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The atomic masses, based on the exact number 12.00000 as the assigned atomic.mass tff the^nV** cipal isotope of carbon, 12C, are the most recent (1961) values adopted by the International Union of Pure and Applied Chemistry. The unit of mass used in this table is called atomic mass Group—*

I

II

III

IV

Period

Series'

I

I

IH 1.00797

2

2

3 Li 6.939

4 Be 9.0122

5B 10.811

6C 12.01115

3

3

11 Na 22.9898

12 Mg 24.312

13 Al 26.9815

14 Si 28.086

4

19 K 39.102

20 Ca 40.08

4

29 Cu 63.54

5

5 I**

;

.

m

55 Cs 132.905

48 Cd 112.40

7

87 Fr [223] ,

10

‘ L anthanide series: ** Actinide series:

Table A—2

57 L a 138.91 89 Ac [227]

.

57-71 Lanthanide series*

80 Hg 200.59

59 P r 140.907 91 P a [2311

50 Sn 118.69 *72 Hf 178.49

81 Tl 204.37

82 Pb 207.19

89-Actinide series**

88 Ra [226.05]

58 Ce 140.12< 90 T h 232.038

40 Zr 91.22

49 In 114.82 r

56 Ba 137.34

79 Au 196.967

32 Ge 72.59

39 Y 88.905

6 9

22 Ti 47.90

31 Ga 69.72

38 Sr 87.62

47 Ag 107.870

7 ^

30 Zn 65.37

37 Rb 85.47

6

21 Sc 44.956

60 N d 144.24 92 U 238.03

61 P m [147] 93 Np [237]

62 Sm 150.35 94 P u [242] .

F u n d a m en tal C on stants

Constant

Symbol

Value

Velocity of light

C

2.9979 X 10® m s * 1

Elementary charge

€

1.6021 X ΙΟ '19 C

Electron rest mass

■

me

9.1091 X 10“ 31 kg

Proton rest mass

mp

1.6725 X IO"27 kg

Neutron rest mass

Wn

1.6748 X ΙΟ"27 kg

h

6.6256 X 10·*34 J s

h = h /2?r e/m e

1.0545 X IO"34 J s

Planck constant Charge-to-mass ratio for electron Quantum charge ratio Bohr radius ,Compton wavelength: of electron I of proton Rydberg constant

h/e ao v

^C,P B

t 1.7588 X IO11 kg” 1 *

4.1356 X IO"15 J s ( 5.2917 X 1 0 -U m 2.4262 X ΙΟ"12 m 1.3214 X 10“ 15m 1.0974 X IO7 m - 1

.

unit (amu): I amii — 1.6604 X 10~27 kg. The atomic mass of carbon is 12.01115 on this scale because it is the average of the different isotopes naturally present in carbon. (For artificially produced elements, the approximate atomic mass of the most stable isotope is given in brackets.) y

Vi

V iI

y in

o

2 He 4.0026 ’

7N . 14.0067 15 P 30.9738

9F 18.9984

10 Ne 20.183

16 S 32.064

17 Cl 35.453

18 A 39,948

24 Cr 51.996

23 V 50.942 33 As 74.9216 : 41 Nb * 92.906

25 Mn 54.9380

34 Se 78.96 42 Mo 95.94

83 Bi 208.980

44 Ru 101.07

75 Re 186.2

65 Tb 158.924 97 Bk [249]

28 Ni 58,71 '

45 Rh 102.905

46 Pd 106.4

53 1 126.9044

84 Po [210]

64 Gd 157.25 96 Cm [245]

27 Co 58.9332

36 K r 83.80

43 Tc [99]

74 W 183.85

73 Ta 180.948

26 Fe 55.847

35 Br 79.909

52 Te 127.60

51 Sb 121.75

63 Eu 161.96 95 Am [243]

80 15,9994

54 Xe 131.30 76 Os 190.2

77 Ir 192.2

78 P t 195.09

85 At [210]

66 D y 162.50 98 Cf [249]

86 Rn [222]

67 Ho 164.930 99 Es [253]

Constant

68 E r 167.26 100 Fm [255]

69 T m 168.934 101 M d [256]

Symbol

Bohr magneton Avogadro constant Boltzmann constant

70 Yb 173.04 102 BTo

71 Lu 174.97 103

Value

MB

9.2732 X IO"24 J T " 1

Na k

1.3805 X IO -23J 0K " 1

6.0225 X IO23 m o l"1

Gas constant

E

8.3143 J 0K " 1 m o l"1

ideal gas normal volume (STP)

2.2414 X IO "2 m3 mol” 1

Faraday constant

V0 F

Coulomb constant

Xe

9.6487 X IO4 C mol” 1 8.9874 X IO9 N m 2 C ” 2

Vacuum permittivity

Co

8.8544 X IO -12N " 1 m " 2 C2

Magnetic constant

Km

1.0000 X IO"7 m kg C " 2

Vacuum permeability

MO

1.3566 X IO "6 m kg C " 2

Gravitational constant

7

6.670

Acceleration of gravity at sea level and at equator

Q

9.7805 m s ” 2

Numerical constants:

*

» 3.1416;

e

» 2,7183;

X l O " 11 N m 2 kg” 2

λ/2 — 1.4142:

\ / 3 «* 1.7320

IN T E R N A T IO N A L A D V IS O R Y B O A R D (for th e S eco n d E d itio n ) Prof. Mario B ertaIoccini P o ly te c h n ic I n s titu te o f Milan, Ita ly Prof. Ju d ith B o sto c k M assachusetts I n s titu te o f T e c h n o lo g y Camhridge, Mo., U.S.A. Prof. C am illo B u ssolati P o ly te c h n ic I n s titu te o f Milan, Italy Prof. M arcello Cresti I n stitu te o f P h y sic s “ G a lile o ” Padua, Italy Prof. M. D aune U niversity o f Strasbourg, France Prof. Kaarle K urk i-S uon io U niversity o f Helsinki, Finland Prof. Peter Laut Danish Engineering A c a d e m y L y n g b y , D en m ark Dr. H. W alther U niversity o f M unich, G e rm a n y Prof. G. Weill University o f Strasbourg, France

FUNDAMENTAL UNIVERSITY PHYSIOS 2nd Edition VOLUME TWO FIELDS AND WAVES M A R C ELO A L O N SO

Florida In stitu te o f Technology M elbourne, Florida F orm er D irecto r D ep a rtm en t o f S cien tific A ffairs Organization o f Am erican S tates Washington, D. C. ED W A R D J. F IN N

D ep a rtm en t o f Physics G eorgetow n University

▲ ▼▼

A D D ISO N -W ESLE Y P U B LIS H IN G C O M P A N Y Reading, Massachusetts · Menlo Park, California Don Mills, Ontario · Wokingham, England · Amsterdam · Bonn · Sydney Singapore · Tokyo · Madrid · Bogota · Santiago · San Juan

L ib rary o f C ongress C ataloging in P u b lic a tio n D ata (R evised fo r V ol. 2) A lonso, M arcelo. F u n d a m e n ta l U n iversity Physics. C O N T E N T S: v. I . M e c h a n ic sa n d T h e rm o d y n a m ic s, - v . 2. F ields a n d Waves, —[e tc .] Includes in d ex . I. Physics. I. F in n , E dw ard J., jo in t a u th o r. II. T itle. Q C 21.A 4 1980 530 7 9 -5 1 8 0 2 ISBN 0 -2 0 1 -0 0 1 6 2 -4 P a p e rb a c k e d itio n ISB N 0 -2 0 1 -0 0 0 7 7 -6 H a rd b o u n d e d itio n

C op y rig h t © 1 9 8 3 , 1967 b y A ddison-W esley P ublishing C o m p a n y , In c., All rig h ts reserved. No p a rt o f th is p u b lic a tio n m ay b e re p ro d u c e d , sto re d in a retriev al sy stem o r tr a n s m itte d , in a n y fo rm o r b y a n y m eans, ele c tro n ic , m ech a n ical, p h o to c o p y ­ ing, reco rd in g , o r o th e rw ise , w ith o u t th e p rio r w ritte n perm issio n o f th e p u b lish er. P rin ted in th e U n ited S ta te s o f A m erica. P ublished sim u lta n e o u sly in C anada.

