t„ = (0.998 X 3 X 10*m/s)(2 X lQ-‘ s) = 599m The iniiial disiance, i , to the eanh, however, is shortencđ because o f the Lorcntz contraciion:

i = io > /l -{i>-’ /í^) = (6 x 10’ m )^ ! -(0 9 9 8 )^ = 3 7 9 m Thus, an obscr\'er on thc íi-meson dctermines that, on ứie average, it will reach the earth, in agreement

with the result o f Problcm 6.3.

<6.5 A pilot in a rocket ship traveling w iứi a velocity o f 0 6 c p asses the earth and adjusts his clock so ứiat

it coincides wiih 12:00 P.M on earth A t 12;30P.M., as determined by ứie pilot, ứie rocket ship

passes a space station that is stationary wiih respect to the earth What time is it at the station when

Ans From ứie time dilation expression,

30 min

y i - (0.6)^

Therefore, the time at the space station is 12;37.5 P.M,

6.6 In Problem 6.5, what is the distance from the earth to the space station as detennined (a ) by ửie pilot? (Ế) by an observer on the earth?

Ans (a) distance = velocity X tim e = (0.6 X 3 X 10* m/s)(30min X óOs/min) = 3.24 X 10" m ( i) dìstance = velocity X dme = (0.6 X 3 X 10*m/s)(37.5min X 60s/min) = 4.05 X lo " m

6.7 Refer to Problems 6,5 and 6 6 W hen the rocket ship passes ứie space station, the pilot reports to

earth by radio W hen does the earth receive the signal, (a) by eartìi tứne? (b) by rocket túĩie?

Ảns (a) According to an earth observer,

distance 4.05 X 10" m 1 min time = — , = —— ^ -X = 22,5 min velocity 3 X 10*m/s 60 s Thus the signal arrives, according to an observer on the earth, at

12:37.5 P.M + 22.5 min = 1:00 P.M.

(6) Accordỉng to the pilot,

distance 3.24 X 10" m 1 min time = - ■, = — -—^— — X , " = 18 min

Ans Thcre exists a sccond observer, ơ \ moving relative to ứie fifst observcr who will ổctcrmỉne that thí

two events occur at the Sãiĩie spatỉal locatỉon The propcr time intcrval betwccn the two cvcnls ỉs thí subtracting two Lorentz transformations

^ 3 6 x 1 0 « m -i;(2 s )

v/1 - ( é ^ ị é )

u = 1.8 X 10* m /s = 0.6c Agaìn subtracting two Lorentt ứ^sformations, wc obtain ihc proper timc inicrval as

= - = 4 = = - = -3 x lO » m /s

Another way to soh e ứiis problem is 10 use u and the time dilation expression;

Aio = (Aí) = (2 s)^ /r^ " {ã 6 7 = 1.6 s

6 9 For obser\'er o , rvvo events are simultaneous and occur 600 km apart What is the time difFerence

ber\veen ihese tsvo events as detennined by O ', w ho measures ứìeir spatial separation to be

The minus sicn denoies that evenl A occurred aíter event B as determined by O'.

Problems 6 ] 0 -6 ỉ 2 illustrate the íamous “nv’in e ữ e ct” in sp ecia l Reìativity.

6 1 0 Observer o \ moving with a speeđ o f 0 8 c relative to a space platíorm, travels to a-Centauri, which,

at a distance o f 4 light years, is the nearest star to the piatform When he reaches the star he

immcđiatcly tums around and retums to the platform at the same speed W hen O ' reaches the space

p!aĩform, compare his age with ửiat o f his twin sister ơ , who has stayed on the platform.

Ans According to o ứie tìme elapsed during ứìe trip from the space pIatform to a-Cenlauri is

^ _ disiance 4 y r X {distance traveled by lighư yr)

velocity 0.8 X (distance traveled by lighưyr) ^ Sincc thc rctum trip lakcs placc wiih ứìc same speed, ứie total time eiapseđ, as measured by ứìe

plaiỉoưn observcr o , is

'^'rounđ trip = 'Oyr

0 ' measures the propcr time inter\'al betwecn thc departurc from Ihe plaiíorm and ứic anrival ai

thc siar Hence, from the timc dilation cxpression.

