the electric field in physics

7/28/2019 The Electric Field in physics

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" T h a t o n e b o d y m a y a c t u p o n a n o t h e r a t a d i s t a n c e t h r o u g h a v a c u u m w i t h o u t t h em e d i a t i o n o f a n y t h i n g e l s e . . , i s t o m e s o g r e a t a n a b s u r d i t y , t h a t I b el i ev e n o m a n , w h oh a s i n p h i l o s o p h i c a l m a t t e r s a c o m p e t e n t f a c u l t y o f t h i n k i n g , c a n e v e r f a l l i n t o i t ."

--S ir Isaac Ne wton (1692)" B y m a g n e t i c c u r v es , I m e a n l i ne s o f m a g n e t i c f o r c e . . . , w h i c h c o u l d b e d e p i c te d b y i r o nf il in g s ; o r th o s e to w h i c h a v e r y s m a l l m a g n e t i c n e e d l e w o u l d f o r m a t a n g e n t . "

~Michael Faraday,Exper imenta lResearches 1831 )

C h a p t e r 3The Electr ic FieldC h a p t e r O v e r v i e wSection 3.2 shows how to measure the electr ic f ie ld, and Sect ion 3.3 shows how todetermine i t by ca lcu la t ion . Sec t ion 3 .4 d iscusses how to d raw f ie ld l ines~Fa raday 'swa y of look ing a t e lectric ity . S ect ion 3.5 em ploys the p r i n c ip l e o f su p e r p o s i t io n to addup the electr ic f ie lds due to more than one point charge, and Sect ion 3.6 does thesame for cont inu ous dis tr ibut ions of charge. S ect ion 3.7 d iscusses f ie ld l ine draw ing inmore detai l . Sect ion 3.8 considers the torque on and the energy of an electr ic d ipolein a uni fo rm electr ic f ie ld. Sect ion 3.9 d iscusses the force on a d ipole in a nonu nifo rmelectr ic f ie ld ( th is is re lated to th e a m ber effect) . S ect ion 3.1 0 considers the de f lect ion ofe lectr ic charges by a uni form electr ic f ie ld, as in o lder TV tubes and com pu ter mo nitors .(Mo dern TV tubes and m onitors use ma gne tic def lect ion, as d iscussed in Cha pter 10.)Sect ion 3.1 1 d iscusses w hy electr ic forces alone are no t en oug h to s tabi lize a wor ldgoverned by New ton 's laws o f mo t ion , w

3ol3~1~I

I n t r o d u c t i o nDeve lop ment o f the E lec tr ic F ie ld ConceptI n e le c t r o st a ti c s , t h e e l e c t ri c f o r c e b e t w e e n t w o c h a r g e s c a n b e t h o u g h t o f a s i n -s t a n t a n e o u s a c t i o n a t a d i s ta n c e , n o m a t t e r t h e s e p a r a t i o n b e t w e e n t h e c h a r g e s.W h e n M i c h a e l F a r a d ay , b e g i n n i n g i n t h e 1 8 3 0s , e s p o u s e d a n a lt e r n a t i v e v i e wb a s e d o n l i n e s o f f o r c e t h a t e x i s t e v e r y w h e r e i n s p a c e , m o s t o t h e r e x p e r t s i n e l e c -t r i c it y t h o u g h t i t s u p e r f lu o u s . N e v e r t h e l e s s , o v e r a c e n t u r y e a r l ie r I sa a c N e w t o n ,des p i t e h i s quan t i t a t i ve s ucces s des c r i b i ng g rav i t y v i a ac t i on a t a d i s t ance , f e l t t ha ta n i n s t a n t a n e o u s r e s p o n s e i s u n t e n a b l e : w h e n a d i s t a n t s ta r m o v e s , b y a c ti o n a t ad i s t a n c e a m a s s l i g h t- y e a r s a w a y w o u l d h a v e t o f e e l a c h a n g e d g r a v i ta t i o n a l f o r c ei n s t an t aneous l y .S i m i l a r l y , F a raday be l i eved t ha t an i n s t an t aneous e l ec t r i ca l r e s pons e was un -t e n a b l e . R e p e a t e d l y t h r o u g h o u t h i s l o n g c a re e r , F a r a d a y t r i e d to d e t e r m i n e t h es peed a t wh i ch changes i n e l ec t r i c fo rces p ropaga t e . B ecaus e , a s we wi l l d i s cus si n C h a p t e r 1 5 , s u c h c h a n g e s p r o p a g a t e a t a v e r y h ig h s p e e d ( t h a t o f l ig h t , a b o u t

1 0 8


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3.1 Introduction 1 0 9

3 x 108 m/s ) , Fa ra da y wa s una b l e t o me a sure t h i s spe e d wi t h t he m e t hod s o f h i st ime.By tho ro ug h and se l f -cr i t ica l exper im enta t ion, Faraday tes ted hi s ideas, re -j e c t i ng some a nd re f in i ng o the r s , u l t ima t e l y e mp l oy i ng t he c o nc e p t o f ma gn e t i cl ine s o f fo rc e in c om pl e t e l y ne w wa ys , a nd e x t e nd i ng t h i s c on c e p t f ro m m a g-ne t i sm to e lec t r ic i ty . Faraday ' s concepts were f i rs t given mathemat ica l formi n 1845 by W i l li a m Thom son . (Th om son wa s l at e r ma de L ord Ke lv i n, fo rsupervis ing the laying of the f i rs t e f fec t ive t rans-At lan t ic t e leg raph cable , in1865 . In th i s p ro je c t , he m a de gre a t p ra c t ic a l a pp l i c a t i on o f h i s know l e dge ofelectrici ty.)Ja me s C l e rk Ma xwe l l , i n 1855 , be ga n h i s own progra m t o de ve l op Fa ra da y ' si de as ma t he ma t i c a ll y . H e de ve l op e d t he c o nc e p t o f t he electric field, th e magneticfield, a n d ( w h e n t i m e - d e p e n d e n t p h e n o m e n a w e r e i n c l u d e d ) t h e electromagneticfield. Because o f M axwe l l , l ines of force a re now ca l led field lines. In 1865 , hefound tha t the resul t ing equat ions uni f ied e lec t r ic i ty , magnet i sm, and l ight . Hispre d i c t i on o f e l e c troma g ne t i c r a d i at ion , p ropa ga t i ng a t t he spe e d o f l igh t , isone o f the grea tes t of any sc ienti f ic achievem ents : radio, TV, and micro wa vec om mu ni c a t i ons a re a ll p ra c t ic a l c onse que n c e s o f t ha t w ork . Thu s Fa ra da y wa scorrec t about e lec t r ic i ty: e lec t r ic forces do propagate a t a f ini te speed. In 1916,Ne wt on wa s shown t o be c or re c t a bout g ra v i t y , whe n Al be r t E i ns t e i n de ve l ope da t h e o r y ~ s i n c e v e ri fi ed e x p e r i m e n t a l l y ~ i n w h i c h t h e g r a v it at io n a l f ie ld p r o p -a ga te s wi t h a f i n it e spe e d t ha t is t he sa me a s t he spe e d of l igh t . The c onc e p t o fthe gravi ta t iona l f i e ld did not a r i se unt i l a f te r the f ie ld concept had ente red thea re a o f e l e c troma g ne t i sm.Th e idea of the e lec t r ic fi e ld (som et imes ca l led the electric force field ) is simp le:one e lec t r ic charge produces an e lec t r ic f i e ld, and another e lec t r ic charge fee l s aforce due to tha t f i e ld.An example of a f i e ld, and of inte rac t ions via a f i e ld, can be seen in the

i n t e ra c t ion o f t wo wa t e r s t ri de r s on a wa t e r su r fa c e. Th e w e i gh t o f e a c h de pre s se st he wa t e r ' s su r fa c e , a nd t he de pre s s i ons p roduc e d by e a c h c a n be fe l t by t heother . See Figure 3.1. T he f ie ld, in tha t case, is th e di s tor t ion of the wa ter ' ssur fa c e . The t wo wa t e r s t r i de r s i n t e ra c t t h rough t ha t f i e l d . Whe n one move s ,t he d i s t o r t ion c ha nge s l oc a ll y, a nd t he re m us t be a t i me de l a y be fore t he c ha ngemak es i t s presen ce fe l t a t the other . Simi la rly, e lec t r ic charge pr odu ces an e lec t r icf i e l d , a nd whe n one c ha rge move s t he re mus t be a t i me de l a y be fore t he s i gna lre a c he s t he o t he r c ha rge .Ind i spe ns i b l e fo r de sc r i b i ng dyna mi c phe nome na , t he e l e c t r i c f i e l d c onc e p ta l so provides ins ights into s ta t i c phenomena . Elec t r ic f i e ld l ines for a conf igura-t ion o f e lec t r ic charges give the p a t te rn of the e lec t r ic f i e ld wi th only minim al

F i g u r e 3.1 Two wa ter striders on a water surface. On ewater strider can detect the presence o f the other by thelatter's deflection of the w ater surface. Th e totaldeflection o f the surface is the sum of the deflections d ueto each.


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110 Chapter 3 ~ The Electr ic Field

c o m p u t a t i o n . T h e e l e c t r i c f i e l d c o n c e p t h a s t w o a d d i t i o n a l a d v a n t a g e s , e v e nin e l e c t ros t a t i c s . F i r s t, be c a use e l e c t r i c c ha r ge i s t he sour c e o f t he e l e c t r i cf i e ld , t he r e i s a de e p r e l a t i onsh ip be twe e n f i e ld l i ne s a nd e l e c t r i c c ha r ge ( r e c a l lF i g u r e R . 9 ) . W e w i l l d e v e l o p t h i s i n C h a p t e r 4 . S e c o n d , e l e c t r i c a l p o t e n t i a l e n -e r gy c a n be e xp r e s se d i n t e r m s o f t he e l e c t r i c fi e ld . We w i l l de v e lop t h i s i nC h a p t e r 5 .

3 .2 Obta in ing the E lec t r ic F ie ld : Exper imentT he m a gne t i c f i e ld ne a r a m a gne t c a n be v i sua l i z e d w i th i r on f i l i ngs . S im i l a r l y ,t he e l e c t r i c f i e ld ne a r a n e l e c t r i c a l l y c ha r ge d body c a n be v i sua l i z e d w i th g r a s sse e ds. Se e F igur e 3 .2 ( a ) .

N e i t h e r i r o n f il in g s n o r g r a ss s ee d s h a v e p e r m a n e n t e l e c tr i c o r m a g n e t i c p r o p -e r t i e s , b u t t h e y a r e m o r e m a g n e t i z a b l e a n d p o l a r i z a b l e a l o n g t h e i r l o n g a x e s .Be c a us e t he a xe s o f bo th i r on f il ings a nd g r as s se e ds ha ve no p r e f e r r e d se nse , t hef ield direction i s a m b i g u o u s . C o m p a r e F i g u r e 3 . 2 ( a ) a n d F i g u r e 3 . 2 ( b ) .An a na logy to g r a v i ty le a ds t o a p r e c i se de f in i t i on o f t he e l e c t r i c fi eld . T hegr a v i t a t i ona l f i e ld ~ i s de f ine d a s t he r a t i o o f t he g r a v i t a t i ona l f o r c e /~m on a t e s tbody to i t s g r a v i t a t i ona l m a ss m :

Fm- - - - . ( 3 .1 )mN e i l A r m s t r o n g , t h e f i r s t p e r s o n t o s e t f o o t o n t h e m o o n , c a n c l a i m t o h a v es e r v e d as a t e s t m a s s o n b o t h t h e e a r t h a n d t h e m o o n . H e f o u n d t h a t ]glmoon ~"( ] / 6 ) l g l e a r t h . A n y m a s s m o n t h e s u r f a c e o f t h e m o o n is s u b j e c t t o a ~ o f t h i sm a gni tude a nd f e e l s a g r a v i t a t i ona l f o r c e m ~. A d i r e c t i ona l c he c k on ( 3 .1 ) i st ha t , s i nc e on t he e a r th ' s su r f a c e / ~m po in t s t o t he e a r th ' s c e n t e r, ( 3 .1 ) s a ys t ha ta l so po in t s t o t he e a r th ' s c e n t e r , a s e xpe c t e d .By a nalogy , t he e l e c t r i c f i e ld E a t a g ive n po in t P is de f ine d a s t he r a t i o o f t hee lec t r ica l force /~q on a tes t body a t P to i t s e lec t r ica l charge q"

-. Fq { d e f in i~ o n o f f i e l d ) f ~ i ~ iqE x p e r i m e n t a l l y , a n y v a l u e fo r t h e t e s t c h a r g e q y i e ld s t h e s a m e v a l u e f o r / ~ . T h a t

O O Electric field line

Grass seed(polarized)(a)

O OCo)

Electric field line

Grass seed(polarized)

Figure 3.2 T h e o r i e n t a t i o n o f g ra s s s ee d s i n r e s p o n s e t o a n a p p l i e d e l e c t r ic f i e ld is t h esa m e i f t he e l e c t r i c f i e ld is r e ve r se d . ( a ) F i e ld i n one d i r e c t ion . (b ) R e v e r se d f i e ld .


