Physics -Gausss Law Flux and Charge 1

7/28/2019 Physics -Gausss Law Flux and Charge 1

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"No w the quan t i t y o f e lect ric it y in a bo dy i s mea sured in t erms , accord ing to Faraday ' s"ideas, by the n u m b e r o f l ines o f for ce . . , wh ich proceed f rom i t. These l ines o f force m us tal l termina te somewhere, ei ther on bodies in the neighborhood, or on the walls and roof ofthe room, or on the earth , or on the heaven ly bod ies , an d wherever they term ina te therei s a qu an t i t y o f elec tr ici ty exac t ly equa l an d oppos ite to tha t on the par t o f the b o d y f r o mwhich they proceeded.

---James Clerk Ma xwe ll,A Trea t ise on E lec t r ic ity an d Ma gne t ism (1873)

C h a D t e r 4Gauss's Law: F lu x and C hargeAre Rela tedC h a p t e r O v e r v i e wSect ion 4.2 def ines elect r ic f lux and e lect r ic f lux dens ity, and o btains the constan t ofproport ional i ty between elect r ic f lux leaving and charge enclosed by a Gaussian sur-face. Sect ion 4.3 discusses Gauss's law in mo re detai l. Sect ion 4.4 presents a nu mb er o fexamples of the calculat ion o f electr ic f lux, f rom wh ich th e charge en closed is deduced.Sect ion 4.5 considers the three symm etr ical cases wh ere a Gaussian surface S can befound for which the f lux per un i t area is constant, and uses a know ledge of the chargeenclosed to f ind the elect r ic f ield. S ect ion 4.6 considers electr ical co nduc tors in equi-l ibr ium, sh ow ing th at the elect r ic f ield is zero inside them (elect rostat ic "scre enin g") ;appl ication of Gauss 's law then shows that any charge on the c ondu ctor must res ideon i ts surfaces. Sect ion 4.7 sh ows that , fo r a cond ucto r in equ i l ibr ium, the local valueof the surface charge is prop ort ional to the f ield just outs ide the surface. Sect ion 4.8discusses charge me asurem ent u sing a charge e lect rom eter and Faraday's ice-pai l con-ductor. Sec t ion 4.9 proves Gauss's law . Sect ion 4.10 discusses elect rostat ic screeningin more detai l , and shows that there can be no analogous gravitat ional screening.Sect ion 4.1 1 discusses some p ropert ies of electr ical co nduc tors tha t depe nd on themicroscopic nature of the conduc tor, s

4.1 I n t r o d u c t i o nElec t r i c it y is d if f icu l t t o co mp rehe nd in pa r t because w e can no t ac tua l ly see th eelect r ic charge that produces speci f ic e lect r ical effects . However, by t racing thepa th o f t he f i e ld l ines due to a se t o f e lec t r ic cha rges , Fa raday wo u ld h ave bee nab le t o l oca t e t he cha rges , and even d e t e rm ine th e i r m agn i tudes . Th i s i s t he bas i so f f ie ld -l ine d rawing ru l e 3 o f C hap te r 3 .For more precis ion, we replace f ie ld l ines by electr ic f lux. (In d iscussing aflu id , wi th the e lect r ic f ie ld replaced by the f lu id veloci ty , the f l u i d f l u x w o u l dbe a measu re o f t he ra t e a t wh ich f l u id vo lume l eaves. ) Gauss ' s l aw s ta t e s t ha t

14 5


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146 Chapter 4 9 Gauss's Law

Figure 4.1 Flux tub e associated with a chargedq on one conductor, and a charge -d q onanother conductor.

the to t a l e l ec t r i c f lux l eav ing aclosed sur face equals 4zrk t imesthe ne t e l ec t r i c cha rge ~.,enc en -closed w i th in tha t s u r f ace . B ecauseof Gauss ' s law, f lux tubes can bed raw n , w hos e s ides a r e pa ra l l e l t othe e l ec t r i c f i e ld , and w h ich ca r ryt h e s a m e a m o u n t o f fl u x a l on gany cross -sect ion . This f lux can bet r a c e d a l o n g t h e t u b e a t o n e e n dto a de f in i t e am ou n t o f pos i t ivecharge , and a t the o the r end to an equa l am ou n t o f nega t ive cha rge . See thef lux tube and i t s a s s oc ia t ed cha rge in F igu re 4 .1 , w h ich i s d r aw n fo r cha rge ont w o c o n d u c t o rs .A cons e quence o f G aus s ' s l aw is tha t , i f t he e l ec t r ic f lux leav ing the s u r f ace o fa n o b je c t is k n o w n , e i t h e r b y c a lc u l a t io n o r b y m e a s u r e m e n t , w e c a n d e t e r m i n e

the e l ec t r i c cha rge w i th in tha t ob jec t .4 .2

4 . 2 . 1M o t i v a t i n g G a u ss 's L a w : Defining Elect r ic F lux (~EThe Nu mber of F ield L ines N L eaving a Closed SurfaceA ccord ing to F a raday , the n um be r N o f f ie ld l ines (o r l ines o f fo r ce ) l eav ing ac lo s ed s u r face ( s uch as the ex te r io r s u r f ace o f a foo tba l l ) i s p rop o r t iona l to thecharge enc los ed (Qenc) by tha t s u r f ace . Tha t i s ,

N = o~ Q~en c. (4.1a)H o w e v e r , t h e p r o p o r t i o n a l i t y c o n s t a n t ~ d e p e n d s u p o n t h e c h o i c e o f n u m b e ro f fi e ld l ines pe r un i t cha rge : F a raday m igh t choos e e igh t l ines pe r u n i t cha rge ,w he reas Maxw e l l mi gh t ch oos e s ix l ines pe r un i t cha rge . O u r goa l is to w r i t ea r e l a tion l ike (4 .1 a ) w i th a new c~ w ho s e va lue ev e ryone w i l l ag ree upon . Th enew r e la t ion i s ca ll ed G aus s ' s l aw .Le t ' s f ind N fo r the i r r egu la r c lo s ed s u r f ace o f F igu re 4 .2 (a ) . B reak up thefu ll s u r f ace in to an in f in it e nu m be r o f in f in i te s ima l a r eas d A. Take ~ to po in ta l o ng t h e o u t w a r d n o r m a l t o d A. M e a s u r e t h e n u m b e r o f f ie ld l in e s d N leavingeach d A and add t he m up . R epres en t ing th i s s um as an in teg ra l , and us ing a l i t tl ec i r cl e to den o te an in teg ra l ove r a c lo sed s u r face , w e have

d N " AN - f d N - f -d -~d . (4.1b)S i n c e t h e n u m b e r p e r u n i t a r e a d N / d A of the f ie ld lines is p ropo r t iona l toIEI, th is sugges ts taking dN/ dA " . . IEI in (4 .1b) . That works for a sur face dAo tow h ic h E i s no rm al ( i. e. , E po in t s a long the n o rm al h0 in Fig 4 .2b ) . T hen

d N ]El, for E alo ng fz0. (4.2a)H ow ever , cons ide r a s u r f ace d A i nc l ined to dAo at an angle O, w h e r e t h es ame n um be r o f f i e ld lines d N p a s s t h r o u g h b o t h dAo a n d dA. N o w E m a k e s


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4 .2 Mot i va t i ng Gauss ' s Law 147

Figure 4 .2 Defining electric f lux ~ . (a) Flux leaving an arbi t rary closedsu rface , in t e rms o f the f lux th roug h a su r face e lem en t d A w i t h o u t w a rdnorma l h . F lux per un i t a rea i s d q ) ~ / d A =/~ 9h . (b) Reorient ing fromsurface e lemen t d A o with no rmal h0 a long /~ to su r face e lemen t d A w i t harb i t ra ry no rmal h0 decreases d ~ / d A .

a n a n g l e 0 r e l a t i v e t o h" c o s 0 - / ~ 9 h . F r o m F i g u r e 4 . 2 ( b ) , d A o - d A c o s 0 , s od A - d A 0 / c o s 0. W i t h / ~ - I/~ I/~, w e h a v e

d N d N d N -. -~ -.= = ~ c o s 0 ~ I EI c o s 0 - I E I / ~ " ~ - E - h . ( 4. 2b )d A d A o / c os 0 d A o

E g u a t i o n ( 4 . 2 b ) i s c o n s i s t e n t w i t h t h e f a c t t h a t , i f / ~ i s p a r a l l e l t o o u r s u r f a c e( E . ~ - 0 ) , t h e n n o f ie l d l i ne s c r o s s t h e s u r f a c e ( so d N / d A - 0 ) . M o re o v e r , i ta l s o i n c l u d e s i n f o r m a t i o n a b o u t w h e t h e r o r n o t t h e f i e l d l i n e s g o i n o r g o o u t :/~ 9 i s p o s i t i v e fo r f i e l d l i n e s le a v i n g , a n d n e g a t i v e fo r f i e ld l i n e s e n t e r i n g . I n d e e d ,/~ 9 ~ , n o t IE l, i s t h e p r o p e r m e a s u r e o f d N / d A .

4 ~ 1 7 6 Def in ing E lect r i c F lux 9 EI n s t e a d o f (4 . 2 b ) f o r d N / d A , d e f i n e

i iiiiiiiiiiiiiiiiiiiii!iiiili!iiiiiiiii!iii !~i iii ! i!iiiiiiiiiii~i::~i!i~iiii!iiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiiiii!iiiiiiiiliiiiiiiiii!iiliiiiiiiiiiiiiiiiliiiiilliiiiiiiiiiiiiiiii!!iiiiiiiiiiiiiiiiiiiiiiiiiiiiii!iiiiiii!iiiiiiiiiiiiiiiiiiili!iiii!iiiii!iiiiiiiiiii!iiiiiiiliiiiiljilT h e n ( 4. 1 b ) i s r e p l a c e d b y

In (4 .4 ) , t h e re m a y o r m a y n o t b e a r e a l p h y s i c a l o b j ec t a s s o c i a t e d w i t h t h e c lo s e dsur face . S u c h a s u r f a ce , b e c a u s e i t i s i n t e n d e d f o r u s e w i t h G a u s s ' s l a w , i s c a l l e da G a u s s i a n s u r f a c e .


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148 C h a p t e r 4 I G a u s s ' s L a w

S o m e a u t h o r s u s e t h e e q u i v a l e n t d e f i n i ti o n

d , ~ = i l d A , (4 .5)

w h e r e d A = I d A I. T h e n o t a t i o n d S - d A is s o m e t i m e s u s e d .W i t h ( 4 . 4) , i n s t e a d o f ( 4. 1 b ) w e w r i t e ,

r E m o l ~ e n c , (4.6)w h e r e c~ w i l l b e d e t e r m i n e d s h o r tl y .

N o t e t h a tE 9 ~ - E x n x + E y n y + E z n z - IE I c o s O, (4.7)

w h e r e 0 i s t h e a n g l e ( o f le s s t h a n o r e q u a l t o 1 8 0 ~ b e t w e e n E a n d f t. E q u a -t i o n s ( 4 .7 ) a n d ( 4 .3 ) g i v e u s m o r e t h a n o n e w a y t o o b t a i n d O E / d A .

~ Surface, ield, and fluxC o n s i d e r a c l o s e d su r f a ce a n d a n e l e m e n t o f a r ea d A = 10 -6 m 2 wh ere t heou t w ar d no rm al i s ~ = 0 .36 )? + 0 .851 + 0 .48s and t he e l ec t r i c f i e l d is E =(2)? - 33? + 4s N/ C at d A . ( a ) V e r i f y t h a t t h e n o r m a l f i i s a u n i t v e c t o r . F i n d[b ) ]E 1, ( c ) t he f l ux per u n i t a r ea , (d ) t he f l ux , and ( e ) t he a ng l e 0 be t w ee nE and f t .Solution: ( a ) B y ( R . 1 1 ) ,

Ifi - V/(0.36) 2 + ( 0.8) 2 + (0.48) 2 1.(b) I /~1 = v/ (2) 2 + ( . -3) 2 + (4) 2 = , /~ = 5 .39 N/C. (c) By (4 .3) , the f lux per un i tarea i s d O E / d A = E . f i = (2 ) (0 .36 ) + ( -3 ) (0 .8 ) + (4 ) (0 .48 ) = 0 .24 N/C a t d A .(d) Since d A = 10 -6 m 2, we ha ve d o e = ( d O E / d A ) d A = 0.24 x 10 -6 N-m ?/C .Th i s i s t he f l ux t ha t l eaves t h rough d A . ( e ) By (4 .7 ) , cos0 = /~ . /~ / I /~ l =0 .045 ,so t ha t 0 = 87 .4 ~ Thus , t he f ie l d l i ne leaves t he su r f ace e l em en t a t near g l anc i ngi nc i dence .