Aío = (Aí)v/1 - = (5 y r)^ ! - (0.8)= = 3 yr

and the toial timc clapsed, as rricasuređ by 0 ', is

= 6 yr

Thcrcfore, O' is 4 ycarĩ younger ứian o when ứiey meet Tliis resuit illustrates the íamous ‘*twin

to get back home ihc ưavcling twin must tum around This tuming arounđ is real (O' cxpcricnccs

mcasurable accelerations), in contras! lo thc apparcnt tuming around that O ' obscrves o f o (who

expencnces no acceleration during her entirc history) Thus the motion o f O' is equivalcnt to that of

two diíTerent inertial obscrvers, One moving wiih V = + 0 8 c and ứie other moving wilh V = —O.Sc

6 U , Refer to Problem 6 Í0 Suppose that every year, as determined by ơ , o sends a iight signal to ỡ '

How many signals are received by O ' on each leg o f his joum ey? (In other words, what would t\vừ

O ' actually s e e if he looked at his sister o through a telescope?) Ans As determined by o , brother O ' reaches a-Centauri at f = 5 yr In order for a light signal to rcach

a-Centauri simultaneously with ỡ ' , it must have been sent by o at an earlier time, determừied by

distance _ 4 yr X (distance traveled by lighưỵr)

= 4yr velocity disiance traveled by lighưyr Thereíore, a signal sent by ơ at í = I yr reaches a-Centauri simultaneously wiứi O'. Since o sends í

toial o f 10 signals, the remaining 9 signals all reach O' on ihe retum joumey.

6.12 Refer to Problems 6 Ỉ 0 and 6 11 Suppose that every year, as determined by O ', O ' senđs a ỉighi signal to o Consider the signal sent by O ' just as he reaches CỂ-Centauri What is the time, aí đetermmed by o , when this signal is received (That is, whaí \vouId twin o see i f she looked at hei

brother O ' through a telescope?)

Ans As determineđ by o , brother O ' reaches a-Centauri at / = 5yr A light signal sent by O' ữorr a-Centauri will reach o in a time interva! (as determineđ by 0 ) o f

^ distance _ 4 yr X (distance traveleđ by light/>T) velocity distance traveled by light/yr ^

Therefore, ứiis signal reaches o at / = 5 yr + 4yr = 9 yr Hence, o f the six signals sent by o three ol

ứiem are received by o during the first nine years (one cvery threc years) and the remaining three ar< receiveđ by o during the lasl year.

A man in the back o f a rocket shoots a high-speeđ buỉlet toward a target in the front o f the rocket, The rocket is 60 m iong and the bullet’s speed is 0 8 c, both as measured by the man Find the time that ửie bullet is in Aight as measured by the man.

velocity 0.8 X 3 X 10* m/s = 2.50 X 10"’ s 6.14, Refer to Problem 6.13 I f the rocket moves with a speed o f 0 6 c relaiíve to the earth, fìnd the time that the bullet is in Aight as measured by an observer on the earth.

Subtracting tw'0 invcrse Lorenlz transformations:

Ans As determined by pilot A,

A/ 5 x l O - ’ s 1-7 « 1.8 x 10'm /s = 0 6c '

6.16 In Problem 6 ,15, what is the time interval, as determined by a pilot in the nose o f B, bctween

ÀỈĨS The relaiive specds are Ihe same as đetermined by each observer Pilot B measurcs ứie lengih of

spaceship 4 to be contracted according 10

L = L o ự l - (r^/c^) = ( 9 0 m ) y i ^ ^ 6 ) * = 72m

The time intcrval as measured by B is Ihen

72 m

V 0.6 X 3 X 10*m /s '

6.17 A rocket ship 90 m long travels at a constant velocity o f 0 8 c relative to the ground A s ứie nose o f

ứie rocket ship passes a ground observer, ửie pilot ừi the nose o f the ship shines a Aashlight toward

ửie tail o f the ship, W hat tứne does ứie signal reach ửie tail o f the ship as recordeđ by (a) the pilot,

(b) the ground observer?

Ẩns (a) Let events A and 3 be deíĩned by the emission o f the Hght signal and the light signal’s striking the

tail o f ứie rocket, respcciively Since the signal travels al speed c in ửie negative đừection,

“ C - 3 X 10“ m /s (t) Subtraction o f tv.'0 inverse LorenLz ưansformations gives

- ( 0 8 )=

6.18 Refer to Problem 6.1 7 VVhen does the tail o f the rocket pass the ground observer, (a) according to

ứie ground observer? (è ) according to ứie pilot?

Ans (a) As determined by the ground observer, the lengứi, L, of the rocket is

6.19 The speed o f a rockct vviứi respect to a space station ịs 2 ,4 X 10* m /s , and observers O ' and 0 in

the rocket and the space station, respectively, synchronize ứ ieừ clocks in the usual íashion (i.e.,

í = / = 0 when JT = y = 0) Suppose that o looks at 0"% clock ửirough a telescope What time

does he see on ỡ ' ’s clock when his ow n clock reads 30 s?