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3 .2 Obtaining the Electric Field: Experiment 11 1

i s w h y t h e f i el d id e a w o r k s : i t i s a p r o p e r t y o f t h e p o i n t i n s p a c e , n o t o f t h e t e s tc h a r g e . T h e t e s t c h a r g e d o e s n o t f e e l a f o r c e d u e t o i ts e lf ; t h e e l e c t r ic f i e ld is d u et o c h a r g e o t h e r t h a n t h e t e s t c h a r g e. T h e u n i t o f t h e e l e ct r ic f i el d is N / C . / ~ isc a l l e d a f i el d b e c a u s e i t is d e f i n e d a t a l l p o i n t s ~ i n s p a ce . T o r e p r e s e n t / ~ a t t h ep o s i t i o n ~ , d r a w t h e v e c t o r r e p r e s e n t i n g E w i t h i t s t a i l a t t h e p o s i t i o n ~ .~ Electric a protonield o f

F in d t h e e l e c t r i c f i el d a c ti n g o n t h e e l e c t r o n d u e t o t h e p r o to n , i n F ig u re 3 . 3 (a ) .T a ke t h e s e p a r a t i o n t o b e 1 . 0 x 1 0 - l~ m .So lu t ion : Fir s t cons ide r the d i rec t ion of /~ . The force on the e lec t ron i s , by@ -- -- Source ~ Tes t chargeT

Test charge @ -- -- Source

Positive charge Neg ative chargemakes outward field makes inward field(a) (b)

Figure 3.3 M easur ing the e lec tr ic f ieldE, w hose ta i l is a t the po in t wh ere i t i smeasured- - the te s t cha rge . ( a ) Pos i t ivesource and negative test charge. (b)Nega t ive source and pos i t ive te s t cha rge.

"opposites a t tract , l ikes repel ," to -w a r d the pro ton , which i s cons id-e red to be the source of the e lec -tr ic f ield acting on the e lectron. SeeFigure 3.3(a). From (3.2), becausethe s ign of the cha rge on the e lec-t r o n is n e g a ti v e (q = - e ) , t h e d i -r ec t ion of the e lec tr ic f ie ld on th eelectron is opposite tha t o f the forceon i t . Hence the e lectr ic f ie ld a t thesite of the e lectron is a w a y f r o m th epos i t ive ly cha rged pro ton . (Notetha t /~ has it s ta i l a t the p o in t wh erei t i s me a s u r e d - - t h e e l e c t r o n . ) B e -cause th e e lectr ic f ie ld is a pro per t yof space, we conclude quite gener-a lly tha t

positive charge produces an electric field tha t points ou twar d from the charge.Similar ly, as shown in Figure 3.3(b), for the e lectron as the source,

negative charge produces an electric field th at points in wa rd to the charge.

N o w c o n s i d e r t h e f i e ld m a g n i t u d e 1/~ 1. S e c t i o n 2 . 4 .1 f i nd s t h a t f o r a n e l e c t r o na n d p r o t o n s e p a r a t e d b y 1 . 0 x 1 0 - l ~ m , t h e f o r c e o n th e e l e c t r o n h a s m a g n i t u d eI F I - 2 . 3 x 1 0 - 8 N . W i t h q - - 1 . 6 x 1 0 - 19 C , ( 3 .2 ) y i e ld s , f o r t h e m a g n i t u d eo f t h e e l e c t r ic f i e ld d u e t o t h e p r o t o n , a t t h e s i te o f t h e e l e c t r o n ,

I E I -Fq

2 . 3 x 1 0 - S N( - 1 . 6 ) x 1 0 - 19 C = 1 . 44 x l 0 1 1 NC

A n y c h a r g e q a t th i s p o s i t i o n is s u b j e c t t o a n / ~ o f t h i s m a g n i t u d e a n d f e el s ane l e c t r i c a l f o r c e q / ~ .

T h e e l e c t r i c f i el d d u e t o t h e e l e c t r o n , a c t i n g o n t h e p r o t o n , h a s t h e s a m e m a g -n i t u d e , b e c a u s e t h e m a g n i t u d e s o f t h e f o r c e o n t h e p r o t o n d u e t o t h e e l e c t ro n ,a n d t h e m a g n i t u d e o f t h e p r o t o n c h a rg e , a re t h e s a m e a s i n t h e p r e v i o u s c as e.


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112 Chapter 3 ~ The Electric Field

~ Electric on a test chargeie ld f rom the forceSect ion 2 .5 (see F igure 2 .5) fo und the force on a charge q = 2 .0 x 10 -9 C ,d u e t o t w o o t h e r c h ar ge s q l = - 4 . 0 x 1 0 - 9 C a n d q 2 = 6 . 0 x 1 0 - 9 C :F x - l . 1 1 2 x 1 0 - 6 N , F y - - 0 . 9 8 3 x 1 0 - 6 N , an d I F l - 1 . 4 8 4 x 1 0 - 6 N .F ind the e lect r i c f ie ld a t the pos i t ion of q .Solution: Use of (3.2) gives

Ex - Fx 5.56 x 102 N Ey = Fy _ - 4 . 9 2 x 102 Nq C ' q C 'P IFI = 7.42 x 102 NI/~1 = q = Iql C "

Because q is posit ive, by (3.2) the direct ion of E is along F. If the test chargeq were negat ive, then F would be in the opposi te direct ion, but E would beunaffected.

3~2ol E xper imenta l C autionT h e t e s t c h a r g e q c a n p r o d u c e e l e c t r o s t a t i c i n d u c t i o n o r p o l a r i z a t i o n i n n e a r b ym a t e r i a l s , i n p r o p o r t i o n t o q . T h e s e i n d u c e d c h a r g e s c a n t h e n c o n t r i b u t e t o t h ee l ec t r i c f i e l d . To e l i m i na t e t h i s e f f ec t t he t e s t cha rge q m us t be ve ry s m a l l :

/~ - lim --.Fq ( 3 . 3 )q--~0 qT o s e e h o w e l e c t r o s t a t i c i n d u c t i o n c a n c h a n g e t h e e l e c t r i c f i e l d , c o n s i d e ra n e v e r y w h e r e n e u t r a l i n f i ni te s h e e t o f a l u m i n u m f oi l. I t p r o d u c e s n o e l e c t ri c

f i e l d : /~ - 6 . A pos i t i ve cha rge q , b ro ug h t up t o t he fo il , a l t e r s t he d i s t r i bu t i ono f c o n d u c t i o n e l e c tr o n s , s h i f ti n g t h e i r o r b i ta l s t o w a r d t h e p o s i t iv e c h a r g e . T h i sp ro duc es an e l ec t r ic f ie l d t ha t , a t t he s i t e o f t he pos i t i ve cha rge , po i n t s t ow ardt h e f o il , w i t h s t r e n g t h p r o p o r t i o n a l t o q . F i g u r e 3 .4 ( a ) d e p i c t s t h e f o r c e F a c t i n go n q ( d a r k a r r o w ) , t h e f i e l d / ~ a t t h e s i te o f q ( s h a d e d a r r o w ) , a n d t h e h u m p l i k e

- q

(a) (b)Figure 3.4 Force on test charges above a neutral, infiniteconduct ing sheet . (a) +q, (b) -q.


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3 . 3 Ob t a i n i n g the E l ect ri c F i e ld : T heory 113

s hape o f the cha rge d i s t r ibu t ion indu ced on the fo i l. (Th i s pa r t o f the f igu rei s s chemat i c ; the induced cha rge dens i ty does no t ac tua l ly r i s e up nea r q , b u tr a the r i t is l a rges t nea r q . ) I f t he s ign o f the cha rge q i s r eve r s ed , th e s ign o fthe in duc ed s u r f ace cha rge w i l l r eve rs e , w h ich t hen r eve r s es the d i r ec t ion o f theinduc ed e l ec t r i c fi eld : i t w i l l now po in t aw ay f rom the fo il . S ee F igu re 3 .4 (b ) . I nbo th cases, t he r e i s a fo r ce on q , o f m agn i tude p rop or t iona l to q2 . There fo re , a sq -+ 0 , ( 3 .3 ) y ie ld s /~ - 0 . (A s imi la r e ff ec t occu r s w h en a cha rge q is b ro ug h tup to a neu t r a l p i ece o f pape r, a s in the am ber e f f ec t. )T h e t e s t c h a r g e m u s t b e o f s m a l l p h y s ic a l d i m e n s i o n , b o t h t o d e f in e t h eo b s e r v a ti o n p o s i t io n o f t h e f i e ld m e a s u r e m e n t a n d t o m i n i m i z e p o l a r i z a ti o n o rinduc t ion e f f ec t s on the t e s t cha rge i t s e l f .

3~ 3 Obta in ing the E lec t r i c F ie ld : TheoryI m a g i n e t h a t y o u h a v e c a l c u l a te d t h e force/~q on a charge q a t a specific pos i t ion ,due to q l and q2 , as in F igure 3 .5(a) . Let ' s say i t i s 2 N, poin t ing a long ~ . ( Inp r inc ip le , t he r e cou ld be many o the r cha rges a l s o con t r ibu t ing to the fo r ce onq . ) N o w m e n t a l l y re p l a c e q b y Q a n d c o n s id e r h o w t o o b t a i n t h e f o r c e F Q o n Q .It is n o t mos t eas i ly ob ta ined by r eca lcu la t ing the ind iv idua l f o r ces and add ingthe m up . A s imp le r ap p roac h i s to t ake the r a t io o f Q to q , a n d t h e n m u l t ip l yb y t h e force /~q o n q . T h u s , if Q - - 2 q , t h e n t h e f o rc e o n Q h a s t w i c e t h em a g n i t u d e o f t h e f o r c e o n q , or 4 N , bu t i s in the oppos i t e d i r ec t ion , o r -~ .A n o t h e r s i m p l e r a p p r o a c h is t o d e t e r m i n e / ~ b y c o n s i d e ri n g q t o b e a t e s t c h ar g e,a n d t h e n t o u s e t ~ Q - Q ~ L e t us d e v e l o p t h e s e id e as .Cons id e r the fo r ce F q on q a t F , due to cha rges q i a t r i . W i th /~ i - r - r i ,(2 .8) g ives

-~ kqq i Ri ]~i ~. [~iF q - ~ i R2 - q ( /~ i po in t s to obs e rva t ion cha rge q ) (3.4)

Figure 3.5 (a) Geom etry for the force on q due to ql andq2. Lowercase vectors refer to distances from the origin,and uppercase vectors refer to relative distances. (b) Fieldat site of q , due to ql and q2.


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114 Chapter 3 9 The Electr ic Field

T h e n the f o r c e t~Q on Q i s g ive n by t he r a t i o~ 9q q Ri (3.5)

P r o c e e d i n g m o r e s y s t e m a t i c a l l y , ( 3 . 4 ) m a y b e r e w r i t t e n , f o r a n y c h a r g e q a tt he obse r va t i on po in t , a s

w h e r e

I n p r inc ip l e , we shou ld wr i t e / ~ ( ? ) be c a use t he e l e c t r i c f i e ld de pe nds uponp o s i t i o n ? .L e t ' s d is c u ss ( 3 . 7 ) i n t e r m s o f i n p u t a n d o u t p u t . T h e i n p u t i s t h e o b s e r v a t i o np o s i t i o n ? a n d t h e i n d i v i d u a l s o u r c e c h a r g e s q i a t t h e p o s i t io n s ? i. F o r t w o s o u r c echarges_r this i s given in Figure 3 .5(a ) . Th e o ut pu t i s the se t of ind ivid ua l e lec t r icfields E i a nd the t o t a l e l e c t r i c f i e ld / ~ a t ~ . Assum ing tha t q l > 0 a nd q2 < 0 ,F igur e 3 .5 ( b ) de p i c t s t he d i r e c t i ons f o r t he f i e lds / ~ ] a nd E 2 . I t al so de p i c t s t he i r

s u m E . T h e r e la t iv e le n g t h s o f / ~ l a n d / ~ 2 c a n o n l y b e d e t e r m i n e d w h e n a c t u a lva lue s f o r q l a nd q 2 a r e g ive n ; t he r e f o r e , F igur e 3 .5 ( b ) i s on ly a sc he m a t i c . T hef ie lds a t ~ a re d r a w n wi th t he i r t a il s a t ~ .A m o r e e x p l i c i t e x p r e ss i o n f o r / ~ , w h i c h is u s e f u l fo r n u m e r i c a l c a l cu l a ti o n s ,

is o b t a i n e d b y u s i n g / ~ i / I / ~ l r a t h e r t h a n / ~ i . T h e n ( 3 .7 ) b e c o m e s-~ k q i R i = ~ i k q i ( ~ - ~ i ) -~E - ~ i R 3 - z ] r - r i ] 3 " ( R i - r - ~i t o o b s e r v a t i o n p o i n t r i )

