physics book about batteries, kirchhoff's rules, and complex circuits

7/28/2019 Physics book about Batteries, Kirchhoff's Rules, And Complex Circuits

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" T h is i nduc e d m e t o ho l d t he an i m a l [ a de ad fr og] in one ha nd and t ak e M r . R i a l p i' s han dwi th the o ther so as to form a sor t o f e lec tr ic c i r cui t [R ia lp i -G alva ni -m eta l hook- frog-s il verpla te] , a nd reques t h im at the same t ime to tap the s i lver p la te or s imp ly touch i t w i th h i sf r ee hand. No t w i thou t wo nde r we saw the usual co nt rac t ions [o f the f rog 's l eg] , whichhow e v e r aga i n d i s appe ar e d and r e appe ar e d w he n w e unc l a s pe d hand s o r r e jo i ne d the m . "

~L uig i Galvani (1791 )" [The apparatus e i ther ] works w i thout s topping , or i t s charge i s again automat ical l yres tored a f ter each spark ing . In o ther words i t has an inexhau s t ib le charge . . . [The p il e]cons i st s o f a p la te o f s i l ver in contac t w i th one o f z inc , an d i s connec ted w i th the nex t pai rthrough a su f f i c i en t layer o f mois ture, which s hou ld be t t er be sa l t water th an p la in . . . . "

Alessandro Volta,announcing his invention of the vo ltaic pile (1800)

C h a p t e r 8

B atteries, K irchh off's Ru les,and Complex Circuits

3 3 6

C h ap t e r Ov er v iewSect ion 8.1 provides a br ief int roduc t ion. Sec t ion 8.2 reviews some of the history ofGalvani's discovery of the g a l v a n i c c e l l (which is now cal led the vo ltaic cel l ) , and ofVolta's invent ion of the vo l ta ic p i le (which is no w c al led the bat tery) . (Ne ither of thesewo rkers unde rstood the science of their very real discoveries.) Sect ion 8.3 discusses abatte ry of identical vo ltaic cells, and Se ction 8.4 discusses the relat ive cost of electricalpo we r obtained f ro m bat ter ies and f rom the elect ric company. Sect ion 8.5 considersthe dist inct issues of m axim izing the actual amo un t of po we r transfer ( relevant toh igh-power appl icat ions) and max imiz ing the ef f ic iency of that power t rans fer ( re l -evant to lo w -po we r appl ications). Se ct ion 8.6 p resen ts Kirchhof f 's rules. Sec t ion 8.7presents some n ontr iv ial appl icat ions of Kirchhof f 's rules, including th e jumpe r-cableproblem . Se ct ion 8.8 considers the sho rt - t ime and long -t ime behavior of c ircui ts withboth capacitors and resistors. This is a prequel for the discussion, in S ect ion 8.9, ofthe cha rge and discharge of RC circuits. Sec t ion 8.10 considers the charge on thesurface of a c ircuit , which produces the elect r ic f ield within the volume of the c ircuit .I t also show s ho w the bui ldu p o f surface charge on a wire can be analyzed in terms ofp a r a s i t ic c a p a c i t a n c e in paral lel with the resistance of the wire; the sharper the turnson the elect ronic superhighw ay, the s lowe r the c ircuit can respond. Se ct ion 8.1 1 con-tains an op t ional discussion of the br idge c i r cu i t . Sect ion 8.12 presents an op t ionaldiscussion of p lasma osc i l la t ions , a col lect ive of mot ion of the elect rons in metal l iccond uctors. S ect ion 8.13 provides a br ief Inter lude, sum ma riz ing the ma ter ial alreadystudied, and indicat ing w ha t yet has to be treated. I t discusses the l imitat ions on theval idi ty of a c ircuit analysis, wh ich assumes instantaneo us act ion at a distance ratherthan inc lud ing the f in i te speed of l ight, and d iscusses wh at happens when we f i rs tthr ow a switch in a c ircuit, o r whe n w e se nd an electr ical s ignal dow n a cable. A ful l


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8.2 D iscovery M ust Include Reproducibility 3 3 7

t reatm ent requires that w e pursue the relationship of e lectr ic i ty to magnet ism, and thatwe study electromagne t ic radiat ion. This provides a lead-in for the remainder of thebook.

I n t r o d u c t i o nT h e c o n c e p t o f a n electric circuit d i d n o t e x i s t w h e n t h e o n l y t y p e o f e m f w a se lec t ros ta t ic . M oreover , no t un t i l V o l ta ' s inven t ion o f the vo l t a i c p i le , o r ba t te ry ,cou ld l a rge e lec t r i c cu r r en t s be s us ta ined . In th i s chap te r , a s w e p rog res s tot h e s t u d y o f c o m p l e x c i rc u i ts , w e f ir st s tu d y t h e p r o p e r t i e s o f t h e b a t t e r y a sa com bina t ion o f ind iv idua l vo l t a i c ce ll s.M os t c i r cu i t s canno t be ana lyzed s o le ly in t e rms o f s e r i e s and pa ra l l e l r e s i s -to r s w i th a ba t t e ry he re o r the re . To dea l w i th b r idge c i r cu it s , o r s ys tems w i thm u l t i p l e a r m s c o n t a i n i n g s o u r c e s o f c h e m i c a l e m f , o r s y st e m s w i t h b o t h r e si s-t o r s a n d c a p a c it o rs , w e m u s t a p p l y t h e i d e a s o f t h e p r e v i o u s c h a p t e r i n a m o r es ys temat ic f a s h ion , u s ing Kirchhoff 's rules, f i rs t s t a t ed a roun d 1850 . Th i s s ys -t e m a t i c a p p r o a c h t o c i r cu i t s u se s s o le l y t h e m a c r o s c o p i c p r o p e r t i e s o f c ir c u it s :r e s i s t ances , capac i t ances , and emfs (bo th e lec t ros ta t i c and none lec t ros ta t i c ) .

p ? D i s c o v e r y M u s t I n c l u d e R e p r o d u c i b i l i t y :I t N e e d N o t I n c l u d e UnderstandingT h e h i s t o r y o f t h e v o l t ai c c e ll a n d o f t h e b a t t e r y i l lu s t ra t e s t h a t a g r e at d i s co v e r y o ri n v e n t i o n c a n b e m a d e e v e n w h e n t h e d i s c o v e r e r o r i n v e n t o r d o e s n o t u n d e r s t a n dh o w t h e p h e n o m e n o n o r d e v ic e w or ks .

T h e a n a t o m i s t G a l v a n i , i n 1 7 8 0 , n o t i c e d t h a t f r o g s ' l e g s w e n t i n t o s p a s m sw h e n s t i m u l a t e d b y a n e x t e r n a l e l e c t r o st a t i c so u r ce . H o w e v e r , i t t o o k h i m m u c he f f o r t t o m a k e t h e w o r k r e p r o d u c i b l e . B y 1 7 8 6 , h e h a d l e a r n e d t h a t f r o g s ' l e g sw o u l d a l s o g o i n t o s p a s m s w h e n m a d e p a r t o f a c i r c u i t w i t h t w o d i s s i m i l a rm e t a l s. H e c o n c l u d e d t h a t t h e f ro g s ' le g s t h e m s e l v e s w e r e a s o u rc e o f " a n im a l "e lec t ri c i ty , o f the s ame n a tu r e a s o rd ina ry e lec tr i c ity . H e th ou gh t in t e rm s o f thed i s c h a rg e o f s o m e t h i n g l ik e a L e y d e n j a r in t e r n a l t o t h e f r o g, w h i c h t h r o u g h a nu n k n o w n b i o l o g i c a l p r o c e s s c o u l d r e c h a r g e . G a l v a n i ' s e x p l a n a t i o n w a s w r o n g ,b u t h e h a d m a d e a g r e a t d i s c o v e r y , w h i c h h e p u b l i s h e d i n 1 7 9 1 . I t h a d t a k e nh i m m a n y y e a r s t o m a k e t h e e f f e c t r e p r o d u c i b l e . G a l v a n i is t h e f a t h e r o f t h e f ie l do f electrophysiology.T h e p h y s ic i st V o l t a ~ a l r e a d y r e n o w n e d f o r t h e d i sc o v e ry o f m e t h a n e , f o rt h e i n v e n t i o n o f t h e e l e c t r o p h o r u s ( se e C h a p t e r 1 ), a n d f o r h i s s t u d ie s o fc a p a c i t a n c e ~ b e g a n t o s t u d y G a l v a n i ' s e f f e c t . H e r e a l i z e d t h a t t h e f r o g s ' l e g sw e re s e rv ing as s ens i t ive de tec to r s , r a the r tha n as s ou rces o f e l ec tr i c ity . By 1792 ,h e h a d e s t a b l i s h e d t h a t t h e t w o m e t a l s ( " d r y c o n d u c t o r s " ) w e r e n e c e s s a r y t ocaus e an e lec t r i c cu r r en t to f low , bu t tha t the f rog 's l eg w as no t . I t s imp ly s e rvedas w ha t he ca l l ed a "mois t conduc to r " ( i . e . , an e lec t ro ly te , l ike s a l t w a te r ) . H ew en t on to d i s cover the e lec t rochemica l s e r i e s ( e .g . , s i lve r e l ec t rodes a r e h igheri n v o l t a g e b y 1 . 5 5 V t h a n z i n c e l e c t r o d e s ) . H e a l s o m a d e t h e o b s e r v a t i o n t h a tp l a c in g i n t e r m e d i a t e m e t a l s i n t h e c i r c u i t (e .g ., b e t w e e n t h e s il v er a n d t h e z i n c )h a d n o e f f e c t o n t h e c u r r e n t p r o d u c e d .


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338 Chap te r 8 R Com plex Ci rcuit s

Figure 8.1 A simple battery. (a) A single voltaic cellconsist ing of a piece o f Zn and a piece of Ag separated bywet pasteboard (which provides ions). (b) A bat teryconsisting of many such voltaic cells in series. Note that theZn electrode on the bottom right , and the Ag electrode onthe bottom left , have no effect on the net emf.

I n 1 8 0 0 , V o l t a i n v e n t e d w h a t c a m e t o b e k n o w n a s t h e v o l t a ic p i le , a b a t t e r yo f vo l t a i c c el ls c onne c t e d i n se r ie s , wh i c h m a de t he e f fe c t s t ha t he h a d be e ns t u d y i n g m u c h m o r e i n t e n s e. S e e Fi g u r e 8 .1 . O n e o f h i s e a r ly p il es h a d 3 2 Z n - A gvo l t a i c c e ll s i n s er ie s , w i t h a bo u t 50 vo l t s a c ross t he t e rm i na l s o f t he p i le , ap o w e r f u l b a t t e r y i n d e e d . A l m o s t i m m e d i a t e l y , o t h e r s c i e n t i s t s u s e d t h e v o l t a i cp i le t o d e c o m p o s e w a t e r, c o l l e ct i n g h y d r o g e n a t o n e e l e c t r o d e a n d o x y g e n a t t h eo t h er . T h u s w a s b o r n t h e s u b j e c t o f e le c t r o c h e m i s tr y .Us i ng h i s c ons i de ra b l e e xpe r i m e n t a l sk i l l , i n 1797 , Vo l t a d i s c ove re d a no t he re f f ec t r e l a te d t o v o l t a ic c e l l s ~ t h e con tact po ten t ia l b e t w e e n t w o d i s s i m i l a r m e t -a ls . Un l i ke t he c he m i c a l e m f , wh i c h i s a s soc i a t e d wi t h t r a ns fe r o f bo t h e l e c-t rons a n d i ons , t he c on t a c t po t e n t i a l i s a s soc i a t e d wi t h t r a ns fe r on l y o f e l e c t rons .T h e c o n t a c t p o t e n t i a l i s a m e a s u r e o f t h e w o r k f u n c t i o n o f a m e t a l . ( R ec a llt h a t t h e w o r k f u n c t i o n is t h e e n e r g y t o r e m o v e a n e le c t r o n f r o m a m e t a l . I tis r e l a t e d t o t h e e n e r g y t o r e m o v e a n e l e c tr o n f r o m a n i so l a te d a t o m o f t h a tm e t a l . )