Ans Let evenls A and B be detined, respectively, by the emission o f the light signal from O ' and the

reception o f ứie same signal by o,Our problem is to find 1^ Applying the inverse Lortntz

This result, and that o f Problem 6.20, point out the distinction betvveen seeing an event and

measuring the coordinates o f the same event

6.20 Refer to Problem 6.19 If O ' looks at 0 's clock through a telescope, what time does his o\vn clock

read when he sees 0 's clock reading 30 s?

Ans Let events A and B be delĩned by the emission o f the light signal from o and reception o f the same

signal by O', respectively Our problem is to find (g Applying the Lorentz transfonriations to event A

gives

0 - ( 3 X 10*m/sK30s) í' = • 0 - (3 X 10'm/sK ■ = - 1 5 0 X 10*m

6.2 1 The equation for a spherical pulse o f light starting from the origin at / = í' = 0 is

+ - c V = 0 Show from the Lorentz transformations that o ' w ill also measure this same pulse to be spherical, in accord with Einstein's second postulate stating that the velocity o f light is ứie same for all observers

Ans From the inverse Lorentz MnsfonnatÌĐns

y 1 - 1 3- 3 - (x'^ + I^r- + 2vx'r')

Substituting, one íìnds that

+>^ +z= - + / Therefore sincc = 0, we also have

x'^ + / “ + 1'^ = 0

6.22 Show ứiat the diíĩerential expression

dx^ + d \^ + - c ' d r

is invariant under a Lorentz transformation.

If Ihe cxpression is ỉnvanant, H will retain the same form when expressed in terms o f the primed

coordinates From the invose Lorentz traJisformation One fìnds

is invariant Iinder a Loreniz tnmsformation.

Ans The equation will be invariant i f it reiains the same form when expressed in terms o f ứie new variables

To express the wave equation in terms o f the primcđ variables we íìrst find from the

Lorentz transformations that

6.24 An unstable particle wiứi a mean liĩetìme o f 4 ịis is fonned by a high-energy acceierator and projected through

a laboratory Wiửì a speed o f 0.6c (a) What is the mean lifetime o f the particle as determined by an observer in

the laboratory? (ố) What is the average distance that the particle travels in the laboratory before đismtegrating? (c) How far does an observer al rest with respect to the panicle determine that he ữ^vels before the particle đisintegrates? Ans {a)5ịiS-, (6)900m ; (c)720m

6.25 A /í-meson with a lifetime o f 8 X 10“^ s is formed lOOOOm high in the upper atmosphere and is movừi^ directly toNvard the eanh íf the /í-meson decays just as it reaches the earth’s surTace, what is its speed relative

to the earth? Ans 0.972c

6.26 A meterstick moves along the x-axis with a velocĩty o f 0.6c The miđpoint o f the meterslick passes ơ at / = 0

As đetennined by 0 , wherc are the enđs o f the meterstick at / = 0? Ans 40 cm and - 4 0 cm

6.27 Obser\'er o measures the area o f a circle at rest in his j;>'-plane to be 12 cm" An observer o ' moving relative

to o at 0.8c also obser\'es the figure area does 0 measure? Ans 7.2 cnr

6.28 As đetennined by observ'er o a ređ light Aashes, and, 10“® s later, a bluc ỉight Aashes 600 m farther out on the

x-axis What are ửie magnitude and direction o f the velocity o f a second observer, 0 \ if he measures the red

and blue Aashes to occur simultaneously? Ans + 0.5c

6.29 Reícr to Problem 6.28 What is the spatial separation o f the ređ and blue Aashes as đeiermined by

ơ '? Àns 520 m

6.30 A rocket ship ! 50 m long travels at a specd o f 0.6f As the tail o f the rocket passes by a man on a staíionary when thc light reaches it? (6) As measured by Ihe observer on the space phiíform, how much time elapses recepiion o f thc sígnal as detemũned by an observer in Ihc nose o f the rocket?

Ans (a)300m : ( è ) ! 0 “^s; (c)0.5 X 10" S

6.31 Two cvcnts occur at the same place and are separatcd by a 4 s time intcrval as deiemiìned by onc obscrver ỉ f a second observer measures the time separation between thcse two events to be 5 s, what is his đetermination of their spaiia! separation? Ans 9 X 10*m

6.32 An observcr sets o ff two Aashbulbs that arc on his x-axis Hc rccords ihat the íìrst bulb is sct o ff at his origin at

1 o*ciock and the second bulb is set o ff 20 s laicr at X = 9 X 10* m A sccond observcr is moving atong ihe common X - / axis with a speed o f —o.óc with respect to the first observcr What are thc úme and spatial separations betwccn Ihe lwo fiashes as measured by ihc sccond observcr?