~ Flux or rotated surfaceF o r t h e s a m e E a n d d A a s i n E x a m p l e 4 . 1, r o t a t e t h e n o r m a l f i u n t i l i t m a k e sa n a n g l e o f 2 5 ~ t o / ~ . F i n d t h e f l u x t h r o u g h d A .Solution: By (4 .4) , the f lux d O E = ( d O E / d A ) d A = IEI co se d A - ~ cos (25 ~9 10 -6 - 4 .88 x 10 -6 N- m ? /C l eaves t h rough d A .

4 . 2 ~ 3 R elating Electric Flux @r and Charge Enclos ed QencT h e c o n s t a n t c~ i n ( 4 .6 ) i s d e t e r m i n e d o n c e a n d f o r a ll b y c o n s i d e r i n g a c a s ef o r w h i c h b o t h O E a n d Qenc c a n b e o b t a i n e d : a p o i n t c h a r g e q a t t h e o r i g in .


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4 . 3 G a u s s ' s L a w 1 4 9

Figure 4.3 Electric flux ~ f o r aconcentr ic Gaussian surface, due to a pointcharge. Here, the flux per unit area isuniform, and easi ly computed, so the totalf lux can be eas ily comp uted. A Gaussiansurface is a closed (and typically imaginary)surface, used for appl icat ion wi th Gauss 'slaw, which relates electric flux to chargeenclosed.

I t ma k e s a s p h e r i c a l l y s y mme t r i c e l e c t r i c f i e l d a t ~ , g i v e n b y / ~ = ( k q / r 2 ) ~ , soE ~ - E . ~ - ( k q / r 2 ) . Fo r a c o n c e n t r i c s p h e r i c a l G a u s s i a n s u r f a c e o f r a d i u s r ,h = ~ , b e c a u s e t h e e l e c t r i c f ie l d p o i n t s a l o n g t h e o u t w a r d n o r m a l . Se e F i g u r e 4. 3 .

T h u s t h e f l u x p e r u n i t a r e a o v e r t h is c o n c e n t r i c s p h e r i c a l s u r f a c e o f c o n s t a n tr a d i u s r t a k e s o n t h e c o n s t a n t v a l u e

d c b e / d A - E . : , - E . ~ - Er - ( k q / r 2 ) .H e n c e t h e t o t a l f l u x i s

9 ~ - ( d ~ / d A ) A s p h e r e - ( k q /r P ) As p h e r e - ( k q / r 2 ) ( 4 z c r 2) - 4 Jr k q .M o r e o v e r , f o r t h i s G a u s s i a n s u r f a c e Qen~ - q . P l ac i n g t h e s e ~ e a n d Qenc i n (4 .6 )y ie lds 4 J r k q - a q , so ~ - 4Jrk, w he re 4Jrk - 60 1.

4o3 Gauss 'sLawUs in g c~ = 4 rck , ( 4 . 6 ) b e c o m e s

ii!i}~}i}i)i}ii?iiiiiiiiiiii}iiiiiiiiili!iiiiiiliiNi!!iiiiiiiiiiiiiiii!)i?i!ili ii }iiiiii}!ii!iiiiiiiiiiii!ilil}i}}iil ?i !!! ! ! ?}}!i}~:i~iii::ili!iiiii)ii!!iiiiiiiiii~;i~iiiiiiiiiili!!ii~ill!ili!i!iiiiii ?ili}iil ?iiii!ili!iiiiiiil!ii!iiiiiii! !iii iiiii ii !iiliilio r equ iva len t ly ,

_ 1Q e n c 4: rk ~e - 60~e , k - 8 9875 109 N-m 2C2 . ( G a u s s ' s l a w )(4.9)

E i t h e r o f t h e s e r e s u l t s is k n o w n a s G a u s s ' s l aw . T h e y a r e t r u e f o r a n y s h a p e o f t h eG a u s s i a n s u r f a c e . Co u l o mb ' s l a w l o o k s s i mp l e r u s i n g k , a n d G a u s s ' s l a w l o o k ss i mp l e r u s i n g ~0 .

E q u a t i o n ( 4 . 8 ) e x p r e s s e s t h e i d e a t h a t o n l y Qenc is r e s p o n s i b l e f o r t h e e l e c t r i cf l u x . I t d o e s n ' t t e l l u s h o w t h e c h a r g e i s a r r a n g e d , o r h o w ma n y c h a r g e s t h e r ea r e ; o n l y t h e n e t c h a r g e . I f t h e c h a r g e i n s i d e i s mo v e d , t h e f l u x d o e s n ' t c h a n g e .I f t h e c h a r g e o u t s i d e i s mo v e d , t h e f l u x d o e s n ' t c h a n g e . O n l y i f c h a r g e e n t e r so r l e a v e s t h e G a u s s i a n s u r f a c e d o e s t h e f l u x c h a n g e . I f w e w e r e t o m o v e t h ee n o r m o u s c h a r g e Q = 1 09 C f r o m i n f in i t y t o j u s t o u t s i d e a G a u s s i a n s u r f a c e , t h eloca l va lues o f d ~ / d A o n t h e s ur f a c e w o u l d c h a n g e co n s id e r a bl y , b u t ~ w o u l dn o t c h a n g e a t all. I f w e w e r e t o m o v e t h i s e n o r m o u s c h a r g e t o t h e i n s i d e o f t h e


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150 Chapter 4 m Gauss 's Law

G a u s s i a n s u r f a c e , n o t o n l y w o u l d t h e l o c a l v a l u e s o f d ~ , / d A c h a n g e c o n s i d e -r a b l y ( i n c l u d i n g s i g n c h a n g e s ) b u t s o w o u l d ~ e . T h e l o c a l v a l u e s o f d ~ . / d Aa r e a f fe c t e d b y b o t h i n t e r n a l a n d e x t e r n a l c h a r g e, b u t ~ e i s a f f e c t e d o n l y b yin te rna l cha rge .~ Charge rom flux

F or Example 4 .1 , f ind the am oun t o f cha rge d q as s oc ia ted w i th the f lux d~ .Solution: Usin g (4.9) in the form d q = d ~ e / ( 4 n k ) , Faraday wou ld trace fromd ~ e t o d q . With d ~e = 4.88 x 10 -6 N-m2/C, this gives d q = 4.31 x 1 0 - 1 7 C.The equat ion d q = d ~ E / ( 4 ~ r k ) also relates the d q and d~e of F igure 4 .1 .

4.3,1 Numerical I ntegration o f Electric FluxF o r a s p e ci fi c e x a m p l e o f e le c t r ic f l u x c o m p u t a t i o n f o r a m o r e c o m p l e x s u r f a c ethan a s phe re concen t r i c w i th a po in t cha rge , cons ide r a po ta to . See F igu re 4 .4 .Imag ine tha t s ome e lves have s e t up a f ine g r id onE" ^ the s u r f ace o f the po ta to (w i th pe r hap s 1000 s u r f ace e l -e m e n t s ) . I n t o a s p r e a d s h e e t ( se e T a b le 4 . 1) , t h e y e n t e ri n f o r m a t i o n a b o u t t h e G a u s s i a n su r fa c e , ta k e n t o b e t h eo u t e r s u r fa c e o f t h e p o t a t o .C o lu m n A g ives the in t ege r s i f r om 1 to 1000. F o rt h e i e l e m e n t , t h e e l v e s m e a s u r e t h e a r e a d A i a n d d e -t e r m i n e t h e t h r e e c o m p o n e n t s o f t h e o u t w a r d n o r m a l~ i. C o l u m n B c o n t a i n s t h e d A i ' s , a n d c o l u m n s C , D ,Figure 4.4 E l e c t r i c E c o n t a i n t h e t h r e e c o m p o n e n t s o f t h e ~ i's . ( S o fa r , t h ef lux ~ e l eav ing an e lves have cons id e red on ly the p rop er t i e s o f the s u r face .a rb it ra ry c lo sed s u r f a ce The s e do no t chang e even w he n th e e l ec t r i c f ie ld( a G auss ian s u r f ac e ) , changes . ) N ow , fo r each s u rf ace e l em en t , t he elves~ me a-w i th the s u rf ace s u re the th r e e co m po nen t s o f the e l ec t r i c f i e ld E i , w h ichb roken in to many a r e en te r ed in co lum ns F, G , H . Th e m eas u re m en t s a r es u rf ace e lemen ts , now com ple te , and w e now can co m pu te the e l ec t r i cflUX.Equ a t ion (4 .2) g ives the f lux pe r un i t a r ea fo r each s u r f ace e l emen t , and i se n t e r e d i n c o l u m n I. ( In t e r m s o f t h e s p r e a d s h e e t , t h e d o t p r o d u c t i s t h e s u mo f t h e p r o d u c t s o f c o l u m n s C a n d F , D a n d G , a n d E a n d I. ) T h e c o r r e s p o n d i n gf lu x is t h e p r o d u c t o f t h e f l ux p e r u n i t a r e a ( c o l u m n I ) a n d t h e a r e a ( c o l u m n B ) ,a n d i s p l a c e d i n c o l u m n J . ( I n T a b l e 4 . 1 , t h e e n t r i e s w e r e m a d e a s s u m i n g t h a tExa m ple 4 .1 co r r es pon ds to i = 1 ) Summ ing a l l 1000 en t r ie s in co lu m n J g ivesa num er ica l va lue fo r (4 .4 ) , t he to t a l f lux l eav ing the s u r face . ( I f a g r id o f 1000p o i n t s i s n ' t fi ne e n o u g h f o r a n a c c u r a t e m e a s u r e , t h e n a fi n e r m e s h s h o u l d b e u s e d .Table4.1 Spreadsheet calculation of flux

i dAi n i x n i x n i z E i x E i x E i z E . ~t E . f i dA1 1.0 x 10 -6 0.36 0.8 0.48 2 - 3 4 0.24 0.24 10 -6


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4.4 Com puting E, and T hen U sing Gauss 's Law to Obtain ~enc 15 1

4~3o2

I n p r inc ip l e , a l l num e r i c a l i n t e gr a t i ons shou ld a lwa ys use a t l e a s t two m e she s ,o n e f i n e r t h a n t h e o t h e r , t o b e r e a s o n a b l y s u r e t h a t t h e r e i s n ' t a n e r r o r d u e t otoo c r u de a m e s h . ) Le t ' s sa y t ha t t he a nswe r , c onv e r ge d t o two de c im a l p l ac e s , isCI)E - - 8 .4 x 10 2 N- m 2 /C. A pply ing ( 4 .9) t o our po t a to - sh a pe d Ga us s i a n su r f a ce ,w i th ~ = 8 .4 x 1 02 N- m 2 /C , y i e lds t ha t i t c on t a ins ~enc-- 7.4 x 10 -9 C, ava lue c ha r a c t e r i s t i c o f s t a t i c e l e c t ri c i ty .Ga uss ' s l a w ha s l e t us de t e r m ine t he t o t a l , o r ne t , e l e c t r i c c ha r ge i ns ide t hep o t a t o - s h a p e d G a u s s i a n s u r f a c e , w i t h o u t a c t u a l l y m e a s u r i n g t h a t c h a r g e . W em e a s u r e d s o m e t h i n g e l s e , t h e e l e c t r i c f l u x , w h i c h b y G a u s s ' s l a w y i e l d e d t h eto t a l c ha r ge e nc lose d . T ha t i s som e th ing o f a m i r a c l e . I n p r inc ip l e , we c ou ldbu i ld a n e x t e ns ib l e su r f a c e ( lik e a ba l l oon , o r a 1 9 50 s c om ic s c ha r a c t e r c a l l e dPla s ti c Ma n , o r t he " sha pe - sh i f t e r s" o f t he 1 9 9 0s t e l e v i s ion sho w D e e p S p a c eN i n e ) t o s u r r o u n d a n y re g i o n , a n d t o d e t e r m i n e t h e e l e c t r ic f lu x t h r o u g h i t. I npr a c t i c e , d i r e c t m e a sur e m e nt s a r e d i f f i c u l t . As w i l l be d i sc usse d , de v i c e s us ingt h e p r i n c i p l e o f F a ra d a y ' s i c e p a i l ( w i t h a n a t t a c h e d e l e c t r o m e t e r ) r e m o v e t h a tdiff iculty.Us eful R esult for U niform Flux through O nly One P artof the Gaussian SurfaceC o n s i d e r a G a u s s i a n s u r f a c e t h a t h a s b e e n d e c o m p o s e d i n t o p a r t i a l s u r f a c e s . I ft he f l ux goe s t h r o ug h on ly on e o f t hose p a r t s ( a b ig i f) , a nd i f t he f l ux is un i f o r mt h r o u g h t h a t p a r t ( a n e v e n b i g g e r i f ) , t h e n

9 E - d A ( u n i f o r m , n o n z e r o f lu x o n l y t h r o u g h Aflux)(4.10)

w h e r e A f l ~ i s t he a r e a o f t he su r f a c e t ha t p i c k s up t h e f l ux . T he r e a r e c er -t a i n i m p o r t a n t g e o m e t r i e s , w h i c h i n v o l v e e i t h e r c o n d u c t o r s i n e q u i l i b r i u m o rve r y sym m e t r i c a l c ha r ge d i s t r i bu t i ons ( sphe r i c a l , c y l i ndr i c a l , a nd p l a na r ) , whe r e( 4 .1 0) doe s a pp ly . For t he se c a se s , ( 4.8 ) a n d ( 4 .1 0 ) c o m b ine t o y i e ldi~iiiiii ~ iiii ~~iiiiiiiiiiiii i i i ii i i iii i i iiii

i iii i i iiii iAs a c he c k , no t e t ha t f o r a Ga uss i a n su r f a c e t ha t i s a sphe r e o f r a d ius r c on c e n t r i cw i t h a p o i n t c h a r g e q , t h e e n t i r e s p h e r e i s t h e p a r t t h r o u g h w h i c h t h e f lu x isun i f o r m . Se e F igur e 4 .3 . Wi th Aflux - 4rrr 2, ( 4 .1 1 ) t h e n y i e l d s / ~ . ~ - k q / r 2, ase x pe c t e d . E q ua t ion ( 4 .1 1 ) is c e n t r a l t o Se c t ions 4 .5 a nd 4 .7 .