A l t h o u g h V o l t a w a s a w a r e t h a t o x i d a t i o n o c c u r r e d a t t h e s i l ve r e l e ct r o d e , h efoc use d on l y on t he e l e c t r i c c u r re n t a spe c t o f h is c el ls , c ons i de r i ng t he a s soc i a t e df l ow o f c he m i c a l s t o be a m e re s i de e f fe c t o f no fu nda m e n t a l s i gn if i ca nc e . Hi sv i e w w a s t h a t t h e c o n t a c t p o t e n t i a l a t t h e m e t a l - m e t a l i n t e rf a c e w a s t h e p o w e rsou rc e t ha t d rove t he e l e c t r i c c u r re n t i n h i s c e l l s , a nd t ha t t he e l e c t ro l y t e s e rve do n l y to b r i n g t h e o t h e r e n d s o f t h e m e t a l s t o t h e s a m e e l e c t ri c a l p o t e n ti a l . I no t h e r w o r d s , t o V o l t a t h e v o l ta i c c el l w a s a p e r p e t u a l m o t i o n m a c h i n e . ( T h i swa s som e 50 ye a rs be fo re c ons e rva t i on o f e ne rgy wa s a n e s t a b l i she d p r i nc i p l e . )H e w a s w r o n g , b u t h is r e p u t a t i o n , j u s t ly d e s e r v e d o n t h e b a s is o f h i s m a n yc on t r i bu t i on s t o t he s c i e nc e o f e l e c tr i c it y , ke p t t he s c i e n ti f ic c om m un i t y f romse r i ous l y c ha l l e ng i ng h i s v i e wpo i n t fo r m a ny ye a r s . E ve n a f t e r Fa ra da y ' s work one l e c t ro l y s i s e s t a b l i she d t ha t t he e l e c t r i c a l a nd c he m i c a l e f f e c t s a re i ne x t r i c a b l e ,m a n y p r o m i n e n t s c ie n t is t s c o n t i n u e d t o a c c e p t th e c o n t a c t p o t e n t i a l a s t h e e n e r g ysource for the vol ta ic ce l l .A s i m p l e e x p e r i m e n t w o u l d h a v e d e m o n s t r a t e d t h a t V o l t a w a s w r o n g a b o u tt he sou rc e o f e ne rgy i n t he vo l t a i c ce ll . As 1915 Nob e l L a u re a t e i n phys i c s


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8.3 Batteries Ar e Com binations o f Voltaic Cells 339

W. L. Bragg ( in Electricity, Macm i l l an , New Yo r k , 1 9 3 6 , p . 5 0 ) wr o te :The energy for dr iv ing the curren t [ i n Ga lvan i ' s f rog exper iment s] comes f romthe s l igh t chemica l ac t ion be tween the meta l s and the muscles or nerves theytouch . In Vo l ta 's P i l e i t comes f rom an ac t ion o f the l iqu id upo n the meta l sthemselves , an d i f yo u look at i l lustrat ions o f his pi le yo u wil l see that there isan ex t ra s i l ver p la te a t one end a nd z inc p la te a t the o ther which are rea l lyunne cessar y an d do not help to increase the s trength o f the pi le.S ee F ig u r e 8 .1 . As Br agg in d i ca t es , r em o v a l o f t h e se ex t r a p l a t e s w o u ld h av ey ie ld ed n o ch an g e in t h e ch em ica l em f , i n co n t r ad i c t i o n to Vo l t a ' s v i ew .Ul t im a te ly , i t was accep ted th a t t h e en e r g y so u r ce f o r t h e v o l t a i c c e l l wasch em ica l i n o r ig in . Ho wev e r , t h e r e was n o f u l l - f l ed g ed th eo r y u n t i l ch em is t r yi t s el f h a d b e c o m e m o r e d e v e l o p e d , t o w a r d t h e e n d o f t h e 1 9 t h c e n t ur y . A fu l ld i scu ss io n o f v o l t a i c ce ll s, i n c lu d in g wh a t h ap p en s w i th in t h e e l ec t r o ly t e an d th ee l ec t r o d es , m u s t co n s id e r b o th t h e i r ch em ica l an d th e i r p h y s i ca l a sp ec t s . Ph y s i -

c i s ts an d ch em is t s a r e s t il l t r y in g to d ev e lo p a q u an t i t a t i v e ly accu r a t e m ic r o sco p ict h e o r y o f t h e v o l t a i c ce ll . T h e h a r d e s t p a r t i s t h e d e s c r i p t io n o f w h a t h a p p e n s a tt h e e l e c t r o d e - e l e c t r o l y t e i n te r fa c e .

3 B a t t e ri e s A r e C o m b i n a t i o n s o f V o l ta i c C e llsI den t i c a l V o l ta i c C e l l s i n Se r i e sCo n s id e r n i d en t i c a l v o l ta i c c e ll s e ach w i th ch em ic a l em f & an d in t e r n a l r e s is -t a n c e r l , i n se r i e s w i th a r e s i s to r R . W e wi sh to f i n d th e cu r r en t an d t e r m in a lv o lt a ge . F i g ur e 8 . 2 ( a) d e p i c t s s ix o f t h e m , a s w o u l d b e f o u n d w i t h i n a c o m m o n9 V b a t t e r y u sed f o r e l ec t r o n ic s ( co n ta in in g s ix 1 .5 V ce l l s i n s id e ; o p en o n e u pi f y o u d o n ' t b e l i ev e t h i s ). As in t h e p r ev io u s c h ap te r ' s d i scu ss io n o f a r e s i s to r Rin se r i e s w i th a s in g l e ce l l , t h i s w i l l b e an a ly zed in two way s . F i r s t , Oh m ' s l aw

/~1 61 61 61 E 1 ~e 1

~V~ I..... a V R --(a)

Pb

....P b O 2(b)

O

O +

Figure 8 .2 Identical voltaic cells. (a) A sche ma tic of identical voltaic cellsconnected in series across a load resistor R. The net em f is proportional to th enum ber of cells in ser ies. (b) A schem atic of the connections for a lead-acidcell, wh ere m any no minally identica l voltaic cells are placed in parallel. Thema xim um current is proportional to the num ber of cells in parallel.


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34 0 C ha p te r 8 i C om ple x C i r c u it s

w i l l b e a p p l i e d t o e a c h c i r c u i t e l e m e n t . N e x t , O h m ' s l a w w i l l b e a p p l i e d t o t h ec i r c u i t a s a w ho le .B e c a u s e e l e c t r ic c h a r g e i s c o n s e r v e d , f o r s t e a d y c u r r e n t f lo w , a ll t h e c e ll s a n dt h e r e s i s to r R h a v e t h e s a m e c u r r e n t I p a s s in g t h r o u g h t h e m . B y s y m m e t r y , e a c hc e ll h a s t h e s a m e v o l t a g e A V i a c r o ss i ts t e rm i n a l s . M o r e o v e r , b y u n i q u e n e s s o ft h e v o l t a g e d i f f e r e n c e , G - G , t h e v o l t a g e d r o p a c r os s t h e u p p e r a r m ( n A 111) i st he s a m e a s t ha t a c r os s t he l ow e r a r m ( A V R), so nA V1 = A V R. A l so , by O h m ' s

l aw , w i th s a nd A V1 d r iv ing c u r r e n t i n oppo s i t e d i r e c t i ons , e a c h c e l l s a t is f i esI = s - A V 1 , A V 1 _ _ 1 ( l ~ - V ~ ) , ( 8 . 1 )

t" 1 t l

a nd the r e s i s to r s a t i s f i e sav RI - R ' A V n - - ( V b - - V a ) - - n A ~ . ( 8.2 )

W i t h t h i s i n f o r m a t i o n , w e c a n o b t a i n t h e c u r r e n t s a n d v o l t a g e s a s s o c i a t e dw i th t he c i r c u i t . S o lv ing ( 8 . 1 ) a nd ( 8 . 2 ) f o r A VR y ie ld sA V e = I R = n A V1 = n ( s - I r l ) . (8.B)

S o l v i n g f o r I a n d t h e n A V R g ive sns nE1RI = A V R = (8.4)n r l + R ' n r l + R "

H e n c e , t h e e f f e c t i v e c h e m i c a l e m f s a n d e f f e c t i v e i n t e r n a l r e s i s t a n c e reff are

s = n s r ef f = n r l . (8.5)T h u s , p u t t i n g t h e c e l l s i n s e r i e s i n c r e a s e s t h e n e t i n t e r n a l r e s i s t a n c e a n d t h e n e tc h e m i c a l e m f . I t a l s o i n c r e a s e s t h e n e t e n e r g y t h a t t h e s y s t e m c a n p r o v i d e .A l t e r n a t iv e l y , c o n s i d e r t h e s y s t e m a s a w h o l e . T h e n e t c h e m i c a l e m f i s d u e t on cells in ser ies, so s = n s T h e n e t r e s i s t a n c e i s d u e t o n i n t e r n a l r e s i s t a n c e sr and R in se r ie s , so R a ft = n r + R . F r o m O h m ' s l a w a p p l i e d t o t h e c i r c u i t a s aw h o l e , I - s ( 8 . 4 ) f o l l o w s i m m e d i a t e l y . T h u s , f o r t h e b a t t e r y a s a w h o l e ,t h e e m f s i s n t im e s l a r ge r t ha n f o r a s i ng l e c e l l . H ow e ve r , t he " c ha r ge " Q e f fi s t he s a m e a s f o r a s i ng l e c e l l . B y ( 7 . 28 ) , t he e ne r gy s to r a ge E c e I ! - E e f f Q e f f is nt i m e s l a r g e r t h a n f o r a si n g l e ce ll , a s e x p e c t e d .

~ Six iden tical cells in seriesCo nsid e r an e lec t ronics ba t te ry o f s ix ident ica l ce l ls in se rie s , w i th s = 1 .5 V,r l - - 0 .12 ~2, and charge Q 1 = 240 C . (a ) F ind s a n d reff. (b) F i nd t h e c u r r e n tl thr ou gh and th e vo l tage A VR across a 2 .28 S2 res is tor co nn ec te d to th isba t t e ry . ( c) F ind the e ne r gy s to r e d b y t h i s ba t t er y .S o l u t i o n : (a) By (8.5), s - - 6 ( 1 . 5 ) = 9 V a nd raft --6(0.12) = 0.72 ~2. (b) By(8.4), I = 3 A and A VR -- 6.84 V. (c) By (7.28 ), E --- 9(2 40 ) - 2, 160 J. Th is issix t ime s the ene rgy storage 1.5(240 ) - 360 J for a single cell.


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8 .3 Ba t t er ies Ar e Co m b in a t io n s o f Vo l tai c Ce ll s 34 1

8~3o2 Vo l t a i c Ce l l s i n P a ra l l e lOp en u p a 1 2 V ca r b a t t e r y an d y o u w i l l f i n d six 2 V l ead s to r ag e ce ll s i n se ri e s.Mo r eo v e r , e ach ce l l r e a l ly co n s is t s o f m an y ( n eg a t iv e ) Pb e l ec t r o d es c o n n ec t e din p a r a l l e l , i n t e r lo ck in g wi th ( p o s i t i v e ) Pb O2 e l ec t r o d es co n n ec t ed in p a r a l l e l .S e e F i gu r e 8 . 2 (b ) , w i t h f iv e P b a n d f o u r P b O 2 , w h i c h c o r r e s p o n d s t o a c o m m o nm o to r cy c l e b a t t e r y . T h i s i s eq u iv a l en t t o e ig h t 2 V ce l ls in p a ra l le l ; b o th s id e s o ft h e f o u r P b O 2 e l e c t ro d e s c o n t r i b u t e . Eight identical cells in parallel

Co nsid er eigh t identical cells (El, r~) in parallel , with a term inal voltage A Vand charge Q 1. (a) F ind the cur ren t I and ef fec tive res is tance r4 f o f t he systemin terms of gl , A V, and r l . (b) For the bat tery as a whole, f ind its effectivecharge Qef f , i t s e ffec t ive e m f Eeff, and i t s energy s to rage E re la t ive to the va luesfor a single cell.Solution: (a) For eight ide ntical cel ls (g~, r~) in parallel, as in Figure 8.2(b),the current adds, so I = 8(E1 - A V ) / r l = (81 - A V ) / ( r l / 8 ) . Thus r 4 f = r l / 8 .(lo) This use of eight c ells in parallel gives an effective e m f that is the same asfor a single cell: Eef f = 8"1. How ever, th e effective cha rge and th e e nergy storedincrease by a factor of eight: Q e f f - - 8 Q 1 , E e f f - - ~ ' e f f Q e f f = 881 Q1 = 8E1.