(3.8)Fr om ( 3 .7 ) , t he e l e c t r i c fi e ld t ha t a s ing l e c ha r ge q l p r od uc e s a t t he s i te o f qis given by

- k q l / ~ l _ q l /~ 1, k - 1 ~ 9 x 1 0 9 N - m 2E1 - R- -}- 4z r~ 0R ~ 4JrE0 C 2 "( /~ ] t o obse r v a t i on po in t ) ( 3 .9 )

I n a g r e e m e n t w i t h t h e p r e v i o u s s e c t i o n , a p o s i t i v e c h a r g e s o u r c e m a k e s a f i e l dt h a t p o i n t s a w a y f r o m t h e s o u r c e ( i . e . , t o w a r d t h e o b s e r v a t i o n p o i n t ) , a n d ane ga t ive c h a r ge sour c e m a k e s a fi e ld t ha t po in t s t o wa r d t he sour c e ( i. e. , a wa yf r o m t h e o b s e r v a t i o n p o i n t ) .F i e l d Ed needed to caus e sparking in air

For a charge ql = 10 -9 C ( typica l of s ta t ic e lec t ric i ty) a t a dis tance of 1 cm,f ind the e lec t r ic f ie ld . Rep ea t for a dis tance of 1 mm . In bo th cases, com parewi th the e lec t r ic f ie ld above which spark ing (e lec t r ica l breakdown) occurs ,


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3 .4 Visua lizing the Electric Field: Pa rt 1 1 1 5

ca l led the dielectric strength Ed. In a i r a t a tmospher ic pressure , Ed i s about3 x 10 6 N/C.Solution: By (3.9), for ql = 10 -9 C and a distance of 1 cm,

1/~11 = (9 x 10 9 N- m 2/ C2) ( 10 - 9 C ) / ( 10 - 2 m ) 2 - 9 x 10 4 N/ C;this is less than Ed, so n o sparking w ould occur . For the same ql - 10 -9 C and adistance of 1 mm , ]E 11 - 9 x 106 N/C ; this exceeds Ea , so sparking wo uld occur.

Ob se r v e t ha t a c a l c u l a t i on o f / ~ a t F , us ing ( 3 .7 ) , d i f f e r s f r om a c a l c u l a t i onof F o n q a t ~ , us ing ( 2 .12) , on ly i n t ha t t he f a c to r q o f ( 2 .12) i s om i t t e d f r om( 3 . 7 ). B e f o r e g e t t in g i n t o a n y d e t a i l e d c a l cu l a ti o n s , w e p r e s e n t a m o r e g e o m e t r i cview of e lec t r ic f ie lds .

3~3o4~I

V i s u a l i z i n g t h e E l e c t r i c F i e l d - P a r t 1Ru les fo r D ra w in g E lec tr ic F i e ld L inesField lines, o r lines of force, a r e use d t o r e pr e se n t p i c to r i a l l y t he e l e c t r i c f i e ld / ~ .T o b e a n a c c u r a t e r e p r e s e n t a t i o n , t h e y m u s t h a v e t h e f o l lo w i n g p r o p e r t i e s:1 . F i e ld l i ne s po in t i n t he d i r e c t i on o f t he e l e c t r i c fi e ld /~ . F i e ld l ine s c a nn o t c r oss .I f t w o l in e s d i d c ro s s, t h e n t h e f o r ce o n a c h a rg e w o u l d h a v e t w o d i re c t io n s ,whic h i s im poss ib l e .2 . T h e areal dens i ty o f f ie ld l ines ( t h e n u m b e r o f l in e s p e r u n i t a r e a in t h e p l a n ep e r p e n d i c u l a r t o t h e f ie l d l in e ) i s p r o p o r t i o n a l t o t h e m a g n i t u d e I /~ l o f t h e

e l e c t r i c f i e ld / ~ . T hus , t h e l a r ge r t he f ie ld , t he h ig he r t he de ns i t y o f t he f i eldl ines , and vice versa .

By de f in i t i on , r u l e s 1 a nd 2 ho ld f o r a n y v e c t o r f i e l d , i n c l u d i n g t h e m a g n e t i cfield.I n a dd i t i on , f o r f i e ld l i ne s due t o e l e c t r i c c ha r ge s a t r e s t ( e l e c t r os t a t i c s ) , t hef o l l owing r u l e s a pp ly :

3 . F i e ld l i ne s o r ig ina t e on pos i t i ve c ha r ge s a nd t e r m in a t e on ne ga t ive c ha r ge s ,t h e n u m b e r o f f ie ld li n e s b e i n g p r o p o r t i o n a l t o t h e c h a rg e . ( T h i s p r e s c r ip t i o ni s n o t u n i q u e b e c a u s e d i f f e r e n t p e o p l e , o r t h e s a m e p e r s o n u n d e r d i f f e r i n gc i r c u m s t a n c e s , m i g h t c h o o s e t o u s e d i f f e r e n t n u m b e r s o f fi el d li n es f o r t h esa m e c ha r ge . )4 . F i e ld l i ne s do no t c lose on t he m se lve s .

R u l e 3 i s m a d e m o r e p r e c i s e i n C h a p t e r 4 ( w h i c h e l i m i n a t e s t h e a m b i g u i t ya b o u t t h e n u m b e r o f li n e s p e r u n i t c h a r g e ) . R u l e 4 is d e r i v e d i n C h a p t e r 5 . T h e s er u l e s a r e s im ple , bu t t he y a r e no t obv ious . T he y ho ld on ly f o r e l e c t r i c f i e lds dueto e l e c t r i c c ha r ge s a t r e s t . M a gne t i c f i e ld l i ne s do no t s a t i s f y r u l e s 3 a nd 4 , bu tr a t h e r h a v e t h e i r o w n s e t o f r u le s , w h i c h w e s t u d y i n C h a p t e r 1 1 .T h e s e r u l e s y ie l d s i m p l e p i c t u r e s o n l y o u t s i d e o f c h a r g e d i s t ri b u t i o n s .E v e r yw he r e w i th in a ba l l o f c ha r ge , f ie ld l i ne s a re o r ig ina t i ng o r t e r m ina t i ng .T h i s c a u s e s c o m p l i c a t i o n s t h a t w e n e e d n o t c o n s i d e r h e r e .


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116 Ch apte r 3 i The Elec t ric F ie ld

3.4.2 App l icat ions to Simple Geometries~ Positive poin t charge q

Figure 3.6 Representa t iono f t h e t h r e e -d ime n s io n a lf ie ld l ines due to a pointcharge.

C o n s id e r t h a t q p r o d u c e s N f ie l d l in e s. T h e nth e f i e l d l i n e s p o in t o u tw a r d a n d a r e u n i f o r mlyd i s t r i b u t e d , w i t h o n e f i e ld l in e f o r e a c h o f t h ec o r r e s p o n d in g p a r t s o f t h e t o t a l s o l i d an g l e o nthe sur face o f a sphe re . See F igure 3 .6 . I t is d if -f i c u lt to r e p r o d u c e t h i s o n a s h e e t o f p a p e r , s oth e f i g u r e mu s t b e c o n s id e r e d t o b e s c h e ma t i c .T h e n u m b e r o f f i el d l i n es N i s f ix e d , a n d t h e a r e a4 r r r 2 o f a c o n c e n t r i c s p h e r e o f r a d iu s r ( t h r o u g hw h ic h t h e l i n es p a s s ) v a r i es a s r 2 . H e n c e t h ed e n s i t y o f fi e ld l i n es ( a n d t h u s t h e e l e c t ri c f i e ldm a g n i tu d e I /~ l , o r i n t e n s i t y )v a r i e s a s N/4:rr 2 ,s o as e x p e c t e d f o r a p o in t c h a r g e, IEI fa l ls off as-2.

~ A charged sphere qA p o in t c h a r g e q - 1 0 - 8 C p r o d u c e s N = 4 fi e ld l in e s. ( a) H o w m a n y f i e ldl i n es a r e p r o d u c e d b y a s ma l l c h a r g e d s p h e r e o f Q - - 2 x 1 0 - 8 C ? ( b ) A t af i x e d p o s i t i o n o u t s i d e t h e s p h e r e , h o w d o e s t h e m a g n i tu d e I /~ l o f t h e e l e c t r i cf ie ld c h a n g e i f t h e r a d iu s o f t h e c h a r g e d s p h e r e d o u b l es ? ( c) H o w d o e s th em a g n i t u d e IE I o f t h e e l e c t r i c f ie l d c h a n g e i f t h e o b s e r v e r d o u b l e s h e r d i s t a n c ef r o m th e c h a r g e ? A s s u m e th a t , s i n c e t h e s p h e r e i s s ma l l , i t r e ma in s s ma l l e v e nw h e n i t s r a d iu s i s d o u b l e d .Solution: (a) By sym metry, th e f ie ld l ines mu st po int radia lly. Since q = 10 -8 Cproduces four f ie ld l ines outward, Q - - 2 q - - 2 x 1 0 - 8 C p r o d uc e s e i gh t f ie ldlines poin ting inward. (b) D o u b l in g t h e s p h e r e ' s r ad iu s d o e s n ' t c h an g e t h e n u mb e rof field lines, or th eir density, so IEI doe sn 't ch ange. (c) S ince I/~1 falls off as r -2,wh en t he observer d oubles her distance, I /~1 decreases by a factor of 4.

~ Infinite line of negative chargeNegat~ll ~ r '~ ~

R adiallynward '~fieldFigure 3.7 R e p r e s e n t a ti o n o fthe three -d im ens iona l f ie ld l inesdue to a l ine charge.

U n l i ke t h e e l e c t ri c f i e ld o f a p o in t c h a rg e ,the e lec t r ic f ie lds of a l ine cha rge can bea c c u r a t e ly r e p r e s e n t e d o n a p l a n e . F o r an e g a t i v e l i n e c h a r g e o f u n i f o r m c h a r g e p e ru n i t l e n g th ~ , l e t N l i n es t e r m in a t e u n i -f o r mly i n a n g l e o v e r a l e n g th l ( d o u b l i n gN d o u b l e s l ) . S e e F ig u r e 3 . 7 . T h e n u m b e rof fie ld l ines N i s f ixed , and th e a rea of ac o n c e n t r i c c y l i n d e r o f r a d iu s r a n d l e n g thl ( t h r o u g h w h ic h t h e l i n e s p a s s ) i s 2rrrl.H e n c e t h e d e n s i t y o f f i e l d l i n es ( a n d t h u sthe f ie ld in tens i ty ]El ) va r ie s a s N/2rrrl , soI /~l fa l ls of f as r - a .


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3 .4 Visua l i z ing the Electric Fie ld: Part 1 117

Figure 3.8 Posit ive sheet of charge intersecting the page.(a) F ie ld l ines above and be low the shee t . (b ) Shee t and tw oplanes , equid is tan t f rom the shee t .

~ Infinite shee t of positive chargeL e t t h e c h a r g e p e r u n i t a r e a b e p o s it i v e, o f ma g n i tu d e o r, a n d c o n s id e r t h a ti t p r o d u c e s N f ie l d l i n es p e r u n i t a r e a . T h e n N / 2 f ie ld l ines pe r un i t a reaw i l l p o i n t o u t w a r d i n e a c h d i r e c t i o n n o r m a l t o t h e s h e e t . ( W h y o u t w a r d ? )S e e F ig u r e 3 . 8 ( a ) , w h ic h d e p i c t s a s i d e v i e w o f a s h e e t t h a t i s n o r ma l t o t h ey - a x is . T h e l o c u s o f p o in t s a d i s t an c e r f r o m th e s h e e t i s a p a i r o f p a ra l l e lp l a n e s. S e e F ig u r e 3 .8 ( b . ) T h e i r a r e a d o e s n o t d e p e n d o n r . H e n c e t h e d e n s i t yo f fi e ld li n es , a n d t h u s t h e f i e l d i n t e n s i t y I /~ l , d o e s n o t d e p e n d o n r , s o IE Ii s c o n s t a n t i n s p a c e ; h o w e v e r , E c h a n g e s d i r e c t i o n o n c r o s s in g t h e s h e e t , a sseen in F igure 3 .8(a ) .

I ~ ~ ~ F ield lines for a uniform ly charged d isk , near and farS e e F ig u r e 3 . 9 ( a ) , w h ic h i s o n ly i n t e n d e d t o b e q u a l i t a t i v e ly a c c u r a t e . T h i sd isk is tak en to be an insu la tor , so the cha rge w i l l s tay in p lace . ( I f i t we re ac o n d u c to r , w e w o u ld h a v e t o ima g in e s o me e x t r a f o r c e h o ld in g t h e c h a r g e i np l a c e .) N e a r t h e d i s k t h e f i el d l i n es a r e n e a r ly u n i f o r m ly s p a c e d t o c o r r e s p o n dto a p l a n a r g e o me t r y . F a r f r o m th e d i sk , t h e f i e ld l in e s c o r r e s p o n d t o t h o s e o fa p o in t c h a r g e . C l e a r l y , a g r e a t d e a l c a n b e l e a r n e d s imp ly b y s ke t c h in g t h ef ie ld l ines . Typica l ly , the f ie ld l ines a re no t a long the normal a s they leave thesurface.