4 ~ C o m p u t i n g E a n d T h e n U s i n g G a u s s ' s L a wt o O b t a i n QencH e r e a r e s o m e e x a m p l e s o f h o w t o c o m p u t e (I)E f o r a c l o s ed s u r fa c e, a n d t h e nde t e r m ine t he c ha r ge e n c lose d Q~n~, by G a uss ' s l aw , ( 4 .9) .


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152 C h a p t e r 4 ~ G a u s s ' s L a w

~ Un iform field and cubeC o n s i d e r a G a u s s i a n s u r f a c e S t h a t e n c l o s e s a c u b e o f s i de a , w i t h o n e c o r n e ra t t h e o r i g in , a n d a n o t h e r c o r n e r a t ( a , a , a ) . I t i s i n a un i fo rm e l ec t r i c f i e l da l o n g t h e x -a x is , s o / ~ - E ~ , w h e r e E i s a c o n s t a n t . S e e F i g u r e 4 .5 . F i n d t h ef l ux t h r o u g h S a n d t h e c h a r g e e n c l o s e d .S o l u t i o n : Si nce E i s cons t an t , everyne t number o f l i nes l eav i ng S i s zero .

Figure4.5 Gauss i an su r f ace t ha t i sa cube .(t~ r~ ~ back ~E ontis , , ~ = E A . T h e t o t a l f lu x is ~ = +e x p e c t e d .

l ine that enters S al so leaves i t , so theT h u s w e e x p e c t t h e e x p l i c i t c a l c u l a t i o nt o y i e l d zero ne t f l ux , o r ~E = 0 . Then ,by Gauss ' s l aw, Qenc "--O. L e t ' s n o w d ot h e a c t u a l c a l c u l a ti o n o f ~ . T h e t o pa n d b o t t o m o f t h is c u b e , w i t h h = + ~ ,h a v e n o f l u x p a s s i n g t h r o u g h t h e mb e c a u s e d ~ / d A = E -h = 0 for thes ecases . There i s a l so no f l ux t h rough t her ight and lef t faces, w i th h = +3? . Fort h e b a c k f ac e , w h e r e h = - 2c , d ~ / d A =E . h = - E , i t i s nega t i ve because t hef i e l d en t e r s t he su r f ace . Thus , t he f l uxt h rou gh t he back f ace i s q~back = - E A ,wh ere A = a 2 . Fo r t he f ro n t f ace , whe reh = Yc, d ~ / d A = P , . h - E . It is pos-i t i ve because t he f i e l d l eaves t he su r -f ace . Thus , t he f l ux t h rough t ha t f ace

= - E A + E A = O, as

N o t e : a u n i f o r m f ie ld c a n b e p r o d u c e d b y a s h e e t o f u n i f o r m c h a r g e d e n s i t yo r , w h e r e

E - 2zrkcr , (3 .26)d e t e r m i n e s or. ( A fi e ld a l o n g + x c a n b e p r o d u c e d e i t h e r b y a s h e e t cr i n a c o n s t a n tx - p l a n e w i t h x < 0 o r b y a s h e e t - c r i n a c o n s t a n t x - p l a n e w i t h x > a . I n d e e d ,t h e r e a r e in f i n it e l y m a n y w a y s t o p r o d u c e a u n i f o r m / ~ f o r 0 < x < a . ) S i n c et h e s h e e t o f c h a rg e d o e s n o t i n t e r s e c t th e c u b e , w e e x p e c t t h a t ~_~enc -- O. B yG a u s s ' s la w , ( 4 .8 ) , w e t h e n e x p e c t t h a t ~ - O, c o n s i s t e n t w i t h t h e d i sc u s s i o na b o v e . ( I f t h e c h a r g e d s h e e t l i e s i n 0 < x < a , s o i t i n t e r s e c t s S, t h e n t h e f l u xi s + E A f o r b o t h f r o n t a n d b a c k fa c e s , f o r a n e t f l u x o f ~ - 2 E A , a n d ~_~enc --( 4 Jr k ) - 1 ~ E - c rA , a s e x p e c t e d . W e w i l l r e t u r n t o t h i s c a s e . )

~ Un iform field and half-cubeN o w c o n s i d e r t h a t t h e c u b e i s s l i c e d d i a g o n a l l y , a s i n F i g u r e 4 . 6 . F i n d t h e f l u xl e a v in g th i s n e w G a u s s i a n s u r f a c e S a n d t h e a m o u n t o f c h a r g e e n c l o s e d b y Sw h e n i t i s i n a u n i f o r m e l e c t r i c f i e l d a l o n g t h e x - a x i s .S o l u t i o n : For th is uni form f ield , every l ine that enters S al so leaves i t , so thene t n um be r o f l i nes l eav i ng S i s zero ; aga i n we expe c t zero n e t f l ux, andzero enc l osed charge . Let ' s now do t he ca l cu l a t i on . As befo re , t here i s no f l ux


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4 . 4 C o m p u t i n g E a n d T h e n U s i n g G a u s s ' s L a w t o O b t a i n ~ e n c 153

IZ I

III " ~ ' - - - ' - a

o

x /f

f

F i g u r e 4 . 6 G a u s s i a n s u r f a c e t h a t i sh a l f o f a c u b e .

t h r o u g h t h e t o p o r b o t t o m ( t h e r e i sn o b o t t o m ) , a n d n o f lu x t h r o u g h t h er igh t o r l e f t , be c a use the f i e ld l i ne sd o n o t p a s s t h r o u g h a n y o f t h e s e s u r -f a c e s . A s b e f o r e , t h e f l u x t h r o u g h t h eb a c k is - E A . F o r t h e f r o n t, t h e n o r -ma l is h = (~ - ~) /~ /2 , so d ~ E / d A -/~ 9 fz - E / v ~ , a d e c r e a s e d f lu x p e r u n i ta r e a . H o w e v e r , f o r t h e f r o n t f a c e t h ea r ea h a s i n c r ea s e d t o ~ A , s o t h a t

E~ -- ( E / v / 2 ) ( v ~ A ) = E A d o e s n o tc ha nge . Th us the t o t a l f l ux is @~ =- E A + E A = 0 , a nd by (4 .9 ) , Q e n c =0 , a s e x pe c te d .

~ Nonuniform field and boxC o n s i d e r a G a u s s i a n s u r f a c e t h a t e n c l os e s a p a r a l l e l o p i p e d w h o s e f r o n t a n d

b a c k fa c e s h a v e x - b a n dx - a , a n d a r e a A . L e t t h ee l e c t r i c f i e l d E p o i n t o n l ya l o n g ~ , w i t h a c o m p o n e n tE x t h a t m a y d e p e n d o n x ,b u t n o t o n y o r z ; E -E x(x )Yc . S e e F i g u r e 4 . 7 .( a ) F i n d t h e n e t f l u x t h r o -u g h t h i s G a u s s i a n s u r f a c e .( b ) F o r A - 4 x 1 0 - 4 m 2,a = 1 . 5 m , a n d b = 2 . 5 m ,a n d E x ( x ) - C x , w h e r e

F i g u r e 4 .7 G a u s s i a n s u r f a c e t h a t i s a C - 2 x 1 06 N / C - m , f i n dr e c t a n g u l a r p a r a l l e l o p i p e d . ~ a n d Q e n c .Solution: ( a ) H e r e t h e f i el d i s n o n u n i f o r m , s o m o r e f i e ld li n e s m a y e n t e r t h ef r o n t t h a n t h e b a c k , m e a n i n g p o s s i b l e n e t f l u x a n d n o n z e r o Q e n c . A s i n t h e p r e -v i o u s e x a m p l e s , t h e r e i s n o f l u x t h r o u g h t h e t o p o r b o t t o m , o r t h e r i g h t o r l e f t.T h e f l u x p e r u n i t a r ea th r o u g h t h e f r o n t t a ke s o n th e u n i f o r m v a l u e d q ) E / d A =/~ 9 - E 9 )~ = Ex (b ) ; t h ro ug h th e ba c k the f l ux pe r un i t a re a t a k e s on th e d i f -f e r e n t u n i f o r m v a l u e d ~ E / d A - E . f~ = E . ( -Y c ) - - E x ( a ) . T h e n e t f l u x t h u s i sg i v e n b y

9 ~ = E x ( b ) A - E x ( a ) A = [ E x ( b ) - E x ( a )] A. (4 .1 2 )I f E x ( x ) > 0 , t h i s c o r re sponds to pos i t i ve f l ux fo r x = b ( l e a v ing ) a nd ne ga t ivef lux fo r x = a ( e n t e r ing ) . (b ) F rom G a u ss ' s l a w a nd (4 .1 2 ) , w e c a n o b t a i n Q e n c .H e r e E x ( a ) = C a = 3 x 10 6 N/ C a nd E x ( b ) = Cb = 5 x 106 N/C , so by (4 .12)9 ~ = 8 x 1 02 N/C . H e nc e , b y (4 .9 ) , Qenc --- 7 .0 7 x 1 0 -9 C . I f de s i r e d , w e c ou lda l so ide n t i fy c ha rge s Q a a n d Q b a s s o c i a t e d w i t h t h e f l u x e s E x ( a ) A a n d E x ( b ) At h r o u g h t h e f r o n t a n d b a c k f a c e s .


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154 C h a p t e r 4 = G a u s s ' s L a w

Figure 4 . 8 S p h e r i c a l l y s y m m e t r i cc h a r g e d i s t r ib u t i o n a n d c o n c e n t r i cG a u s s i a n s u r f a c e S t h a t e n c l o se s aspher i ca l she l l o f f in i t e t h ickness . T husS h a s b o t h a n i n n e r s u r f a c e a n d a no u t e r s u r f a c e . T h e i n n e r ( o u t e r ) n o r m a lp o i n t s a l o ng t h e o u t w a r d ( i n w a r d )rad i a l d i r ec t i on . Fo r each su r f ace , t hef l ux dens i t y is un i fo rm .

Spherical symm etryC o n s i d e r a G a u s s i a n s u r f a c e S t h a t e n c l o s e s a s p h e r ic a l s h e l l w h o s e i n n e ra n d o u t e r r a d i i a re a t r = a a n d r - b , w h e r e a < b . ( T h u s S h a s a n i n n e ra n d a n o u t e r s u r f a c e .) L e t t h e r e b e a n o n u n i f o r m r a d ia l fi el d, s o E - ~ ' E r ( r ) ,w h e r e E r ( r ) m e a n s t h a t E r is a fu n c t i o n o f r . S e e F i g u re 4 .8 . F i n d th e n e t f l u xt h r o u g h S .S o l u t i o n : Th e f l ux per un i t a r ea l eav i ng t he i nner su r f ace a t r = a , wh ereh = -~ , i s

( d @ E / d A ) l i n n e r = E . h = E . ( - ~ ) = - E r ( a ) ;fo r t he ou t er su r f ace a t r = b , wh ere f i = ~ , t he f l ux per u n i t a r ea i s

( d ~ E / d A ) ]o u t e r - E . n - - E " ~ " - E r ( b ) .T h e f lu x t h r o u g h t h e i n n e r s u rf a c e, o f a r e a A i n n e r -- 4zra 2, is - E r ( a ) ( 4 r r a 2 ) , a n dt h e f lu x t h r o u g h t h e o u t e r s u r f a c e, o f a r e a Aouter = 4 ;rb 2, is E r ( b ) ( 4 r r b 2 ) , so t hene t f l ux leav i ng t he spher i ca l she l l i s g i ven by

@ 6 = E r ( b ) ( 4 7 r b 2 ) - E r ( a ) ( 4 r r a 2 ) 9 (4 .13 )I f E ( a ) > 0 , t he n f l ux en t er s a t r = a , and i f E (b) > 0 , then f lux leaves at r - b .