T h e e l ec t r i c e e l u se s a d i f f e r en t d e s ig n th an a c a r b a t t e r y : i n s t ead o f c el ls i n ser ie s ,e ach ce l l h av in g m an y su b ce l l s i n pa r a l le l , t h e ee l h a s m an y ce l l s i n p a ra l l e l, e achce l l h av in g m an y su b ce l l s i n se r i e s . Mo r eo v e r , t h e so u r ce o f en e r g y in t h e ee l ' sspec ia l ized ce l l s , ca l led e l ec t ro cy t es , i s n o t d u e to ch e m ica l r e ac t io n s . Ra th e r , t h eye m p l o y i o n p u m p s t h a t se l ec t iv e ly an d ac t iv e ly t r an s f e r c e r t a in i o n s ac r o ss t h ecell wall . I o n c h a n n e l s a l so se l ec t iv e ly b u t p a ss iv e ly p e r m i t c e r t a in i o n s t o c r o ssthe ce l l wal l .8...~.3~3 T e r m in a l V o l t a g e o f a B a t te r y i n U s e

As for an ind iv idual vo l ta ic ce l l , the t e r m i n a l v o l ta g e A V T o f a b a t t e r y i s ~ n o ts u r p r i s i n g l y ~ d e f i n e d a s t h e v o l t a g e a c ro s s t h e t e r m i n a l s o f t h e b a t t e r y. T h e s ig ni s ch o sen so t h a t A V T o p p o s e s t h e c h e m i c a l e m f g . S e e F i g u r e 8 . 3 . B y O h m ' sl aw , i f p o s i t i v e cu r r en t f l o w is t ak en to b e cau sed b y th e em f g , t h en

i i i i i i i i i i i i i i i i l i l i i i ! i i i i i i l i i i i i i i i i i i! i i i i i i i i ! i i i i i i i i i i i l iSO

A V T = 8 - I r . (8.7)E q u a t i o n ( 8 .6 ) is e q u i v a l e n t to ( 7 . 2 5 ) o f t h e p r e v i o u s c h a pt e r, w i t h A V r e l a b e l e dA V T . I t h a s a lso b een u sed a s ( 8 .1 ).On o p e n c i r cu i t , so t h e r e i s n o cu r r en t f lo w , A V T = g . On d i sch a r g in g , wh e r eI > 0, by (8.7) A V T i s l e ss t h an g . O n ch a r g in g , wh e r e I < 0 , b y ( 8 .7 ) A V Tex ceed s g . Ho wev e r , t h e t e r m in a l v o l t ag e i s n o t a lway s d e sc r ib ed b y ( 8 .7 ) . T h i s


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3 4 2 Chapter 8 9 Com plex Circu its

+- - - NI ro l

m

Figure 8.3Terminal voltageA Vr of a battery.

i s b ecau se t h e v o l t a i c c e l l s w i th in b a t t e r i e s r e a l l y a r e m o r eco m p lex in t e r n a l ly , a s d i scu ssed in Ch ap te r 7 .Ba t t e r i e s t h a t c an b e r ech a r g ed ( su ch a s c a r b a t t e r i e s ) a r eca l led secondary b a t t e r i e s ; t h o s e t h a t c a n n o t b e r e c h a r g e d( su ch a s Zn - Ag ce l l s ) a r e c a l l ed primary b a t t e r i e s . C h e m i -ca l reac t ions a re no t a lways, in p rac t ice , revers ib le (e .g . , a gasm i g h t f o r m o n a n e l e c t r o d e ) , o r t h e b y p r o d u c t a t a n e l e c -t r o d e m i g h t g e t c o n s u m e d b y a n o t h e r r e a c t a n t w i t h i n t h ee l ec t r o ly t e , so n o t a l l b a t t e r i e s a r e r ech a r g eab le . I n r ecen ty ea r s , t h e sh e l f l i f e o f a lk a l in e b a t t e r i e s h a s d r am a t i ca l ly i n -c r e a s e d , i n p a r t b e c a u s e i m p r o v e m e n t i n s e p a r a t o r m a t e r i a l s w i t h i n t h e e l e c -t r o ly t e ( Mn O2 ) h a s d ec r ea sed in t e r n a l ch em ica l r e ac t io n s .

T h e " C h a r g e " o n a B a t t e r y , a n d I ts C o s tW e n o w e s t i m a t e t h e a m o u n t o f "c h a rg e " t h a t a n a u t o m o b i l e b a t t e r y c a n p r o v id e .( Rem em b er , t h i s is n o t t h e t r an s f e r o f ch a r g e f r o m o n e m a te r i a l t o an o th e r , a so ccu r s w i th r u b b in g , b u t r a th e r t h e t r an s f e r o f ch a r g e a r o u n d an e l ec t r i c c i rcu i t . )W h e n t h e t w o a u t o m o b i l e h e a d l i g h t s a re o n (u s i ng a b o u t 3 6 W e a c h ) , b y (7 . 3 1 )t h e c u r r e n t t h a t t h e y d r a w i s a b o u t 2 - 3 6 W / 1 2 V = 6 A . If o u r a u t o m o b i l e b a t t e r ycan b e d i sch a r g ed a f t e r ab o u t f o u r h o u r s , we d ed u ce th a t i t s ch a r g e i s ab o u t6 A - 4 h r = 6 A - 1 4 , 4 0 0 s = 8 6 , 4 0 0 C . Si n c e t h e c h a r g e s h o u l d n o t d e p e n d o nh o w th e ce l l s a r e co n n ec t ed , i t sh o u ld b e t h e sam e wh en th ey a r e i n p a r a l l e l ,i n wh ich ca se it is c l e a r t h a t e ach o f t h e 4 8 ce ll s h a s a ch a r g e o f 1 8 0 0 C . Nowo n d e r t h a t , wh en v o l t a i c c e l l s we r e f i r s t i n v en ted , t h ey we r e su ch a m ar v e l ;co n t r a s t t h a t w i t h t h e 1 0 - 9 C a sso c i a t ed w i th s t a t ic e l ec t r i c i t y f r o m a co m bp u l l ed th r o u g h c l ean , d r y h a i r . ( T h e o r ig in a l b a t t e r i e s d id n o t h av e a ch a r g e a sl a rg e as 1 8 0 0 C , b u t e v e n a c h a rg e o f 1 0 C is e n o r m o u s c o m p a r e d t o w h a t c o u l db e o b ta in ed b y s t a t i c e l ec t r i c i t y . ) I t wo u ld t ak e a co n s id e r ab l e t im e to p r o d u ce1 8 0 0 C o f ch a r g e b y e l e c tr o s t a ti c m e t h o d s , e v e n b y s o m e o f t h e s o p h i s t i c a t e de l ec t r o s t a t i c i n d u c t io n - b a sed " ch a r g e - d o u b l in g " d ev ice s o f t h e l a t e 1 8 th cen tu r y .Mo r eo v e r , su ch ch a r g e wo u ld b e s to r ed a t h ig h v o l t ag es , an d th u s b e su b jec t t oe l ec t r i c a l b r eak d o wn . F ig u r e 8 .4 p r e sen t s t e r m in a l v o l t ag e v e r su s t im e f o r s lo wan d r ap id d i sch a rg e , f o r a l e ad - ac id ce l l. No t o n ly is t h e d i sch a r g e t im e sh o r t e r o na fa s t d i sch a r g e ( b y d e f in i t i o n ) , b u t t h e t e r m in a l v o l t ag e i s l es s, f o l l o win g ( 8 .6 ) .I n co m p ar in g v o l t a i c c e ll s o f t h e sam e ty p e , a g o o d r e l a t i v e m eas u r e o f h o w lo n g

tVoltage (V)2.01.81.6 Rapid1.4

11

dischargeSlow discharge

I I I I2 3 4 5Discharge tim e (hours)

Fig u r e 8 . 4 Characteristic discharge properties of avoltaic cell.


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8.5 Ma ximizing Power Transfer 343

t h e y w i l l l a s t ( th e i r c h a r g e ) i s t h e i r r e l a t i v e m a s s . T h u s t h e r e a r e 1 . 5 V A A A( 11 g ) , A A ( 23 g ) , C ( 66 g ) , a nd D ( 138 g ) a lka l i ne c e l ls w i th c ha r ge s o f a bo u t1 .2 , 2 .5 , 6 . 8 , a n d 1 4 . 0 A - h r o f l o w - c u r r e n t ( 1 0 m A ) u s a g e.

~ E lectr ic i ty cost : bat ter ies versus the po wer com panyF ind the c os t pe r hou r t o r un a 100 W l igh tbu lb b o th f o r 80 D c e l ls in s e ri e s( to provide 120 V) , a t $0 .50 p e r D ce l l, and for 1 20 V ac pow er , a t $ . 16 pe rki lowat t -hour . Neglec t in te rna l r e s is tance . ( In prac t ice , the in te rna l r e s is tanceo f t he ba t t e r i e s w o u ld dom ina t e t h e l i gh tbu lb r e s i s t anc e , so t he l i gh tbu lbp r o b a b l y w o u l d n o t e v e n l i g h t. )Solution: For 100 W a t 120 V, the cur rent i s I - 100 W /1 20 V = 0. 8 3 A. 80 Dcells in ser ies are equivalent to a 120 V dc cell . They will last a t ime 5.2A-hr /0 .83A = 6 .24 hr, a t a cos t of (80) ($0 .50) = $40. On the o th e r hand, a t $ . 16 pe rki lowat t -hour , the e lec t r ic com pany wi l l cha rge for (100 W) (6 .24 hr ) = 0 .624 kW -hr, or $1.00. T hus, for these pr ices battery p ow er is 40 t im es as expens ive as po we rf rom the e lec t r ic company; about $12.80 pe r kW-hr .F i n a l l y , n o t e t h a t h e a v y - d u t y a l k a l i n e c e l l s l a s t n e a r l y t w i c e a s l o n g f o r h i g h

c u r r e n t u s a g e a s d o g e n e r a l - p u r p o s e a l k a l i n e c e l l s a l t h o u g h t h e g e n e r a l - p u r p o s ec e l l s l a s t ne a r ly a s l ong f o r l ow c u r r e n t u sa ge . Typ ic a l a lka l i ne c e l l s u se a Z nc a t h o d e , an M n O 2 a n o d e, a n d a K O H e l e ct r o ly t e .

8~ M a x i m i z i n g P o w e r Tr an s fe rv e rs u s M a x i m i z i n g E f fi c ie n c y o f P o w e r T ra n s f erN o w c o n s i d e r a b a t t e r y o f e m f g a n d i n t e r n a l r e s i s t a n c e r i n s e r ie s w i t h a r e s i s to rR , a n d c o n s i d e r t h e p o w e r t r a n s f e r t o R , c a l l e d t h e load, o r load resistor. ( T h eu s a g e c o m e s f r o m m e c h a n i c s , w h e r e w e s p e a k o f m a c h i n e s d r i v i n g a m e c h a n i c a ll o a d . ) S e e F i g u r e 8 . 5 . T w o d i f f e r e n t q u e s t i o n s o f t e n c o n s i d e r e d a r e ( 1 ) h o w t og e t t h e m a x i m u m p o w e r t ra n s fe r to R ( i n d e p e n d e n t o f h o w m u c h p o w e r is l o sti n t h e i n t e r n a l r e s i s t a n c e r) ; ( 2 ) h o w t o g e t t h e m o s t e f f i c i e n t p o w e r t r a n s f e r t oR ( t h u s m i n i m i z i n g t h e p o w e r l o s s i n t h e i n t e r n a l r e s i s t a n c e r ) .

I m p e d a n c e M a t c h (R ---- r ) f o r M a x i m u m P o w e r T r a n s f erI s i t p o s s i b l e t o p l a y t e n n i s w i t h a p i n g - p o n g p a d d l e , o r p i n g - p o n g w i t h a t e n n i sr a c k e t ? Y es . H o w e v e r , n o c h a m p i o n s h i p t e n n i s p l a y e r u se s a p i n g - p o n g p a d d l e

I ! Rw

Figure 8.5 A batt ery in ser ies with a load resistor , tode t e r m ine how to m a x im iz e pow e r t r ans f er a nd how toma xim ize effic iency.


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344 C h ap t e r 8 m C o m p l ex C i r cu i t s

( e v e n o n e m a d e o f s t e e l, s o it w o n ' t b r e a k ) . F o r a g i v e n b a ll , w e m u s t a s k t h eq u e s t i o n " W h a t r a c k e t g iv e s t h e m a x i m u m p o w e r t ra n s fe r ? " ( T h i s n e g l e c t s t h ei s s u e o f control.) T h e a n s w e r is t h a t w e s h o u l d m atch the im pedance o f t h e r a c k e tt o t h a t o f t h e b a l l. T h e p r i n c i p l e o f im pedance m a tch ing is o n e o f t h e m o s tu s e f u l d e s i g n p r i n c i p l e s t h a t a s c i e n t is t o r e n g i n e e r c a n e m p l o y . M o r e g e n e r al l y ,i m p e d a n c e m a t c h i n g t e ll s u s t o m a t c h a p r o p e r t y o f t h e p o w e r s o u r c e ~ t h ei m p e d a n c e ( w h a t e v e r t h a t is i n a g i ve n s i t u a t i o n ) ~ t o t h e c o r r e s p o n d i n g p r o p e r t yo f t h e o b j e c t r e c e i v i n g t h a t p o w e r . W e n o w e s t a b li s h t h is p r i n c i p l e f o r a n e l e c t ri cc i r c u i t w i t h r e s i s t o r s , w h e r e im pedance m e a n s electrical resistance, a n d w e t h e na p p l y t h i s p r in c i p l e t o e s t i m a t e t h e e l e c tr i c a l re s i s ta n c e o f b o t h t h e s t a r ti n g m o t o ra n d t h e b a t t e r y o f a c a r.