Figure 3.9 F ie ld lines for two-dim ens iona l d is t r ibu t ions ofcharge. (a) A uniformly charged insula tor . (b) A neutra lcondu c tor in the presence of a po in t cha rge .


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118 Cha pte r 3 9 The Electric Field

~ Field lines and electrical conductorsConsider a posi t ive charge near the center of a neut ra l , f ini te conduct ingshee t . See Figure 3 .9(b) , which i s only intended to be qua l i ta t ive ly accura te .I t was drawn using the fac t tha t , i f an e lec t r ic f ie ld l ine has a componenta long the sur face of an e lec t r ica l conductor , then e lec t r ic charge wi l l moveunt i l the f ield l ines becom e no rmal to the sur face . Seven f ie ld l ines ente r andseven leave , each normal to the shee t . This example i l lus t ra tes how muchinf or m a t ion c a n be ob t a ine d b y sk e t c h ing t he f i eld l ine s, w i th ou t pe r f o r m inga single calculation.

3~ Finding E" Pr inciple of Superposi t ionfor Discrete ChargesH a v i n g c o m p l e t e d o u r g e o m e t r i c a l d e t o u r , l e t ' s c a l c u l a t e s o m e e l e c t r i c f i e l d s ,us ing (3 .7) or (3 .8) .

3 5 , 1 The E lectr ic Dipo leC o n s i d e r a s e t o f c h a r g e s t h a t s u m s t o z e r o , fo r w h i c h t h e c e n t e r o f t h e n e g a t i v ec h a r g e ( o f t o t a l - q ) is a d i s t a n c e l f r o m t h e c e n t e r o f t h e p o s i t iv e c h a r g e ( o f t o t a lq). I ts e le c tr ic d i p o l e m o m e n t ~ p o i n t s f r o m t h e c e n t e r o f t h e n e g a t i v e c h a r g e t ot h e c e n t e r o f t h e p o s i t i v e c h a r g e a n d h a s m a g n i t u d e

i o m e n t m a ~ i ~ ~ ~ { 3 ~q l . ( d i p o l e mL i k e m as s , t h e d i p o l e m o m e n t } is a q u a n t i t y a s s o c i a t e d w i t h a g i v e n o b je c t .T he se t o f c ha r ge s i s c a l l e d a n e lec t r i c d ipo le . Se e F igur e 3 .10( a ) f o r t he c a seo f a w a t e r m o l e c u l e , s a i d t o b e a p o l a r m o l e c u l e b e c a u s e i t h a s a p e r m a n e n t

(a)

C h a r g e s o u r c ee r

Co)Figure3.10 Exam ples of dipole mom ents. (a)Permanent dipole m om ent of water molecule .(b) Induced dipole mom ent on a neut ra lpiece of paper . Note: The charge on paper isfrom polar ized atoms and molecules.

e l e c t r i c d i p o l e m o m e n t . P o l a rm o l e c u l e s f i g u r e p r o m i n e n t l y i nphys i c s , c he m is t r y , a nd b io logy .W i t h o u t th e m , o r g a ni s m s w o u l dno t be a b l e t o f o r m c e l l s . Wa te ri s a good so lve n t be c a use o f i t sp e r m a n e n t d i p o l e m o m e n t . S e eF igur e 3 .10( b ) f o r t he c a se o f pa -p e r , w i t h a d i p o l e m o m e n t o n l yw h e n p o l a r i z e d b y a n a p p l i e d e le c -t r i c f i e ld . A l l m ole c u l e s c a n bep o l a r i z e d b y a n a p p l i e d e l e c t r i cfield.N o w c o n s i d e r a d i p o l e c o n s i s t -i n g o f t w o s e p a r a t e d c h a r g e s + q a t+ a a long the y - a x i s , a s i n F igur e3 .11 ( a ). S inc e t he c ha r ge s a r e se p a r a t e d by t = 2 a , t h e d i p o l e m o m e n t m a g n i -t u d e is g iv e n b y p - q l - q ( 2 a ) .


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3 .5 Finding E: Princ ip le o f Superpo si t ion for Discre te Charges 1 19

Figure 3.11 A pair of equal and opposite charges.(a) Individual contributions and the total f ield along thex- and y-axes . (b) F ie ld l ines f rom + q to -q .

T h e f i e l d- l in e p a t t e r n i s s k e t c h e d i n F i g u re 3 . 1 1 (b ) . W e n o w s h o w t h a t e l e c tr i cd ipo les have an e l ec t r i c f i e ld tha t f a l l s o f f a t l a rge d i s t ances a s the inve r s e cubeo f the d i s t ance .Field along Dipole Axis. L e t t h e o b s e r v a t i o n p o i n t b e a l o n g t h e p o s i ti v e y - ax i s.S ince the pos i t ive cha rge i s nea re r to the obs e rva t ion po in t , i t s upw ard e l ec t r i cf ie l d d o m i n a t e s , i n a g r e e m e n t w i t h t h e f i e ld d i r e c ti o n a t t h e c o r r e s p o n d i n g p o i n tin F igu re 3 .11 (b ) . The un i t vec to r to the obs e rve r is ~ fo r bo th 4 -q. The n by (3 .7 )w i th p = q (2a ) ,

kq .. k ( - q) .. kq (y + a) 2 _ kq (y - a) 2E - - ( y _ a ) 2 j - ]- ( y _ q _ a ) 2 j - - ( y 2 _ a 2 ) 2

4 k q a y .. 2 k p y ..= (y2 _ a 2 )2 J - ( y 2 _ a 2 )2 J" (3.11)

In the l a rge d i s t ance l imi t , w here y ~ a , w e m a y n e g l e c t t h e a 2 t e r m i n t h ed e n o m i n a t o r , s o ( 3 . 1 1 ) b e c o m e s

2kpAE - - y 3 J , Y ~ a . (d ipo le f ie ld a long axis of d ipo le) (3 .12)F or a w a te r m o lecu le , a s in F igu re 3 .10 (a ) , p ~ 6 .0 x 10 -3o C-m . T hus , a t1 0 0 n m ( n m - 1 0 -9 m ) , a d i st a n ce o f a b o u t 1 0 0 0 t i m e s t h e s iz e o f a w a t e rmo lecu le , ( 3 .12 ) g ives IE I - 1 0 8 N / C . T h i s is c o m p a r a b l e t o t h e f i e ld in t h ee a r t h ' s a t m o s p h e r e .Field Normal to Dipole Axis. N o w l e t t h e o b s e r v a t i o n p o i n t b e a l o n g t h ex-axis . I t i s a d is tan ce r ~- ~ /a 2 + x S f rom eac h ch arge, so the f ie lds du e to eac hc h a rg e h a v e t h e s a m e m a g n i t u d e k q / r 2 . S ee F igu re 3 .1 l ( a ) . H ow ever , t he un i tv e c t o r s t o t h e o b s e r v a t io n p o i n t d i ff er . T h e h o r i z o n t a l c o m p o n e n t s o f t h e t w of ie ld s cance l , and the ve r t i ca l componen t s add . Thus the ne t e l ec t r i c f i e ld po in t s


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3 .6 Finding E" Pr inciple of Superposi t ion for Contin uou s Charge Dis tr ibut ions 121

Figure 3.12 A pa i r o fequal charges. Individualcont r ibu t ions and thetota l f ie ld a long they-axis.

T h e v a l u e o f ~ d e p e n d s u p o n t h e a t o m o r m o l e c u l e . I t ca n b e w r i t t e n a s ~ -V ~ / k , w h e r e V~ i s a " v o lu m e . " ( A c h a r a c t e r i s t i c a t o m i c v o l u m e i s ( 1 0 - l ~ m ) 3 -

1 0 - 3o m 3.) A s c a n o f t h e p o l a r i z a b i l i ti e s o f t h e a t o m si n t h e p e r i o d i c t a b l e r e v e a l s t h a t t h o s e w i t h t h el a r g e s t a n d s m a l l e s t " v o l u m e s " a r e , n o t s u r p r i s i n g l y ,t h e v e r y r e a c t i v e a l k a l i m e t a l c e s i u m ( w i t h V ~ s =5 9 . 6 x 1 0 - 3 ~ m 3 ) a n d t h e v e r y i n e r t n o b l e g a s h e l i u m( w i t h V ~ e - 0 . 2 0 5 x 1 0 - 3 ~ m 3 ) . T h e 1 / r 5 d e p e n -d e n c e o f ( 3 .1 7 ) i n c r e a s e s r a p i d l y a s r d e c r e a s e s ; p e r -h a p s y o u h a v e o b s e r v e d , o n r u b b i n g a c o m b t h r o u g hy o u r h a i r a n d b r i n g i n g i t c lo s e r t o a s m a l l p i e c e o f p a -p e r , t h a t a t s o m e p o i n t t h e p a p e r s u d d e n l y " j u m p s "u p t o t h e c o m b . C o n s i d e r a n a t o m o s c a r b o n , w i t h- 1 . 7 6 x 1 0 - 3 ~ m 3 a n d m - 2 0 . 0 x 1 0 - 27 k g . L e ti t b e a c t e d o n b y a c h a r g e Q - 1 0 - 9 C . T h e j u m p i n gp o i n t f o r t h i s c a r b o n a t o m is f o u n d , a p p r o x i m a t e l y ,b y e q u a t i n g ( 3 . 1 7 ) t o rag . T h i s l e a d s t o a d i s t a n c er = 1 . 0 7 7 m a t w h i c h t h e c a r b o n a t o m j u m p s u p t ot h e c h a r g e !

~ Field to two equal charge su eF in d t h e f i e l d a t a d i s t a n c e y o n t h e y - a x i s , d u e t o tw o e q u a l c h a r g e s q ] =q2 = q p la c e d o n t h e x -a x i s a t S e e F ig u r e 3 . 1 2 .Solution: By symmetry the x-components of /~1 and /~2 cance l , and they-components add . Thus the to ta l f ie ld po in ts a long ) . By (3 .7) , i t has magni -tude 21/~11 cos0 - 2[k q/ ( a 2 + y2) ] cos0 , w here cos0 - y / ( a 2 + y2)1 /2 . Thu s

2 k q yEy = (a 2 + y2)3/2" (3 .1 8)As expec ted, t he f ie ld is zero a t the or igin (y = 0), w here t he f ie lds of the in di-vidual charges should cancel .

3 ~ F i n d i n g E : P r in c i p le o f S u p e r p o s i t i o n f o r Co n t in u o u sCharge Dis t r ibut ionsG e n e r a l i z a t i o n o f ( 3 . 7 ) t o t h e e l e c t r i c f i e ld / ~ a t o b s e r v a t i o n p o s i t i o n ~ d u e t o ac o n t i n u o u s d i s t r i b u t i o n o f s o u r c e c h a rg e d q a t p o s i t i o n ~ ' g i v e s

i i~i~i~iii!ii~=iiiii!iiii~iiiii!iiiiiiiiiiiii~iiiiiii~ii~iiiii~iiiii!i~i!iiiii~i~i~i!i!i!i~i~i~iii!iiiii~ii~i~!iiiiii~i~iiiiiiiiii~i~iiiiiiiiiiiiiiiiii~iiiiiiii~!~i =i ! ! ! ! !

H e r e d q i s t h e c h a r g e a t t h e s o u r c e p o i n t ~ ' , R - ~ - ~ ' is t h e v e c t o r f r o m t h eo b s e r v a t i o n p o i n t ~ t o t h e s o u r c e p o i n t , a n d R - ] /~ ].


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122 C ha p te r 3 m T he E le c t r i c F i e ld

3,6.1

A s d i s c u s s e d i n d e t a i l in C h a p t e r 1 , t h e r e a r e t h r e e t y p e s o f c o n t i n u o u s d i s -t r i b u t i o n s o f c h a r g e :1 . d q = ( d q / d s ) d s = ~ .d s f o r l i n e c h a r g e d e n s i t y )~ = d q / d s . T h u s t h e c h a r g e

o n a l i n e s e g m e n t i s q - f ( d q / d s ) d s - f ~. d s, w h e r e t h e i n te g r a l e x t e n d so v e r t h e l i n e s e g m e n t .

2 . d q = ( d q / d A ) d A = ~ d A f o r s u r f a c e c h a r g e d e n s i t y ~ = d q / d A . T h u s t h ec h a r g e o n a n a r e a is q - f ( d q / d A ) d A - f cr d A , w h e r e t h e i n t e g ra l e x t e n d so v e r t h e a r e a .