FGIIIJKLMN-MA-99876543210

PREFACE TO THE SECOND EDITION

Physics is a fu n d a m e n ta l science th a t has a p ro fo u n d in flu e n c e on all o th e r sciences. Since this is so, n o t o n ly physics m ajo rs and en gineering stu d e n ts, b u t an y o n e w ho plans a career in science (inclu d in g stu d e n ts m ajo rin g in b io lo g y , ch e m istry and m a th em a tic s), m ust have a th o ro u g h u n d e rsta n d in g o f th e fu n d a m e n ta l ideas o f physics. T h e p rim ary p u rp o se o f a gen eral physics course (a n d p e rh a p s th e o n ly reaso n it is allow ed a place in th e c u rric u lu m ) is to give th e stu d e n t a unified view o f physics. This should be d o n e w ith o u t bringing in to o m a n y details. A u n ifie d view o f physics is atta in e d by analyzing th e basic p rin cip les, develo p in g th e ir im p lic atio n s, and discussing th e ir lim itatio n s. T h e s tu d e n t will learn sp ecific a p p lic a tio n s o f th e basic p rinciples In th e m ore specialized courses th a t follo w . C o n se q u e n tly , th is te x t p rese n ts w h at w e believe are th e fu n d a m e n ta l ideas th a t c o n stitu te th e co re o f to d a y ’s physics. We have given careful c o n sid eratio n to th e re c o m m e n d a tio n s an d suggestions o f p revious users o f th e te x t and th e In te rn a tio n a l A dvisory B oard o f E d ito rs in selecting th e su b je c t m a tte r and th e o rd er a n d m e th o d o f its p re se n ta tio n . In m any courses physics is ta u g h t as if it w ere a c o n g lo m eratio n o f several sciences, m ore o r less re la te d , b u t w ith o u t an y real unify in g view. T h e tra d itio n a l division o f physics in to (th e “ scien ce” o f) m ech a n ics, h e a t a n d k in e tic th e o ry , so u n d , o p tics, e le c tric ity and m agnetism , and m o d e rn p h y sics n o lo n g er has a n y ju s tific a tio n . We have d e p a rte d from th is tra d itio n a l a p p ro ac h . In ste a d , we fo llo w a logical and unified p resen ­ ta tio n , em phasizing th e c o n serv atio n p rin cip les, th e c o n ce p ts o f fields and waves, an d th e a to m ic view o f m a tte r. T h e special th e o ry o f re la tiv ity is used e x ten siv ely th ro u g h o u t th e te x t as o n e o f th e guiding p rin cip les th a t m ust be m e t by a n y p h y sical th e o ry . M any ideas o f q u a n tu m physics are in tro d u c e d ra th e r early. F o r convenience, th e te x t ap p ears in th re e vo lu m es and th e su b jec t m a tte r has been divided in to five p arts: ( I ) M echanics, (2 ) In te ra c tio n s and F ields, (3 ) Waves, (4 ) Q u a n tu m P hysics, (5 ) S tatistical Physics. In V olum e I we p resen t m ech a n ics in o rd e r to estab lish th e fu n d a m e n ta l principles n eeded to describ e th e m o tio n s we observe aro u n d us. In clu d e d in th is volum e, in o rd e r to ad a p t to th e re q u ire m en ts o f m an y sch o o ls, we have in c o rp o ra te d an elem en tary in tro d u c tio n to th e rm o d y n a m ic s a n d sta tistic a l m echanics. AU p h e n o m en a in n a tu re are th e re su lt o f in te ra c tio n s, a n d in te ra c tio n s are an alyzed in term s o f fields. P art 2, in V olum e II, considers n o t o n ly th o se k in d s o f in te ra c tio n s we u n d e rsta n d b est (th e g rav itatio n al a n d e lec tro m ag n etic in te ra c tio n s, w hich are th e in te r­ actio n s responsible fo r m o st o f th e m acro sco p ic p h e n o m en a we observ e), b u t also inclu d es a discussion o f th e n u clea r in te ra c tio n . F o r th e sak e o f co n v en ien ce, th e dis­ cussion o f th e g rav itatio n al in te ra c tio n has been placed in V o lu m e I; in V o lu m e II we discuss electro m ag n etism in co n sid erab le d etail, c o n clu d in g w ith th e fo rm u la tio n o f M axw ell’s eq u a tio n s. P art 3, w hich deals w ith wave p h en o m en a as a co n seq uen ce o f th e field c o n c e p t, is also in clu d e d in V o lu m e II. It is h e re th a t w e have in clu d ed m u ch o f th e m aterial usually covered u n d er th e headings o f aco u stics an d o p tics. The em phasis, h o w ev er, has been place d o n e le c tro m a g n e tic w aves as a n a tu ra l e x te n sio n o f M axw ell’s v