C o n s i d e r a c h e m i c a l e m f g w i t h i n t e r n a l r e s i s t a n c e r i n s e r ie s w i t h a r e s i s t o rR , a s i n F i g u r e 8 .5 . F r o m ( 8 .4 ) w i t h n = 1 , g = E l , an d r = r ~ , t h e cu r r e n t i s

CI - . ( 8 . 8 )r + R

T h e n t h e r a t e o f h e a t i n g 7~ o f R i se ~ 2 R e ~ 2 R / r7 - I 2 R - = ~ (8.9)( r + R ) 2 r ( l + R / r ) 2"

F i g u r e 8 . 6 p l o t s t w o d i m e n s i o n l e s s q u a n t i t i e s , 7 2 / ( g 2 / r ) v e r s u s R / r , s h o w i n gt h a t a m a x i m u m o c c ur s n ea r R / r = 1 . S u c h a m a x i m u m m a k e s s e ns e : f o r s m a l lR / r t h e l o a d ( R i n 1 2 R ) is s o sm a l l t h a t i t d o e s n o t u s e v e r y m u c h p o w e r , a n df o r l a r g e R / r t h e l o a d i s s o la r g e t h a t t h e c u r r e n t ( I i n 1 2 R ) is n o t v e r y l a rg e . F o rs o m e i n t e r m e d i a t e v a l ue o f R / r w e t h u s e x p e c t a m a x i m u m , w h i c h o c c u rs w h e ni t s d e r i v a t i v e w i t h r e s p e c t t o t h e " l o a d " R i s z e r o . U s i n g t h e s e c o n d e q u a l i t y i n( 8 . 9 ) ,

d /~ o 2 ~ e 2 R E 2 ( r - R )d R = ( r + / ) 2 - 2 ( r + R) = ( r + R) 3 " ( 8 . 1 0 )

C l e a r l y t h e m a x i m u m o c c u r s fo r r - R : w h e n t h i s c o n d i t i o n i s s a ti sf ie d , w e s a yt h a t t h e r e is i m p e d a n c e m a t c h i n g . I n t h i s c a s e I 2 R = I 2 r s o t h a t t h e p o w e rp r o v i d e d b y t h e b a t t e r y g o e s e q u a l l y to t h e l o a d a n d t h e i n t e r n a l r e s i st a n c e.

l 0 . 25 -p 0 . 20

( e 2 / r ) 0 . 1 5 -0 . 1 0 -0 . 0 5 - /

0 I i I I j0 0 .5 1 1.5 2 2.5R / r

I I I3 3.5 4

Figure 8.6 No r m a l i zed p o w e r t r an sf e r to t h e l o adresis tor , as a func tion of the rat io of load resis tance tointernal resis tance.


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8 . 5 M a x i m i z i n g P o w er T r a n sf e r 345

~ Pow er consumption by a starting mo torC o n s i d e r a 1 2 V c a r b a t t e r y t h a t p r o v i d e s I = 6 0 0 " c o l d - c r a n k i n g a m p s "( C C A ) o n s t ar t u p . F in d t h e i n t er n a l r e si s ta n c e r t h a t g iv e s m a x i m u m p o w e rt o th e s t a r t in g m o t o r , a n d fi n d t h a t m a x i m u m p o w e r .Solution: S t a r t i n g t h e c a r i s t h e g r e a t e s t p o w e r u s e o f t h e b a t t e r y , s o f o r m a x -i m u m p o w e r t h e s t a r t i n g m o t o r r e s i s t a n c e R s h o u l d b e i m p e d a n c e m a t c h e d t ot h e b a t t e r y r e s i st a n c e r : R - r . W i t h g - 1 2 V a n d I = 6 0 0 c o l d - c r a n k i n g a m p s ,(8 .8) gives g = I ( R + r ) = I ( 2 R ) , o r 1 2 = 6 0 0 ( 2 R ) , s o R = r = 0 . 0 1 o h m f o rb o t h t h e b a t t e r y a n d t h e s t a r t i n g m o t o r . T h e p o w e r c o n s u m e d b y t h e s t a r t i n gm o t o r , t a k e n t o b e i m p e d a n c e m a t c h e d , i s 72 = 1 2 R = 3 6 0 0 W . A r u l e o f t h u m bi n t h e c a r i n d u s t r y i s t h a t a b a t t e r y s h o u l d h a v e a b o u t 1 .5 C C A p e r c u b i c i n c ho f e n g i n e. N o t e t h a t a n e l e c tr i c a l m o t o r i n o p e r a t i o n p r o d u c e s a s o - c a l le d b a c ke m f t h a t o p p o s e s t h e d r i v i n g e m f , a n d t h u s c a u s e s t h e c u r r e n t t o d e c r e a s e a s t h em o t o r c o m e s i n t o o p e r a t i o n . T h i s b a c k e m f is a c o n s e q u e n c e o f F a r a d a y ' s l aw , t ob e d i s c u s s e d i n C h a p t e r 1 2 . I t h a s n o t h i n g t o d o w i t h c h e m i c a l e m f s .

8,5,~2 L o w I n te rna l R es i s tance ( r ~ R )f o r E f f i c i en t P o w e r T r an s f erT h e r e i s a n o t h e r, d i s t i n ct , p r i n c i p l e i n v o l v e d w h e n w e w a n t m a x i m u m e f f i c i e n c yo f p o w e r t ra n s fe r , s o t h a t n e a r l y a ll t h e e n e r g y p r o v i d e d b y t h e c h e m i c a l e m f g o est o t h e l o a d R r a t h e r t h a n t o t h e i n t e r n a l r e s i s ta n c e r . H e r e , w e w a n t t o m a x i m i z et h e r a ti o o f t h e p o w e r t o t h e l o a d ( 1 2 R ) r e la t i v e to t h e t o t a l p o w e r ( I 2 ( r + R ) ) .T h a t is, w e w a n t t o m a x i m i z e~ ~ ~ > ~ ~ ...................................' ~ ......................................i.......................................................................................................i ~ ...........................................................................~ 8 4 1 8 48 4 i i i i i i i i i i i

iiii ii iiiii iiiiw i t h r e s p e c t t o r , f o r f i x e d R . I t d o e s n o t t a k e a r o c k e t s c i e n t i s t ( i . e . , c a l c u l u s )t o d e t e r m i n e t h a t t h i s o c c u r s f o r r - ~ 0 . T h i s p r i n c i p l e i s u s e d i n l o w - p o w e ra p p l i c a t i o n s .

C u r r e n t d raw by a headlightC o n s i d e r a h e a d l a m p , c o n s u m i n g 7 2 = I 2 R = 3 6 W . (a ) W h a t c u r r e n t m u s tt h e b a t t e r y p r o d u c e t o r u n t h e h e a d l a m p , a s s u m i n g m a x i m u m e f f i c i e n c y o fp o w e r t r a n s f e r ? ( b ) F i n d t h e r a t e o f h e a t i n g o f t h e i n t e r n a l r e s i s ta n c e .Solution: A s s u m e t h a t R ) > r . T h e n ( 8 . 8 ) g i v e s I , ~ E / R , so 72 ~ s T h u s Rg 2 /7 2 = 1 4 4 / 3 6 = 4 s2 . A s a s su m e d , t h i s i s i n d e e d m u c h m o r e t h a n r = 0 . 0 1 s2 .T h e b a t t e r y c a n e a si l y p r o v i d e 3 6 W , a n d t h e r a t e o f h e a t i n g o f t h e i n t e r n a lre s i s t ance , I 2 r = I 2 R ( r / R ) , ~ 3 6 ( . 0 1 / 4 ) = 0 . 0 9 W , is n e g l ig i b le . N o t e t h a t I =C / R = 12 /4 = 3 A . S ince ba t t e r i e s a re l e ss e f fec t ive a t l ow t em pe r a tu r e s , i n co ldc l i m a t e s s o m e p e o p l e t u r n o n t h e h e a d l i g h t s b e f o r e t r y i n g t o s t a r t t h e i r c a r . A st h i s e x a m p l e s h o w s , t h a t c a n h a r d l y w a r m t h e b a t t e r y . T h e m o s t e f f e c t i v e w a y t oh e a t t h e b a t t e r y i s t o s h o r t i t s t e r m i n a l s s o t h a t a l l i ts p o w e r g o e s in t o t h e b a t t e r yi t s e l f . T ha t w i l l p rov ide I - C / R = 1 2 / 0 . 0 1 = 1 2 0 0 A , a h u g e c u r r e n t t h a t c a nc a u s e s p a r k i n g a n d c a n b e p r o v i d e d f o r o n l y a f e w s e c o n d s a t a t i m e .


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346 Chapter 8 u Complex Circu i ts

8 .6 K i r c h h o f f ' s R u l es Te ll Us H o w t o A n a l y z eC o m p l e x C ir c uit sPr o p e r ly sp eak in g , K i r ch h o f f ' s r u l e s a r e a se t o f two r u l e s t h a t d e sc r ib e an y so r to f c i r cu i t w i th co n n ec t in g wi r e s o f z e r o r e s i s t an ce , r e s i s to r s R , cap ac i to r s C ,an d n o n e lec t r o s t a t i c so u r ce s o f em f ~ '. Ho w ev e r , b e f o r e t h e y can b e ap p l i ed toan y c i r cu i t , t h e cu r r en t s an d v o l t ag es f o r t h a t c i r cu i t m u s t b e d e f in ed p r ec i se ly .T h e r e f o r e , we p r eced e K i r ch h o f f ' s r u l e s w i th r u l e O .Rule D r a w a s c h e m a t i c o f t h e ci rc u it , i n c l u d ip o s it iv e m r r e n t f l o w I t h ro u g h e a & a ~

I n ad d i t i o n , i f sp ec if i c v o l tag es w i l l l a t e r ap p ea r i n y o u r an a ly s is , d e f in e t h e mex p l i c i t l y . W e h av e a l r ead y ap p l i ed th e ze r o th r u l e i n F ig u r e 8 .2 ( a ) , F ig u r e 8 .3 ,an d F ig u r e 8 .5 , b u t i t m a y h av e s l i p p ed p a s t y o u th a t s o m e o n e h a d t o d e c i d e w h a tto ca l l p o s i t i v e an d n eg a t iv e . T h i s i s l i ke se t t i n g u p a co o r d in a t e sy s t em , w h e r e y o uh av e to ch o o se w h ich d i r ec t io n i s p o s i ti v e . ( P r ec i s io n i s v a lu ab le ev en in o r d in a r yc o n v e r sa t i on : p r o p e r n a m e s l e a d t o l es s c o n f u s i o n t h a n " he , " " . . . . . . . .sne, or ~t ;sp ec if icd i r ec tio n s , l ik e " u p " o r " d o wn ," a r e b e t t e r t h an " th is way " an d " th a t way ." )W e n o w ap p ly Ru le 0 t o t h e b a s i c c i r cu i t i n F ig u r e 8 .7 ( a ) , wh e r e o n ly t h en o n e le c t r o s t a t i c em f an d th e r e s i s t an ces a r e g iv en . Becau se a b a t t e r y o f p o s i ti v ee m f d r i v e s c u r r e n t u p t h r o u g h t h e b a t t e r y a n d d o w n t h r o u g h t h e r e s i s t o r s , w etak e th e co n v en t io n s f o r p o s i t i v e cu r r en t f l o w a s i n F ig u r e 8 .7 0 9 ) . F o r a g iv ena r m o f th e c i r c u i t th e d i r e c t io n d r a w n f o r t h e c u r r e n t d o e s n ' t m e a n t h a t t h ecu r r en t a c tu a l ly f l o ws in t h a t d i r ec t io n , o n ly t h a t a p o s i t i v e cu r r en t wo u ld f l o wi n t h a t d i r ec t i o n. S im i la rl y, b e c a u s e w e e x p e c t t h a t t h e b a t t e r y w i l l p u m p p o s i ti v ech a r g e to t h e t o p , t h e p o s i t i v e v o l t ag e d i f f e r en ce i s d e f in ed a s i n F ig u r e 8 .7 ( b ) .T h e p o s i t i v e s id e d o e sn ' t m ean t h a t t h e v o l t ag e i s h ig h e r t h e r e , o n ly t h a t a p o s i t i v ev o l t a ge w o u l d b e h i g h e r t h e re . I f t h e b a t t e r y w e r e r e v e r se d , w e c o u l d s t il l e m p l o yth e sam e s ig n co n v en t io n s , b u t t h e cu r r en t s an d v o l t ag e d i f f e r en ce wo u ld a l l h av enegat ive s ign .F o r t h e c i r cu i t o f F ig u r e 8 .7 ( a ), t h e r e a r e two way s to d e f in e p o s i t i v e f o reach o f t h e t h r ee cu r r en t s a s we l l a s f o r t h e o n e v o l t ag e d i f f e ren ce . H en c eth e r e a r e a t o t a l o f 2 4 - - 1 6 p o ss ib i l i t i e s f o r d e f in in g th e s ig n co n v en t io n s ;F ig u r e 8 .7 ( b ) is o n ly o n e o f t h em . Al l 1 6 p o ss ib i li t i es w i l l y i e ld t h e sam e ( co r r ec t )an swer s o n ce th e d i f f e r en t s ig n co n v en t io n s a r e acco u n ted f o r . ( W e co u ld ev en

Bare circuit Circuit with sign conventions+e 1 i 2 r l~ e l ~ l~ i v

(a) (b)Figure 8.7 Circ uit to illustrate R ule 0 of KArchhoff's rules"define your sign conventions. (a) Basic circuit, w ith n o signconventions given. (b) Same circuit, but with a set of signconventions chosen.