Estimat ing the charge on the earth!A s s u m e t h a t a m e a s u r e m e n t o f I EI a t t h e s u r f a c e o f t h e e a r t h ( b - 6 . 3 7 x106 m) y i e l d s E r ( b ) = - 1 3 0 N / C , w h i c h p o i n t s to w a r d t h e e a rt h , in d i c at -i n g t h a t t h e e a r t h i s n e g a t i v e l y c h a r g e d . T a k i n g t h i s f i e l d a s c h a r a c t e r i s t i co f t h e e n t i r e e a r t h a n d e l i m i n a t i n g t h e i n n e r s u r f a c e b y s e t t i n g a = 0 i n( 4 . 13 ) y i e l d s C E = - 6 . 6 3 x 1 0 16 N - m 2 / C . G a u s s ' s la w , ( 4. 9 ), t h e n y i e ld sQ e a r t h ~ - - 5 . 8 6 X 10 5 C . T h i s i s q u i t e a l o t o f c h a r g e . I t c o r r e s p o n d s t o ac h a r g e p e r u n i t a r e a r h - - Q e a r t h / 4 7 r b 2 - - 1 .15 x 10 -9 C /m 2. Ove ra l l , t hee a r t h a n d i ts a t m o s p h e r e a r e n e u t r a l ; a n e q u a l a n d o p p o s i t e a m o u n t o f p o s i t iv ec h a r g e r e s i d e s i n t h e a t m o s p h e r e .

ylindrical sym m etryC o n s i d e r a c y l in d r i c a l g e o m e t r y w i t h a n o n u n i f o r m r a d ia l fi el d, s o E - i" E r ( r ) ,a n d e m p l o y a c o n c e n t r i c G a u s s i a n s u r f a c e o f l e n g t h L t h a t i s a c y li n d r ic a l


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4 . 5 D e t e r m i n i n g E b y S y m m e t r y 1 55

Figure 4.9 Cyl indr ica l ly symm etr ic cha rgedis t r ibu t ion and co ncent r ic Gauss ian sur facetha t is a f ini te cylinder . Flux passes only thro ug hthe round s ide , wh ere the f lux dens i ty isu n i f o r m.

s h e l l o f r a d iu s r . ( S o m e t im e s t h e s y m b o l p = v /x 2 + y 2 i s u s e d f o r t h e r a -d iu s i n c y l i n d r i c a l c o o r d in a t e s , b u t w e h a v e a l r e a d y u s e d p f o r t h e c h a r g ep e r u n i t v o l u m e . T h e r e j u s t a r e n ' t e n o u g h s y m b o l s t o g o a r o u n d ! ) T h e f u l ls u r f a c e h a s t h r e e p a r t s : t h e t o p , b o t t o m, a n d r o u n d p a r t s . S e e F ig u r e 4 . 9 .( a) F in d t h e n e t f l u x t h r o u g h t h i s G a u s s i a n s u r fa c e . ( b ) If L - . 2 m , r = . 02 m,a n d E r ( r ) = 4.00 x 104 N/C, f ind ~enc.S o l u t i o n : ( a ) The f luxes for the top and bot tom a re ze ro because the outwardnormals are a long z; this is perpendicular to/~, which is radia l . The f lux per unita rea lea~ng the s ide a t r , w here f i - -~ , takes on the uni form va lue d ~ E / d A =F, 9 = E 9 - E r ( r ) . Th e f lux thro ug h th e ro un d surface, of area A = 2Jrr L, isE r ( r ) ( 2 z r r L ) , so the net f lux is given by

* E = E r ( r ) ( 2 J r r L ) . (4.14)(b) For our specific values of L, r , an d E r ( r ) , (4.14) gives ~E = 1.005 x103 N-m 2/C. From Gau ss 's law, (4.9), w e then d educ e that Qenc = 8.89 X 10 -9 C.This pos i t ive s ign is cons is ten t wi th th e f ac t tha t the f ield po in ts ou tward .

4 .5 De term ining E by Symm etry" The Three CasesC h a r g e d i s t r i b u ti o n s o f t h r e e a n d o n l y t h r e e t y p e s p r o d u c e a f lu x t h a t is u n i -f o r m o r z e r o f o r a ll p a r t s o f an a p p r o p r i a t e l y c h o s e n G a u s s i a n s u r f a c e : c e n t r o -s y m m e t r i c c h a r g e d i s t r i b u t io n s w i t h s p h e ri c a l, c y l i n d ri c al , a n d p l a n a r s y m m e t r y .G a u s s ' s l a w a n d a k n o w l e d g e o f t h e c h a r g e t h e n e n a b l es u s to d e d u c e t h e m a g -n i t u d e o f t h e e l e c t r ic f i el d ; w e s i m p l y a p p l y ( 4 . 11 ) . ( A c u b e t h a t i s c o n c e n t r i cw i t h a p o i n t c h a r g e h a s s o m e s y m m e t r y ; e a c h o f it s fa c es p i ck s u p t h e s a m e f l ux ,w h i c h is o n e - s ix t h o f t h e t o t a l f lu x . H o w e v e r , t h e / ~ f ie ld is n o t u n i f o r m o v e re a c h f a c e , s o t h a t w e c a n o n l y o b t a i n t h e a v e r a g e f l u x f o r e a c h f a c e , r a t h e r t h a nth e e l e c t r i c f i e l d it s e lf . )


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1 5 6 Ch apte r 4 I Gauss 's Law

v Ia ?"(6 )

Figure4 .1 0 (a) Spherical ly symmetric charge distribut ion and concentricGaussian surface for which the flux density is uniform. (b) Electric fieldmagnitude when the charge densi ty is uniform for r < a and zerof o r r > a .4.5.1 Spherical Symmetry

C o n s i d e r a s p h e r ic a l d i s t r i b u t i o n o f c h a r g e, w h i c h p r o d u c e s a n e l e c tr i c f ie ld t h a tpo i n t s r ad ia l ly . Take as Gaus s i an s u r f ace S, a s phe re o f r ad i u s r t h a t i s con cen t r i cwi t h t he cha rge d i s t r i bu t i on . S ee F i gu re 4 .10 (a ) . In gene ra l , t he cha rge d i s t r i bu -t i o n m a y e x t e n d b e y o n d r .H e r e d ~ E / d A i s un i fo rm over S , and t h e a rea t ha t p i cks up t he f l ux i s Aflux -

4 z r r 2 . T h u s , b y ( 4. 11 ) , w i t h / ~ , h - / ~ . ~ - E r ,4 ;r kQ, enc kQ, encE r - 4 Jr r 2 = r2 . ( s phe r i ca l l y s ym m et r i c cha rg e d i s t r i bu t i on )

(4.15)Th i s l ooks l i ke C ou l om b ' s l aw fo r a po i n t cha rge , bu t i t app l i e s t o a n y s pher -i ca ll y s ym m e t r i c cha rge d i s t r i bu t i on . I t i s dec ep t i ve l y s i m p l e . Equ a t i on (4 .15 )

s ays t ha t (1 ) if cha rg e i s w i t h i n t h e Ga us s i an s u r f ace o f r ad i u s r , i t c o n t r i b u t e sas i f i t we re a t t he cen t e r , and (2 ) i f cha rg e i s ou t s i de t he s u r f ace , i t does no tcon t r i bu t e a t a l l . Th i s r e s u l t i s r e l a t i ve l y d i f f i cu l t t o ob t a i n by d i r ec t ca l cu l a t i onf r o m E - k f ( d q / r 2 ) ~ . E q u a t i o n ( 4 . 1 5 ) r e p r o d u c e s t h e r e s u l t f o r a s i n g l e p o i n tcha rge q , E - kq / r 2 .

Since gravi ty satis fies an inverse square law, there is a Gauss's law fo r gravi tat ion, wh ichwo u ld have saved Newton a g rea t dea l o f e f fo r t . I t was n o t easy , even fo r the g rea tNew ton, to d irect ly calculate the grav i tat ional f ie ld due to a bal l of u ni form mass densi ty.At a great d istance, i t is not too bad an approximat ion to consider the earth to be apoint , but for a satel l ite in near earth orb i t , the ea rth is not a point . By the g ravi tat ionalanalog of G auss's law, the ea rth 's gravi ty can indeed be treated as if i t is due to a l l theear th 's mass concent ra ted a t the geo met r ica l cen ter o f the ear th .

U n i f o r m ball of chargeCon sider a spher ical ball o f rad ius a and to ta l charge Q tha t i s uni form lyd i s t r i bu t ed ove r i t s vo l um e ( t he nex t s ec t i on s hows t ha t t h i s canno t be acondu c t o r i n equ il i b r ium ) . F i nd E r for al l r .


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4 . 5 D e t e r m i n i n g E b y S y m m e t r y 157

Solution: A conce ntr ic sphe r ical Gaussian surface S wi th r > a has Qenc-Q.By (4.15) , this implies E r - k q / r 2. (This is the large r par t of Figure 4.10b.) Forr < a, the re i s a cha rge pe r un i t vo lum e

Q Q (4 16)all 4=aThen for r < a ,fff 4r (r)e n c -~- d Q = p d V - p d V - p V = T - - - - ~ r c = Q - . (4.17)~ rc a 3 a

Pl a c i n g Qenc in (4.15) then gives E r - - k Q ( r / a ) 3 / r 2 = k Q r / a 3. This falls to zeroat the or igin, as expected by symmetry; i t a lso takes the value k Q / a 2 at r = a ,as expec ted because then a l l the cha rge i s enc losed . Of course , a conduc tor inequi l ib r ium cannot produce such a vo lume cha rge d is t r ibu t ion; a conduc tor inequi l ib r ium would have a l l the cha rge Q on the oute r sur face a t r = a . F igure4.10( b) plo ts I/~] = ]Er[ for all r . We can also obta in (4 .17) by the r a t io o f theappropr ia te vo lumes: Qenc/Q-- ( 4 r c p r 3 / 3 ) / ( 4 n p a 3 / 3 ) = r 3 /a 3 .

4o5o2 Cylindrical SymmetryC o n s i d e r a c y l i n d r i c a ll y s y m m e t r i c d i s t r i b u t i o n o f c h a r g e , in f i n i te i n e x t e n t . B ys y m m e t r y , i t p r o d u c e s a n e l e c t r i c f ie l d t h a t p o i n t s r a di a ll y . S e e F i g u r e 4 .1 1 (a ) . A st h e G a u s s i a n s u r f a c e S , t a k e a c y l i n d e r c o n c e n t r i c w i t h t h e c h a r g e d i s t r i b u t i o n ,o f ra d i u s r a n d f i n it e l e n g t h L . ( E r s h o u l d n o t d e p e n d o n L . ) H e r e , d ~ E / d A isz e r o o v e r t h e t o p a n d b o t t o m , a n d i s u n i f o r m o v e r t h e c y l i n d r i c a l s u r fa c e , so t h ea r e a t h a t p i c k s u p t h e f l u x i s A f lu x - 2 r c r L . B e c a u s e Qenc is p r o p o r t i o n a l t o L ,

Figure 4.11 ( a) Cyl indr ica lly symm etr ic cha rge d is t r ibu t ion and concent r ic Gauss iansurface that is a f ini te cylinder . The f lux density through the round side is uniform.(b) E lec tr ic f ield ma gni tude wh en the cha rge dens i ty i s un i form for r < a and ze ro for/ ~ > a .


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1 5 8 C h a p te r 4 9 Gauss ' s Law

)~enc ~ ~_~enc/ L i s i n d e p e n d e n t o f L . F r o m ( 4 .1 1 ), w i t h E 9 z - E 9~ - E r ,4 r r k ~ e n c 2 k Xe n c= . ( c y l i n d r i c a l c h a r g e d i s t r i b u t i o n ) ( 4. 18 )E~ = 2rr r L r

E q u a t i o n ( 4 . 1 8 ) i m p l i e s t h a t , i f c h a r g e i s w i t h i n t h e G a u s s i a n s u r f a c e o f r a d i u s r ,i t c o n t r i b u t e s a s i f i t w e r e o n t h e a xi s, a n d i f i t i s o u t s i d e t h e s u r fa c e , i t d o e s n o tc o n t r i b u t e a t al l. F o r a l i n e c h a r g e )~, ( 4 . 1 8 ) r e p r o d u c e s ( 3 . 2 2 ) , w i th ~.enc ~ ~:t h a t i s , E r - 2k)~ / r .