33 The relalivc spced o f o and O' is 0.8c At í' = 2 X 10"’ s, a super bullet is fired from = 100 m Traveling

in the negative y-direciion with a constant specd, it strikcs a target at ihe origin o f O' at r" = 6 X lO '’ s As

determincd bv o , what is the speed of the bullet and how far did it travel?

Ans 3 X 10’ m /s ; 6 6 7 m

34 A ground obsener determincs that it takes 5 X 10"'’ s, for a rocket 10 travel betvveen two markers in th e

ground that are 90 m apan \Vhat is the speed of the rocket as detennineđ by the ground obser%'er?

35 Refer to Problem 6,34 As determined by an observer in the rocket, what is ửie distance between the two

markers and ửie time mterval between passing the two markers? Ans 72m; 4 X 10“'' s

,36 A laser beam is rotated ai 150 rev/min and ứirow5 a beam on a screen 50000 miles away What is the svveep

speed o f ứie bearaacross the screen? Ảns 7.85 X 10^ m i/s (note; since c = 1.86 X 10’ mi/s, the sweep

speed is larger than c.)

.37 Shovv that ửie expressions r + T - írr^ m d dx^ + d}~ + dz^ - c ĩd r arc not invariant under Galilean

transformations.

Relativistic Velodty

Transformations

velocity ữansfonnations, \ve consider an arrangement identical lo that for the Lorentz

■ansformations (Fig, 1-1) O ne obser\’er, O ', m oves along ihe com m on x - x ' axis at a constant [iửi respect to a second observer, 0 Each observer measures the velocity o f a súigle particle, ding (u ,, u,„ Uj) and Q ' recording (i4 , u'ỵ, u Ị) for the com ponents o f the particle’s velocity entz coordinate transformations One finds the follow ing L o ren l: velocity ư a n sfo rm a tio n s (see

1 + (v/c^X

» ĩV l - (v^/c^)

l+ ( r / c ^ K

(7 2 )

i with the Lorentz coordinate transforTnatíons, the inverse velocity ừansformations can be

: velocity transfomiations (7 1 ) by interchanging primeđ and unprimed variables and

This is to be expected from symmetry, since from Postulate 1 o f Section 2 4 both Bmpletely equivalent, and observer o m oves with a velocity o f - V with respecl to O'.

E NTZ V E L O C IT Y T R A N S P O R M A T IO N S A N D T H E S PE E D 0 F L IG H T the experiment discussed in Section 3.1 where a light signal is sent in the x - x !

j com mon origin when o and o ' pass each other at f = /' = 0 If o measures the signars

|n ts to be u , = c, Uy = = 0 then, by ( 7 / ) , O' w ill measurc

, u , ~ V c — V , , ^

7.2 G E N E R A L C O N S ID E R \T IO N S IN SOLVĨNG V E L O C IT Y PR O B L E M S

In v elocity problems there are three objects involved: two observers, oand O ', and a particle, p.The

particle phas tw o velocities (and, hence, six numbers) associated with it: its velocity vvith respect to o ,

(u ,, u ,) its velocity with respect to 0 ‘, (u^, uỊ) The quamity V appearing in ứie velocity

transformations is the veIocit>' o f O'wilh respect to o.

\M ien anacking a velocity problem, one should íìrst determine which objects in the problem are to be

identiíied with o O ', and p.Sometim es this identification is dicated; other times the identiíìcation can be

made arbiưarily (see, for exam ple, Problem 7.3) O nce the identiíìcation has been made, one ữien uses ửie

appropriate Lorentz velocity transfomiations to achieve the answer.

In dealing vvith velocity problems, the best way to avoid mistakes is not to forget the phrase “ with

respect to.” The phase “ velocity o f an object" is m eaningless (both classically and relativistically) because

a v elocity is always measured with respect to something.

7.3 T H E R E L A T IV IST IC D O PPL E R E FFE C T

Consider a source that emits electromaanetic radiation with a Irequency V(| as measured by an observer

w ho is at rest with respect to the source Suppose this same source is in motion wiih respect to anoứier

obser\er, who measures the ữequency V o f the radiation received from the source \Vith the angie 0 and

velocit>' V o f the source as detìned in Fig 7-1, the ử equency V as measured by observer ois given by the

Because all observers measure the speed o f light as c, the above equations also aIlow ứie change in

wavelength to be obtained via À = c /v

S o lv ed P ro b lem s7.1 Derive the Lorentz velo city ơansfonnation for the x-direction.