Preface

eq u atio n s. P art 3 and V olu m e II co n c lu d e w ith a discu ssio n o f M aiterW av es as an in tro ­ d u ctio n to th e m a th em atic a l fo rm u la tio n o f q u a n tu m m echanics. T h u s, V olum es I and I I cover th e usual m aterial in m ost in tro d u c to ry general physics courses. V olum e III includes th e fin al tw o P arts o f th e te x t. In P art 4 we an aly z e th e stru c tu re o f m a tte r th a t is, ato m s, m o lecu les, n uclei and fu n d a m e n ta l p artic le s—an analysis preceded b y th e necessary b ack g ro u n d in q u a n tu m m echanics. This p a rt c o n stitu te s an elem en tary in tro d u c tio n to th e q u a n tu m th e o ry o f m a tte r. F in ally , in P art 5 w e talk a b o u t th e p ro p e rtie s o f m a tte r in b u lk . T h e prin cip les o f sta tistic a l m ech a n ics are first p resen ted an d th e n applied to so m e sim p le, b u t fu n d a m e n ta l, cases. We discuss th e rm o ­ dynam ics from th e p o in t o f view o f sta tistica l m echanics. P art 5 concludes w ith a stu d y o f the th erm al p ro p ertie s o f m a tte r w hich explains h o w th e principles o f sta tistic a l m echanics and th e rm o d y n a m ic s m ay be ap p lied . T h e re fo re , V olum e III covers th e subject m a tte r in clu d ed in m o st in tro d u c to ry M odern P hysics courses, w ith th e advantage th a t it c o n stitu te s a logical e x te n sio n o f V olum es I a n d II. T his te x t differs fro m stan d a rd university-level physics te x ts n o t o n ly in its a p p ro a ch , b u t also in its c o n te n t. We have in clu d ed a n u m b e r o f fu n d a m e n ta l to p ics n o t fo u n d in m an y books a n d we have d ele te d o th e r to p ic s th a t are tra d itio n a l. T h e level o f m a th e ­ m atics used in th e te x t assum es th a t th e stu d e n t has h ad a m inim al in tro d u c tio n to calculus and is c u rre n tly en ro lled in th e in tro d u c to ry co u rse o f th a t su b ject. A lso, it is highly desirable th a t th e stu d e n t have had a physics co u rse in high school. M any a p p li­ catio n s o f fu n d a m e n ta l prin cip les, as well as a few m o re advanced to p ics, a p p e a r in th e form o f w o rk ed -o u t exam ples; th ese m ay be discussed at th e in s tru c to r’s conven ien ce o r p roposed to in d iv id u al stu d e n ts o n a selective basis. T h e m aterial in th e exam ples th u s allow s fo r fle x ib ility in designing th e course in acco rd a n c e w ith b o th th e w ishes o f th e in s tru c to r and th e b ack g ro u n d o f th e stu d e n ts. T h e p ro b lem s at th e end o f each c h a p te r are divided in to tw o groups: basic p ro b lem s and challenging p ro b lem s. T h e basic p ro b lem s are designed to d rill th e s tu d e n t and assist him in m asterin g th e m a tte r. T he m ajo rity o f these p ro b lem s sh o u ld be solved w ith o u t to o m u ch e ffo rt. T h e challenging p ro b lem s, o n th e o th e r h a n d , sh o u ld serve to stim u la te th e stu d e n t, testin g his u n d erstan d in g an d in itiativ e. A n u m b e r o f th e challenging p ro b le m s have b een ta k e n from th e free-response sectio n o f th e A dvanced P lacem en t P hysics E x a m in a tio n w ith th e perm ission o f th e C ollege E n tra n c e E x a m in a tio n B oard and th e E d u c a tio n a l T esting Service. T hese are id e n tifie d a t th e end o f th e p ro b le m ; e.g., (A P-B , 19 7 5 ) id e n tifie s a pro b lem fro m th e 1975 B (n o n -calcu lu s) E x am , w hile (A P-C; 1975) is a p ro b lem fro m th e calculus-based ex am in atio n o f th e sam e y ear. U niversities have been u n d e r g reat pressure to in c o rp o ra te in to th e cu rricu la fo r all sciences new subjects th a t are m o re relev an t th a n th e tra d itio n a l to p ic s. We ex p e c t th a t this te x t will relieve som e o f th is p ressu re by raising th e s tu d e n ts ’ level o f u n d e rsta n d in g o f physical co n cep ts an d increasing th e ir a b ility to m an ip u la te th e co rre sp o n d in g m a th e ­ m atical relatio n s. T his will p e rm it an u pgrading o f in te rm e d ia te courses p re se n tly o ffered in the u n d erg rad u ate c u rricu lu m , from w hich th e tra d itio n a l courses in m ech an ics, elec tro m ag n etism an d m o d ern physics will b e n e fit m o st. T h u s th e physics s tu d e n t will finish u n d erg ra d u a te e d u c a tio n a t a h igher level o f k n o w led g e th a n fo rm erly p o s s ib le -a n im p o rta n t b e n e fit fo r th o se w ho te rm in a te th e ir fo rm a l tra in in g at th is p o in t. A lso, th ere will now be ro o m fo r n ew er (a n d p erh ap s m ore ex citin g ) courses at th e g ra d u a te level. It is gratifying to e n c o u n te r this sam e tre n d o f u pgrading in th e m o re re c e n t basic te x tb o o k s in o th e r sciences. T he te x t is designed fo r a th ree-sem ester o r fo u r-q u a rte r general physics course. It m ay also be used in th o se curricula in w hich th e gen eral physics co u rse, using V o lu m e I and 1 1 , is follow ed b y a o n e- o r tw o -sem ester co u rse in m o d e rn p h y sics, w hich w ould use V o lu m e III. In eith er case, th e seq u en ce w ould o ffe r a un ified p re se n ta tio n to th e s tu d e n t.

Preface

We h o p e th a t th is te x t will be o f assistance to th o se progressive physics in stru c to rs w ho are c o n sta n tly struggling to im p ro v e th e courses th e y te a c h . We also e a rn e stly h o p e th a t it will stim u la te th e m an y s tu d e n ts w h o deserve a p re s e n ta tio n o f physics th a t is m ore m a tu re th a n th a t o f m o st tra d itio n a l courses. We w ant to express o u r g ra titu d e to all th o se w hose assistance has m ade th e co m ­ p letio n o f this w o rk and its revision possible. We reco g n ize o u r d istin g u ish ed colleagues, Professors D. L azarus and H. S. R o b e rtso n , w h o read th e o riginal m a n u sc rip t, and Dr. R. G. H ughes, w ho solved all th e p ro b lem s. We also w ish to express o u r deep a p p reciatio n to th e m an y users th ro u g h o u t th e w orld o f th e first e d itio n o f th is te x t. T heir help fu l c o m m en ts, w hich reach ed us in any one o f th e te n languages in w hich th a t ed itio n has b een p u b lish e d , w ere resp on sib le fo r a n u m b e r o f c o rrec tio n s an d revisions. In p articu lar, th e en co u rag e m e n t and suggestions o ffered b y th e In te rn a tio n a l A dvisory B oard o f E d ito rs, w hose m em b ersh ip is listed o p p o site th e title page, have g reatly assisted us in im proving th e clarity o f p re s e n ta tio n . T h eir help has been invaluable, H ow ever, we rem ain solely responsible fo r th e d eficiencies in th e te x t. We are also g ratefu l fo r th e ab ility and d e d ic a tio n o f th e s ta ff o f th e In te rn a tio n a l D ivision o f A ddison-W esley. Last, b u t certain ly n o t least, we sin cerely th a n k o u r wives, w ho have b een so p a tie n t w ith us. Washington, D. C. March, 1 9 7 9

M. A. E. J. F.

NOTE TO THE STUDENT

T his is a b o o k a b o u t th e fu n d a m e n ta ls o f physics, w ritte n fo r s tu d e n ts m ajo rin g in science or engineering. T h e c o n ce p ts and ideas y o u learn fro m it w ill, in all p ro b a b ility , b eco m e p art o f y o u r p ro fe ssio n a l Life and y o u r w ay o f th in k in g . T h e b e tte r y o u u n d e rsta n d th e m , th e easier the rest o f y o u r u n d e rg ra d u a te and g rad u ate e d u c a tio n will be. T h e course in physics th a t y o u are a b o u t to b eg in is n a tu ra lly m o re advanced th a n y o u r high-school physics course. Y o u m u st be p rep a re d to tack le n u m e ro u s d ifficu lt puzzles. T o grasp th e law s and tec h n iq u e s o f physics m ay be, a t tim es, a slow an d p a in fu l process. B efore y o u e n te r th o se regions o f physics th a t ap p eal to y o u r im ag in atio n , y o u m ust m aster o th e r, less ap p ealin g , b u t very fu n d a m e n ta l o n es, w ith o u t w hich y o u c a n n o t use o r u n d e rsta n d physics p ro p e rly . Y ou should k ee p tw o m ain o b jectiv es b efo re y o u w hile tak in g th is course. F irst: becom e th o ro u g h ly fam iliar w ith th e h a n d fu l o f basic law s an d principles th a t c o n stitu te th e core o f physics. S eco n d : develop th e ab ility to m a n ip u la te th ese ideas and a p p ly th e m to co n c re te situ a tio n s; in o th e r w o rd s, to th in k and a c t as a p h y sicist. Y o u can achieve th e first o b jectiv e m ain ly by reading a n d re-reading th o se sectio n s in large p rin t in th e te x t. T o h elp y o u a tta in th e seco n d o b jec tiv e , th e re are m a n y w o rk e d -o u t ex am p les, in sm all p rin t, th ro u g h o u t th e te x t, an d th e re are th e h o m e w o rk p ro b le m s at th e end o f each ch ap te r. We stro n g ly re c o m m e n d th a t y o u first read th e m ain te x t a n d , once y o u are a cq u ain ted w ith it, p ro c e ed w ith th o se ex am p les and p ro b le m s assigned b y th e in s tru c to r. T h e exam p les e ith e r illu stra te an a p p lic a tio n o f th e th e o r y to a c o n c re te situ a tio n , o r e x ­ ten d th e th e o ry b y co n sid erin g new asp ects o f th e p ro b le m discussed. S o m etim es th e y provide so m e ju s tific a tio n fo r th e th e o ry . The p ro b lem s at th e end o f each c h a p te r vary in degree o f d iffic u lty a n d have b een b ro k en in to tw o categ o ries: basic p ro b lem s and challenging p ro b lem s. T h e basic p ro b lem s are m o stly o f th e ty p e th a t sh o u ld be solvable a fte r reading th e te x t m a teria l; th e y are m ade available so th a t y o u m ay a p p ly w h a t y o u have re a d to a given p a rtic u la r situ a tio n . T h e challenging p ro b lem s, o n th e o th e r h a n d , sh o u ld fo rce y o u to p e rfo rm a series o f step s b efo re th e an sw er can be o b ta in e d ; in o th e r w o rd s, y o u m ay be req u ired to re tu rn to m aterial p reviou sly in tro d u c e d in o rd e r to w o rk a p ro b lem . T hose challenging p ro b lem s follow ed b y , fo r ex am p le, (A P-B; 1 9 7 0 ) o r (A P-C ; 19 7 0 ) are ta k e n fro m th e free-response sectio n o f th e A dvanced P lacem ent P hysics E x a m in a tio n ; th e B-exam s are no n -calcu lu s based and th e C -exam s are calculus-based; th e y e a r sta te d is th e y ea r th e q u e stio n ap peared on th e given test. In general, it is a good idea to try to solve a p ro b lem in a sy m bolic o r algebraic fo rm first, an d in sert n u m erical values o n ly a t th e end. If y o u c a n n o t solve an assigned p ro b lem in a reaso n ab le tim e, lay th e p ro b le m aside and m ake a seco n d a tte m p t la te r. F o r th o se few p ro b lem s th a t refuse to yield a s o lu tio n , y o u should seek help. O ne so u rc e o f self-help th a t w ill te a c h y o u th e m e t h o d o f p ro blem -solving is th e b o o k H o w to S o lv e 11 (seco n d e d itio n ), b y G. Polya (G ard e n C ity , N. Y .: D o u b led ay , 1957). Physics is a q u a n tita tiv e science th a t req u ires m a th e m a tic s fo r th e ex p ressio n o f its ideas. All th e m ath e m a tic s used in th is b o o k can be fo u n d in a sta n d a rd calculus te x t,