~ Uniform cylinder chargefL e t t h e c h a r g e p e r u n i t v o lu m e p b e a c o n s t a n t f o r r < a , a n d ze r o o th e r w i s e .F in d E r for a l l r .S o l u t i o n : A concent r ic cy l inde r of rad ius r and length L conta ins cha rge Q~encp r r r 2 L , so ~enc ~ ~ . e n c / L - - p z r r 2 . Equa t ion (4 .18) then g ives E r = 2 z r k p r fo rr < a . As a check, note that , as expected by symmetry, the f ie ld is zero a t theor igin ( r = 0) . For r > a , we have Xenc = pr ra 2. Equa t ion (4 .18) then g ives EF =2 z r h p a 2 / r for r > a . As a check , no te tha t ou ts ide the cha rge d is t r ibu t ion thecharge behaves l ike a l ine charge on the axis. Figure 4.11 (b) plots ] E l- IE F ] forall r.

4o5~3 P lanar SymmetryF o r a p l a n a r d i s t r i b u t i o n o f c h a rg e t h a t i s u n i f o r m i n th e y z - p l a n e , t h e e l e c t r i cf i e l d m u s t p o i n t a l o n g t h e x - d i r e c t i o n . If , i n a d d i t i o n , t h e r e i s a c e n t e r o f s y m -m e t r y a t t h e o r i gi n , t h e e l e c t r i c fi e ld w i l l b e t h e s a m e i n m a g n i t u d e a t b o t h xa n d - x . W e n o w e n c l o se t h e c h a r g e d i s t r i b u t io n w i t h a G a u s s i a n s u rf a c e S s h a p e dl ik e a t i n y r i g h t - c i r c u l a r c y l i n d e r ( a " p i ll b o x ") . W i t h n o r m a l s a l o n g + ) ~ , a n d a r e aA f o r e a c h o f t h e r i g h t a n d l e f t si d es , w e h a v e A f l u x - 2 A . ( F o r t h e p i l l b o x , w h i c hc a n b e m a d e a s f l at as a p a n c a k e , t h e r o u n d s i d e p a ra l l e l to E 3 ~ ic k s u p n o f l u x . )S e e F ig u r e 4 . 1 2 . L e t ( Ye nc ~ Q e n c / A . F r o m ( 4 . 1 1 ) , w i t h E = E 9h ,

2 r r k Q ~ e n cE - A = 2rrkcr en c. ( s y m m e t r i c a l p l a n a r c h a r g e d i s t r i b u t i o n ) ( 4. 19 )

Figure4.12 Plana r sym metr ic cha rge d is t r ibu t ion andconcentr ic Gaussian surface that is a f ini te cylinder . Thef lux dens i ty th roug h th e ro und s ide is ze ro , and th e f luxdens i ty th roug h each of the caps is un i form.


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4.6 Elec trical Conductors in Equi l ibrium 159

F o r a s h e e t o f c h a r g e d e n s i t y o r, ( 4 . 1 9 ) r e p r o d u c e s ( 3 . 2 6 ) w i t h O'e n c = ( 7 ; t h a t i s ,E = 2J rka .~ Un iform slab of charge

L e t t h e c h a r g e p e r u n i t v o lu m e b e g iv e n b y p f o r - a < x < a , a n d ze r oothe rwise . F ind Er for a l l r .Solution: Conside r a Gauss ian sur face S tha t i s a symmetr ic p i l lbox of a reaA tha t ex tends f rom -x to +x . I t conta ins cha rge ~ ) _ ~ e n c - - f iA(2x) , so ( Te n c - -~enc/A= p( 2 ) c ) . Placed in (4.19), this gives E = 4zckpx for -a < x < a . As acheck, note tha t , as expect ed by sym metry, th e f ie ld is zero a t the or igin (x = 0) .For x > a , we have ( T en c . ~ - / ) ( 2 a ) . Placed in (4.19), this gives E = 4Jrkpa fo rx > a . As a check , no te tha t ou ts ide the cha rge d is t r ibu t ion the cha rge behaveslike a sheet charge o n the axis. More generally, for planar sy mm etry, if the reis a region where E var ies l inear ly with distance, that region contains a uniformcharge density p. For example , if Ex = cx + d , then c = 4ztkp re la tes c and p.The cons tan t d could be due to d is tan t shee ts or d is tan t s labs of cha rge . Anindica t ion of a un i form cha rge dens i ty i s tha t the s lope of the f ie ld ve r sus d is -tance is a constant; here axEx = c = 4zrkf i is constan t . An app roxim ately uni-form p lana r cha rge dens i ty can be produced by bombarding a p la s t ic s lab wi thions.

4~ Elect r i ca l Conductors in Equi l i b r iumC o n s i d e r a n e l e c t ri c a l c o n d u c t o r o f a r b i t r a r y s h a p e ( e .g ., a p o t a t o , a n a l u m i n u m -p a i n t e d r u b b e r d u c k , o r t h e o b j e c t i n F i g u r e 4 . 1 3 ) .

I n i s o l a ti o n , t h e c o n d u c t o r c o n t a i n s a l a r g e n u m b e r ( a " se a ") o f m o b i l e , n e g a -t i v e l y c h a r g e d c o n d u c t i o n e l e c t r o n s ( t h e " e l e c tr i c fl u id " ), a n d a fi x e d b a c k g r o u n do f p o s i t i v e l y c h a r g e d n u c l e i , w i t h o v e r a l l n e u t r a l i ty . H o w d o e s t h a t e l e c t r i c f l u idr e s p o n d w h e n w e b r in g a c h a r g e d ro d n e a r t h is c o n d u c t o r ? W h e n w e a d d s o m ee l e c t r i c c h a r g e t o t h e c o n d u c t o r ? I f t h e c h a r g e is e x t e r n a l t o t h e c o n d u c t o r , t h e n( as d i sc u s s e d i n C h a p t e r 1 ) i t c a u s e s e l e c t r o s t a t i c i n d u c t i o n o f t h e e l e c t r i c fl u i d ( ap o l a r i z a ti o n o f t h e c o n d u c t o r ) ; i f t h e a d d i t i o n a l c h a r g e is p l a c e d o n t h e c o n d u c -t o r i t s e l f ( p e r h a p s a s a d d i t i o n a l e l e c t ro n s , o r a s p o s i t i v e l y c h a r g e d i o n s t h a t s t i c kt o t h e s u r fa c e ) , t h e e l e c t r i c f l u id r e d i s t r i b u t e s i t s e l f t h r o u g h o u t t h e c o n d u c t o r .

Figure4 . 1 3 Con duc to r in a un i form f ie ld . ( a ) The mo bi le cha rgeo n t h e c o n d u c to r n o t p e r m i t t e d t o m o v e ( n o e le c t ro s t a ti cinduc t ion) . (b) The mobi le cha rge on the conduc tor pe rmi t ted tomove (full e lectrosta t ic induction) . The f ie ld l ines enter negativecharge and leave posit ive charge.


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160 Ch apte r 4 a Gauss's Law

.~ 1 1 [ . . . . . . ] : /I ~ . II111 II1 I111 i n l ~ 1 11[ I I 1111 n fl~ il l _ L , w Z i l ~ _ ~ _ _ .....~ ! ~..1-*if[~ ~q:,~ i~4~ii I~w~[.~ilqil~..l~.~lI:t*. [ ~ < q a ~ . ] ~:.z.~ L, ~ lRecal l tha t the mobi le e lec t rons are in orb i ta ls tha t ex tend over the ent i re conductor .W hen the e lec tr ic f lu id red is t r ibu tes i t se l f th ro ug ho ut the conductor , i t i s not v ia ind i -v i dua l mob i l e e lec t rons . Rather , a l l the mobi le e lec t rons on the conductor ad jus t the i rorbi tals sl ight ly, co l lect ive ly red is t r ibu t ing the i r to ta l charge in the same way as wo uldthe classical electr ic f lu id.

4 . 6 . 1

A n e x a m p l e o f ch a r g e r e d i s t r ib u t i o n c a n b e s e e n b y c o m p a r i n g F i g u re 4 . 13 ( a )to F igu re 4 .13 (b ) . F igu re 4 .13 (a ) s how s a conduc to r in a un i fo rm e lec t r i c f i e ldbefore t h e c o n d u c t o r h a s h a d t i m e t o r e s p o n d . F i g u r e 4 . 1 3 ( b ) s h o w s t h e c o n -duc to r and the to t a l e l ec t r i c f i e ld after t h e c o n d u c t o r h a s c o m e t o e q u i l i b -r ium. I t demons t r a t e s tha t in equ i l ib r ium ( a ) the r e a r e no f i e ld l ines w i th in thecon duc to r ( /~ - 0 ) ; ( lo ) excess cha rge r e sides on the c ond uc to r ' s s u r f ace (nocharge l ie s w i th in the m a te r i a l o f the c ondu c to r ) ; and ( c ) ju s t ou t s ide the con -duc to r the e l ec t r ic f i e ld is no rm al to the s u rf ace . I n the r em aind er o f th i s s ec tion ,w e w i l l de r ive each o f thes e r e s u lt s , and then d i scus s w h a t hap pen s w hen theconduc to r has a cav i ty .

W i th in Conducto rs in Eq u i l i b r ium, E = 0We now es tab l i s h tha t , f o r an e l ec t r i ca l conduc to r in equ i l ib r ium, the e l ec t r i cf i eld i s ze ro w i th in th e m a te r i a l o f the conduc to r . Th i s r e s u l t depend s on ly upo ntw o p roper t i e s o f e l ec t r i ca l conduc to r s :1 . I n equ i l ib r ium , the re i s no e l ec t r i c cu r r e n t f low ing anyw here w i th in an e l ec -t r i ca l conduc to r . I n con t r as t , t he f i l amen t o f a f l a s h l igh t bu lb connec ted to aba t t e ry ca r r i e s an e l ec t r i c cu r r en t and i s not i n equ i l ib r ium . H ow ever , a f t e r

t h e p o w e r h a s b e e n t u r n e d o f f a n d t h e f i la m e n t h a s c o o le d d o w n t o r o o mt e m p e r a t u r e , i t is i n equ i l ib r ium. N o e lec t r i c cu r r en t then f low s th rough i t .2 . In an e lect r ical co nduc tor , such as a wire , i f the re is an e lect r ic f ie ld , the n th echarge car r iers feel an e lect r ical force , which dr ives an e lect r ic cur rent . Sincein equ i l ib r ium the re i s no e l ec t r ic cu r r en t , /~ - 0 in equ i l ib r ium. ( In f ac t, t hecu r r en t i s p ropo r t iona l to the e l ec t r i c f i e ld . Th i s i s a ve r s ion o f O hm' s l aw . )

Cons id e r a conduc to r , s uch as tha t in F igu re 4 .13 . Le t ex te rna l cha rge qextproduce a f i e ld Eext, and l e t e l ec t r i c cha rge qcoM d i s t r i b u t e d o v e r t h e c o n d u c t o ri t s e l fp rodu ce a f i eld Econd. Th en th e to t a l f ie ld E = Eext + Econd. I f t h e c o n d u c t o ri s in equ i l ib r ium , t hen by p rop er ty 1, the e l ec t r ic cu r r e n t m us t be ze ro . F u r the r ,by p roper ty 2 , w i th in the conduc to r (bu t no t ou t s ide i t ) , t he to t a l e l ec t r i c f i e ldE - 0 . H e n c e , w i t h i n t h e m a t e r i a l o f t h e c o n d u c t o r i ts el f, E~ond m u s t c a n c e lEext. Tha t i s ,

E = ~ E ~ + E ~o ~," s o E ~ o ~ = ~~: ~ :".'.!~exti i!~!i !~ ~i! i~~ii!i:

Th e f i e ld in s ide the con duc to r i s ze ro becaus e th e f i e ld o f the con duc to r cancelst he ex te rna l f i e ld , not b e c a u s e t h e c o n d u c t o r s o m e h o w p r e v e n t s t h e e x t e r n a lf i e ld f rom en te r ing ("screening"). Th e r es u l t o f th i s s ubs ec t ion i s t r ue r ega rd less


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4 .6 E l ec t ri ca l C ond uc t or s i n E q u i l i b r i um 161