À n s Taking the differentials o f the Lorentz coorđinate transformations one fìnđs

Dividing dx! by dt' gives

, d y f d x — V d í

d x

7.2 At what speeds w ill the Galilean and Lorentz expressions for điffer by 2%?

1 - (u/c2)u 1

-Rearranging,

Thus, if the product vUg cxceeds 0.02c^, ửíe error in using ưie Galilean tnmsformation instead o f ứie

Lorentz ữansíormation will excceđ 2%.

7.3 Rocket A ữavels to the right and rocket B travels to the left, with velocities 0 ,8 c and 0.6c,

respectively, relative to the earth What is the v elocity o f rocket A measured from rocket B I

Ans Lct observere ỡ , O ' and ứie panicle bc associated with the earth, rocket B, and rocket A, respective!y

respect to rocket A.)

7 4 Repeat Problem 7.3 i f rocket yi travels with a velocity o f 0 8 c in the +.v-direction relative to the

Lei observers o , O ' and the panicle be associaieđ with the earth, rocket B, and rocket A, respeciively.

7.5 A panicle moves with a speed o f O.Str at an angle o f 30° to the JT-axis, as determúied by o. What is

the velocity o f the particle as determined by a sccond observer, 0 \ moving with a speed o f - 0 ,6c

along the com mon x - x ' axis?

/Ins For observer o we have

u , = (O.Sc) cos 30“ = 0.693C = (0 8 c) sin 30° = 0.4 0 0 c

Using ứie Lorenc velocity ữansformations, we have for observer o '

u , - v _ 0.693C - ( - 0 6 c )

= 0.913c

(0 4 c )v /l-(0 6 )^ , , , ,

='( 0 6 9 3 c )

The speed measured by observer O ' is

u' = J u Ị + uỹ = V(0.913c)ỉ + (0.226c)2 =: 0.941c

and the anglc ệ ' thc velocity makes with the y-axis is

0,226cian<p = “ =

u ; 0 9 13c= 0.248 ò' = 13.9“

7.6 Consider a radioactive nucleus ihat moves vvith a constant speed o f O.Sc relative to the laboratory

The nucleus decays and emits an electton with a speed o f 0 9 c relative to ửie nucleus along the

direction o f motion Find the velocity o f ứie electron in the laboratory frame.

Àns Lct the laboraiory observer, Ihe mdioactỉve nuclcus and ư>e electron be respectively associaled wiih o ,

O ' and the panicle Then

_ < + f _ 0.9c + 0.5c _ ^

(0 5 c)(0 9 c) ■

7.7 Refer to Problem 7.6 Suppose that the nucleus decays by emitting an electron with a speed o f 0.9 c

in a direction perpendicular to the direction o f (the laboratory’s) motion as determined by an

observer at rest with respect to the nucleus Find the velocity o f the electron as measured by an

j4ns Wiưi ửie same assocỉation as in Probỉem 7.6, O ne has

7 8 At í = 0 observer o em its a photon ư aveling at speeđ c in a dừection o f 60° with ứie x-axis A

second observer, o ữ a v e l s wiứi a speed o f 0 6 c along the com m on x - x ' axis What angle does the

photon make with the y -a x is o f 0 "ì

7 9 The speed o f light in still water is c / n , where the index o f retiaction for water is approxúnately

n - 4 /3 Fizeau, in 1851, found that the speed (relative to the laboratory) o f light in water m oving

with a speed y (relative to the laboratory) could be expressed as

u = - + i f ' n

where the “dragging co eíỉìcien t” was measured by him to be * s: 0.4 4 , Determine the value o f k

••Íní- An observer ai rest relative to the water will measure the speed o f light to be u’^ = c/n Treating the

light as a particle, ihe laboratory observer vviil find its speed 10 be

For small values o f y ứie approximation

yields

vvhere lerms o f order y ~ /c have been neglected Thus

which agrces wiih Fizeau’s experimental result.

(4/3)-7.10 Esaluate the Doppler equation to íìrst order in v / c when the source and obser\’er are receding from

each other.

which is Ihe cỉassỉcal expression for the Doppler efFect vvhen the receíver is stationary with respect to

the medium.