N ote to the Student

a n d y o u sh o u ld co n su lt su c h a te x t w h en ev er y o u do n o t u n d erstan d a m a th e m a tic a l derivation. B ut b y no m eans sh o u ld y o u feel d isco u rag ed b y a m a th e m a tic a l d iffic u lty ; in case o f m a th e m a tic a l tro u b le , c o n su lt y o u r in s tru c to r o r a m ore advanced s tu d e n t. F o r th e phy sical sc ien tist and engineer, m a th e m a tic s is a to o l, and is seco n d in im p o rta n c e to u n d erstan d in g th e ph y sical ideas. F o r y o u r c o n v en ie n ce, so m e o f th e m o st useful m a th em atical relatio n s are listed in an a p p en d ix at th e en d o f th e b o o k . All p h y sical calcu latio n s m u st be carried o u t using a c o n siste n t set o f u n its. In th is b o o k th e SI sy stem is used. Y ou m ay find it u n fa m iliar at first; how ever, it req u ires very little e ffo rt to b eco m e a cq u a in ted w ith it. A lso, it is che sy stem th a t is used in all m ajo r g overn m en t lab o ra to rie s th ro u g h o u t th e w orld and is beco m in g sta n d a rd in all th e m ajo r scien tific p u b licatio n s. It is a good idea to use a m ech a n ical o r e le c tro n ic slide ru le fro m th e s ta rt; th e accu racy o f th ese in s tru m e n ts and th e ir a b ility to h o ld in te rm e d ia te resu lts w ill save y o u m an y h o u rs o f c o m p u ta tio n . M echanical slide ru les, even th e sim p lest, have three-place accu racy an d th is is a lm o st alw ays su ffic ien t fo r p ro b lem s in this te x t. T he electro n ic slide ru le /c a lc u la to r has co n sid erab ly g re ate r a cc u ra cy and ap p ears to be th e indispensable to o l fo r th e scien tist o f th e fu tu re . T h e te x t does n o t stress th e h isto ric a l asp ects o f p h y sics. F o r th o se s tu d e n ts in te re ste d in th e ev o lu tio n o f ideas in p h y sics in a h isto rica l c o n te x t, th e re are a n u m b e r o f in fo rm a ­ tive te x ts available. In p a rtic u la r, we w ould reco m m en d th e fin e b o o k b y H o lto n and R oller, F o u n d a tio n s o f M odern P hysical S cien ce, seco n d e d itio n , (R ead in g , Mass.: A ddison-W esley, 1973),

CONTENTS 1 Electric Interaction, 5 1.1 introduction. 6; 1.2 Electric charge, 7; 1.3 C oulom b’s law, 8; 1.4 Electric field, / / ; 1.5 The quantization o f electric charge, 77; 1.6 Electric potential, 19: 1.7 Electric potential o f a point charge, 22; 1.8 Energy relations in an electric field. 25; 1.9 Electric current. 27; 1.10 Electric dipole. 29; 1.11 Higher electric m ultipoles, 26 2

Static Electric Field. 49

2.1 Introduction, 50; 2.2 Elux o f a vector field, 50; 2.3 G auss’s law for the electric field, 52; 2.4 G auss’s law in differential form. 58; 2.5 The polarization o f matter, 62; 2.6 Electric displacem ent, 66; 2.7 Calculation o f electric sus­ ceptibility, 6 8 : 2.8 Electric capacitance; capacitors. 75; 2.9 Energy o f the electric field. 78 3

Electric circuits, 91

3.1 Introduction. 92; 3.2 Electrical conductivity; Ohm ’s law, 92; 3.3 Origin o f electric resistance. 94; 3.4 The Joule effect, 97: 3.5 Conductors, insulators, and sem iconductors, 99: 3.6 Electrom otive force, 103; 3.7 Nonohm ic conduc­ tors. 108 4

Magnetic Interaction, 115

4.1 Introduction, /76; 4.2 Magnetic force on a moving charge, 717: 4.3 Motion o f a charge in a magnetic field, 120: 4.4 Examples o f motion o f charged particles in a magnetic field, /27; 4.5 Magnetic flux o f a moving charge (nonrelativistic), 134; 4.6 Electrom agnetism and the principle of relativity, 136; 4.7 The elec ­ tromagnetic field o f a moving charge (relativistic). 138: 4.8 Electrom agnetic interaction betw een two moving charges, 141 5

Magnetic Fields and Electric Currents, 155

5.1 Introduction, 756; 5.2 Magnetic force on an electric current, 756; 5.3 Magnetic torque on a closed electric circuit, 158; 5.4 Magnetic field produced by a closed current loop, 161; 5.5 Magnetic field o f a rectilinear current, /62; 5.6 Forces betw een currents, 166; 5.7 N ote on SI units, 767; 5.8 Magnetic field of a circular current loop, /69

Contents

6

The Static Magnetic Field, 181

6.1 Introduction. 182: 6. 2 Am pere's law for the magnetic field. 182: 6.3 Am ­ pere's law in differential form. 187: 6.4 Magnetic flux. IS9: 6.5 Magnetization o f matter. 190: 6.6 The magnetizing field, 191: 6.7 Calculation o f magnetic susceptibility. 194: 6.8 Summary o f the laws for static fields. 199 I

I he Electrical Structure of Matter, 203

7.1 Introduction. 204: 7.2 Electric interactions in atom s and m olecules. 204: 7.3 Atom ic structure. 207: 7.4 Electron energy levels: the Bohr theory. 215: 7.5 Magnetic dipole moment caused by the orbital motion o f a charged particle, 219: 7.6 Torque and energy o f a charged particle. 221 8