3 . d q = ( d q / d V ) d V = p d V f o r v o l u m e c h a r g e d e n s i t y / 9 = d q / d V . T h u s t h ec h a r g e w i t h i n a v o l u m e is q - f ( d q / d V ) d V - f p d V , w h e r e t h e i n t e g r a le x t e n d s o v e r t h e v o l u m e .W e w i l l w o r k o u t e x a m p l e s o f t h e e l e c t r i c f i e ld s d u e t o l i n e a n d s u r f a c e

c h a r g e d e n s i t i es . F o r s i m p l i c i ty , i n o u r p r e v i o u s e x a m p l e s w i t h d i s c r e t e c h a r g e sw e c o n s i d e r e d v e r y s y m m e t r i c a l s i t u a t io n s . H e r e w e w i ll f i rs t c o n s i d e r a v e r yn u m e r i c a l a p p r o a c h . I t is a d a p t e d t o t h e c a s e w h e r e t h e c h a r g e d i s t r i b u t i o n isc o m p l e t e l y a r b i t r ar y ( a n d p o t e n t i a l l y c o m p l i c a t e d ) .Nu merical AnalysisB e c a u s e t h e e l e c t r i c f i e l d i s a v e c t o r , a n d v e c t o r s i n t h r e e - s p a c e h a v e t h r e e c o m -p o n e n t s , a n d b e c a u s e ( 3 . 1 9 ) i n v o l v e s an i n te g r al , s t u d e n t s o f t e n c o n c e n t r a t e o nt h e i n t e gr a l p a r t, a n d n e g l e c t o r o v e r s i m p l i f y t h e v e c t o r p a r t . T o c o n c e n t r a t e o nt h e v e c t o r a s p e c ts , w e f i rs t d i s c u s s t h e i n t e g r a l a s a s u m .

F r o m t h e v i e w p o i n t o f a s p r e a d s h e e t a n al ys is , i m a g i n e t h a t s o m e e l v es a p -p r o x i m a t e t h e s o u r c e b y m a n y t i n y e l e m e n t s o f s o u r c e c h a r g e d q i , f o r e x a m -p le , N - 1 0 0 0 . T h e y d e t e r m i n e th e dq i a n d t h e i r m i d p o i n t s r i. T h e y a ls o d e -t e r m i n e t h e o b s e r v a t i o n p o s i t i o n ~ . L e t R~ - F - F ~ b e g i v en b y i ts c o m p o n e n t sX i = x - x 4, a n d s o o n , a n d l e t , b y a n a l o g y t o ( 2 . 1 2 ' ) ,

dE'[ = k d q i

( d E * c a n b e n e g a t i v e , s o i t i s n o t t h e s a m e a s ] d E i ] . ) F o r i - 1 , T a b l e 3 . 1 g i v e sa s e t o f q u a n t i t i e s l e a d i ng u p t o t h e t h r e e c o m p o n e n t s o f d E i i n c o l u m n s 1 0 t o1 2. S u m m i n g c o l u m n s 1 0 t o 1 2 w o u l d t h e n y i e l d E x, Ey, a n d Ez . ( N o t e t h a t t h es u m o f c o l u m n 9 is m e a n i n g l e s s . ) F r o m t h is E , t h e f o r c e o n a c h a r g e q w o u l d b eg i v e n a s q E .

T h e a b o v e p r o c e d u r e is w e l l d e f i n e d a n d , f o r c o m p l e x n u m e r i c a l p r o b l e m s ,n e c e s s a r y . O n c e t h e e l v e s m a k e t h e s o u r c e e n t r i e s i n c o l u m n s o n e t o f o ur , a n dm a k e t h e o b s e r v e r e n t r ie s e l s e w h e r e , a ll t h e o t h e r r e s u l t s a re ca l c u l a t e d a u t o m a t -i c al ly b y th e s p r e a d s h e e t . I f t h e o b s e r v a t i o n p o i n t i s c h a n g e d , w h e n t h e o b s e r v e r -p o s i t i o n i s c h a n g e d , t h e s p r e a d s h e e t a u t o m a t i c a l l y r e d o e s a l l c a l c u l a t i o n s .

T a b l e 3.1 Spre a dshe e t e n t r i e s fo r f i e ld c a l c u l a t ion!iiiiiiiiiiiiii:iiii~iiii!~i~i !!~iiiiiiiiilliiiliilllliiii!llli!il~iiiiiiiiiiii!~i ~i::iiii~:iiiiiiiiiiiiiilii~iiliillif!filliiiiliiiiiiiiiiiiili}iiiiiiiiii~i~i~i~iiii iii~:~i!!!~iiiiiiii!iii~ililiii!iiiiliililiiiililiiiiiiiiililiiiiiiiiiiiiiiii~iiii:i~!!~iiiiiiiiiiiiiiiiiii!i!ililiii!i!iiiiiiiiii!!liiiiiiili iiii::iii:iiiiliiiilliiii:;::iiiiiiiii!iiiii!ililiii!i!!iliilf!f!iiii!iiiiiiiiiiliiiiiiiii:iiii!~iiii~i!iiiiiiiliiiii!iiiiiii!iiiiiiii!li!ii!ili!!iiiiiiiiiiii!iiiiiii i!!~i~:ii~i~i~i!iiiiiiiiliiiiiiiiiiiiiiiiii~iiliii~iiii!iiliiiiii!llllf!ii ~iiii:ii!iii:ii~i~i~i~iiiiiiiiiiiiiiiliiiiiiiiiiiiii!iiiilil!

X1 Y1 Z1dql xl y l z l X 1 Y1 Z1 R1 dE{ dE{(--K-i , dE{(N) dE{(N)


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3 .6 Finding E" Principle of Superposition for Con tinuous Charge Distributions 1 2 3

3 ~ 1 7 6

A v e r y s i m i l a r c a l c u l a t i o n h a s a l r e a d y b e e n p e r f o r m e d : t h e s p r e a d s h e e t e x -a m p l e i n C h a p t e r 2 , f o r t h e f o r c e o n a c h a r g e q a t t h e o r i g i n , d u e t o a l i n e c h a r g e)~. F o r q - 1 0 - 9 C , p e r f o r m i n g t h e s u m f o r N = 1 2 y i e l d e d a f o r c e o f m a g n i t u d e1 2 . 3 6 x 1 0 - 9 N . U s e o f ( 3 . 2 ) t h e n y i e l d s 1/771- I f : / q l - 1 2 . 3 6 N / C . I f w e w e r et o m o v e t h e o b s e r v e r p o s i t i o n ~ f r o m t h e o r i g in , t h e s p r e a d s h e e t w o u l d r a p i d l yc o m p u t e t h e n e w / ~ a n d E .

Calculus AnalysisT h e s e e x a m p l e s , i m p o r t a n t i n t h e m s e l v e s , a l s o c a n b e u s e d a s t e s t s t h a t a n u m e r -i c a l c a l c u l a t i o n g i v e s t h e c o r r e c t r e s u l t .

~ Fie iddueto uni form ine charge densityF i n d t h e e l e c t r i c f i e l d a t t h e o r i g i n d u e t o a u n i f o r m l i n e c h a r g e d e n s i t y )~ t h a ti s p a r a l l e l t o , a n d a d i s t a n c e a f r o m , t h e y - a x i s . S e e F i g u r e 3 . 1 3 .Solution: A p r e v i o u s d i s c u s s io n o f t h is c a s e c o n c l u d e d t h a t IEI ~ r -~ 9 T h e p r e -v i o u s c h a p t e r c o n s i d e r e d t h e r e l a t e d p r o b l e m o f t h e f o r c e / ~ o n a c h a r g e q a tt h e o r i g i n , d u e t o a c h a r g e Q u n i f o r m l y d i s t r i b u t e d o v e r a r o d o f l e n g t h l , a t ad i s t a n c e a a l o n g t h e x - a x i s . ( H e n c e t h e c h a r g e p e r u n i t l e n g t h i s ;v = Q~ l . ) B y( 3 . 2 ) , d i v i d i n g F b y q y i e l d s / ~ . T a k i n g t h e l i m i t w h e r e l - + o o t h e n y i e l d s t h ef i e ld d u e t o a n i n f i n i te l i n e c h a r g e . S p e c i f i c al l y , F i g u r e 3 . 1 3 d e p i c t s t h e f o r c e d Fo n q d u e t o t h e c h a r g e d Q - ) ~ d y i n an e l e m e n t o f l e n g t h dy .

T h e r e s u l t o f s u m m i n g t h e d F ' s d u e t o c h a r g e s d Q o n e a c h o f t h e l e n g t he l e m e n t s d y i s t h a t F y = 0 , a n d F x i s g i v e n b y ( 2 . 2 1 ) , r e p r o d u c e d h e r e a s

kq QF x = - (3 .2 0)av /a 2 + ( / / 2 ) 2T o o b t a i n Ex, b y ( 3 . 2 ) , w e d i v i d e ( 3 . 2 0 ) b y q , w h i c h l e a d s t o

E ~ = - k Q . ( 3 . 2 1 )a v / a 2 + ( / / 2 ) 2A s a c h e c k , n o t e t h a t , a s I /a -+ O, ( 3 . 2 1 ) g o e s t o t h e r e s u l t f o r a p o i n t c h a r g e Qa t a d i s t a n c e r - a .

N o w t a k e t h e l i m i t w h e r e l / r - + o c . U s i n g E = ] E l = I E x l , a n d r in p l a c e o fa , ( 3 . 2 1 ) y i e l d s E - + k Q / [ r ( l / 2 ) ] - 2 k Q / l r . T h u s~:~:~~~~:~iiii~~~~!~~ii!ii~ii~ii ~~iiiil~i~iiii:iiiii~i~i~~!~!~!!a!i~i~!~!~i!!iii~i~ii~ill i~i!~i~iii i~i~i~iii~~~!:ii:i iii:iiiii~ii~i~i~iiliii~ili:i!!iiiiilii!i ~i~i~iili!iiiiiii!i~iiiiiiiillii i!iii~iii~iii~i~i!~iii~i~i!ii~ili:iii i~i~~~~lii!i~i~!~!~~i~iiii ~i~ili:iii ~i~ilii~!liliiiliiiiii!!!;ilSiliiiii!il~iiiii ~i!!~ii!iilili!!!iililiii~i!~iiiiiii!i):ili~iiiiiiilii~!iiiii!~i~i!iiiiliiiii!iiiiiiiii!ii!iif!!~i~i!i!iili!!iiiiiill!!! ~if!~i~i: iiiilii!~i! !!!;ii!~!ilili i!~i~i!!~i!~ii!~i~ii!ii! !iiiiil !i!il iiililiili:iil!iiili~iiii!iiiiililiiiiiliiiiiiiiiiiiJiiiiili!!! !!~ii:iiiii:i!ii;ii:~i;~i~i!ii~!i~i~ii~iiii~i~i~ii~iiii~ii~i!ii~iiiii~i;ii~i~i~i~iii~ii~i!ii{iii~iii~iiiii!iiii!iii~!iiiiiiii~ii~i~iiiiii~iiiiii!ii~iiiiii~iiiiiiiiiii~iii!i~i~i!iii~iii~ii~ii~i!iii~iiiiiiii~ii~i~iiiii~iiii

dF

ld Q = ( Q / l ) d y

X ---- ~Figure3.13 F i n d i n g t h e f i e ld a t t h e o r i g i n d u e t o a l i n ec h a r g e : f o r c e d / ~ o n t e s t c h a r g e q a t o r i g i n d u e t o a ne l e m e n t d Q .


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12 4 C h a p t e r 3 = T h e E l e c t ri c F i e l d

w he r e )~ = Q / l i s t h e c h a r g e p e r u n i t l e n g t h . A s i n d i c a t e d e a r l i e r , t h i s v a r i e si n v e r s e l y w i t h r . T o o b t a i n a sp a r k i n g f i e ld i n a ir ( E d = IE ! = 3 x 106 N /C ) a t ad i s t a n c e o f 1 m m , ( 3 . 2 2 ) g i v e s ;~ = 1 . 6 7 x 1 0 - 7 C / m .

T h i s i m p o r t a n t g e o m e t r y i s u s e d i n p a r ti c l e d e t e c to r s , s u c h a s G e i g e r c o u n t e r s .W h e n t h e r e i s i o n i z i n g r a d i a t i o n , a l a r g e e n o u g h f ie l d c a u s e s e l e c t r o n s t o a v a l a n c h et o w a r d a p o s i t i v e l y c h a r g e d w i r e . T h e f i e l d i s l a r g e s t a t t h e w i r e .

Fie ld du e inf in i te sheet of uni fo rm sur faceo an chargedens i tyF i n d t h e e l e c t r i c f i e ld a t t h e p o i n t P = ( 0 , a , 0 ) , d u e t o a n in f i n i t e s h e e t o fu n i f o r m s u r f a c e c h a r g e a i n t h e y = 0 p l a n e . S e e F ig u r e 3 . 1 4 .Solution: T h i s p r o b l e m c o u l d b e s o l v e d u s i n g d q = a d A , w h e r e d A = d x d z, a n di n t e g r at i n g o v e r b o t h d x a n d d z . H o w e v e r , t h e i n t e g r a l o v e r a u n i f o r m l y c h a r g e d

l i n e ( h e r e i n v o l v i n g d z r a t h e r t h a n d y o fF i g u re 3 . 1 3 ) h a s a l r e a d y b e e n d o n e i n( 3 . 2 2 ) . H e n c e t h e a r e a c a n b e b u i lt u p a sa n in f i n i te n u m b e r o f t h i n s tr i ps , e a c h o fw h i c h c a n b e t r e a t e d a s a l i n e c h a r g e . N o t et h a t t h e d i r e c ti o n s o f t h e f i e ld s p r o d u c e db y e a c h l i n e a r e d i f f e r e n t .