7.1 1 A car is approaching a radar speed trap at 8 0 m i/h r I f the radar set works at a írequency o f

20 X 10’ Hz, what frcqucncy shift is observeđ by the palrolman at the rađar set?

Ans To first order in v/c, the frequency received by the car is

c

The car ihen acts as a moving source with this frequency The frequency received back al the rađar set

from which (80m i/h r= :35m /s)

X ‘0’

7 1 2 A star is receding from the earth at a speed o f 5 X 1 0 -’ c What is the vvavelength shift for the

sodium Dj line (5890 Ả)?

RELATIVISTIC VELOCITY TRANSPORMATIONS

Ans The Doppler equation gives

T = • J - X /q Y £T + y o r -l = •*!!

Hence, AX = 5920 Ả — 5890 Ằ=: 30 Ằ The shift is to a greater wave]engứi {red shifỉ).

7.13 Suppose that the Doppler shift Ln the sodium D 2 ĩine (5 8 9 0 Ả) is lOOẢ when the light is observed from a distaní slar Determine ứie star’s velocity o f recession.

/ l - ( ư j c ) 3 X 10*m/s /1 - 0.946 „ „14 , ,

‘'“v ì + w / c ) ~ 5 x 10- ^ m V Ì + 0'946

í

S u p p le m e n ta ry P ro b lem s7.16 A rocket moves with a velocity o f c/3 with respect lo a man Holding a lantem The pilot o f the rocket ưansformations Ans c

7.17 The pilot o f a rocket m oving a t a velocity o f0 8 c r e la tiv e to the earth observes a second rocket approaching in

the opposite directíon at a velocity o f 0.7c What does an observer on earth measure for the sêrônd rockẻtls vclocity? Ans 0.227c

7.18 An observer in rocket A finds that rockets c and B are moving away from him in opposite đirections at spccds

o f 0.6c and 0.8í, respectively What is the speed o f c as measured by

7.19 An obser\'er O ' is moving along'ứìc x - x ỉ axis wiih a speed o f c/2 with respecl lo anoứier observer, o

Observer o measures a panicie moving in the positive 7 -đirection wiứi a speed cỊy/ỹ Calculaie the velocity

o f ứie parxicle as measured by O' Àns c Ị-J Ĩ, 135*

7.20 A man stanđing on the platform o f a space station observes two rocket ships approaching hừn from opposite

directions ai speeds o f 0.9c and 0.8c At what speed đoes one rocket ship move with respect to ứie

7.21 Derive the Lorentz velocity tnmsfoimaiions for ihe V- and r-directions.

7.22 Starting ÍTom the Lorcntz velocity ơansformations, (7.7), oblain the ìnverse Lorentz velocity transformations,

(7.2):

7.23 A K°-meson, at rest, decays into a yf''-meson and a 7r~-meson, each wiửi a speed o f 0.827c When a K°-

meson travcling at a speed of 0.6c decays, what is the greaiest speeđ ứiat One o f the 7t-mesons can

7.24 VVìiai is ứìe Dopplcr shiít in 5500 Ả lighl if the source approaches the observer wiih velocity of

7.25 Suppose that the largest wavelength visible to the eye is 6500 Ả How fast must a rocket move in order ứiat a

green lighi = 5000 Ả) on Ihe rocket shalỉ be invisible to an observer on the earth?

Ans 0.257c avvay from the observer

7.26 How fast must a star receđe from ứie earth in orđer that a given wavelcngth shall be shiíteđ by

0.5%? Ans 4 9 9 x l 0 - ’c

Mass, Energy, and Momentum ìn

Relativity

[ỈEED T O RE D E K IN E C L A S S IC A L M O M E N T U M

ỉ major developments to com e out o f ứie Special Theory o f Relativity ís ứiat the m ass o f a ' with its velocity A heuristic argument for this variatión can be givén as follow s.

| a ballistics experiment where an observer, say O ', íìrcs a bullet in th ey -d ứ ectio n into a block

)ve relative to him It is reasonable to suppose ứiat the amount the bullet penetrates into the

túned by the y -c o m p o n e n t o f the m om entum o f tíie bullet, given by p'y - n iiíy , w here m' is

l e bullet as measured by o

lid e r the sam e experiment from the point o f view o f observer owho sees observer O'

pommon jT -y direction with a v elocity u Since the tunnel left by the bullet is at right angles

1 o f relative motion, o w ill agree with o ' as to the đistance that ứie bullet peneừates into the eforc would expect to íìnd the sam e value as O ' for the >>-component o f the buliet's ped by 0 , P y = mUy where m is the m ass o f the bullet as measured by o. From ứie Lorentz

im ation s w é find, since = 0 , that

ự l - Since from above p'y = it is seen that i f both observers assign the

I bullet, so that m' = m , ứiey w ill find Ị Ỉy Ậ P y , contraiy to what is expected.