The Time-Dependent Electromagnetic Field, 229

8.1 Introduction. 230: 8.2 the Faraday-Henry law, 230: 8.3 The betatron. 234: 8.4 Electrom agnetic induction caused by relative motion o f conductor and mag­ netic field. 237: 8.5 Electrom agnetic induction and the principle o f relativity. 239: 8.6 Electric potential and electrom agnetic induction. 240: 8.7 The Faraday-Henry law in differential form. 241: 8.8 The principle o f conservation o f charge. 243: 8.9 The Am pere-M axwell law. 244: 8.10 The Am pere-M axwell law in differential form. 247: 8.11 M axwell's equations, 249 9

Time-Dependent Electric Circuits. 259

9.1 Introduction. 260: 9.2 Self-induction. 260: 9.3 Energy o f the magnetic field. 265: 9.4 Free electrical oscillations in a circuit. 268: 9.5 Forced electrical oscillations in a circuit. 270 : 9.6 Coupled circuits, 274 10

Wave Motion: Elastic Waves, 287

10.1 Introduction, 288: 10.2 Mathematical description o f wave m otion. 289: 10.3 Fourier analysis o f wave m otion. 293: 10.4 Differential equation o f wave m otion. 297: 10.5 Elastic w aves in a solid rod. 299: 10.6 Pressure w aves in a gas colum n. 304: 10.7 Transverse w aves in a string. 308: 10.8 Surface w aves in a liquid. 313: 10.9 What propagates in wave m otion. 317: 10.10 Group velocity, 321: 10.11 The Doppler effect. 323: 10.12 W aves in two and three dim ensions. 327: 10.13 Spherical w aves in a fluid. 332 II

Electromagnetic Waves, 341

11.1 Introduction. 342: 11.2 Plane electrom agnetic w aves. 342. 11.3 Energy and momentum o f an electrom agnetic w ave. 347: 11.4 Radiation from an o scil­ lating electric dipole, 351: 11.5 Radiation from an oscillating magnetic dipole.

Contents

356: 11.6 Radiation from higher-order oscillating m ultipoles, 359-, tion from an accelerated charge. 360 12

11.7 Radia­

Interaction of Electromagnetic Radiation with Matter, 373

12.1 Introduction, 374: 12.2 Absorption o f electrom agnetic radiation, 374: 12.3 Scattering o f electrom agnetic radiation by bound electrons. 376: 12.4 Scattering o f electrom agnetic radiation by a free electron; Compton effect, 378: 12.5 Photons, 383: 12.6 More about photons: the photoelectric effect, 387: 12.7 Propagation o f electrom agnetic w aves in matter: dispersion. 390: 12.8 Doppler effect in electrom agnetic w aves, 394: 12.9 The spectrum o f electrom agnetic radiation. 399 13

Reflection and Refraction, 409

13.1 Introduction. 410: 13.2 H uygens’s principle. 410: 13.3 M alus's theorem . 413: 13.4 Reflection and refraction o f plane w aves, 414: 13.5 Reflection and refraction o f spherical w aves, 419: 13.6 More about the laws o f reflection and refraction. 421: 13.7 Reflection and refraction at metallic surfaces, 427: 13.8 Propagation in a nonhom ogeneous medium; Fermat's principle. 428 14

Reflection and Refraction of Electromagnetic Waves. Polarization, 435

14.1 Introduction, 436: 14.2 Reflection and refraction o f electrom agnetic w aves, 436: 14.3 Propagation o f electrom agnetic w aves in an anisotropic medium, 441: 14.4 Dichroism , 447: 14.5 Double refraction. 449: 14.6 Optical activity, 454 15

W'ave Geometry, 463

15.1 Introduction, 464: 15.2 Reflection at a spherical surface, 464: 15.3 Re­ fraction at a spherical surface, 475: 15.4 L enses, 480: 15.5 The m icroscope, 486: 15.6 The telescop e, 488: 15.7 The prism, 490: 15.8 Dispersion. 492: 15.9 Chromatic aberration. 495 16

Interference, 505

16.1 Introduction, 506: 16.2 Interference o f w aves produced by two synchronous sources. 506: 16.3 Interference o f several synchronous sources, 512: 16.4 Standing w aves in one dim ension, 518: 16.5 Standing w aves and the w ave equation. 521: 16.6 Standing electrom agnetic w aves. 527: 16.7 Standing w aves in two dim ensions, 530: 16.8 Standing w aves in three dim ensions; resonating cavities, 536: 16.9 W ave guides, 538 17

Diffraction, 553

17.1 Introduction. 554: 17.2 Fraunhofer diffraction by a rectangular aperture, 555; 17.3 Fraunhofer diffraction by a circular aperture, 561: 17.4 Fraunhofer

Contents

diffraction by two equal, parallel slits. 563; 17.5 Diffraction gratings, 565; 17.6 Fresnel diffraction, 570; 17.7 Scattering, 576: 17.8 X-ray scattering by crystals. 577 18

Quantum Mechanics, 589

18.1 Introduction, 590; 18.2 Particles and fields. 590; 18.3 Scattering o f parti­ cles by crystals, 592; 18.4 Particles and wave packets. 595: 18.5 H eisenberg's uncertainty principle for position and momentum. 597: 18.6 Illustrations o f H eisenberg's principle. 598: 18.7 The uncertainty relation for time and energy, 600: 18.8 Stationary states and the matter field. 601: 18.9 Wave function and probability density, 604; 18. IO The Schrodinger equation, 606; 18.11 T h ew a v e function o f a free particle, 608: 18.12 The w ave function o f a particle in a potential box, 609: 18.13 The wave function o f the simple harmonic oscillator. 610: 18.14 The hydrogen atom. 612

PART TWO

ELECTRO­ MAGNETIC FIELDS

2

Once the general rules governing m otion have been grasped, the next step in develop­ ing an understanding o f physics is to investigate the interactions responsible for such m otions. There are several kinds o f interactions. The gravitational interaction manifests itself in planetary motion and in the motion o f matter in bulk. Gravitation, although the weakest o f all known interactions, was the first interaction to be studied carefully because agricultural and other forecasring purposes provoked an early interest in astronomy and because many phenomena caused by gravitation affect people’s lives directly. The electromagnetic interaction is the best-understood interaction and perhaps the most important for daily life. M ost o f the phenomena observed every day, including chemical and biological processes, are the result o f electromagnetic interactions between atom s and molecules. The strong, or nuclear. interaction is responsible for holding protons and neutrons (known as nucleons) within the atom ic nucleus, and for other related phenomena. In spite o f intensive research, knowledge o f this interaction is still incomplete. The weak interaction is responsible for certain processes, such as beta decay, among the fundamental particles. Our understanding o f this interaction also is still very meager. The relative strengths o f these above interactions, measured against the strong interaction as I. are electromagnetic ~ 10“ 2. weak ~ 1 0 5, and gravitational ~ 1 0 “ 38. Among the as-yet-unsolved problems o f physics are why there appear to be only four interactions and why there is such a wide difference in their strengths. Alternatively one might ask w'hy there is not only one interaction that, in various limits, gives the appearance o f the four interactions experimentally identified. It is interesting to see what Isaac N ew ton said about interactions 200 years a g o : Have not the small Particles of Bodies certain Powers, o r Forces, by which they a c t . .. upon one ano ther for producing a great P art o f the Phenom ena o f N atu re? F o r it’s well know n, th at Bodies act one upon an o th er by the A ttractions o f G ravity. M agnetism , and Electricity . . . an d m ake it not im probable but th at there may be m ore attractive Powers th an these — H ow these a ttra c ­ tions may be perform ed. I d o n o t here c o n sid e r. . . . The A ttractions o f G ravity, M agnetism , and Electricity, reach to very sensible distances . . . an d Lhcre may be others which reach to so small distances as hitherto escape observation . . . . (Opticks. B ook 111. Q uery 31)

To describe these interactions, the concept o f a field is introduced. By field we mean a physical entity that extends over a region o f space and is described by a func­ tion of both position and time. Introducing this concept to describe the interaction between two particles is appropriate since the interaction between them depends on their relative positions and m otions. Accordingly, for each interaction a particle is assumed to produce around it a corresponding field This field, in turn, acts on a second particle to produce the required interaction. The second particle, o f course, has its own field, which acts on the first particle and results in a mutual interaction. The electromagnetic interaction is customarily described in terms o f two fields; the electric field and the magnetic field. However, it should be emphasized that these two fields are not independent entities but rather are intimately related to each other.