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8.6 Kirchhof f ' s Rules 347

c hoose s i gn c onve n t i ons by f l i pp i ng a c o i n . W ha t c oun t s i s s t i c k i ng t o t he s i gnc o n v e n t i o n . )In a c om pl e x p rob l e m , wi t h 20 r e s i s t o r s a nd 10 ba t t e r i e s , t he re i s no a pr ior iwa y t o t e l l how a ny o f t he c u r re n t s f l ow o r t he s i gn o f a ny o f t he r e l a t i ve vo l ta ge s .I t is s a t is f a c t o ry t o a rb i t r a r i ly c hoose a s e t o f s ign

Recal l that, for a bal l start ing at the ori -gin, at rest, and fal l ing, i f posi t ive y is1upw ard , then y - - ~ g t 2" and i f posi t ive

1 2y i s dow nwa rd, then y = ~ g t . In eachcase, the bal l fal ls down, even though"d ow n" is negat ive in one case and pos-i t ive in the other.

c o n v e n t i o n s a n d t h e n s t i ck t o t h e m . I f / 9 t u r n so u t t o b e n e g a t i v e , t h e n / 9 a c t u a l l y f l o w s o p p o -s i te t o t he d i r e c t i on o f pos i t i ve /9 . I f A V7 t u rn sou t t o b e ne ga t i ve , t he n t he a c t ua l s i gn o f A 1/7 i sopp os i t e t o pos i t i ve A V7.For a r e l a t e d p rob l e m ( t r a f f i c f l ow) , de f i n i ngt he d i r e c t i on o f pos i t i ve c u r re n t f l ow is l i ke de -c i di n g , fo r a n o r t h - s o u t h s t r ee t , w h e t h e r t o c a lln o r t h w a r d o r s o u t h w a r d t h e p o s i t i v e d i r e c t i o n ;i f n o r t h w a r d is p o s it iv e , t h e n s o u t h w a r d is n e g a -t ive. B e c a u s e y o u r l a b p a r t n e r m i g h t n o t c h o o s e t h e s a m e c o n v e n t i o n s , i t i s i m p o r t a n tt o s h o w i n y o u r f i g u r e y o u r c o n v e n t i o n s f o r p o s i t i v e d i r e c ti o n s.

ii i ii ~iiii ii ..................== == = = i= i= i= = i= i i i i= i = = ii = = i = = i i = i i = i i................................! i= = = i= = == i= i= i= i i= i i i i i= i= i= = = J = i= =......(Th i s c a n be done on l y a f t e r de f i n i ng t he s i gn c onve n t i ons fo r c u r re n t , i nR u l e 0 . ) Fo r e a c h n o d e (o r j u n c t i o n ) o f t h e c i rc u i t, t h i s a m o u n t s t o e n s u r i n g t h a t

t h e c u r r e n t i n t o e a c h n o d e e q u a l s t h e c u r r e n t o u t o f th a t n o d e . T h i s i s k n o w n a sK i rc hhof f ' s f i r s t ru l e , a nd i s o f t e n c a l l e d t he noda l ru l e , o r t h e j unc t ion ru l e . T h a ti s , a t each node ,

F o r F i g u re 8 . 7 ( b ) , ( 8 . 1 2 ) a p p l i e d t o t h e t o p n o d e i m p l i e s th a t I = 11 + / 2 ; a p p l i e dt o t h e b o t t o m n o d e , i t i m p l i e s th e e q u i v a l e n t r e s u l t I1 + 12 = I . [M ore ge ne ral ly ,fo r a c l o se d c i r c u i t w i t h n node s , t he r e a re on l y n - 1 i nd e p e nd e n t a pp l i c a t i onso f (8 .12 ) . ] Fo r F i gu re 8 .8 (a ) , (8 .12 ) i m p l i e s t ha t 11 + h = /3 .

1 1 t I = d Q / d t t I = - d Q / d tI~

(a) (b) (c) (d)F i g u r e 8 . 8 Circuit to illustrate Rule 1 of Kirchhoff's rules: applycharge conservation. (a) Cu rre nt in = cur ren t out (11 = 12) at thenode where three a rms mee t . (b) Current in= current out ( /1 =/2)for a node that separates two parts of a single wire. (c) Relationshipbetween current and charge when posi t ive current enters posi t iveplate of a capaci tor. (d) Relat ionship be twee n curren t and chargewh en posi t ive cu rrent enters negative plate of a capacitor.


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34 8 Chapte r 8 a Complex Ci rcui t s

~ Current leaving one enteringesistor an d anotherBetw een any two c i rcui t e lements , w e can take an a rb i tra ry point to be a node .See Figure 8.8(b), which shows the two resistors R1 and R2. Relate 11 and 12.Solution: Here, (8.12) implies that 11 = 12.

~ Cu rren t cha rge a capacitornd fo rRela te I in a wi re leading to a capac i tor having charge Q, to d Q / d t for thecapacitor, in Figure 8.8(c) and Figure 8.8(d).Solution: For Figure 8.8(c),

I - d Q (8.13)dtgives the re la t ionship be tween posi t ive I and the charge on the positiveplate ofthe capacitor. For Figure 8.8(d) I = - d Q / d t gives this re la t ionship because aposi t ive current decreases he charge on the posi t ive plate.

R u l e 2 Apply the path independence of the voltage: ~nce you r e ~ ~t h~ ~poi nt, the voltage change on circulatingarou nd each oop in::the~~ i~ i~us i b~(This i s of ten ca l led the loop rule.) That i s , for each c i rcui t ,

~ A V 0 .l o o p

( p a t h i n d e p e n d e n c e o f v o l ~ g e ) :

E qu i va l e n t l y , t he vo l t a ge d i f f e re nc e be t we e n t wo po i n t s , v i a a ny g i ve n pa t h ,is i n d e p e n d e n t o f t h a t p a t h .F i g u re 8 . 2 ( a ) a n d F i g u r e 8 .7 u s e d t h e p a t h i n d e p e n d e n c e o f t h e v o l t a g e inde f i n i ng t he vo l t age s . App l i e d t o F i gu re 8 .9 , t he pa t h i n de p e nd e nc e o f t he vo l t -a g e m e a n s t h a t b e g i n n i n g a t d a n d e n d i n g a t b w i l l g i ve t h e s a m e a n s w e r w h e t h e rw e c ir cu la te c l o c k w i s e ~ I 4 - ~ = ( 1 4 - V ~ ) - ( V d - V c ) ~ o r c o u n te r cl o ck -w i s e ~ 1 4 - Vd = ( 1 4 - V ~ ) - ( I / ~ - V a).To use R u l e 2 , we m us t know, fo r e a c h c i r c u i t e l e m e n t , t he r e l a t i onsh i pb e t w e e n t h e v o l t a g e d i f f e r e n c e s a c r o s s i t a n d t h e c u r r e n t t h r o u g h i t . F o r c o m -p l e t e ne s s , he re i s a sum m a ry .

b ~I ~ cR1 ,, R2

R3

Figure8.9 Circuit to i l lustra te Rule 2 of K irchhoff ' srules" apply uniqueness of the voltage so that the netvoltage change on going around a c ircui t is zero.


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8.6 Kirchhoff's Rules 34 9

( a ) F o r a r e s i s t o r R ( o r a c i r c u i t a r m w i t h r e s i s t a n c e R ) , a p p l y O h m ' s l a w , ( 7 . 3 8 ) ,R ' (8.15 a)

w h e r e ~ C i s t h e s u m o f a ll t h e e m f s a s s o c i a t e d w i t h t h a t r e s i st o r : t h e e l e c -t r o s t a ti c v o l ta g e a c ro s s t h e a r m ( w h i c h m a y b e u n k n o w n ) , c h e m i c a l e m f s, o ro t h e r e m f s . T h i s c o m m o n t r e a t m e n t o f b o t h b a t t e r i e s a n d re s is t o rs im p l i e st h a t a r e s is t o r c a n b e t h o u g h t o f a s a b a t t e r y w i t h n o c h e m i c a l e m f .

I f a n e m f t e n d s t o c a u s e p o s i t iv e I , t h e e m f i s g i v e n a p o s i t i v e s ig n ; i f i t t e n d st o c a u s e n e g a t i v e l , t h e e m f is g i v e n a n e g a t i v e s i g n . T h e r e f o r e , i f i n o n e c i r c u i t ab a t t e r y e m f is r e c k o n e d a s p o s i t iv e i n ( 8 . 1 5 a ) , o n r e v e r s in g t h e b a t t e r y t e r m i n a l st h e e m f m u s t n o w b e r e c k o n e d a s n e g a t i v e i n ( 8 . 1 5 a ) . F o r a s i m p l e r e si st o r, t h eo n l y e m f i s i t s v o l t a g e d i f f e r e n c e A V , s o if A V t e n d s t o d r i v e c u r r e n t i n t h ep o s i t i v e d i r e c t i o n , I = A V~ R . F o r t h e b a t t e r y o f F i g u r e 8 . 7 ( b ) , ( 8 . 1 5 a ) g i v e s

a v a v ( c - a v )I1 -- 12 -- an d I -R I ' R2' rF o r F i g u r e 8 . 9 , ( 8 . 1 5 a ) g i v e s I = C / ( r + R ~ + R 2 + R3).C u r r e n t for, and voltage around, a circuit

In Fig ure 8.9 , let C = 6 V, r = 4 ~2, R1 = 1 ~2, R2 = 2 ~2, R3 = 3 ~2, an dV~ = 3 .4 V . F ind the cu r re nt I , an d 14, V~, ~ .Solution: r + R1 + Rz + R3= 10 f2. He nce I = ( 6 V ) / l O S 2 = 0 . 6 A. Since If lows from b to a , by Ohm's law G is higher than V~. Specif ical ly,G = V~ + I R1 = 3.4 + 0.6 = 4.0 V. Similarly, ~ = V~ - I R3 = 3.4 - 1.8 = 1.6 V,and Vc = ~ - I R2 = 1 .6 - 1 .2 = 0 .4 V . Al te rna t ive ly , for the emf ,I =[ E - A V ] / r =[ 6 - ( I 4 - V ~) ] / 4, so V ~ = I 4 - 6 + 4 1 = 4 . 0 - 6 . 0 + 2 . 4 =0 . 4 V .

A s s h o w n i n S e c t i o n 7 . 1 0 , f o r a n i d e a l b a t t e r y ( i n t e r n a l r e s i s t a n c e r - + 0 ) , t h et e r m i n a l v o l t a g e is t h e s a m e a s i ts c h e m i c a l e m f , t h e h i g h - v o l t a g e s i d e a s s o c i a t e dw i t h t h e l a r g e r p l a t e . T h e c u r r e n t t h r o u g h a n i d e a l b a t t e r y h a s n o e f f e c t o n i t sv o l t a g e s i n c e i t s l r v a l u e i s z e r o .

I t is conve nt iona l to assume tha t the conn ect ing wi res in a c i rcu i t have neg l ig ib le re-s is tance. Th is is cer ta in ly n ot l i te ra l ly t rue ; i t is very wron g i f we use many exten s ioncords in ser ies ( i .e. , to br ing power to a place far f rom one's house). In such cases, wireres is tance must be inc luded.