\T I O N O F M A S S W IT H V E L O C IT V

h ỉ w e have two ch oices We can assum e that momentum principles— in particular,

^momentum— do not apply at large v elocities Or we can look for a way to redelĩne the

I body in order to make mom entum principles applicable to Special Relativity The latter

em ative was chosen by Einstein He showed that all obser\'ers w ill find classical mom entum principles

hold if the mass m o f a body varies vvilh its speed u according to

" - ỹ r í t a

ìen nĩQ, ửie rest m a ss, is ửie mass o f ứie body measured when it is at rest wiửi respect to the observer See

oblem 8 1.

J N E W T O N ’S S E C O N D LAW IN RE L A T IV IT Y

The classical expression o f N ew to n ’s second law is that the net force on a body is equal to the rate o f

ange o f the b o d y ’s momentum To incỉude relativistic eíĩects, allowance must be mađe for ứie fact ứiat

ĩ mass o f a body varies with its velocity Thus ửie relativistic generaỉization o f N ew to n ’s second law is

In relativistic m echanics, as in classical m echanics, the kinetic energy, K , o f a body is equal to the

jrk done by an extemal force in incrcasing ửie speed o f ứie body from zero to som e value u, i.e.,

^ ° F ưs

u=0 iing N ewton's second law, F = d { m u ) /d i, One íìnds (Problem 8 21) that this expression reduces to

K = m<p — mpC^

The kinetic energy, K , represents the difĩerence between the rota/ energy, E, o f the m oving particle and

e resl energ}’, Eịị, o ỉthe particle when at rest, so that

E — E q = mc^ —

the rest cnergy is choscn so that E ịị = mo<~, we obtain Einstein's famous relation

E = mc^

hich show s the equivalence o f mass and energy Thus, even w hen a body is at rest it still has an energy

intent given by £ o = niQC^, so that in principle a m assive body can be com pletcly converted into another,

ore familiar, form o f energy.

s M O M E N T U M A N D E N E R G Y R E L A T IO N S H IP

S in ce momentum is conserved, but not velocity, it is often useful to express the energy o f a body in

rms o f its momentum rather than its velocity To this end, i f the expression

V i

-squaied and bolh sides are multiplied by c^[l — ( u ^ / c r ) l one obtains

MASS, ENERGY, AND MOMENTUM IN RELAXrVITY

Using ứie results £ — m c^, E(Ị — m^cp-, and Ipl = mu, we find the đesired relationship between E a n d p to

= ( p c ỷ + or (K + mỊịC^Ý = i p c ỹ +

8.6 U N IT S F O R E N E R G Y A N D M O M E N T U M

T h e e le c tr o n -v o h (eV ) is the kinetic energy o f a body w hose charge equals the charge o f an electron,

aner it moves through a potential diíTerence o f One volt.

l e V = (1 6 0 2 X 1 0 - ” C ) ( 1 V ) = 1.602 X 1 0 - ” j

l M e V = 1 0 ‘ eV l G e V = 1 0 ’ eV The relationship 1.602 X 10“ ‘’ J = 1 eV can be looked at as a conversion factor between two different units o f energy.

The Standard units for momentum are kg ■ m /s In relativisíic calculations, units o f M e V /c are frequently used for momentum These units arise from the energy-m om entum expression

The conversion íactor is

w h ile th e m a s s o f the b u lle t a s m e a s u i e d b y o , s in c e u i = V is

= mu;v'l - (i-^/c^) = ( y i , ^ ^ ^ 1 - ( 1^ / ^ ) = =p'y

s 2 From Uie rest m asses listed in the Appendix calculate the rest energy o f an elecư on in jo u les and

^ „5, We have £o = = (9.109 X IQ -^' kgì{2.998 X 1 0 » m /s )’ = 8.187 X and

(8.187 X = 0.511 M eV

8 3 A body at rest spontaneously breaks up into two parts w hich move in opposite directions The parts

have rest m asses o f 3 kg and 5.33 kg and respective speeds o f 0 8 c and 0 6 c Find the rest m ass o f

ứie onginal body.

8 4 VkTiat is the speeđ o f an electron ứiat is accelerated through a potential difference o f 10^ V?

0.1 MeV = , — - mo<r

Substituting moC^ = 0.511 MeV (Problem 8.2) and solving, we find c = 0.548c.