3

and the separation o f the electromagnetic field into these two components is dictated by the relative m otion o f the electric charges and the observer. On the other hand, the concept o f a field is not used exclusively to describe inter­ actions but also to describe other physical phenomena. For example a meteorologist may express both the atmospheric pressure and the temperature as functions of the latitude and the longitude on the earth’s surface as well as the height above it. We have come to recognize that the key features o f a field, in order that the field properly describe an interaction between particles, are that the field, itself, must possess energy and momentum and that the field must be able to transport both o f these properties from one particle to another. Gravitational interaction and the gravitational field were discussed in Chapter 13 of Volume I. In this volume the electromagnetic interaction will be considered in depth. The remaining two interactions, the weak and the strong (nuclear)interactions, will be discussed descriptively in this volum e; a detailed investigation into their properties is reserved for Volume III.

CHAPTER ONE

E LE C T R IC IN TERA CTIO N

Electric Interaction

1.1

Introduction

Consider a very simple experiment: Run a comb through a person’s hair on a very dry day: when the comb is brought close to tiny pieces of paper, the paper scraps are swiftly attracted by the comb. Similar phenomena occur if a glass rod is rubbed with a silk cloth or an amber rod is rubbed with a piece o f fur. We describe these experiments by saying that as a result o f rubbing, materials may acquire a new1 property, which is called electricity (from the Greek word elektron, meaning amber), and that this electrical property gives rise to an interaction much stronger than gravitation. Several fundamental differences exist between electrical and gravitational inleiactions. First there is only one kind o f gravitational interaction, resulting in a universal attraction between any two masses. However, experiment shows that there are two kinds o f electrical interactions. When an electrified glass rod is placed near a small cork ball hanging from a string, the rod attracts the ball (Fig. I-fa). Ifthe experiment is repeated with an electrified amber rod. the same attractive effect on the ball is observed (Fig. 1-lb). However, if both rods approach the ball simultaneously as showm in Fig. 1-1 (c), instead o f a larger attraction, a smaller attraction on the ball or no attraction at all is observed. These simple experiments indicate that although both the electrified glass rod and the amber rod attract the cork ball, they do it by opposite physical processes. When both rods are present, they counteract each other to produce a smaller or even null effect. Therefore there are twro kinds o f electrified states; one glasslike and the other amberlike. The first is called positive and the other negative. Suppose now that two cork balls are touched by an electrified glass rod. It may be assumed that the two balls also becom e positively electrified. When the balls are brought together, they repel each other (Fig. I-2a). The same result occurs after the balls are touched by an electrified amber rod and acquire negative electrification (Fig. I-2b). However if one ball is touched by the glass rod and the other by the

Fig. 1-1. Experim ents with electrified glass and am ber rods.

Electric Charge

(a)

(It)

(c)

Fig. 1-2. Electric interactions between like and unlike charges, amber rod. the balls attract each other (Fig. I-2c). Thus, although the gravitational interaction is always attractive, the electrical interaction may be either attractive or repulsive. Two bodies with the same kind o f electrification (either positive or negative) repel each other, but if they have different kinds o f electrification tone positive and the other negative), they attract each other. This statement is indicated schematically in Fig. 1-3. Had the electrical interaction been only attractive or only repulsive, the existence o f gravitation might never have been noticed because the electrical interaction is much stronger. However since most bodies seem to be com posed o f equal amounts o f positive and negative elec­ tricity. the net electrical interaction between any two macroscopic bodies is very small or zero. Thus as a result o f a cumulative mass effect, the dominant macroscopic interaction appears to be the much weaker gravitational interaction

1.2

E Ie ctricC h a rg e

In the same way that the strength o f the gravitational interaction is characterized by attaching to each body a gravitational mass, the state o f electrification o f a body is characterized by defining an electric mass, more com monly called electric charge (or simply charge) and represented by the symbol q. Thus any piece of matter or any particle is characterized by two independent but fundamental properties; mass and charge. Since there are two kinds o f electrification, there are also two kinds o f electric charge: positive and negative. A body exhibiting positive electrification has a positive j 0

. _

- . 0

t .

J 0 -— Θ £-

Oi

Fig. 1-3. Forces between like and unlike charges.

FD

Electric Interaction

Reiercnee body

I i f f e r e i H 1C

Inxly

Fig. 1-4. C om parison o f electric charges q and q ' . and th eir electric interactions with a third charge Q.

electric charge, and a body with negative electrification has a negative electric charge. The net charge o f a body is the algebraic sum o f its positive and negative charges. A body having equal am ounts o f positive and negative charges (i.e.. zero net charge) is called electrically neutral. On the other hand, a particle having a non­ zero net charge is often called an ion. Since it does not exhibit gross electrical forces, matter in bulk is assumed to be com posed o f equal am ounts o f positive and negative charges. For an operational definition o f the charge o f an electrified body, we adopt the following procedure. Choose an arbitrary charged body Q (Fig. 1-4) and. place the charge q al a distance cl from Q. Measure the force Fexerted on q. Next, place another charge q' at the same distance d from Q and measure the force P . Then define the values o f the charges q and q' proportional to the forces F an d F'. That is.

Ifa value o f unity is arbitrarily assigned to the charge q ‘. the value o f q can be obtained. This method o f comparing charges is very similar to the one for comparing the masses o f two bodies. This definition o f charge implies that, all geometrical factors being equal, the force o f electrical interaction is proportional to the charges o f the particles. It has been found that in a l physical processes thus far observed in nature, the net charge o f an isolated system remains constant. In other words the net or iota! charge does not change fo r any process occurring within an isolated system. N o exception has been found to this rule, known as the principle o f conservation o f charge. We shall have occasion to discuss this rule later w'hen we deal with processes involving fundamental particles.

1.3

Coulom b's L a w

Consider the electric interaction between two charged particles at rest in the observer’s inertial frame o f reference or. at most, in m otion with a very small velocity; the results of such an interaction constitute electrostatics. The electrostatic interaction for two charged particles is given by Coulomb's law. named after the French engineer Charles A. de Coulomb (1736 1806), who was the first to stale this law in the following manner:

Coulomb's Law

Fig. 1-5. Cavendish torsion balance for verifying the law o f electric interaction between two charges.