4o6~2

o f t h e e l e c tr i c a l f o r ce l a w . E v e n i f t h e f o r c e l a w w e r e a ra d i a ll y o u t w a r d i n v e r s ec u b e ( f o r w h i c h G a u s s ' s l a w d o e s n o t a p p l y ) , t h e e l e c t r i c f i e l d w o u l d b e z e r ow i t h i n t h e m a t e r i a l o f a c o n d u c t o r i n e q u i l i b r iu m .O n e m i g h t e x p e c t t h e c h a r g e a s s o c i a t e d w i t h t h e e l e c t r i c f l u i d t o b e d i s -t r i b u t e d t h r o u g h o u t t h e c o n d u c t o r . T h e n e x t s e c t i o n s h o w s t h a t , b e c a u s e o fG a u s s ' s l a w ( w h i c h f o l l o w s f r o m C o u l o m b ' s l a w ) , t h e c h a r g e a s s o c i a t e d w i t ht h e e l e c t ri c f l u id r e s i de s o n l y o n t h e s u r f ac e s o f t h e c o n d u c t o r .For a Co nductor in Equi l ibr ium, A ny N et Charge CanReside Only on Its SurfacesC o n s i d e r a c o n d u c t o r w i t h o u t e r s u r f ac e Soute r ( s uch as i n F i gu re 4 .13b ) and ,i f i t is ho l l ow, w i t h i nne r s u r f ace Sinner . D r a w a n a r b i t r a r y G a u s s i a n s u r f a c e St h a t t o t a l ly p a s s es t h r o u g h o n l y c o n d u c t i n g m a t e r i a l . B e c a u s e / ~ - 0 w i t h i n t h econduc t o r , t he e l ec t r i c f l ux dens i t y d ~ E / d A = E , 9 h i s zero for any par t of S , sot he t o t a l f l ux t h r ou gh S i s ze ro . By Ga us s ' s l aw , t he e l ec t r i c c ha rge w i t h i n t h ev o l u m e e n c l o s e d b y S m u s t a l so b e z e r o. B y b r e a k i n g u p t h e c o n d u c t o r i n t o t in yv o l u m e s , w e c a n p i e c e w i s e e l i m i n a t e t h e p o s s i b i l i t y t h a t t h e r e i s e l e c t r i c c h a r g ei n a n y o f t h o s e v o l u m e s . T h u s , f o r a n e l e c t r ic a l c o n d u c t o r i n e q u i l i b r i u m , n e t e l ec t ri cc h a r g e c a n r e s i d e o n l y o n i ts s u rf a c e s. T h i s r e s u l t d e p e n d s c r u c ia l ly o n G a u s s ' s l aw .The e l ec t r i c f i e l d due t o t he e l ec t r i c f l u i d a r i s e s on l y f rom s u r face cha rge . Ingenera l , t h i s s u r f ace cha rge d i s t r i bu t i on i s com pl ex and d i f f i cu l t t o de t e rm i ne .

" .oncept Qu iz 4 .I f an object has a nonzero volum e charge d i s t r ibut ion (as in Exam ple 4 .1 O)and i t i s in equi l ibr ium, can the object be a conductor?S o l u t i o n : No. It must be an insulator. For example, charged ions might be heldin place within an insulator by a nonelectrostat ic interact ion with the moleculesof the object .

: o n c e p t Q u i z 4 . ,I f an object has a non zero v olum e cha rge d i s t r ibut ion and i t i s n o t in equil ib-r i um , m us t t he ob j ec t be a conduc t o r?Solution: No. It also could be an insulator. For example, ch arged ions within theinsulator m ight be in the p rocess of rearranging.

. 'oncept Quiz 4. ' .I f an object has a non zero surface charge d i s t r ibut ion , m us t the object be acondu c t o r i n equ i l ib r i um ?Solution: No. The ob ject could be ei ther an insulator or a conductor, a nd i t couldbe e i ther in or out of equi libr ium. For example, a comb rubbed through yourhair has a surface charge distribut ion, but the comb is an insulator. (Moreover,the distribut ion may or may not be in equil ibrium.) Or, a piece of aluminumfoil could have a surface charge distribut ion due to a sudden discharge, but notenough t ime has passed for the foi l to reach equil ibrium.


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16 2 Chapter 4 ~ Gauss's Law

4 . 6 . 3 _a,J u s t O u ts i de a U n i fo rm Co nduc tor E Is No rma lto the SurfaceCon s ide r an e l ec t r ica l cond uc to r tha t i s un i fo rm (o r loca l ly un i fo rm ) a long i t ssur face . (Here , uni form m e a n s t h a t t h e a t o m i c o r d e r is u n i f o r m . ) I f t h e e l e c tr i cf i eld had a co m po ne n t tha t i s pa r a l l el t o the s u r face , an e l ec t ri c cu r r e n t w ou ldf low a long the s u r f ace , con t r a ry to the a s s um pt ion o f equ i l ib r ium . H ence , inequ i l ib r ium, the e l ec t r i c f ie ld is no rm al to th e s u r face . See F igu re 4 .14.Fie ld i s no rm a l _~ ]~t o s ur fa ce ~ E \

E l e c t r ic a l c o n d u c t o ri n e q u i l i b r i u m

Figure 4.14 E lectric flux ~ leaving anelectrical conductor in equilibrium, with thesurface broken into many surface elements.The field l ines are normal to th e surface.

T h e r e s u l t t h a t E i s n o r m a l t othe s u r f ace can a l s o be ob ta inedf rom F araday ' s v i ew po in t . Ima-g ine f r eez ing in p lace the mob i l echa rge ( the e l ec t r i c f lu id ) on a neu -t r a l conduc t ing ob jec t , and thenplacing i t in an external f ie ld . SeeF igu re 4 .13 (a ) . Wi th the cha rgeon the conduc to r f rozen in p lace ,these f ie ld l ines , due only to the ex-t e rna l f i e ld , pas s th rough the con -d u c t o r u n a f f e c t e d . A c c o r d i n g t oF araday ' s v i ew , on un f r eez ing themobi le charge, the f ie ld l ines , tow h o s e e n d s t h e m o b i l e c h a r g e i sa t t ached , s ho r t en b ecaus e o f f i eld -l ine t ens ion ( s ub jec t to the con -s t r a in t tha t the l ines no t ge t too dens e , becaus e o f f i e ld - l ine p r es s u re ) . Th e m o t ions t o ps w h e n t h e l i ne s a re n o r m a l t o t h e c o n d u c t o r a n d h a v e t h e a p p r o p r i a t e s e p -a r a t ion . See F igu re 4 .13 (b ) , w here th r ee l ines t e rmina te on nega t ive cha rge and

th ree l ines o r ig ina te on pos i t ive cha rge , the conduc to r hav ing ze ro ne t cha rge .B e c a us e o f ch a r g e r e a r r a n g e m e n t o n t h e c o n d u c t o r , t h e total electr ic f ield is nor-ma l to the s u r f ace . I n F igu re 4 .13 (b ) , t he f i e ld due to the ex te rna l cha rge i s thes ame as in F igu re 4.13 (a ) ; th e d i f f e r ence in the tw o cas es i s s o le ly due to cha rger e a r r a n g e m e n t o n t h e c o n d u c t o r .~ l i _ L , : ~ . I . ; L ! X i l I i ~ J i ]: .l i ! it'~l.~l ~ T g , S r . q l ! ] r ~ - i ~ - ii ~ 1~ _ i T , T d l

The idea of freezing and unfreezing the charge distr ibution isn't theoretical. Certaintypes of paper can take m any seconds to fu l ly respond. For short t imes only the polar-ization of localized electrons is noticed, bu t for longer t imes io ns can move, to give aresponse more l ike that of a metal.

4 . 6 . 4 E = 0 w i th in a Co nductor ' s Emp ty C av i tyCons ide r an e l ec t r i ca l conduc to r w i th ou te r s u r f ace Souter t ha t i s s o l id excep t fo ra cav i ty w i th s u r f ace S inner , a s i n F igu re 4 .15 . Co ns ide r w ha t h appe ns i f t he r e i sex te rna l cha rge o r cha rge on the conduc to r , bu t no cha rge in tha t cav i ty .To s tudy th is cas e, w e us e an a lm os t tr iv i a l bu t ve ry imp or ta n t r e s u l t t ha t m aybe ca l l ed the scooping-out theorem , w h i c h s t at e s t h a t r e m o v a l o f m a t e r i a l f r o m aconduc to r in equ i l ib r ium does no t d i s tu rb the e l ec t r i c f i e ld anyw here .


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F igure 4 .1 5 An electr ical conduc tor

T h e p r o o f o f t h e s c o o p i n g - o u t t h e -o r e m i s s t r a i g h t f o r w a r d . T h e r e a r e t w ow a y s t h a t s c o o p i n g - o u t m i g h t p r o d u c ean e l ec t r ic f ie ld : (1 ) Scoop ing -o u t mig h tl e a v e b e h i n d v o l u m e c h a r g e i n t h e i n -t e r i o r , w h i c h w o u l d p r o d u c e a n e l e c t r i cf ie ld. H ow ever , t he in t e r io r o f the con -d u c t o r i s n e u t r a l , s o t h a t s c o o p i n g o u tt h e c a v i ty d o e s n o t r e m o v e v o l u m e e l ec -w i th a cav ity , t r i c cha rge . (2 ) I f t he r em ove d m a te -r i a l w e r e p o l a r i z e d , s c o o p i n g - o u t m i g h tl e a v e b e h i n d s u r f a c e c h a r g e , w h i c h w o u l d p r o d u c e a n e l e c t r i c f i e l d . H o w e v e r ,t h e i n t e r i o r o f a c o n d u c t o r c a n o n l y h a v e a n i n d u c e d p o l a r iz a t io n , p r o p o r t i o n a lto E . S ince /~ - 0 , t he in t e r io r i s no t po la r ized , and the re fo re s coop ing ou t thecav i ty does no t l eave beh ind s u r f ace cha rge .W i t h i n t h e c a v i ty o f t h e c o n d u c t o r i n F i g u re 4 . 1 5 , s i n c e th e e l e c tr i c f ie ld w a sze ro be fo re the re w as a cav i ty , t he e l ec t r i c f i e ld i s ze ro a f t e r s coop ing ou t thecavi ty . Ther e i s no cha rge a nyw here on the inne r s u r f ace o f the conduc to r , s incet h e r e w a s n o c h a r g e t h e r e b e f o r e t h e c a v i ty w a s s c o o p e d o u t .

4~ -+Ju s t O u t s i d e a U n i f o r m C o n d u c t o r , Eo ut. h V a r ie sa s t h e L o c al S u r f a c e C h a r g e D e n s i t y ~ sCo ns ide r a con du c to r in equ i l ib r ium, a s in F igu re 4 .13 (b ) . S ince E - 6 in s ide aconduc to r , a ll f i e ld l ines and e l ec t r i c f lux en te r ing o r l eav ing pa r t o f a con duc to rc a n b e a t t r i b u t e d o n l y t o t h e c h a r g e o n t h a t p a r t o f t h e c o n d u c t o r . T h i s p e r m i t su s to d e t e r m i n e t h e s u r fa c e c h a r g e d e n s i t y ~s .Very c lo s e to an e l ec t r i ca l conduc to r , t he s u r f ace appea r s to be f l a t . (Wen e g l e c t t h e a t o m i c n a t u r e o f m a t t e r. ) D r a w a G a u s s i a n s u r fa c e S t h a t i s a sm a l lp i l lbox o f a r ea A enc los ing a p iece o f the s u r f ace , o r i en ted w i th i ts f la t ou te r f acepa ra l l e l t o the s u r face . See F igu re 4 .16(a ) . T he e l ec t r i c f lux ~E com es on ly f romt h e f la t o u t e r f a c e b e c a u s e E = 0 in s i d e a n d b e c a u s e t h e t h i n r o u n d p a r t o f t h ep i l lbox may be t aken to have neg l ig ib le a r ea . F o r a p i l lbox s o s ma l l tha t the r e i sa u n i f o r m n o n z e r o f l ux t h r o u g h o n l y o n e p a r t o f t h e s u rf a ce , ( 4 .1 1 ) a p p l i e s w i t h

Figure 4.16 Locally planar charge dis tr ibutions, and associated pillbox-shapedGaussian surfaces. (a) At surface of a condu ctor, only the outside circular facepicks up flux. (b) Infinite sheet of charge, the circular surfaces pick up equalam ounts o f flux.