T h e s t r i p i n F i g u r e 3 . 1 4 , w i t h t h i c k -nes s d x , i s pa r a l l e l t o t he z - ax i s , a t ad i s t a n c e r = ~ / a 2 + x 2 f r o m t h e p o i n t P .A s a s t ri p o f c h a r g e ( w i t h a v e r y l a r g el eng t h l ) , i t has a r ea d A - l d x , c h a r g ed q -- ( d q / d A ) d A = a ( ld x ) , a n d c h a r g epe r l eng t h d )~ = d q / l = a d x . Since d)~ is

Figure 3 . 1 4 F in d i n g t h e f ie ld a t i n d e p e n d e n t o f l , w e m a y l e t l ~ ~ . I nt h e o ri g i n d u e to a s h e e t o f c h a r g e : ( 3 . 2 2 ) , l e t u s r e p l a c e E b y d E a n d k b yf ie ld d E d u e t o a s t r ip o f c h a r g e , d k . T h i s l e a d s t o

d E = I d f 2 1 - 2 k d ) ~ _- 2 k a d x , ( 3 . 2 3 )T T

w h i c h p o i n t s a l o n g t h e n o r m a l f r o m t h e s t ri p to t h e o b s e r v a t i o n p o i n t . B y s y m -m e t r y , o n l y E y w i l l i n t e g r a t e t o a n o n z e r o v a l u e , w h i c h r e q u i r e s t h e d i r e c t i o nc o s i n e c o s 0 = a / r . T h u s

2 k a d xd E y = d E cos 0 - cos~ ). ( 3 .24 )TT h i s c a n b e i n t e g r a t e d m o s t e a s il y b y e l i m i n a t i n g x i n f a v o r o f 0 . S i n c e x = a t a n O,w e h a v e d x = a sec 20dO . A l so , a = r cos 0 , so r = a s ec 0 . Th en

2 k a ( a s e c 2 0 d 0 )d E y = c o s 0 = 2kr~dO. ( 3 . 2 5 )a sec 0S in c e J-~/2f~/2dO = r r, ( 3 . 2 5 ) i n t e g r a t e s t o E y - 2 k ~ z r . T h u s , w i t h E i n p l a c e o f E y ,w e h a v e


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3 . 6 F i nd i ng E : P r inc i p l e o f Superpos i t i on f or C ont i nu ous C harge Dis t r i bu t i ons 125

A s i n d i c a t e d e a rl ie r , t h i s is i n d e p e n d e n t o f p o s i t i o n . I t is n o r m a l t o t h e s h e e t a n dc h a n g e s d i r e c t i o n o n c r o s s in g f r o m o n e s i d e o f t h e s h e e t t o t h e o t h e r . T o o b t a i n aspa rk ing f i e ld i n a ir (E d - 3 x 10 ~ N / C ) , (3 .26 ) g ive s cr - 5 .31 10 -S C / m 2.

T h i s i m p o r t a n t g e o m e t r y w a s e m p l o y e d i n s i d e o l d e r T V t u b e s t o f o c u s t h ee l e c tr o n s . I t i s a l so u s e d i n e l e c t r o s t a t i c p r e c i p i t a to r s , w h e r e i t is u s e d t o d r a w o f fp a r t i c l e s o f s m o k e t h a t h a v e b e e n e l e c t r i c a l l y c h a r g e d .T h e p r e s e n t m e t h o d c a n b e a p p l i e d to c o m p u t e t h e f i el d d u e t o a s h e e t t h a tis f in i te a long x , b y c h a n g i n g t h e l i m i t s o f i n t e g r a t i o n . I n t h a t c a se , t h e f i e ld m a ya ls o h a v e a n x - c o m p o n e n t . T o a n o b s e r v e r n e a r t h e c e n t e r o f s u c h a f i n it e s h e e t ,t h e f i el d w i ll a p p e a r t o b e d u e t o a n i n f i n i t e s h e e t.

~ Fie ld axis of of uni form l inearn a ring charge densi tyand radius aF i n d t h e e l e c t r i c f i el d a d i s t a n c e z a l o n g t h e a x i s o f a c ir c u l a r r i n g o f u n i f o r ml i n e a r c h a r g e d e n s i t y )~ a n d r a d i u s a .S o l u t i o n : L e t t h e c i r c le l ie a t t h e c e n t e r o f t h e c o o r d i n a t e s y s t e m , w i t h i t s no r -ma l a long z . Fo r a n obse rva t ion po in t on the a x i s , a l l c ha rge i s a t a d i s t a nc er - ~ fa 2 + z 2 . S e e F i g u r e 3 . 1 5 .

B y s y m m e t r y , o n l y E z i s n o n z e r o . A n e l e m e n t o f c h a r g e d q p r o d u c e s a f i e l do f m a g n i t u d e d E - Id/~[ - k d q / r 2, a n d t h e d i r e c t i o n c o s i n e a l o n g t h e z - a x i s i scos~) - z / r fo r e a c h d q . T h u s

k d q z k z d q ( 3 . 2 7 )dE z = ]dE ] cosO - . r2 r = 7 "Sinc e k z / r 3 i s a c on s t a n t on the a x is , a nd s inc e q - f d q - f ( d q / d s ) d s - f ) ~ d s =)~ f ds - ~ ( 2 7 r a ), in t e g r a t i o n o v e r d q y ie ld s

kz f kz ;L(2zra)E z - - r7 dq - ( a 2 + z 2 ) 3 /2 . ( 3 . 2 8 )C a l c u l u s w a s u n n e c e s s a r y t o d e r i v e t h i s r e s u l t : q - f d q - f ) ~ d s - )~(2z~a) iss t r a i g h t f o r w a r d .

H e r e a r e t w o s i m p l e c h e c k s .( T h i n k o f t h e S e s a m e S t re e t c h a r a c -t e r G r o v e r , w h o l ik e s t o i l lu s t r a t en e a r a n d far.) N e a r t h e r i n g ( h e r e ,a t i ts c e n t er , z - 0 ) , b y s y m m e t r yt h e f i e l d m u s t b e z e r o ; i n d e e d , s e t -t ing z - 0 in (3 .28 ) g ives Ez - 0 .F a r f r o m t h e r i n g ( z ) ) a ) , t h e f ie l ds h o u l d b e t h e i n v e r s e s q u a r e l a w o fa p o i n t c h a r g e ; i n d e e d , u s i n g q -)~ (2~ a ) , (3 .28 ) g ive s E z - + k q / z 2.Figure 3 .15 F ie ld d / ~ due to pa r t o f a r i ng Fo r a n o f f -a x i s obse rve r , t he

o f c h a r g e, p r o b l e m w o u l d b e v e r y d i f f ic u l t b e -c a u s e r a n d ~ w o u l d b e d i f f e r e n t f o r e a c h d q , a n d t h e r e w o u l d b e t h r e e c o m p o --+n e n t s o f E t o c o m p u t e . W i t h o u t a d v a n c e d t e c h n iq u e s , o f f- a xi s i t w o u l d b e e a s i ert o a p p r o x i m a t e t h e i n t e g r a l a s a s u m o n a s p r e a d s h e e t .


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1 2 6 C h a p t e r 3 a T h e E l ec t r ic F i e ld

! x a m p l e 3 . F i el d d u e t o a d is k o f u n i f o r m c h a r g e d e n s i t y r a n dra d iu s a

F i n d t h e e l e c t ri c f i e ld a t a p o i n t z o n t h e a x is o f a d i s k o f u n i f o r m s u r f a c ec h a r g e d e n s i t y a a n d r a d i u s a . S e e F i g u r e 3 . 1 6 . T h e f i e ld li n e s h a v e a l r e a d yb e e n s k e t c h e d i n F ig u r e 3 . 9 ( a) .Solu t ion: L e t u s b u i l d u p t h e d i s k o u t o f ri n g s t o t a k e a d v a n t a g e o f t h e r e s u l t s o ft h e p r e v i o u s e x a m p l e . S i n c e o n ly t h e z - c o m p o n e n t w i l l b e n o n z e r o , w e a d d u p

th e d E z ' s d u e t o e a c h o f t h e r i n g st h a t m a k e u p t h e d i s k . T h u s , c o n -s ide r a typ ica l r ing , o f rad ius u anda n n u l a r t h i c k n e s s du , c e n t e r e d a tt h e o r i g i n w i t h n o r m a l a l o n g t h ez - a x i s . T a k e a n o b s e r v a t i o n p o i n ttha t i s a d i s t ance z a long the z -ax is .S e e F i g u r e 3 . 1 6 . T h e n t h e o b s e r v e ri s a d i s t ance R - ~ /u 2 + z 2 f rom a llt h e c h a r g e o n t h e r i n g . M o r e o v e r ,Figure 3.16 F i e ld d /~ d u e t o a n a n n u l u s t h e r in g h a s c h a r g e d Q = r ~ d A -of cha rge . ~ (2z ru d u ) .

T o a p p l y t h e r e s u l t s o f ( 3 . 2 8 ) , w e l e tEz - - , dEz , q - )~ (2z ra ) -~ dO . = a ( 2 r r u d u ) ,

Th us 2zr ;~a - . 2s t ~ u d u , s o ( 3 . 2 8 ) b e c o m e sr = v / a 2 nu z 2 ~ R .

k zd E z - - ~ ( 2Jr ~ u d u ) . ( 3 . 2 9 )

N o t e t h a t d /~ = dEz~ f o r t h e t o t a l f i e ld o n t h e a x is o f t h e a n n u l u s , b e c a u s e d E x =d E y = 0 , b y s y m m e t r y .N o w c h a n g e t o t h e v a r ia b l e R , s o R 2 = u 2 + z 2 . N o t e t h a t d ( R 2 ) =[ ~ R ( R 2 ) ] d R = 2 R d R , a n d s i m i l a rl y d ( u 2 ) = 2 u d u . S ince z i s cons tan t , we have2 R d R = 2 u d u . T h e n ( 3 .2 9 ) b e c o m e s

2rr kcr z d RR 2

k zd E z = ~-g(2zrcr R d R ) =T h e t o t a l Ez is due to a l l the r ings , so

R+

R_

( 3 . 3 0 )

( 3 . 3 1 )f 2rr ka zEz = I dEz =J RS ince , by F igure 3 . 16 , R+ = ~ / a ? + z ? and R_ = z , ( 3 . 3 1 ) b e c o m e s

E z = 2 z r k o z ( lz ~ /a 21+ z ? ) = 2zrker (1 - v ia 2 + z ? ) . ( 3 . 3 2 )T h e i n te g r al s c a n n o t b e d o n e w i t h e l e m e n t a r y m e t h o d s f o r a n o b s e r v a ti o n p o i n tof f the ax is .

H e r e a r e t w o s i m p l e c h e c k s . N e a r t h e d i s k ( a / z ~ e c ) , t h e f i e l d s h o u l d l o o kl ik e t h a t o f a n i n f i n i t e s h e e t. I n d e e d , f o r a / z - + ~ , (3 . 32) goes to Ez = 2zrkcr. Th isa g r e es w i t h ( 3 . 2 6 ) f o r a n i n f in i t e s h e e t b u i l t u p o u t o f li n es , r a t h e r t h a n r in g s. F a rf r o m t h e d i sk ( a / z ~ 0 ) , t h e f ie l d s h o u l d l o o k l i k e t h a t o f a p o i n t c h a r ge , k Q / z 2,


20/37

3 . 6 F i n d i n g E ,: Pr i n c i pl e o f S u p e r p o s i t io n f o r C o n t i n u o u s C h a r g e Di s t r ib u t i o n s 127

w h e r e Q = ~ J r a 2 . T o s h o w th i s, c o n s i d e r t h e p a r e n t h e s e s i n ( 3 .3 2 ) , a n d d i v i d en u m e r a t o r a n d d e n o m i n a t o r b y z. T h a t g iv e s

1 - ~/a 2 + z 2 = 1 - X/(za)2 + 1 = 1 - 1 + . ( 3 . 3 3 )A s a / z - + O , t h i s goes t o zero . Bu t does t h i s app roach zero as 1 /z2? To see t h i s ,set x = ( a / z ) 2 and fo r n - - 89 app l y t he r esu l t

(1 + x ) n ~ 1 + n x , I n x l


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128 Ch apte r 3 i Th e Electr ic Field

This case is impo rtant b ecause parallel plate capacitors have this geometry. N oteth a t /~ and/~2 add vec tor ia l ly in each region by the pr inc iple of superposi tion;the f ie ld due to one shee t i s independent of the presence of the other shee t .