The electrostatic interaction between two charged particles is proportional to their charges and to Ihe inverse o f the square o f the distance between them, and its direction is along the line joining the two charges. This law may be expressed mathematically by

where r is the distance between the two charges q and q'. F is the force acting on either charge, and K e is a constant to be determined by our choice o f units. This law' is very similar to the law for gravitational interaction. Thus many mathematical results proved in Chapter 13 o f Volume I are applicable here simply by replacing ymm' by K cXjq'. We can experimentally verify the inverse-square law (1.2) by measuring the force between two given charges placed at several distances. A possible experimental arrangement, indicated in Fig. 1-5, is similar to the Cavendish torsion balance used to verify the law o f gravitation. The rod AB with a charged sphere B at the end is suspended from the fiber OC, Then another charged body D is brought near, As a result of the forces between spheres D and B. the rod AB experiences a torque and twists the fiber OC. The force between the charge at B and the charge at D is found by measuring the angle fi through which the fiber O C is rotated to restore equilibrium. The constant K'c in Eq. (1.2) is similar to the constant y in the law o f gravitation except that in the gravitational case the units o f mass, distance, and force were already defined and the value o f y was determined experimentally. In the present case although the units o f force and distance have already been defined, the unit o f charge is as yet undefined. If a definite statement about the unit o f charge is made, then the value o f K 1, may be determined experimentally. Alternatively if the value o f K c is given. Eq. (1.2) may be used to define the unit o f charge. In SI units the value o f K e is assigned the numerical value o f 10“ 1 c2 = 8.9874 x IO9 where c is the velocity o f light in vacuum.* For practical purposes, w:e may say that K e is equal to 9 x IO9. *Thc choice of a particular value for K e will be explained in Section 5.7.

Electric Interaction

Then when the distance is measured in meters and the force in newtons, Eq. (1.2) becomes F = 9 x l0 9 ^ - .

(1.3)

The unit o f charge is called a coulomb and, by application o f Eq. (1.3), is defined in terms o f the force experienced: when placed one meter from an equal charge in vacuum, the coulomb o f charge experiences a repulsive force q/'8.9874 x IOy newtons. Formula (1.3) holds only for two charged particles at rest relative to the observer and in vacuum ; that is. for two charged particles in the absence o f any other charge or matter (see Section 2.6). According to Eq. (1.2), K e isexpressed in N m 2 C 2 or m 3 kg s 2 C 2. For practical and computational reasons, it is more convenient to express K e in the form K e= ~ . 4ne0

(1 .4 )

where the new physical constant e 0 is called the vacuum permittivity. According to the value assigned to K e. e 0 has the value IO7

C0 = —

4nc

8.854 x 10 12 N l H i-2 C2.

(1.5)

Accordingly. Eq. (1.3) is normally written in the form

4ne0r

0 -6)

The signs of the charges q and q' must be included when Eq. (1.6) is used. A negative value o f Fcorresponds to attraction and a positive value corresponds to repulsion

Example 1.1. Given the charge arrangem ent of Fig. 1-6. in which q t = + 1,5 x IO" 3 Γ , q2 — —0.5 x IO 3 C. i/3= + 0.2 x IO 3 C. and ,4C = Ci = 1.2 m. BC = C2 = O 5 m, find the resultant force on charge q v

Fig. 1-6. Resultant electric force produced by q, and q2 on r/3.

Electric Field

f The force F 1 between q, and q3 is repulsive while the force F 2 between q2 and q} is attractive, prom Eq. (1.6) their values are F 1- ^ 3- , = 1.875 x IOj N . 4πε0τϊ

F1

Ane0V2

- 3.6 x IO3 N.

Therefore the m agnitude of the resultant force is F = v F f + F j —4.06 x IO3 N. A

1.4

E Ie c tric F ie Id

Any region in space in w hich an electric charge experiences a force is called an electric field. The force is due to the presence o f other charges in that region. For example, a charge q placed In a region where there are other charges cy. q2, q 3, etc. (Fig. 1-7) experiences a force F - F 1-PF2 F F 3+ · · ■, and is in an electric field produced by the charges q ^ q ^ Q3 (The charge q o f course also exerts forces on q i f q 2, q 3......... but we ignore these forces for the present.) Since the force that each change q i . q 2, q 3, . . . produces on the charge q is proportional to q. the resultant force F is also propor­ tional to q ; that is. the force on a particle placed in an electric field is proportional to the charge o f the particle. Therefore to determine whether an electric field is present in a certain region, a small test charge must be brought into the region and the force experienced by the test charge must be analyzed. By definition the intensity o f the electric field. at a point equals the force per unit charge experienced by the test charge placed at that point. Thus F δ —-

q

or

F= qS

(1.7)

The electric field intensity S is expressed in new tons, coulom b or N C 1 or m kg s 2 C '.in fundamental units. N ote that in view o f the definition (1.7) if q is positive, the force F acting on the charge has the same direction as the field δ ; but if q is negative, the force F has the direction opposite to S (Fig. 1-8). Therefore if there is an electric field in a region

Electric field Positive charge

(ΐ)"

F'-ο 'ε

F qE

\

Negative charge —·-------------- ( — )

Fig. 1-7. The electric Torces acting on a positive charge at P. The resultant force on q is the vector sum o f all the forces.

Hg. 1-8. D irection o f the force produced by an electric field on a positive and a negative charge.

Electric Interaction

/

//

\ V

I '

(b)

Fig. 1-9. Electric field produced by a positive and a negative charge. where positive and negative particles or ions are present, the field will tend to move the positively and negatively charged bodies in opposite directions, and produce a separation o f charge. This effect is sometimes called polarization. Writing Eq. (1.6) in the form F = q '(q /4 n ( 0r2) gives the force produced by the charge q on the charge q' placed a distance r from q. From Eq. (1.7) we may also say that the electric field S at the point where q is placed is such that F = q S . There­ fore comparing both expressions o f F. we conclude that the electric field at a distance r from a point charge q is S = q 4ne0r2, or in vector form

Fig. 1-10. Lines o f force (solid lines) and equipotential surfaces (dotted lines) o f the electric field o f a positive and a negative charge.

Electric Field

where η, is the unit vector in the radial direction, away from the charge q. Expression 1 1 8) is valid for both positive and negative charges; the direction o f S relative to js given by the sign o f q. Thus S is directed away from a positive charge and toward a' negative charge. In the corresponding formula for the gravitational field, the negative sign was written explicitly because the gravitational interaction is always attractive. Figure I-9(a ) indicates the electric field at points near a positive charge, and Fig. I-9(b) shows the electric field at points near a negative charge. just as in the case o f a gravitational field, an electric field may be represented by lines (called field lines or lines offorce) that at each point are tangent to the direction of the electric field at the point. The lines o f force in Fig. l-10(a) depict the electric field o f a positive charge, and those in Fig. I -10(b) show the electric field o f a negative charge. In each case the lines of force are radial lines passing through the charge. These electric field lines are called lines of force because they define the direction in which a positive test charge q would tend to move when placed at that point in the field. When several charges are present as in Fig. 1-7. the resultant electric field at any point is the vector sum o f the electric fields produced at the point by each charge. That is, *· ·= ^ . i Figure 1-11 shows the resultant electric field in the case o f a positive and a negative

Fig. l - l l . Lines o f force and equipotential surfaces o f the electric field o f tw o equal b u t opposite charges.

Electric Interaction

Fig. 1-12. Lines o f force and equipotential surfaces ot the electric field o f tw o identical charges.

charge o f the same magnitude, such as a proton and an electron in a hydrogen atom. Figure 1-12 shows the lines o f force for two equal positive charges, such as the two protons in a hydrogen molecule. In both figures the lines of force o f the resultant electric field produced by the two charges have also been represented. A uniform electric field has the same intensity and direction everywhere and is represented by parallel lines o f force (Fig. 1-13). The best way o f producing a uniform electric field is by charging, with equal and opposite charges, tw'o parallel metal plates. Symmetry indicates that the field is uniform, but this assertion is Iaterverified in Section 2.3.

Fig. t-13. U niform electric field.

Electric Field

I.·»»

ε. Fig. 1-14. R esultant electric field produced by r/i and q 2 at C.