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Affix ~ A. H e r e Q e n c - - ~ s A - ~ s A f l ~ , whe r e ~s i s t he su r f a c e c ha r ge de ns i t y .W i t h Eout t h e e l e c tr i c fi el d j u s t o u t s i d e t h e c o n d u c t o r , a n d f f t h e o u t w a r d n o r m a lf r o m t h e c o n d u c t o r ( a n d f r o m t h e o u t s i d e o f t h e p i l l b o x ), ( 4 .1 1 ) y ie l d s

Eo ut - /z = 4zrk~s. ( f ie ld jus t ou t s id e a ny c o ~ t o ~ ~ :i ~ ~I ) :T h i s c a n b e r e w r i t t e n a s

E o u t 9 l"t~ s - 4 r rk " (4.22)I f /~ p o in t s a w a y f r om th e su r f a ce , t he n ~s is pos i t i ve ; i f /~ p o in t s t ow a r d t h esur f ac e , t he n ~s is ne g a t i ve . T h e se a r e a s e x pe c t e d .B y m e a s u r i n g Eout, o n e c a n d e d u c e ~ s . F o r e x a m p l e , i f ] E ] - 2 5 0 N / C a t P,j us t ou t s ide pa r t o f a su r f a c e whose l oc a l c ha r ge de ns i t y i s ne ga t i ve , t he n by( 4 .2 2) ~s - - 2 . 2 1 x 1 0 - 9 C/m 2 ne a r P , a nd E po in t s i n to t he su r fa c e . No te t hef i e ld l i ne s e n t e r ing t he c onduc to r i n F igur e 4 .1 3( b) .

Cou lomb developed a device cal led a proo f plane, a smal l th in conduct ing d isk w i th aninsulating handle, to dete rmine c~s directly. On part of the condu cting surface he wouldplace the proof plane and then l i f t i t off. The amount of charge on the proof planewas p ropo rtiona l t o c~s so tha t he could obta in relativ e charge dens ities. By suitablecalibration, absolute measurements can be obtained with a proof plane.For ~s = ~ , ( 4.2 1 ) f o r t he e x t e r io r o f a c on du c to r ha s a n e x t r a f a c to r o f two

r e l a t i ve t o t he c a se o f t he e l e c t r i c f ie ld E s h e e t = 2 r r k ~ p r o d u c e d b y a n i s o l a t e ds h e e t o f c h a r g e d e n s i t y e . T h i s f a c t o r o f t w o o c c u r s b e c au s e , f o r t h e c o n d u c t o r , a l lt he e l e c t r i c f l ux l e a ve s one s ide ( t he e x t e r io r ) o f t he c on du c to r p i l l box , whe r e a sf o r a n i so l a t e d she e t o f c ha r ge , on ly h a l f t he e l e c t r i c f lux l e a ve s e a c h s ide o f t hec or r e sp ond ing p i l l box , a s i n F igur e 4 .1 6 ( b) . T hus , f o r t he c ond uc to r , a ll t he f l uxge t s c onc e n t r a t e d on one s ide . E q uiva l e n t l y , i n t e r m s o f ( 4 .1 1 ) , A f l ~ i s ha l f asl a r ge whe n a pp l i e d t o a c onduc to r , so t he f l ux i s tw ic e a s l a r ge .~ A n infinite sheetonducting

An inf ini te conduct ing shee t ini t ia l ly has charge per uni t a rea c r on only onesur face . ( a ) Find the e lec t r ic f ie ld . (b) Th e sys tem n ow com es to equi l ibr ium.Find the charge dis t r ibut ion and the e lec t r ic f ie ld outs ide the conductor .Solu t ion: (a) On each side of the charged surface (even inside the con ductor) ,the field has magnitude 2rrkc~ and points away from the surface. Because/~ 4= 0within the conductor , i t is not in equil ibr ium. (b) The charge rearranges so that ,by symm etry, in equilibriu m each surface has ~s = ~/2. T his giv es zero fieldinside the conductor . By (4.22), the f ield outside each surface of the conduc-tor has magnitude 4rrk~s = 4rrk(cr/2)= 2zrk~, the same as before the chargerearranged.E q u a t i o n s ( 4 .2 1 ) a n d ( 4 .2 2 ) h o l d j u s t o u t s id e a n y c o n d u c t o r i n e q u il i b r iu m .B y i n t e g r a t in g o v e r ( 4 . 22 ) , w e o b t a i n t h e r e s u l t th a t t h e t o t a l c h a r g e o n t h e


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4 . 7 J u s t O u t s i d e a U n i f o r m C o n d u c t o r 165

s u r f a c e i s

4 rr k 4 rr k - - Q e n c , (4.23)

a s e x p e c t e d . T h a t i s , Q e n c , t h e t o t a l c h a r g e e n c l o s e d b y a G a u s s i a n s u r f a c e e n -c l o s i n g t h e c o n d u c t o r , e q u a l s t h e t o t a l c h a r g e Q s o n t h e o u t e r s u r f ac e o f t h ec o n d u c t o r , e v e n i n t h e p r e s e n c e o f c a v it i es .

~ T w o parallel charged conducting sheetsC o n s id e r tw o i n f in i t e p a r a l le l c o n d u c t i n g s h e e t s o f n e t c h a r g e / a r e a ( t o p a n dbo t to m of each she e t ) ~1 > ~2 > O, wi th ~1 = (3 /2)~2. F ind the equi l ib r iu mc h a r g e d e n s i t i es o n t h e i n s id e a n d o u t s i d e o f e a c h s h e e t . F ig u r e 4 . 17 p r e s e n t sshee ts o f f in i te s ize ; th i nk o f thes e a s in f in i te , so tha t the re a re no edges .Solution: For the p lana r geometry , un l ike the geom etry of F igure 4 .13 , th e r ea r-rangem ent of cha rge on a conduc tor has no e f fec t on the e lec tr ic fie ld ou ts ide tha tconduc tor . However , a s in F igure 4 .13 and Example 4 .12 , the r ea r rangement ofcha rge does cause the f ie ld to be ze ro wi th in each conduc tor . In equi l ib r ium, the reis charge only on th e surfaces of the con duc ting sheets. Since the f ie ld for a she etdoesn ' t f al l o f f wi th d is tance , we can com pute the f ie ld as the sum over the to ta lcha rge on each shee t. Su perpos i t ion appl ied to (3 .26) , o r ] E s h e e t l - - 2:rkl~l, yieldsthe f ie lds outside and between the sheets. See Figure 4.13. Including direction,above both shee ts

E o u t = 2 7 r k ( ~ y l - [ - o - 2 ) { - 5 y r k o 2 { .With h = { just above sh eet 1, (4.22) the n yie lds _top =Inc luding d i rec t ion , be tween both shee ts

Ebetween -- 2:rk(~l - ~ y2 ) j , - J r k f f 2$ .With ~ - ~ just belo w sheet 1 (4.22) the n yie lds ~ o t t o m _ 1- - X~2. Th e tota l charge

Figure 4.17 Two inf inite sheets of charge, and theelectr ic f ie ld in the regions above, between, andb e lo w th e m.


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166 Chapter 4 w Gauss 's Law

_ t o p 0"b~176 = 3~2 = thus satisfying charg e conserva-ensity on shee t 1 is o 1 + ~1,t ion. A similar analysis can be ma de for sheet 2. The charges on the inner surfacesr t o p _ _ _ _ o .b o t t o m )are equal and opp osite t~2 so, as expe cted, the field lines start onposi t ive charge and end on equal negat ive charge. The charges on the outer sur-faces are equal (a~ ~ - ( y b o t t o m ) and produce field l ines that extend to infini ty sothat each outer surface produces the same flux.

4 .8 C h a r g e M e a s u r e m e n t a n d F a r a d a y ' s I c eP a i l E x p e r i m e n tC o n s i d e r a c h a r g e e l e c t r o m e t e r a s d i s c u s se d i n Se c t i o n 2 . 4 .3 , a t t a c h e d t o t h ee x t e r i o r o f a c o n d u c t o r i n t o w h i c h o b j e c t s c a n b e p l a c e d . T h e c o n d u c t o r i t se l f isp l a c e d o n a n i n s u l a te d s t a n d t o p r e v e n t t h e e s c a p e o f e le c t r ic c h a r g e . W i t h t h i se l e c t r o m e t e r w e c a n d e t e r m i n e t h e c h a r g e o n a n o b j e c t , w i t h o u t t e a r i n g a p a r tt h a t o b j e c t a n d w i t h o u t e v e n m e a s u r i n g t h e e l e c t r i c f l u x p a s s i n g t h r o u g h t h es u r f a c e o f t h a t o b j e c t. S e e F i g u r e 4. 1 8 . ( W e a s s u m e t h a t t h e s i m p l e e l e c t r o s c o p ei n F i g u r e 4 . 1 8 h a s b e e n c a l i b r a t e d , t h e r e b y m a k i n g i t i n t o a n e l e c t r o m e t e r . )T h e e l e c t r o m e t e r g iv e s a r e s p o n s e t h a t i s d i r e c t ly p r o p o r t i o n a l t o t h e l o c a l c h a r g ed e n s i t y o n t h e o u t s i d e o f t h e c o n d u c t o r : e i t h e r i t m e a s u r e s t h e e l e c tr i c f ie ld ,w h i c h b y ( 4 . 2 2 ) i s p r o p o r t i o n a l t o t h e s u r f a c e c h a r g e d e n s i ty , o r it m e a s u r e s t h es u r face cha rge dens i t y d i r ec t l y .

4 . 8 , 1 Noncontact ExperimentsF a r a d a y h u n g a n e le c t r ic a l ly c h a r g e d o b j e c t A , w i t h c h a r g e + Q , by an i n s u l a t i ngs tr in g . O n l o w e r i n g A i n t o t h e u n c h a r g e d i c e p a il , w i t h o u t A t o u c h i n g t h e s id e s,t h e e l e c t r o m e t e r r e a d i n g i n c r e a s e d , b u t o n c e A w a s a b o u t 4 i n c h e s b e l o w t h et o p o f t h e i c e p a il , th e e l e c t r o m e t e r r e a d i n g s t ab i li ze d , e v e n w h e n A w a s m o v e da b o u t . W h e n A w a s l if t e d b a c k o u t o f t h e i c e pa il , t h e e l e c t r o m e t e r r e a d i n g

Figure 4.1 8 Faraday ice pai l experimen t . T heelectroscope responds to charge that goes to theouter surface o f the ice pai l.


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4.8 ChargeMeasurement 167

r e t u rne d t o z e ro . More ove r , whe n A wa s wi t h i n t he i c e pa i l , t he e l e c t rome t e rre a d i ng wa s t he sa m e a s i f a ll t he c ha rge on t he ob j e c t ha d be e n p l a c e d on t h e i c epa i l. Th i s i nd i c a t e d t ha t (1) i f t he c ha rge d obj e c t A is f a r e nou gh wi t h i n t he i c epa i l, t he n a c ha rge d i s t r i bu t i on goe s t o t he ou t e r su r fa c e t ha t i s t he sa me a s i f + Qha d be e n p l a c e d d i re c t l y on t he i c e pa il ; (2) t he c ha rge + Q on A a nd t he c ha rge- Q r e m a i n i n g o n t h e i n n e r s u r fa c e t o g e t h e r p r o d u c e e l ec t ri c f ie ld s t h a t h a v e n oe f fe c t on t he c ha rge d i s t r i bu t i on on t he ou t e r su r fa c e o f t he i c e pa i l. Re c a l l t ha t(4.22) re la tes the e lec t r ic f i e ld a t the surface and the surface charge densi ty .4~8~2 Co ntact E xperiments

On l owe r i ng a c ha rge d conduct ing obje c t A i n t o t he i c e pa i l , t he e l e c t rome t e rre sponse i nc re a se d , a s be fore . Touc h i ng A t o t he i ns i de o f t he i c e pai l, t e mp ora r i l ym a k i n g a n i c e p a i l - o b j e c t c o m b i n a t io n , y i e l d e d n o c h a n g e i n t h e e l e c t r o m e t e rre a d i ng . Howe ve r , l i f t i ng ou t A, t h e e l e c t r o m e t e r r e a d in g r e m a i n e d a t t h e s a m eva l ue a s whe n A wa s wi t h i n t he i c e pa i l . More ove r , p l a c i ng A wi t h i n a no t he ri c e pa i l e l e c t rome t e r ga ve no re sponse . Th i s i nd i c a t e d t ha t , whi l e A a nd i c epa i l we re i n c on t a c t (1) the re wa s no c ha rge on t he i nne r su r fa c e o f t he i c epa i l -ob j e c t c ombi na t i on ; (2) t he re wa s no c ha rge i n t he vo l ume a s soc i a t e d wi t hA , t h o u g h t o f as p a r t o f t h e i c e p a i l - o b j e c t c o m b i n a t io n . H e n c e , t o u c h i n g t h ec ha rge d c ond uc t i ng ob j e c t t o t he i n t e r i o r o f t he i c e pa il ma d e A t ra ns fe r it scharge to the ice pa i l .Al t hough A or i g i na l l y wa s a t t r a c t e d t o bo t h t he i n t e r i o r a nd t he e x t e r i o rof t he ne u t ra l i ce pa il , a f t e r c on t a c t w i t h t h e i n t e r i o r i t wa s a t t r a c t e d on l y t ot he e x t e r i o r o f t he n ow c ha rge d i c e pa il . (Th i s wa s no t e d e a r l ie r by Fra nkli n .)Thi s be ha v i or a f t e r c on t a c t wi t h t he i n t e r i o r c a n be e xp l a i ne d us i ng e l e c t ros t a t ici ndu c t i on i f now A is unc ha rg e d , a nd t he i c e pa il is c ha rge d o n t he o u t s i de bu tno t on t he i nsi de.