3.73.7.1

V i s u a l i z i n g t h e E l e c t r i c F i e ld : P a r t 2More on Fa raday ' s and Maxwe l l ' s Ways o f Th ink ingF a r a d a y c o n s i d e r e d t h a t t h e s o u r c e o f t h e l in e s o f f o r c e ( t h e c h a r g e ) i s s u b j e c t t oa ne t f o r c e due t o t he f o r c e on a l l i t s f i e ld l i ne s , us ing t he r u l e s t ha t1 . E a c h l in e o f f o r c e is u n d e r t e n s io n . T h e r e f o r e t w o o b j e c t s at o p p o s i t e e n d s o fa l i ne o f f o r c e ( a nd t h us h a v ing op po s i t e c ha r ge s ) a t t r a c t . Se e F igur e 3 .11 ( b ) .2. A d j a c e n t l i n e s o f f o r c e re p e l o n e a n o t h e r. T h e r e f o r e t w o a d j a c e n t o b j e c t s

w i th t he sa m e c h a r ge a r e t he sour c e f o r a d j a c e n t r e pe l l i ng l ine s o f f o rc e , a ndh e n c e t h e o b j e c t s r e p e l .M a x w e l l s h o w e d t h a t t h e s e i d e as a r e v a li d , a n d r e la t e d t h e t e n s i o n a n d p r e s -s u r e o f t h e l in e s o f f o r c e ~ o r f ie ld l i n e s ~ t o t h e e l e c t ri c fi el d s t r e n g t h . H e d e v e l -ope d a n a na logy whe r e t he f i e ld l i ne s l e a d ing f r om a pos i t i ve c ha r ge t o a ne ga t ivec ha r ge , a s i n F igur e 3 .1 l ( b ) , a r e a na logou s t o t he f l u id fl ow (flux) f r o m a s o u r c e( e.g. , a sp igo t ) t o a s i nk ( e .g . , a d r a in ) . T h i s l e d M a xw e l l t o t h in k i n t e r m s offluxtubes, whose s ide s a r e pa r a l l e l t o t he e l e c t r i c f i e ld , w i th f l ow d i r e c t i on a long / ~ .T he f i e ld l i ne s de t e r m ine t he f l ux t ube s , a nd v i c e ve r sa .I n t he f l u id c a se , e qua l f l ux t ube s c a r r y e qu a l a m ou nt s o f f l u id pe r u n i t t im e ,a n d t h e u n i t f lu x t u b e m a y b e d e f i n e d a s o n e t h a t t r a n s p o r t s a m 3 o f fl u id p e r

se c ond . I n t he e l e c t r i c a l ca se, t he u n i t f l ux t ub e i s de f ine d so t ha t i f t he r e a r e 300u n i t f l u x tu b e s p e r u n i t a r e a ( m 2 ), t h e n t h e e l e c tr i c fi e ld m a g n i t u d e i s 3 0 0 N / C .I t i s no t poss ib l e t o d r a w a l l t he un i t f l ux t ube s f o r a g ive n p r ob l e m ; t hus ,t h e f l u x t u b e s o f t e n a r e n o t t a k e n t o b e u n i t t u b e s . D o u b l i n g t h e c h a r g e c a ne i t h e r d o u b l e t h e n u m b e r o f t u b e s o r d o u b l e t h e f i el d s t r e n g t h a s s o c i a t e d w i t he a c h t u b e . T h u s , t h e p r o d u c t o f t h e n u m b e r o f t u b e s a n d t h e f i el d s t r e n g t h i sr e l a t e d t o t he e l e c t r i c c ha r ge t ha t p r oduc e s t he t ube . I n ge ne r a l , i t i s d i f f i c u l t t or e p r e s e n t e i t h e r f l u x t u b e s o r f i e l d l i n e s f o r t h r e e - d i m e n s i o n a l g e o m e t r i e s . W em u s t o f t e n s e t t l e f o r a s c h e m a t i c r e p r e s e n t a t i o n . T h e e x a m p l e s t h a t f o l l o w a r es i m p l e e n o u g h t h a t t h e i r f ie ld l in e s c a n b e d r a w n e x a ct ly .

~ Field ines for two equal line chargesDiscuss and sk e tch th e f ie ld lines for a dipole of two equal l ine charges . Takee ight l ines per )~. F ind the p osi t ion wh ere /~ - 0 .Solution: In two dimensions, l ines give an accurate representation of f lux tubes( the oth er sides of the tubes are perpendicu lar to the p aper). Eight l ines leaveeach positive charge (at equal angles of 36 0/8 =4 5~ See Figure 3.18(a). Farfrom the se charges, this com bination of two charges looks l ike a single l ine chargewith net charge 2)~. This co rresponds to 16 f ield lines leaving, at equal angles of36 0/1 6 = 22.5~ See Figure 3.18(a). Th e boxed po int is a posit ion where/~ = 0. I t


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3 . 7 V i su a l i z i n g t h e E lec t r ic F ield : P a r t 2 12 9

(a)

III+III

(b)Figure 3.18 Rep resen ta t ion s o f t he f i e ld li nes due to two equ a lp o s i ti v e ly c h a r g e d r o d s n o r m a l t o t h e p a g e. ( a) R e p r e se n t a t i o n w i t hf i eld li nes go ing f rom one ro d tow ard th e o the r . These f i eld li nesa p p e a r t o e n d a t t h e o r ig i n ( su r r o u n d e d b y a b o x ), w i t h t w o l in e sl eav ing ve r ti ca lly . Thus the ne t nu m ber of fi e ld l i nes l eav ing the or ig ini s ze ro . (b ) Repres en ta t ion wi th f i eld l i nes tha t do n o t go f rom one rodt o w a r d t h e o t h er .

has two l ines en te r ing and tw o l ines l eav ing so tha t i t enc loses no ne t cha rge. F i e ldl ines don ' t r ea l ly c ross where /~ = 0 s ince the re i s no f i e ld the re . F igure 3 .18(b)has been drawn wi th the f i e ld l i nes a t ano the r angle . Fa raday would imagine thecha rges repe l l i ng because of t he pa ra l l e l f ie ld l i nes pus h ing away f rom each o the r .

~ Fie ld l ines for tw o l ine charges on the x-axisDi sc u s s a n d sk e t c h t h e f i e l d l i n e s f o r 2 ) ~ a t t h e o r i g i n a n d - ) ~ a t a d i s t a n c e lt o t h e r i g h t . T a k e e i g h t l i n e s p e r )~. T h e l i n es a r e o r i e n t e d n o r m a l t o t h e p a g e ,a s i n F i g u r e 3 . 1 9 .S o l u t i o n : Six teen l i nes l eave the pos i t i ve cha rge 2)~ (a t equa l ang les o f 360 /16 = 22 .5 ~ and e igh t l ines en te r t h e nega t ive cha rge - )~ (a t equa l ang les o f

~ - - - l ~ s - - - - ~Figure 3 .19 Fie ld l i nes for two rods o fcha rge 2)~ and - )~ ; t he rods a re no rma lt o t h e p a g e . T h e b o x d e n o t e s t h e p o i n twh ere the f i eld is ze ro .

3 6 0 / 8 - 5 5 ~ F ar a wa y, t h is c o m -bina t ion of two cha rges looks l i kea s ing le l i ne cha rge wi th ne t cha rge2)~ + (- )0 - )~. This corr esp ond s toeight f ie ld l ines leaving a t equala n gl es o f 3 6 0 / 8 - 4 5 ~ T h e b o x e d~poin t on the x-ax is , wher e / ~ -0 , i s a d i s t ance s t o the r igh t o f- )~ , and l + s f ro m 2)~. By (3 .23) ,t he f i e lds due to each cha rge can-c e l w h e n 2 k ( 2 ) ~ ) / ( l + s ) - 2 k ) ~ / s , giv-ing s - t . Two f i eld l i nes l eave th i spo in t t o r igh t and l e f t . In o rde rt h a t t h e n e t n u m b e r o f l in e s e n -t e r ing th i s po in t be ze ro , two f i e ldl i n e s e n t e r i t f r o m t o p a n d b o t t o m .

Faraday would imagine the cha rges a t t r ac t ing because the f i e ld l i nes t ry tocont rac t .


23/37

13 0 Chap ter 3 ~ The E lec t r i c F ie ld

II

(a)Figure 3.20 Geomet ry assoc ia ted wi th d ipo lefield l ines . (a) Sphere defining the polar angle 0relat ive to the dipole axis . (b) Field l ines for anelectric d ipole.~ Flux tubes and solid angle for a point charge

F i n d t h e s o l i d an g l e s u b t e n d e d b y a c o n e fo rm e d b y ro t a t i n g a l i n e a t a n a n g le0 re la t ive to the po lar ax i s , as in F igure 3 .20 (a) . Re la te th i s to f lux tubes .Solution: Your ca lcu lus course may have shown tha t the a rea A p ro jec ted on asphe re of radius r of a cone of angle 0 is A - 2zrr2(1 - cos0 ) . By defini t ion, thesol id angle fa subtended by th is cone is g iven by fa = A / r 2 so that

~2 = 2zr(1 - cos 0).Th e larger the sol id angle, the larger the flux of the associated flux tube.

(3 .37 )

~ Field l ines and flux tubes for a dipoleD i s c u s s a n d s k e t c h t h e f ie l d li n e s a n d f l u x t u b e s fo r a d i p o l e o f e q u a l a n do p p o s i t e p o i n t c h a rg e s .Solution: In Figure 3 .20( b) , the l ines represe nt f lux tubes of equal sol id anglego ing f rom the pos i t ive charge to the nega t ive charge . In th ree d imens ions , th eyare ro ta ted abou t the po lar ax i s to fo rm the f lux tubes . From (3 .37 ) , the ang lesleav ing the pos i t ive charge cons tan t separa t ions in cos 0 . Thus , mea sur ing w i th re -spec t to the po lar ax i s , the ang les o f the tubes en te r ing and leav ing the charges a reat O, 60, 90, 120, 180, and so forth . The objects enclosing equal f lux are the con-ical shel ls that are the differences between successive conical f lux tubes . For thatreason, the y are farther ap art for angles near the polar axis. (Related d is tort ions areseen in maps o f the w or ld , wh ich a lso invo lve rep resen t ing a th ree-d imens iona ls i tua tion in two d imens ions .) Faraday wou ld imag ine the charges a t t rac t ing be-cause the field l ines t ry to contract . At large dis tances , the fields vary as (3 .12) onthe d ipo le ax i s , and as (3 .14) n o rmal to tha t ax is .

F o r c e , T o r q u e , a n d E n e r g y o f a D i p o l ein a U n i f o r m F i e ldC o n s i d e r a n e l e c t ri c d i p o l e i n a c o m p l e t e l y u n i f o r m e l e c t r i c fi e ld , a s i n F i g-u r e 3 . 2 1 . T h e f o r c e q E o n t h e p o s i t i v e c h a r g e i s e q u a l a n d o p p o s i t e t o t h e


24/37

3 .8 Force, Torque, an d Energy of a Dipole "1~'1

fo rc e -q /~ on t he ne ga t i ve c ha rge , so t he re is no n et fo rc e on t he d i po le . A w e l l -known ma gne t i c a na l og i s a n o rd i na ry c ompa ss ne e d l e i n t he e a r t h ' s ma gne t i cf ie ld: the needle rota tes , but the compass i s

E

Figure 3.21 Electric dipole ofmo men t ~ in a uni form appl iedelectric field/~. Tw o equal andopposite charges are separatedby l .