( b ) F o r a c a p a c i t o r , a p p l y V o l t a ' s l aw , w h i c h c a n b e w r i t t e n a sA V = ~ QC "

C i r c u i t s w i t h c a p a c i t o r s a r e d i s c u s s e d i n S e c t i o n 8 . 8 .( 8 . 1 5 b )


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3 5 0 Cha pter 8 a Com plex Ci rcu i ts

We often consider ideal batteries, whic h have neglig ib le internal resistance r. As show nin the previous chapter, the voltage difference A V associated wit h suc h an idealized

T T RFigure 8.10 Circu i t wi th idea lbat teries (having no internalresis tance). If the em fs ofthese ba t t e r ies a ren ' t t hesame, analysis of this circuitwil l lead to co ntradict ions.

battery eq uals its chemical emf E. H owever, theinternal resistance of a battery c annot always beneglected. C onsider tw o ideal batteries with differ-ent em fs, in paral le l with one another, as in Figure8 .1 0. T h e em f s g' l = 1 2 V a n d E 2 - 1 0 V w o u l dgive 12 V across the left arm and 10 V acrossthe m idd le arm. Th is wou ld contrad ict R u le 2 -the un iqueness of the voltage. Including either orboth of the internal resistances wo uld resolve th isproblem. Typically, batteries o f very differen t emfsare not connected in paral le l with one another:that w il l cause the higher em f battery to dischargerapidly, due to the low internal resistances.

W i t h o u t h a v i n g g iv e n a n e x p l ic i t s t a t e m e n t o f t h e m , w e u s e d K i r c h ho f f 'sru les in ou r ana lys i s o f res i s to rs in ser i es and para l l e l , a vo l t a i c ce l l in ser i esw i t h a r e s is t o r, an d a b a t t e r y o f v o l t a i c c e ll s i n se r ie s . T h e n ex t t w o s ec t i o n s w i l la n a l y z e s u c c e s s i v e l y m o r e c o m p l e x c i r c u i t s . T h e c h a p t e r c o n c l u d e s w i t h a s t u d yo f a c i r cu i t co n s i s t i n g o f a b a t t e ry , a r e s i s t o r , an d a c ap ac i t o r . A l a t e r ch ap t e rw i l l c o n s i d e r w h a t h a p p e n s w h e n t h e r e i s a n e l e c t r o m a g n e t i c a l l y i n d u c e d e m f( F a r a d a y ' s l a w ) , a s s o c i a t e d w i t h w h i c h i s a c i r c u i t e l e m e n t c a l l e d a n inductor .No t e : T h e n u m b e r o f u n k n o w n ( v ar ia b le s ) m u s t e q u a l t h e n u m b e r o f e q u a t i o n s( c o n s t r a in t s ) , o r e l s e t h e p r o b l e m is u n so l v a b le .

8~7~1

A p p l i c a t io n s o f K i r c h h o f f 's R u le sA c i r c u i t i n t h e l a b o r a t o r y c e r t a i n l y d o e s n o t l o o k l i k e t h e c l e a n s c h e m a t i c sp r e s e n t e d h e r e . I n s t e a d o f p e r f e c t l y s t r a i g h t w ir e s, r e a l c i r c u it s h a v e s l o p p i lyp l a c e d w i r e s t h a t r u n o v e r a n d u n d e r o n e a n o t h e r . I t c a n b e a d i f f i c u l t t a s k t od i s e n t a n g l e a n d t r a c e t h e i r c o n n e c t i o n s . B u t t h e s e c o n n e c t i o n s a r e p r e c i s e l y w h a ta r e n e e d e d t o a p p l y K i r c h h o f f ' s r u l e s . T h e d i s c u s s i o n t h a t f o l l o w s a s s u m e s w eh a v e a l r e a d y a n a l y z e d t h e c o n n e c t i o n s o f o u r c i r c u i t.T h e e x a m p l e s o f t h e p r e s e n t s e c t io n c o n s i d e r s o m e g e o m e t r i e s th a t c a n, i ns o m e s e n s e , b e t h o u g h t o f a s s e r i e s o r p a r a l l e l c i r c u i t s , e v e n t h o u g h t h e r e a r eres i s to rs a n d b a t t e r i e s i n t h e a r m s . J u s t as se r ie s r e si s to r s h a v e a c o m m o n c u r r e n t ,s o a s e ri e s o f b a t t e r i e s h a s a c o m m o n c u r r e n t ; for ser ies c i rcu i t prob lems , focus ing onthe comm on curren t he lps us to so lve them . J u s t a s p a r a l le l r e s is t or s h a v e a c o m m o nv o l t a g e d i f fe r e n c e , s o b a t t e r i e s i n p a r a ll e l h a v e a c o m m o n v o l t a g e d i f fe r e n c e ; fo rpara l l e l c i rcu i t prob lems , focus ing on the com mo n vo l tage d if f erence he lps us to so lvet h em . T h i s s e c t i o n w i l l c o n s i d e r b o t h t y p e s o f c ir c u it s.B a tt er i es i n S e r ie s ( C o m m o n C u r r e n t I )E a r l i e r i n t h e c h a p t e r , s y m m e t r y c o n s i d e r a t i o n s w e r e u s e d t o s t u d y t h e c a s e o fm a n y iden t i ca l v o l t a ic c e ll s i n se ri es . W h a t h a p p e n s w h e n t h e b a t t e r i e s a r e n o t


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8 . 7 A p p l i c a t i o n s o f K i r c h h o f f 's R u l e s 3 5 1

E2

81

tR

k

i +

(a) (b)Figure 8 .1 1 Two batteries in series" (a) connected to aload resistor, (b) batteries and their terminal voltages.iden t i ca l? C ons ide r an a rm o f a c i r cu i t w i th tw o no n iden t i ca l ba t t e r i e s , each w i thi ts ow n in te rnal res is tance, in ser ies . See Figure 8 .11 (a) , whic h appl ies Rule 0 a ndRule 1 fo r the cu r r en t , b y ind ica t ing the d i r ec t ion o f cu r r en t f low and tha t the r ei s a co m m on cu r r e n t I pas s ing th ro ug h each c i r cu i t e l eme n t . T ha t is, de f in ing 1 1 ,1 2 , and I a s the cu r r en t s th rou gh th e ba t t e r i e s and the r es i s to r R , Rule 1 g ives

I - 1 1 - 1 2 . ( 8 . 1 6 )Figure 8 .11 ( a) a ls o ind ica tes the tw o nodes + and - tha t can be us ed todef ine the vo l t age d i f f e r ence A V = V + - V _. A l tho ugh I and A V canno t bekno w n w i tho u t s pec i fy ing the r es t o f the c i r cu i t (w h ich he re i s the r es i s to r R ) ,a r e l a ti o n s h i p b e t w e e n t h e t w o c a n b e f o u n d j u s t b y a n a l y z in g t h e a r m w i t h t h eba t t e r i e s . To do th is , cons ide r F igu re 8 .11 (b ) . By O hm ' s l aw , w i th th e b a t t e ryemfs and the vo l t ages d r iv ing cu r r en t in oppos i t e d i r ec t ions ,

& - A V ~ 8 2 - / x V 211 - , 1 2 - , ( 8 . 1 7 )/'1 /'2w hich l eads to

A 1/1 - 81 - Ir~ , A 1 /2 - E2 - l r2 . (8.18)B y R u l e 2 - - t h e u n i q u e n e s s o f t h e v o l t a g e - - t h e n e t v o l ta g e a c ro ss t h e t w oba t t e r i e s i s the s um o f the ind iv idua l vo l t ages , s o

A V -- AV ~ + A 8 9 (8.19 )N ow , us e o f (8 .18 ) in (8 .19 ) l eads to

A V - (El q- E2) - I (r~ + r 2 ) , (8.20)so solving for I

I - ( , 5'~ + ~ 2 ) - - A V .r l + r 2 (8.21)


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35 2 C h a p t e r 8 9 C o m p l e x C i r c u i t s

T h u s , f or r eal bat t er i es in s eries, the e f f ec t ive em f E i s the su m of the emfs, a ndthe e f f ec t i ve r es i s tance r i s the same as the sum of the res i s tances . T h i s i s t r u e f o rm a n y b a t t e r i e s i n se ri es , n o t j u s t t w o ; i t is t r u e e v e n w h e n t h e a r m c o n t a i n s i d e a lb a t t e r i e s ( b a t t e r i e s w i t h z e r o i n t e r n a l r e s is t a n c e ) a n d r e s i st o r s ( b a t t e r i e s w i t hz e r o e m f ) .

E l i m i n a t i o n o f 11 a n d 12 i n f a v o r o f A I71 a n d A 1/2 b y e q u a t i n g t h e e x p r e s s i o n si n ( 8 .1 7 ) w o u l d h a v e l e d, w i t h ( 8 . 1 9 ), t o t w o s i m u l t a n e o u s e q u a t i o n s i n A II1a n d A V 2. U s i n g c u r r e n t a s t h e v a r i a b l e , t r e a t i n g f o u r o r f i ve b a t t e r i e s i n s e ri e s ise a s y ; u s i n g v o l t a g e a s t h e v a r i a b l e , i t w o u l d b e a w f u l .

Batteries in series II n F ig u r e 8 .1 1 ( b ) , l e t 8"1 = 1 2 V , r l - - 0 .0 l f 2, 8 2 = 8 V , r 2 - - 0 . 0 1 ~ , an dl e t 1 0 V b e a c r o s s a n u n k n o w n r e s i s t o r R . F i n d t h e c u r r e n t I a n d t h e r e s i s -t a n c e R .S o l u t i on : Ru le 0 t e ll s u s to es tab l i sh a s ign co n v en t io n f o r th e cu r r e n t an d v o l tag ed i f f e r en ce f o r each c i r cu i t e lem en t . Po s i t iv e v o l tag e d i f fe r en ces a r e t ak en as sh o w nin F ig u re 8 .1 1 ( b ) , an d p o s i t iv e c u r r en t i s t ak en as c i r cu la t in g c lo ck w ise a r o u n dth e c i r cu it , a s in F ig u r e 8 .1 1 ( a ). C h ar g e co n se r v a t io n , ex p r esse d b y Ru le 1 , i sau to m at ica l ly sa t i sf ied fo r F ig u re 8 .1 1 ( a ), b ecau se th e cu r r e n t in to each c i r cu i te lemen t ( tw o b a t te r ie s an d o n e r es i s to r ) eq u a ls th e cu r r en t o u t . Ru le 2 , th a tt h e v o l t a g e i s p a t h i n d e p e n d e n t , m e a n s t h a t t h e A V fo r t h e l e f t a rm , t o w h i c h( 8 .2 1 ) i s ap p r o p r ia te , i s th e same as A V fo r th e r ig h t a r m. To o b ta in th e cu r r en t ,(8 .21) y ields I = [ (& + C2) - A V ] / ( r ~ + r2) - - [ (12 + 8) - - 10] / (0 .01 + 0 .01) - -5 0 0 A. No w , f o r th e r es i s to r R, A V d r iv es ! so , acco r d in g to Oh m' s l aw ,

A VI = (8 .22)RThis g ives R = A V / I = 1 0 V /5 OOA = 0 .0 2 ~ .

~ Batteries in series IIN o w l e t A V b e u n k n o w n , a n d l e t t h e b a t t e r i e s b e c o n n e c t e d t o a k n o w nr es i s to r R = 0 .0 6 f 2, as in F ig u r e 8 .1 1 ( a ) . F in d I an d A V .S o l u t i on : Rules 0 , 1 , and 2 were already sat isf ied in the ear l ier d iscussion . All thati s l e f t i s ap p l ica t io n . Su b s t i tu t in g n u mer ica l v a lu es in to ( 8 .2 0 ) an d ( 8 .2 2 ) g iv es

A V = ( 1 2 + 8) - I ( 0 . 0 1 + 0 . 0 1 ) a n d A V = I ( 0 . 0 6 ) .E l i m i n a t i n g A V t h e n g i v es 2 0 - I ( 0 . 0 2 ) = I ( 0 . 0 6 ) , s o I = 2 0 / 0 . 0 8 = 2 5 0 A .Th en A V = I R = 2 5 0 ( 0 .0 6 ) = 1 5 V .

~ Batteries in series Ill---reversing a bat teryI n E x a m p l e 8 . 9 , l e t C ? b e r e v e r s e d , a n d r e p l a c e R b y a 1 0 V b a t t e r y o f n e g l i g i b l ei n t e r n a l r e s i s t an c e ( a n id e a l b a t t e r y ) . F i n d t h e c u r r e n t I .S o l u t i on : Rules 0 , 1 , and 2 were already sat isf ied in the ear l ier d iscussion . Wecan use (8 .21) i f we ch ang e the s ign of E2. This g ives I = (12 - 8 - 10 ) /0 .02 =- 3 0 0 A . T h e s ig n c h a n g e m e a n s t h a t t h e c u r r e n t a c t u a l l y fl o w s c o u n t e r c l o c k w i s e


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8.7 Applications of Kirchhoff's Rules 3 5 3

in Figure 8.11 (b). As long as we include the sign of the current, w e d on' t have toredraw the figure.