4~8oS Interp retation in Terms of F ield LinesOnc e A i s f a r e nou gh i ns ide t he i c e pai l, a l l t he f ie l d l ine s p rodu c e d by + Q t e rmi -na t e on t he i nne r su r fa c e so t ha t a ne t su r fa c e cha rge - Q ha s be e n a t t r a c t e d (bye lec t ros ta t ic indu ct ion ) to th e inn er surface . S ince the ice pa i l it se l f i s a con du c-t o r i n e qu i l i b r i um, i t ha s z e ro vo l ume c ha rge de ns i t y . By c ha rge c onse rva t i on ,t h e c h a r g e - Q o n t h e i n n e r s u r f a c e m u s t h a v e b e e n a t t r a c t e d f r o m t h e o u t e rsur fa c e , wh i c h t he re fo re ha s a c ha rge +! 2- Movi ng + Q a rou nd i ns i de t he c a v i t yma y c ha nge t he f i e l d l i ne s a nd t he d i s t r i bu t i on o f c ha rge de ns i t y on t he i nne rsur fa c e , bu t i t doe s no t c ha ng e t he va l ue - Q of t he t o t a l su r fa c e c ha rge . More -ove r , movi ng +Q a round i ns i de t he c a v i t y doe s no t c ha nge t he c ha rge de ns i t yon t he ou t e r su r fa c e , nor i ts i n t e gra t e d va l ue o f Q , nor the exter ior f i e ld l ines .~ ReproducingFaraday's esults

Many o f Faraday 's resul ts can be re prod uced us ing only (1) a charged plas t iccomb as the charge source A; (2) a small food can (or soft drink can) withi ts top remov ed, having insulat ing handles of folded-over s t i cky tape; (3) alarge food can with i ts top removed (which serves as the ice pai l ); and (4) a


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168 Cha pter 4 u Gauss ' s Law

Figure 4.19 A mo dem , ho me vers ion o f the Faraday i cepa i l exper imen t .p l a s t i c - b a g - t w i s te r v e r s o r i u m a s t h e e l e c t r o me t e r . Se e F i g u re 4 . 19 . T h e l a rg e rc a n s h o u l d b e m o u n t e d o n a n i n s u l a ti n g s ur f ac e , s u c h a s a s t y r o f o a m c u p . T h ev e r s o r i u m s h o u l d b e p l a c e d n e a r a n e d g e o f t h e l a r g e ca n , w h e r e t h e e l e c tr i cf i e ld i s l a rges t . The comb can be charged and used as an insu la t ing charges o u r c e w i t h i n t h e c a n . P l a c i n g t h e c h a r g e d - u p c o mb w i t h i n t h e s ma l l c a n ,a n d t h e n t o u c h i n g t h e s ma l l c a n c h a r g e s t h e s ma l l c a n , w h i c h t h e n c a n b eu s e d a s a c o n d u c t i n g c h a r g e s o u r c e w i t h i n t h e l a r g e c a n . T h e s ma l l c a n a n dt h e c o m b a r e o p p o s i t e l y c h a r g e d , a s s h o w n b y t h e l a c k o f v e r s o r i u m r e s p o n s ew h e n b o t h a r e w i t h i n t h e l a r g e r c a n .

B y d e t e r m i n i n g h o w m u c h c h a r g e g o es to t h e i n t er i o r o f a h o l l o w c o n d u c t o r ,a n d c o m p a r i n g to t h e p r e d i c t io n s o f n o n - C o u l o m b ' s l a w - b a s e d t h e o ri e s, w e c a ns e t a l i m i t o n h o w w e l l C o u l o m b ' s l a w ( a n d t h u s G a u s s ' s l a w ) i s s a t i s f i e d . T h i sw a s t h e b as is o f C a v e n d i s h ' s e x p e r i m e n t s . M e a s u r e m e n t s m a d e i n 1 9 7 0 sh o wt h a t , i f C o u l o m b ' s l a w i s a s s u m e d t o v a r y a s r - (2 +~) , t h e n ~ < 1 . 0 x 1 0 - 16 .

4 .9 P r o o f o f G a u s s ' s L a wW e n o w p r o v e G a u s s ' s l a w b y u s i n g t h e c o n c e p t o f so l id a n g le . T h i s is p e r h a p sm o s t r e a d il y a p p r o a c h e d f r o m t h e v i e w p o i n t o f a st r o no m y . I f w e l o o k u p a t t h es k y a n d i m a g i n e t h a t i t is a g r e a t s p h e r e o f v e r y la r g e r a d iu s R , t h e n b y d e f i n i t i o nt h e s o l i d a n g l e fa t a k e n u p b y , sa y, t h e c o n s t e l l a t i o n O r i o n , i s g i v e n b y t h e a r e aA o f O r i o n , p r o j e c t e d o n t o t h i s g r e a t s p h e r e , d i v i d e d b y R 2. M o r e g e n e r a l l y , f o ra s m a l l p r o j e c t e d a r e a dAt_ p e r p e n d i c u l a r t o t h e l i n e o f s ig h t, w e h a v e a s m a l lso l id ang le

d A ~ R . ~ d Ad ~ - r2 - - R2 . (4 .24)H e r e /~ is t h e d i r e c t i o n i n w h i c h t h e o b s e r v e r i s lo o k in g , a n d f i is t h e n o r m a lt o t h e s m a l l a r e a d A. N o t e t h a t / ~ - f i , a n d t h u s d f a , c a n b e e i t h e r p o s i t i v e o rn e g a t i ve . W e n o w d e v e l o p s o m e p r o p e r t i e s o f t h e s o l i d a n g l e t h a t a r e n e e d e d f o rt h e p r o o f .


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4.10 Con ductors with Cav ities 169

F or the en t i r e g r ea t s phe re t he a r ea is 4Jr R 2, s o the to t a l s o l id ang le o f a s phe reis ~2 = (4; r R 2 ) / R 2 = 4Jr , i nd ep en de n t o f the (unkn ow n) r ad ius R. Moreover , i fw e m o v e o r d e f o r m t h e g r e a t s p h e re , s o l on g a s i t c o n ta i n s t h e o b s e r v e r w i t h i ni t, t he to t a l s o l id ang le w i l l r em ain 4J r. F u r the r , i f w e cons ide r the m oon , i ts to t a ls o l id ang le, f ron t and back , i s ze ro , becau s e the pos i t ive s o l id ang le o f any pa r t ont h e b r i g h t s i d e is c a n c e l e d o u t b y t h e n e g a t i v e s o li d a n g le o f t h e c o r r e s p o n d i n gp r o j e c t i o n o n t o t h e d a r k si de . F in al ly , if w e m o v e o r d e f o r m t h e m o o n , t h e t o t a ls o l id ang le w i l l r ema in ze ro .H e r e is t h e p r o o f o f G a u s s ' s la w . C o n s i d e r a n a r b i tr a r y G a u s s i a n s u r f ac e . L e tt h e r e b e a p o i n t c h a r g e q a t t h e o r i g in , so @e - J / ~ . ~ z d A = f ( k q [ ~ / R Z ) 9~zdA =k q f d ~ - k q ~ . Tha t i s , Ce - kq~2, w h ere the s o l id ang le ~2 enc los e d by theG a u s s i a n s u r f ac e i s m e a s u r e d r e l at i v e to t h e p o s i t i o n o f t h e c h a r g e q. F r o m t h ed i s cus s ion o f the g r ea t s phe re a nd o f the m oon , e i the r ~2 = 4 j r (w here Qen~ = q)or ~2 = 0 (w he re ~P_~enc 0), S O (~PE---4zrkQen~. B y s uperpos i t ion , th i s r e s u l tcan be e s t a b l i s hed fo r a s m any cha rges a s need ed , s o it is t r ue in gene ra l . Th i ses t ab l i s hes G au s s ' s law .W h e n c h a r g e s m o v e , t h e e l e c tr i c f ie l d i s n o l o n g e r g i v e n b y C o u l o m b ' s l a w .N e v e r t h e l e s s , G a u s s ' s l a w c o n t i n u e s t o h o l d . T h a t f i e l d l i n e s a r e p r o d u c e d b ye lec t r i c cha rge i s a more genera l ly va l id idea than tha t the r e i s ac t ion a t ad i s t ance w i th an inve r s e s quare l aw .

4 . 1 0 Conductors w i th Cav i t i es : E lec t r i ca l S creen ingW e n o w e x t e n d o u r e a r l i er d i s c u ss i o n o f c o n d u c t o r s . F o r o u r p r e s e n t p u r p o s e s ,w e m a y c o n s i d e r t h e F a r a d a y ic e p a il t o b e e q u i v a l e n t t o a c l o s e d c o n d u c t o r w i t ha cav i ty ; the l a rge ope n ing o f the i ce pa i l s e rves on ly to pe r m i t u s to b r ing ob jec t si n a n d o u t .

4.10 .1 Conductors with Cavit iesC o n s i d e r a n e l e c t ri c a l c o n d u c t o r w i t h o u t e r s u r f ac e Souter t ha t i s s o l id exc ep t fo ra cav i ty w i th s u r f ace Sinner, a s i n F igu re 4 .15 . I n the mos t gene ra l cas e , the r e i sc h a r g e i n , on , and ou t s ide t he condu c to r . We w i l l be ab le to t r ea t th i s gene ra l caseb y u s i n g t h e p r i n c i p l e o f s u p e r p o s i t i o n a p p l i e d t o t h r e e c a se s:1 . Th ere i s a ne t cha rge Qo,, o n t h e m a t e r i a l o f t h e c o n d u c t o r . T h i s c a s e w a sa l r e a d y c o n s i d e r e d in S e c t i o n 4 .6 . I t s d is t r i b u t i o n d e p e n d s o n l y o n t h e s h a p e

o f Souter a n d t h e a m o u n t o f Qou.2 . There i s a cha rge Qout o u t s i d e t h e c o n d u c t o r , a n d t h e c o n d u c t o r h a s n e tcha rge Qo n = - Qo u t . In th i s cas e , the cha rge Qo n i s on ly on the ou t e r s u r f ace ,w i t h a d i s t r i b u t i o n t h a t d e p e n d s o n t h e p o s i t i o n a n d a m o u n t o f Qout , a n d o nt h e s h a p e o f Souter.

3 . There i s a cha rge Qin i n s i d e t h e c a v i t y , a n d t h e c o n d u c t o r h a s n e t c h a r g eQon --'- --Qin . In th i s cas e , the cha rge Qon i s on ly on the inne r s u r f ace , w i tha d is t ri b u t io n t h a t d e p e n d s o n t h e p o s i ti o n a n d a m o u n t o f Qin , a n d o n t h es h a p e o f Soute .


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170 Chapter 4 ~ Gauss 's Law

Figure 4.20 An isolated charged co nduc tor in equil ibrium, w ith field l ines.(a) Withou t any internal cavi t ies. (b) With an internal cavi ty.

Each o f t hes e cas es has an e l ec t r i c f i e ld t ha t i s ze ro i n t he bu l k and is no rm al t ot h e s u r fa c e s. T h e r e f o r e w e c a n s u p e r i m p o s e l in e a r c o m b i n a t i o n s o f t h e s e t h r e ec a s e s b e c a u s e s u p e r p o s i t i o n w i l l n o t c a u s e a n y d i s t u r b a n c e e i t h e r i n t h e b u l k o ron e i t he r s u r f ace . We wi l l u s e t hes e cas es t o ana l yze t he Faraday cage a n d t h eFarada y ice pail .W e n o w a p p l y th e s c o o p i n g - o u t t h e o r e m t o t he s e t h r e e c a se s.1. Condu ctor with net charge Qo n

F i rs t c o n s i d e r a n u n c h a r g e d , s o li d c o n d u c t o r w i t h ( o u t e r ) s u r f a c e Souter t h a ti s g i ven a ne t cha rge Qon. S ee F i gu re 4 .20 (a ) .P r e v i o u s c o n s i d e r a t i o n s s h o w t h a t Qo n m u s t b e d i s t r i b u t e d o v e r t h e s u r -face S ou t e r . S in c e t h e i n t e r i o r o f t h e c o n d u c t o r is n e u t r a l , b y t h e s c o o p i n g - o u tt h e o r e m w e c a n s c o o

Physics -Gausss Law Flux and Charge 1

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