sub je c t t o no ne t fo rc e . Th i s i s be c a use t hee a r t h ' s m a gne t i c f ie l d is near ly un i form i n t hev i c i n i ty o f t he c omp a ss ne e d l e , so it s nor t h a ndsou t h po l e s f e el e qua l a nd oppo s i t e fo rce s .Ne ve r t he l e s s , a ma gne t p l a c e d i n t hene a r l y un i fo rm ma g ne t i c f i el d o f t he e a r t hfe el s a t o rq ue ~ t ha t t e nd s t o m a ke i t po i n tt owa rd t he nor t h po l e . We wi l l l a t e r de sc r i bet he ma gne t a s a ma gne t i c d i po l e i n a un i fo rmma gn et ic f i e ld, so we wi l l say tha t the magnet icdipole feels a torque that tends to al ign i t wi th themagnet ic f ield.No w c ons i de r t he a na l ogous c a se o f a pe r m a ne nt e l e c tr i c d i po l e ;~ i n a un i -

fo rm e l e c t r i c f i e l d /~ (The d i po l e c ou l d be t he wa t e r mol e c u l e i n F i gure 3 .10a ~ .The electr ic dipole ~2 feels a torq ue tha t ten ds to al ign i t wi th the electr ic f ield E.To de t e rm i ne t h e va l ue o f the t o rqu e ~ , c ons i de r t ha t t he r e a re t wo c ha rg~es + qc o n n e c t e d b y a r o d o f l e n g th l, the charges a t pos i t ions +~ /2. (Th us 26 = q l . ) SeeFi gure 3 .21 . Le t t he re be a un i fo rm f i e l d E t ha t po i n t s t o t he r i gh t . The fo rc eon e a c h of t he c ha rge s is + q /~ . He nc e t he t o rqu e ~ , me a sure d f rom t he i r mi d-point , i s

r - ( 2 ~ ) x ( q /~ ) + ( - 1 [ ) x ( -q / ~ ) -(qh x/~.W i t h ~ - q~, ~ b e c o m e s!iiiiiiiiiiiiiiiiiiiiii!iii!iiiiiiiiii!~iliiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!iiiiiii!i!iiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!iiii!iiiiii!iiiiiiiiiiiiliiii!iiiiiiiiiiliii!i!ii!iiiii!!iili!iiii!liiiiiiliiiii!iiiiii!!iiiiii!i!iii!ii!!iiiii!iiiiiiiii~ii!~!i~ii~iiiiiiiiii!i!iiiiiii!iiiiiiili~!i!i!!!ii!iiiii!iiiiiii!iiiii~iiii ............................. ii ii i iiii ! i i i ............................................i i i i i . . . . .For t he d i po l e a nd f i e ld in F i gure 3 .21 , use o f t he v e c t or p ro du c t r i gh t -ha nd ru l ey i e l ds a t o rque t ha t i s i n t o t he pa ge , whi c h t e nds t o c a use a c l oc kwi se ro t a t i onof t he d i po le .We can obta in the energy by us ing an ana logy to gravi ty. There , a s ingle massm a t pos i t ion ~ in a un i for m gravi ta t iona l f ie ld ~ has gravi ta t iona l poten t ia l en ergyUgr~v -- m g y or, w i th ~ - -g 56

Ugr~v - - - - m( ~ . ~ ) . (3.39)By ana logy, a s ingle charge q a t pos i t ion ~ in a uni form e lec t r ic f i e ld/~ has e lec-t r ic a l po t e n t i a l e n e rgy

U el - - - q ( E . ~ ) . (3.40)I f w e n o w a p p l y th i s to o u r t w o c h ar g es , o n s u m m i n g o v e r b o t h w e h a v e

lU d i p - - I q ( E " ~ ) ] - I ( - q ) ( / ~ " ( - ~ ) ) ] - - q / ~ ' ~ , (3.41)


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132 Chapter 3 ~ The Electr ic Field

3 .9

3.9.1

o r, w i t h ~ - q l ,

W he n } a nd /~ a re a l i gne d , so iv" E - I~11 /~1, t he e ne r gy i s m in im iz e d , a se x p e c t e d .F o r c e o n a D i p o l e i n a N o n u n i f o r m F i e l dI n t he a m be r e f f e c t , t he r e i s a f o r c e on a n i nduc e d d ipo l e i n a nonuni f o r m f i e ld .E qua t ion ( 3 .17) a pp l i e s on ly whe n the e l e c t r i c f i e ld i s due t o a po in t c ha r ge .

|O O E2 |

Figure 3.2 2 Electric dipolemom ents in a nonuni formapplied electric field E.

M o r e g e n e ra ll y , t h e r e c a n b e a p e r m a n e n t d i p o l e( e .g . , due t o a wa te r m ole c u l e ) i n a n e l e c t r i c f i e ldd u e t o m a n y c h a r g e s . B e f o r e g e t t i n g i n t o m a t h e -m a t i c a l de t a i l , we p r e se n t som e ge ne r a l c ons ide r a -t ions .Since th e force 1fi on a dip ole ~ in a uni f orm f ie ldi s z e r o , i n ge ne r a l t he f o r c e on t he d ipo l e m us t de -p e n d u p o n t h e s t r en g t h o f t h e d i p o l e m o m e n t a n do n how E var ies in space. C o n s i d e r F i g u r e 3 . 2 2 ,d e p i c t i n g a n o n u n i f o r m e l e c tr i c f i e l d /~ a n d t h r e ed ipo l e s i n d i f f e r e n t o r i e n t a t i ons . F r om our e a r l i e rc ons ide r a t i ons a bo u t f i eld li nes , t he e l e c t r i c f ie ld iss t r onge r whe r e t he f i e ld l i ne s a r e c lose r . Fur the r ,

t h e f o r ce o n t h e e n d o f t h e d i p o l e i n t h e l a rg e r fi el d d o m i n a t e s . T h u s d i p o l e 1f e e l s a ne t f o r c e t o t he r i gh t , d ipo l e 2 f e e l s a ne t f o r c e t o t he l e f t , a nd d ipo l e 3fee ls no ne t force .Quantitat ive Considerat ionsCo ns ide r a d ipo l e w i th q a t F + [ a nd - q a t F . By ( 3 .6 ) , t he f o r c e on t he c o m b i -na t i on i s g ive n by

p = q ~ ( F + 2 / ) _ q / ~ ( F _ 1 [ ) , (3 .4 3)wh e r e E i s e va lu a t e d a t F + 8 9 W e w i s h t o e v a l u a t e t h i s w h e n / ~ v a r i e s s l o w l yin space .F i rs t c o n s i d e r t h e s t r a i g h t- l in e a p p r o x i m a t i o n a p p l i e d t o a f u n c t i o n o n l y o fx. I f a i s smal l , the n f ( x + a ) - f ( x ) i s ne a r ly g ive n by t he s l ope m - d f / d xt i m e s t h e c o o r d i n a t e d i f f e r e n c e a : f ( x + a ) - f ( x ) ~ ( d f / d x ) a . N o w i n c l u d ethe c oor d ina t e d i f f e r e nc e s i n a ll t h r e e d i r e c t ions , a n d l e t f = q E x , so f (x) -+f ( x , y , z ) = q E ~ (x , y , z ) . T h e n f ( x + a ) ge ne r a l i z e s t o q E x ( x + l x , y + l y , z + l z ) , sof ( x + a ) - f ( x ) ge ne r a l i z e s t o

d E x d E x d E xFx - q l x T x + q l y T v + q l z d z-~ d d dv - + + k d zx JJ ~ y

= ( ~ . V ) E x , D - - q [,(3.44)


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3.10 Motion of Charges 133

with s imi la r equa t ions for Fy and F~. (The quant i ty V i s ca l led the g r a d i en toperator.) For a dipole } a l igned wi th E, thi s says tha t the dipole i s a t t rac tedto regions of l a rger E . M agne t ic dipoles have ana log ous behavior . Ver i fy tha tt h i s e qua t i on a gre e s wi t h t he qua l i t a t i ve c ons i de ra t i ons fo r t he t h re e d i po l e s i nFigure 3.22.I t i s n o t i m p o r t a n t t h a t y o u r e m e m b e r t h i s e q u a t i o n ~ i t i s b e t t e r t h a t y o udo n' t ~. W ha t i s im po rta nt i s tha t , jus t as we argue d above , th i s force i s propor t ionalto the d ipo le mom ent an d to how the e lec t ric f i e ld var ies in space .

3 . 1 0 M o t i o n o f C h a r g e sFigure 3.23 depic t s a pos i t ive ly charg ed par t i c le (e.g. , an a lpha par t i c le , w hich i sa he l i um nuc l eus , w i t h c ha rge q l = 2e ) movi ng pa s t a no t he r pos i t i ve ly c ha rge dpar t ic le (e.g. , the nuc leu s o f an a tom,wi t h c ha rge q2 = Z e , whe re Z i s usu-

a l ly much bigger than 2) tha t i s f ixedin place . The a lpha par t i c le bends awayf rom t he nuc l e us be c a use i t i s r e pe l l e d ,Scattering f r o m n u c l e u s ( q2 = Ze) by " l ikes repe l . " Finding the orbi t o f thea lpha par t i c le i s a solvable but mathe-ma t i c a l l y c ompl i c a t e d p rob l e m. I t i s no tne c e s sa ry t o so l ve suc h a c ompl e x prob-lem to learn how e lec t r ic f i e lds causec ha rge d pa r t i c l e s t o be de f l e c t e d . How-e ver , t h i s p rob l e m l e d to t he d i sc ove ry o ft he a t omi c nuc l e us .Jus t as the mot ion of a par t i c le in auni form gravi ta t iona l f i e ld i s re la t ive lys t ra i gh t fo rwa rd t o ob t a i n , so i s t he mot i on o f a pa r t i c l e i n a un i fo rm e l e c t r i cf ie ld . The m ot i on o f a n ob je c t de pe nds up on i ts a c ce l e ra t ion , no ma t t e r w ha tforces cause tha t acce le ra t ion.

Figure 3.23 An alpha particle (ahelium nucleus) scattering o ff thenucleus o f an atom of nuclear chargeZe. No t depicted are the very l ightelectrons, wh ich c anno t effectivelyscatter the alpha particle because o f themass mismatch.

In 1910 Rutherford, win ner of the 1908 Nobel Prize in Physics for h is discovery ofalpha particles, had a stud ent m easu re the alpha particles tha t are back-scattered bynuclei. The then c urrent "p lum -pu ddi ng" model of the atom (due to J. J. Thomson) hadpositive charge, with most of the mass of the atom unifor mly distr ibuted th rou gho utthe atom. For th is model Rutherford expected to f ind neglig ib le large angle-scattering.When appreciable back-scattering (nearly 180 ~ ) was found, Rutherford developed amodel where the posit ive charge was concentrated at a massive point. Working out thetheory of the alpha part icle orbit, he obtained agreement with his experimental results.

M o t i o n a l o n g a U n i f o r m E l ec t ri c F i e l dTo draw e lec t rons f rom i t , a ca thode i s both hea ted and placed in an e lec t r ic f i e lddue to a pos i t ive ly charged gr id sc reen. At a di s tance , to the e lec t ron the sc reenlooks l ike a plane.


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134 C h a p t e r 3 9 Th e E l ec t r ic F i e l d

C o n s i d e r a u n i f o r m l y c h a r g e d s h e e t i n th ey z - p l a n e , w i t h p o s i t iv e c h a r g e d e n s i t y ~ , a si n F i g u r e 3 . 2 4 . A b o v e i t se l f i t p r o d u c e s au n i f o r m u p w a r d e l e c tr i c fi el d o f m a g n i t u d eE = 2 z r ~ . A n e g a t i v e l y c h a r g e d p a r t i c l e - qw o u l d f e e l a c o n s t a n t f o r c e d o w n w a r d ; t h em o t i o n w o u l d b e v e r y l ik e t h e m o t i o n o f ab a s e b a l l t h r o w n d i r e c t l y u p w a r d a g a i n s t g r a v-F i gu re 3 . 2 4 M o t i o n o f a n e g a ti v e i ty . S e e F i g u r e 3 . 2 4 . L e t u s a n a l y z e t h is s i t u -

c h a rg e - q a b o v e a p o si ti ve , a t i o n n e g l e c t i n g t h e f o r c e o f g r a v it y .u n if o rm l y c h a rg e d s h e et + ~ t h a t T h e f o r c e is d o w n w a r d , a n d o f m a g n i t u d ei n te r s ec t s n o r m a l t o t h e p a ge . F - q E . L e t u s c a l l t h i s t h e y - d i r e c t i o n . B yN e w t o n ' s s e c o n d la w , t h is f o rc e p r o d u c e s a u n i f o r m d o w n w a r d a c c e l e r a ti o n a y,w h e r e m a y = F = q E . T h e n

d v y q E- - - - ( 3 . 4 5 )d t - a y m 'w h i c h i n t e g r a t e s o v e r d t t o y i e l d

d y q Ev y = d t = v o + t , (3 .46)mw h e r e v 0 is t h e i n it ia l s p e e d . A s e c o n d i n t e g r a t i o n o v e r d t l e a d s t o

1 qE 2- ty o + v o t + - ; ,L m (3 .47 )

w h e r e y 0 is t h e i n it ia l p o s i t io n . T h i s i s v e r y l ik e w h a t w e h a v e f o r th e g r a v i typ r o b l e m , e x c e p t t h a t t h e a c c e l e r a t io n n o l o n g e r is g , b u t q E / m .

~ Electron and shee t of positive chargeC o n s i d e r i n F ig u r e 3 . 2 4 a n e l e c t r o n ( q - - e - 1 . 6 x 1 0 - 19 C , m = 9 . 1 x1 0 -3 1 k g ) w i t h s p e e d 1 07 m / s a t th e m o m e n t i t p a s s e s th r o u g h a t i n y h o l e ina p o s i t i v e l y c h a r g e d s h e e t o f ~ - 5 x 1 0 - 6 C / m 2 . F i n d ( a ) t h e f ie l d , ( b ) t h ea c c e l e r a ti o n , ( c ) t h e t i m e f o r t h e e l e c t r o n t o r e t u r n t o t h e c a t h o d e a n d( d) t h e m a x i m u m d i s ta n c e o f t h e e l e c t r o n f ro m t h e s h e e t.S o l u t i o n : (

the electric field in physics

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