8 5 Calculate the mom entum o f 1 MeV elecơon

(1 MeV + 0.511 MeV)’ = ( p c f + (0.511 MeV)^

8.6 Caỉculate the kinetic energy o f an elecừ on w hose mom entum is 2 M e V /c

8.9 The rest m ass o f a ;i-m eson is 207m o„ vvhere mQ, is ứie rest mass o f an electron, and its average lifetim e when at rest is 2 X 10’ * s What is the m ass o f a ;j-m eson i f its average liíetứne in the laboratoo' is 7 X 1 0'* s?

Ans From the time dilatỉon exprcssion,

Ans The rest mass o f a atom, in terms of the UTỉiíìed atomic mass unit (u), is approximatcly 235 u

Using the conversion 1 u = 931.5MeV, we have

Total avaiiable energy = rest energy o f u = (235 u) = 219 X l o ' MeV

‘ 2 !9 X lO^MeV ■

8 1 3 An elcctron is accelerated from rest to a velocity o f 0 5 c Calculate its change in energy.

8.1 5 An electron’s velocit>' is 5 X lũ ’ m /s How much energy is needed to double ứie speed?

Changc in encrgy = 0.024 MeV

8 1 6 A 1 M eV photon collides vviứi a stationary elecơ o n in ứie vicinity o f a heavy nucleus and is

absorbed (A free electron cannot capture a phoion.) I f ửie recoil energy o f the nucleus can be

neglected, what is ứie velocity o f ứie elecữ on after the collision?

Àns From £„,„.1 =

£ + mo,íT + m„„r = — + mo,c^ or I MeV 4- n.Sl 1 MpV =

v^l Solving, 1> = 0.94 Ic.

-8 1 7 An elecữ on m oves in the laboratory with a speed o f 0 6 c An observer m oves with a vclocity o f 0 -8 c

along ửie direction o f motion o f ứie elecư on \\Tiat is the energy o f the electron as determined by

( 6 x 1 0 ’ MeV)^ = [(3 X lo ’ MeV/c)c]’ + Eo'

Solving, £„ = 5.2 X 10’ MeV, and (see Problem 8.12)

8.20 The K“-m eson decays at rest into two n“-m esons l f the rest energy o f the K.° is 498 M eV and o f the

is 135M eV, what is the kinetic energy o f each Tt"?

,4ns Since Ihe initial and final momenia must be equal in the laboratory the K°'s move off in

oppositc directions with equal amounts o f kinetic energy ■

^inilùỉ “ ^fmal 498M eV = 2(135meV) + 2A'

A '= 114MeV

8.21 For one-dimensional motion show that

K =

F ■ í/s = mc^ - mQr

d(mu) — dt

The left-hand side of (2) is exactly the imegrand o f (/) so we obtain

K = \ ỷ d m = í ? ( m - m „ )

8.22 Show ữorn the binomial expansion that £■ - £o reduces to when u /c « 1.

8 23 What is ứie maximum speed ứiat a particle can have such that its kinetic energy can be vvritten as

jm ( ,r w ith an error no greater ứian 0.5%?

At ứie maximum speed.

8.2 4 Suppose that a force í^acts on a particle in the same direction as its velocity Find ứie coiresponding

expression for Newton's second law.

Ans The forcc F is the time derivative o f ứie momentum:

8 2 5 Using N ew lon ’s second law, find an expression for ứie relativistic velocity o f a particle o f charge q

m oving in a circle o f radius R at right angles to a magnetic fielđ B.

Ans In vector fomi NewIon's second law is

1.26 Two identical bodies, each with rest m ass mo, approach each other with equal velocities u , collide, and stick together in a períectly inelastic collision Determine the rest mass o f the com posite body.

Ans Since the initìal velocities are equal in magnitude, and the final momentum must be zero,

^iniljaJ — ^final

2m^c‘

w„ = 2mn

ự l

-1.27 What is the rest mass o f the com posite bođy in Problem 8.26 as detcrmined by an observer w ho is

at rest wiứi respect to One o f the initial bodies?

Ans CoRsider the body, A, that moves in the +AT-dừection The velocity II o f the obscrver ơ at rest with respect to /4 is equal to A's velocity, V = u The second bođy, B, has a velocity = - 1/ as measured

by o Its velociiy as measured by ơ u'ị, is obtaincd fiom Ihe Lorentz velocity ưansfonnation;

Since the composile body, c, is at rcst with respect to the laboratory (observer 0 ), its velocity with

respect to ơ is u'ị- = - u From conservation o f momentum, as deiermineđ by ơ