T h e s e e x a m p l e s y i e ld r a t h e r l a r ge c u r r en t s , c h a r a c t e r is t i c o f t h o s e p r o d u c e d o nstar t ing a car .

8~7~:2 Batter ies in Para l le l (Common Vol tage A V )S ec t io n 8 .3 .2 co n s id e r e d th e ca se o f m an y identical v o l t a i c c e l ls i n p a r a ll e l . W h a th ap p en s wh en th e ce l l s i n p a r a l l e l a r e n o t i d en t i c a l ? Co n s id e r an a r m o f a c i r -cu i t w i th two n o n id en t i ca l b a t t e r ie s , o f k n o wn em f s an d in t e r n a l r e s is t an ces, i np a r a l l e l w i th each o th e r . S ee F ig u r e 8 .1 2 ( a ) . As u su a l , o u r g o a l is t o r e l a t e I an dAV .

B y t h e p a t h i n d e p e n d e n c e o f t h e v o l t a g e ~ R u l e 2 ~ e a c h a r m h as t h e s a m evol tag e d ro p A V across i t, soA V -- AV1 - - AV2. (8 .23)

N e x t , b y O h m ' s l a w a p p l i e d t o e a c h b a t t e r y ,~ 1 - - A V ~ 2 - A V

I ~ - , 12 -- . (8.24 )/ ' 1 / ' 2T h e n o d a l r u l e ~ R u l e l ~ a p p l i e d t o ei t he r of t h e tw o c o m m o n n o d e s y ie ld s

I = Ia + 12. (8.25 )Us in g ( 8 .2 4 ) i n ( 8 .2 5 ) y i e ld s

( ~1 - - A V ) ( c ~ 2- - A V )l - la 4-12- +r l / ' 2 _ C 1 + ~ - A V ~ + ./ ' 2 / ' 1 (8.26)Elim inat io n of A V1 and A 89 in favor o f Ia an d 12 by e qu at in g th e expre ss ion sin ( 8 .2 4 ) wo u ld h av e l ed , w i th ( 8 .2 5 ) , t o two s im u l t an eo u s eq u a t io n s i n I a an d

l I1 1 I 2

6 1 t r l 6 2 t r 21 { i

O +

TVO ~

1 1r 1 8 t Rr2

(a) (b)Figure 8 .12 Two batteries in parallel" (a) with terminalvoltage A V; (b) co nne cted to a load resistor.


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35 4 C h a p t e r 8 ~ C o m p l e x C i rc u i ts

12. U s i n g v o l t a g e a s t h e v a r i a b l e , t r e a t i n g f o u r o r f iv e b a t t e r i e s i n p a r a l l e l i s e a s y ;u s i n g c u r r e n t a s t h e v a r i a b l e w o u l d b e a w f u l .

~ Batteries n parallel IC o n s i d e r t h e s a m e b a t t e r i e s a n d r e s i s t o r s a s i n E x a m p l e 8 . 9 : E l = 1 2 V , r l =0 .0 1 ~2, C2 = 8 V , r 2 - 0 .0 1 ~2. No w tak e A V = 1 0 V . ( Th is 1 0 V c an b ep r o d u c e d b y a 1 0 V e m f , o r b y a n e m f o f m o r e t h a n 1 0 V i n se r ie s w i t h ar e si s to r , b u t n o t b y a n y r e s i s to r a l o n e .) F i n d t h e c u r r e n t I t h a t f l o w s t h r o u g ht h e c i r c u i t. F i n d t h e c u r r e n t t h r o u g h e a c h b a t t e ry , a n d i n d i c a t e i f t h e y a r ec h a r g i n g o r d i s c h a r g i n g .S o l u t i o n : Th e an a ly s i s o f ( 8 .2 3) th r o u g h ( 8 .2 6 ) , w h ich sa t i s f ie s Ru les 0 , 1 , an d2 , ap p l ies h e r e . Eq u a t io n ( 8 .2 6 ) r eq u i r es th e q u an t i t i e s

g2 12 8El t - t - 2 0 0 0 A a n d 1 1 1 1- - t = ~ -4- = 20 0 ~ -1 .r l r2 0 .01 0 .01 r l r2 0 .01 0 .01U s i n g t h e s e i n (8 . 2 6 ) g i ve s I = 2 0 0 0 - 1 0 ( 2 0 0 ) - 0 , s o t h e r e i s z e r o n e t c ur -r en t f lo w . No te th a t I i = 1 2 0 0 - ( 1 0 /0 .0 1 ) = 2 0 0 A an d 12 = 8 0 0 - ( 1 0 /0 .0 1 ) =- 2 0 0 A ; b a t t e r y 1 i s d i s c h a rg i n g a n d b a t t e r y 2 is ch a r gi n g .

~ Batteries n parallel II n E x a m p l e 8 . 1 2 , l e t A V b e u n k n o w n , b u t d u e t o a k n o w n r e s i s t o r R -0 . 0 0 5 f a, a s i n F i g u r e 8 . 1 2 ( b ) . F i n d t h e c u r r e n t t h r o u g h t h e r e s i st o r , t h e v o l t -a g e a c r o s s t h e r e s i s t o r , a n d t h e c u r r e n t t h r o u g h e a c h b a t t e r y . V e r i f y c u r r e n tc o n s e r v a t i o n .S o l u t i o n : Th e an a ly s i s o f ( 8 .2 3) th r o u g h ( 8 .2 6 ) , w h ich sa t i s f ie s Ru les 0 , 1 , an d2 , ap p l ies h e r e . Su b s t i tu t in g n u mer ica l v a lu es in to ( 8 .2 6 ) an d ( 8 .2 2 ) g iv es

I - 2 0 0 0 - 2 0 0 A V a nd A VI - = 2 0 0 A V .0 . 0 0 5E li m in at in g I giv es 2 0 0 0 - - 2 0 0 A V = 20 0 A V , s o A V - - 2 0 0 0 / 4 0 0 - - 5 V .T h e n I = 5 / 0 . 0 0 5 = 1 0 0 0 A , 11 = 1 2 0 0 - ( 5 / 0 . 0 1 ) - 7 0 0 A , a n d 12 = 8 0 0 -( 5 /0 .0 1 ) = 30 0 A. No te th a t 11 + I 2 = I = 1 0 0 0 A; cu r r e n t i s in d ee d co n se r v ed .

F o r b a t t e r i e s i n p a ra l l e l , i t i s u s e f u l t o t h i n k o f e a c h b a t t e r y a s a c u r r e n t s o u r ce ,w h e r e b a t t e r y 1 i n ( 8 . 2 4 ) p r o v i d e s a s o u r c e c u r r e n t J 1 - & / r l , a n d s o o n . W i t ht h i s i n t e r p r e t a t i o n , t h e t w o b a t t e r i e s i n p a ra l l e l h a v e a n e ff e c t i v e s o u r c e c u r r e n ti f , a n d e f f e c t i v e r e s i s t a n c e 7 ~, w h e r e

1 1 )f f - J 1 + J 2 - - g ] + - - , - + ( 8 .2 7 )r-T r2 ~ ~2 "N o t e t h a t a l a r ge e m f w i t h a l a r ge r e s i s t a n c e c a n b e l e ss e f fe c t i v e a t p r o d u c i n gc u r r e n t t h a n a m o d e r a t e e m f w i t h a s m a l l r e si s ta n c e .


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8.7 Applications of Kirchhoff 's R ules 355

U s i n g ( 8 . 2 7 ) , r e w r i t e ( 8 . 2 6 ) a sA V1 - j - ~ . ( 8 . 2 8 )T4

T h u s , for rea l ba t ter ies in para l le l , the s t reng th J o f the e f fec t ive curren t source is thesu m o f the e f fec tive curren ts produ ced by each emf , an d i t s e f fec tive res i s tance T4 i s thes a m e a s t h e e m f r e si s ta n ces i n p a r a ll e l . T h i s i s t r u e f o r m a n y b a t t e r i e s i n p a r a l le l ,n o t j u s t t w o .

~ T h e jumper-cableproblemC o n s i d e r t w o b a t t e r ie s i n p a ra l le l , w i t h t h e n e t c u r r e n t I g o i n g th r o u g h t h er e s i s t an ce R , w h i c h r e p r e s e n t s t h e s t a r ti n g m o t o r o f t h e c a r w i t h t h e " b a d"b a t t e r y . F i g u r e s 8 .1 3 ( a ) an d 8 .1 3 ( b ) g i v e eq u i v a l en t c i r cu i t s t h a t c an b o t hr ep r e s en t t h i s p r o b l em . On e o f t h e b a t t e r i e s i s " g o o d" ( 1 ) an d o n e i s " b ad "( 2 ). T h e c a r w i t h t h e b a d b a t t e r y i s s t a r t e d b y u s in g j u m p e r c a b le s t o p u tt h e g o o d b a t t e r y i n p a r a l le l w i t h t h e b a d b a t t e r y . T h i s co n f i g u r a t i o n i s s i m i l a rt o t h a t o f E x a m p l e 8 . 1 3, b u t f o r c o m p u t a t i o n a l p u r p o s e s w e w i ll us e v a lu e sm o r e a p p r o p r i a t e t o t h e j u m p e r - c a b l e p r o b l e m .N o t e t h a t t h e l a s t j u m p e r - c a b l e c o n n e c t i o n s h o u l d b e t o th e c a r - b o d yg r o u n d , w h i c h is c o n n e c t e d t o t h e g r o u n d p o s t o f t h e b a t te r y . I f t h e r e is a s p a r ko n co n n e c t i o n , i t w i l l b e f a r fr o m t h e b a t t e r y , w h e r e o u t g as s i n g o f (ex p l o s i v e )H 2 c a n o c c u r w h e n t h e b a t t e r y b e c o m e s o v e r c h a r g e d o r u n d e r c h a r g e d .An ea r l i e r e s ti m a t e , f o r a g o o d ca r b a t t e r y o f em f & - 1 2 V , g av e 0 .0 1 s2f o r t h e i n t e r n a l r e s i s t an ce o f t h e b a t t e r y ( an d o f t h e s t a r t i n g m o t o r o f t h e ca r ),s o l e t r l - R = 0 .0 1 S2. T o m o d e l t h e b ad b a t t e r y , n o t e t h a t w h e n a b a t t e r yg o es b ad i t l o s e s s o m e em f , b u t m o r e i m p o r t an t i s t h a t i t s i n t e r n a l r e s i s t an ceg o es u p s i g n i f i c an t l y . ( Of co u r s e , i t l o s e s ch a r g e , b u t t h a t d o es n ' t ap p ea r i nour c i r c u i t equat io ns . ) T o be spec i f ic , t ake r2 - 1 .0 s2 . Also take ba t t e ry 2 toha ve on ly f ive good 2 V cel ls , so g2 - 10 V.A p p l y i n g ( 8 .2 7 ) t o t h i s p r o b l e m , J - 1 2 1 0 A a n d g - 1 _ 1 01 m h o s .[A m h o is a n ( o h m ) - 1 . ] T h e n ( 8 . 2 8 ) yi e ld s I - 1 2 1 0 - 1 0 1 A V , a n d ( 8 . 2 2 )y ie lds I - (A V /O .O 1) = 1 O0 A V . E l i m i n a t i o n o f I g i ve s 1 2 1 0 - 1 0 1 A V =1 0 0 A V , so A V - 1 2 1 0 / 2 0 1 - - 6 . 0 2 V ( a c c u r a t e t o t h r e e d e c i m a l pl a c e s) .T h e n ( 8 . 2 2 ) y i e l d s I = 6 . 0 2 / 0 . 0 1 = 6 0 2 A , a n d ( 8 . 2 4 ) y i e l d s 1 1 - 1 2 0 0 -6 . 0 2 / 0 . 0 1 - 5 9 8 A a n d 1 2 - 1 0 - 6 . 0 2 / 1 - 4 A . N o t e t h a t 11 + 12 I -6 0 2 A , s o t h a t cu r r en t i s co n s e r v ed .

( a ) ( b )Figure8.13 Two c i r cu i t s w i th th re e arms . They look d i f f er en t , bu t fo r R2 = 0they are iden t ica l f rom th e p o in t o f v iew of c i r cu i t ana lys is . They areequ iva len t to the c i r cu i t used when jumper cab les a r e used to s ta r t a car w i tha "dead" ba t te ry . In tha t case , the load r es i s to r R r epresen ts the r es i s tance o fthe s ta r t ing motor . ( a ) R in the m idd le a rm . (b ) R in the r igh t a rm.