C

'1J ·'

% Cl

Example 1.2. D eterm ine the electric field produced at C (in Fig. 1-6) by charges q t and q2. which are defined in Exam ple 1.1. j There are tw o m ethods of solution. Because the solution to E xam ple 1.1 gave the force F on charge at C. using Eq. (1.7) gives < f= — = 2.0 3 x IO7 N C - 1.

Ql A nother procedure is first to use Eq. (1.81 to com pute the electric field produced at C (Fig. 1-14) by each of the charges. E quation (1.8) gives S1

91 = 9.37 XlO6 N C - 1,

4 Hf0T1

and Qi r= 18.0 x IO6 N C 1. 2 4 n e „ rf Since the electric fields are perpendicular, the m agnitude o f th e resu ltan t field is ZT= ^ S 21+ S i = 2.03 x IO7 N C - 1 . The two results are identical. A

Example 1.3 T he m otion of an electric charge in a uniform field. ▼ The equation of m otion o f an electric charge in an electric field is found from the equation F ~ m u —q&

or

a = ■ S.

m

The acceleration of a body in an electric field depends therefore on the ratio qjm. Since this ratio >s generally different for different charged particles o r ions, their accelerations in the sam e electric field are also different. If the field S is uniform , the acceleration a is constant and the path of the electric charge is a p arabola ju st as in the case of projectile m o tio n in a uniform gravitational field. An interesting case is that of a charged particle passing through an electric field occupying a limited region in space (Fig. 1-15). F o r simplicity assume th at the initial velocity u0 of the particle u hen it enters the field is perpendicular to th e direction o f the electric field. The λ -axis is placed Parallel to the initial velocity of the particle and the V-axis is placed parallel to the field After

Electric Interaction

Fig. 1-15. Deflection o f a positive charge by a uniform electric field. crossing the field, the particle resum es rectilinear m o tio n bu t with a different velocity o and in a different direction. Therefore, the electric field has produced a deflection, m easured by the angle x While the particle is m oving through the field with an acceleration af ={q/m)£. the coordinates of the particle are given by X = V0I.

y= ^(q /m )S r.

W ith the lim e t elim inated the equation of the path is

verifying th at the p ath is a parabola. The deflection a is found by calculating the slope dy/dx o f the p ath at x = a . T he result is tan x = td y/d x)I ^„=qSa/mi'l. ΙΓ a screen S is placed at L. the particle with given q/m and velocity t>0 will reach a po in t C on the screen. N oting th at tan a. is approxim ately equal to d>L because the vertical displacem ent BD is small com pared w ith d when Lis large, we have q&a ^ d mi>n —L

(1.9)

By m easuring d. L. a. and S. we may obtain the velocity U0 (or the kinetic energy) given th e ratio q/m: conversely, the ratio q/m may be found given u0. Therefore when a stream o f particles, all having the sam e ratio q/m, passes through the electric field, they are deflected by an am ount inversely p roportional to their entering kinetic energies. A device such as the one illustrated in Fig. 1-15 m ay be used as an energy analyzer, separating identical charged particles m oving with different energies. F o r exam ple (J-rays are electrons em itted by som e radioactive m aterials; if a beta em itter is placed at 0 . all the electrons will co n ­ centrate at the sam e spot on the screen only if they have the sam e energy. If they are em itted with different energies, the electrons will be spread over a region of the screen. T his second situation is found experim entally and has great im portance from the point o f view of explaining nuclear structure. By using tw o sets of parallel charged plates, we can produce tw o m utually perpendicular fields, one horizontal along H H 1and an o th er vertical along VW as show n in Fig. '-16. By adjusting the relative intensity of the fields, we can o btain an arb itrary deflection of the electron beam to any

Tlie Quantization of Elet tric Charge

P la t e s for F o c u s in g h o riz o n ta l d eH cction T lalt^ I o r . ano d e v e r t ic a l d e fle ctio n C o n tro l \ A c c e le ra tin g n rid \ \ anode

E le c tro n b e a m M e ta llic coa t i ng

H e a te r

F lu o re s c e n t

screen""

E le c tro n g u n

Fig. 1-16. Motionofa charge under crossed electric fields. Electrons are emitted from the cathode and accelerated by a large electric field. A hole in the accelerating anode allows the electrons to pass out of the electron gun and between the two sets oF deflection plates. The metallic coating inside the tube keeps the right end free of electric fields by shielding the external sources and by conducting away the electrons of the beam, spot on the screen. If either or both of the two fields are variable, the spot on the screen can be made to describe various curves or patterns. Practical applications of this effect occur in ie/enisioii tubes and in oscilloscopes, k

1.5

The Q uantization of Electric Charge

Many experiments have been devised to resolve the question o f whether the electric charge on a body is an integral multiple o f a finite quantity or whether the charge may be subdivided continuously. The classical experiment to show that electric charge appears not just in any amount, but as a multiple of a fundamental unit, or quantum, is that o f the American physicist Robert A. Millikan (1869-1953). For several years during the early part o f this century he repeatedly performed what is now know'n as the oil-drop experiment. Millikan set up, between two parallel horizontal plates A and B (Fig. 1-17), a vertical electric field i that could be switched on and off.

Fig. 1-17. Millikan oil-drop experiment. The motion of the charged oil drop q is observed through the microscope M.

i (Pj______________________ _____

F,Ietlric Inteiaction

At its center the upper plate had a few small perforations through which oil drops, produced by an atomizer, could pass. M ost o f the oil drops were charged by friction with the nozzle o f the atomizer. This experiment will be first analyzed from a theoretical standpoint. Call m the mass and r the radius o f one oil drop. For this drop the equation o f motion for free fall (i.e., with the electric field switched ofT) when its downward velocity is v is ma =■mg —βπητν where the second term on the right is due to the viscous friction of air. [See Eqs. (7.19) and (7.20) in Volume I.] The terminal velocity V 1 of the drop, when a —0. is W

J g

i

, 1. 10)

οπητ where p represents the oil density and the relation m = (fnr3)p has been used. (To be precise, the buoyancy o f the air must also be taken into account by writing p —p a instead o f p where pa is the air density.) If the drop has a positive charge q, when the electric field is applied, the equation of motion of the oil drop in the upward direction when its velocity is v is ma = q S —mg —βπητν; and its terminal velocity in the upward direction v2, when cj = 0, is q S —mg βπητ Or solving for q and using Eq. (1.10) to eliminate mg gives the charge on the drop: 6πητ(ν. T b ,) 9=— -· 0

(1.11)

The radius o f the drop may be found by measuring v , and solving Eq. (1.10) for r. The charge q is obtained by measuring v2 and applying Eq. (1.11). (If the charge is negative, the upwrard m otion is produced by applying a downward electric field.) A different procedure is foliowed in actual practice. The upw'ard and downward m otion o f the drop is observed several times by successively switching the field on and off. The velocity u, remains the same; but the velocity V 1 occasionally changes, suggesting a change in the charge o f the drop. These changes are due to the occasional ionization of the surrounding air by cosm ic rays. While m oving through the air, the drop may pick up some o f these ions. Changes in charge can also be induced by placing near the plates a source o f x- or y-rays that increase the ionization o f the air. According to Eq. (1.11), the changes Aq and AtJ2 of charge and upward velocity are related by όπ/ρ· A q = —£ - t o v t . (1.12) Ύ Sometimes Aq is positive and at other times negative, depending on the nature of the charge modification. Repeating the oil-drop experiment many times with different

Electric Potential

Table 1-1. M ass and Charge of the Electron, Proton and N eutron. Particle

M ass

Charge

Electron P roton N eutron

m„ = 9.1091 x 10“ 31 kg /Mp = 1.6725 x 10' 21 kg «!„ = 1.6748 x 10 27 kg

—e Fe 0

drops allows us to conclude that the changes Aq are always multiples o f a fundamental charge e (that is. Aq = ne)\ the value of e is