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356 C h a p t e r 8 e C o m p l e x C i rc u i ts

B y t h e p r e v i o u s c h a p t e r, t h e b a t t e r ie s p r o v i d e p o w e r a t a r a t e 7 = g Iw he n I > 0 co r re spo nd s t o d i sch a rge . T hu s ~Pl = g l I1 = 717 6 W and ~P2 =g212 = 4 0 W , g i v i n g a n e t d i s c h a r g e r a t e o f 7 2 1 6 W . T h e r a t e s o f h e a t i n g o ft h e r e s i s t o r s a r e I21rl = 3 5 7 6 W , I~r? = 16 W , and 12 R = 362 4 W , so t he n e tr a t e o f h e a t i n g i s 7 2 1 6 W . In o n e s e c o n d , t h e b a t t e r i e s u s e 7 2 1 6 J o f c h e m i c a le n e r g y , a n d 7 2 1 6 J o f h e a t i s p r o d u c e d . E n e r g y , t o o , is c o n s e r v e d .Al t e rna t i v e ana l y s i s : W e c a n a ls o s o l v e fo r t h e c i r c u i t o f F ig u r e 8 . 1 3 ( a )b y u s i n g c u r r e n t c o n s e r v a t i o n a t t h e u p p e r n o d e (11 + 12 = I ) a n d r e q u i r -i n g z e r o v o l t a g e c h a n g e o n g o i n g a r o u n d e a c h l o o p . G o i n g c l o c k w i s e a r o u n dthe l e f t l oop be g inn ing i n t he l ow er - l e f t co rne r y i e lds 0 = g l - 11 r l - I R , o r0 = 12 - I1 (0 .0 1) - I (0 .01 ) . [H e re a vo l t ag e g l = 12 V is ga ine d on c ross ingthe e l ec t r ode s o f t he b a t t e ry , 11 r l = 11 (0 .01 ) i s l o s t on c ross ing t he i n t e rn a lr e s i s t a n c e r , a n d I R = I ( 0 . 0 1 ) i s l o s t o n c r o s s i n g t h e r e s i s t o r R . ] S i m i l ar l y,g o i n g c o u n t e r c l o c k w i s e a r o u n d t h e r i g h t l o o p g i v e s 0 = - I 2 ( R + r2) + g2 -I R , o r 0 = - 12 (0 .01 + 1 ) + 10 - I (0 .01 ) . Th i s y i e lds a t o t a l o f t h r ee equ a -t i o n s fo r th e t h r e e u n k n o w n s , a n d t h u s i s a w e l l - d e f i n e d p r o b l e m .

~ A mixed circuitN o w c o n s i d e r t h e c i r c u i t g i v e n i n F i g u r e 8 . 1 4 . H e r e , t h e r e s i s t o r s a r e g i v e na n d o n e o f t h e e m f s i s g i ve n . O u r g o al is t o f in d t h e u n k n o w n e m f a n d t h ec u r r e n t t h r o u g h e a c h r e si st or .B y R u le 1 - - c h a r g e c o n s e r v a t i o n B t h e c u r r e n t l r le a vi n g p o i n t a, p a s s i n gc o u n t e r c l o c k w i s e t h r o u g h t h e r i g h t ci r cu i t, a n d t h e n r e e n t e r i n g a f r o m a b o ve ,m u s t b e t h e s a m e t h r o u g h o u t t h e c i r c ui t. A p p l y i n g R u l e 1 a g ai n, n o c u r r e n tc a n e n t e r o r le a v e a f r o m t h e l e ft . A g a i n b y R u l e 1 , n o c u r r e n t e n t e r s o r le a v e sp o i n t b fr o m t h e r i g h t . H e n c e t h e s a m e c u r r e n t m u s t e n t e r b f r o m t h e l e f t a n dl e av e i t g o i ng u p w a r d . T h i s m e a n s t h a t t h e c i r c u it o n t h e r i g h t ( w h i c h i n c l u d e sp o i n t a ) a n d t h e c i r c u i t o n t h e l e f t ( i n v o l v i n g p o i n t b ) a r e i n d e p e n d e n t .C i r c u l a t i n g c l o c k w i s e a r o u n d t h e c i r c u i t o n t h e r i g h t g iv e s a v o l t a g e c h a n g eA V = 0 , b y R u l e 2 , s i n c e w e r e t u r n t o o u r s t a r t i n g p o i n t . B u t w i t h c u r r e n t Iri n t h i s c i rc u i t , t h is v o l t a g e d r o p m u s t b e 8 Ir . H e n c e / r = 0 ; t h e r e is n o c u r r e n tf l ow t h r o u g h t h e c i r c u i t o n t h e r i g h t .

16f~

2Ae 1 - 12 1~ 8f~ 2~

4 ~5 , ) / 2 a

3~ 2~Figure 8.14 Complex c i r cu i t t o i l l u s t r a t e t he use o f a l l o fKi rchh of f ' s ru l e s.


22/48

8 . 8 S h o r t- a n d L o n g - T i m e B e h a v i o r o f C a p a c i to r s 3 5 7

T o an a l y ze t h e c i r cu i t o n t h e l e f t ( w h i ch i n c l u d es p o i n t b ) , f ir s t n o t e t h a tthe vo l tage d rop across the 8 s2 r es i s to r i s A V8 = 2 (8 ) = 16 V , and th a t th i sm u s t b e t h e s am e a s ac r o ss t h e 1 6 S2 r e si s to r , s o 1 6 V = 1 6 1 1 6 . H e n ce t h ecu r r e n t t h r o u g h t h e 1 6 S2 r e s i s to r is 1 A , an d t h e t o t a l c u r r en t c i r cu l a t i n gco u n t e r c l o ck w i s e t h r o u g h t h e l e f t c i r cu i t is II = 2 + 1 = 3 A .B ecau s e t h e w i r e i n p a r a l le l w i t h t h e 5 s2 r e s i s t o r is t ak en t o h av e ze r o r e -s i st an ce, t h e f u ll 3 A cu r r en t f l ow s t h r o u g h t h e w i r e ( p a t h o f l e a s t re s i s t an ce ) .Th us I5 = 0 , and th e 5 S2 r es i s to r has no e f f ec t on th e c i r cu i t . For the 6 V ,6 ~2 b a t t e r y i n p a r a l l e l w i t h t h e 4 s2 r e si s to r , t h e co m m o n v o l t ag e i s

A ~ = 414 = 616 - 6 = 6(3 - 14) - 6 .(H ere R ule 1 says th a t /~ = 14 + /6 . ) So lv ing fo r 14, we ob ta in 14 = 1 .2 A,so A 89 = 414 = 4 .8 V across the 4 S2 r es i sto r . Th e n /6 = I t - 14 = 3 - 1 .2 =1 . 8 A .

C i r cu l a t i n g c l o ck w i s e a r o u n d t h e l e f t c i r cu i t b eg i n n i n g a t p o i n t b , t h esuccess ive vo l tag e increases ac ross the 3 , 1 , 8 , and 4 f2 r es i s tances sumt o 3 ( 3 ) + 3 ( ~ ) + 2 ( 8 ) + 4 . 8 = 3 2 . 8 v . T h is m u s t b e c o m p e n s a t e d b y a n e t3 2 .8 V v o l t ag e i n c r ea s e o n c i r cu l a t i n g co u n t e r c l o ck w i s e ac r o s s t h e e l ec t r o d e -e l ec t r o l y t e i n t e r f ace s o f E~ an d E . B ecau se , f r o m C h ap t e r 7 , t h e v o l t ag e d i f -f e r e n c e a c r os s a n e l e c t r o d e - e l e c t r o l y t e i n t e rf a c e e q u a l s t h e e m f fo r t h a t i n -t e rf a c e, t h e r e m u s t b e a n e t c o u n t e r c l o c k w i s e e m f o f 3 2 . 8 V f r o m E1 a n d E.S i n ce E~ p r o v i d es a 1 2 V c l o ck w i s e em f , g m u s t p r o v i d e a 3 2 .8 + 1 2 = 4 4 .8 Vc o u n t e r c l o c k w i s e e m f .

S oS S h o r t - a n d Lo n g - T im e B e h a v i o r o f Ca p a c ito rsC h a p t e r 6 c o n s i d e r e d c a p a c i t o rs t h a t w e r e c h a r g e d b y u n s p e c i f i e d m e a n s .C h a p t e r 7 s h o w e d t h a t a c ap a c it o r, w h e n c o n n e c t e d t o a b a t t e r y o f e m f g, w o u l dd e v e l o p a v o l t a g e d i ff e r e n c e A V = g . W e n o w c o n s i d e r h o w a b a t t e r y c h a r g e s ac a p a c i t o r i n r e a l c i r c u i t s t h a t c o n t a i n r e s i s t a n c e , a n d h o w a c a p a c i t o r d i s c h a r g e st h r o u g h a r e s is t o r. A c i r c u i t w i t h a r e s i s to r a n d a c a p a c i t o r i s c a l l e d a n R C c i r -cu i t . T h e p r e s e n t s e c t i o n c o n s i d e r s t h e c h a r g e a n d c u r r e n t f o r a n R C c i r c u it a ts h o r t t i m e s ( w h e r e w e s t a r t) a n d a t l o n g t i m e s ( w h e r e w e e n d ) . T h e n e x t s e c t io nc o n s i d e rs t h e d e t a il s o f t h e t i m e d e v e l o p m e n t f r o m s h o r t to l o n g ti m e s .

A l t h o u g h a c a p a c i t o r i n it ia l ly m a y b e u n c h a r g e d ( Q = 0 ) , f o r s h o r t t i m e sa f t e r a b a t t e r y is s w i t c h e d i n t o a c i r c u i t t h e c a p a c i t o r i m m e d i a t e l y c a n s t a r tt h e c h a r g i n g p ro c e s s. C h a r g e c a n e n t e r o n e p l a t e a n d l e a v e t h e o t h e r, y i e l d i n g an o n z e r o c u r r e n t ( I # 0 ) a s so c i a t e d w i t h t h e c a p a c i t o r.

A t l o n g ti m e s , i f t h e p o w e r s o u r c e i s s t e a d y a n d t h e r e a r e re s i s t o rs t o a b s o r be n e rg y , t h e s y s t e m w i l l c o m e t o a n e q u i l i b r i u m , a n d a n y p o i n t o f t h e c i r c u i tw i l l s e tt l e d o w n t o a c o n s t a n t v o l t a g e . T h u s , t h e v o l t a g e A V a c r o ss a n y c i r c u i te l e m e n t w i l l b e a c o n s t a n t . F o r a r e s i s t o r R , t h i s m e a n s t h a t t h e c u r r e n t l -A V ~ R b e c o m e s a c o n s t a n t . F o r a c a p a c it o r, t h is m e a n s t h a t t h e c h a r g e Q = C A Vb e c o m e s a c o n s t a n t , s o d Q / d t ~ O . H e n c e , s i n c e I = d Q / d t , a t lo n g t i m e s t h e r ei s n o c u r r e n t t o o r f r o m a c a p a c i t o r . T h a t i s ,i:i:iiiii:ii~iiiiiiijiiii:~:,,,,,,,,,,,,,,,,,,,,,~,,,,,,,~,~,,,,,~,,,,:,,,,,,,,,,,,,,,,,,,,,i:Jiliii!i~:iii!!~iiiiiiiiiiii:iiiii~ii!i~ii:~:i:i~i:iii:iiiiii:iii:iii:~:ii:iiiiiiil~.,~,,,,,,,~"~;: .,,,,,,~,~,~,,,,0,,,,,,,,,~,,,,,,,,,,,,,,,,,,,,,~o ~g ~m e,,,B e,N ,a~o ~iiiiiiiii!iiiiiiii iiiiiiiiiii!iiii~iiiiiii~i~2~i~iiiii~i~iii~i9~ili9iiii~iii~ii~iii~i;i!ii!~iY~J~iiii~iiilii~i~iiii~i;iliii~iiisliiiiiiii:!i~ii!ii~i;i:iiiiii~i~i~ii~i~iii~Jiii~ii!iiiiiiii;ii~i~i~ili~ijiiiiiii~iil;iii~:ii~i!ii~iliN i!;i!ii~~i~i ~i!~i~i ~!i~i~i ~;~ p ~ e ~ o r~~iii~iii~i~i~i~iii~iii!ii!i! i : : ; : iN i~ : i : i iN :TN~ r r e ~ ] i~i':ii'i"~ii'i~iiiiK:iii8:~ii!i~i~i~i=~i~i~i~il!i~iii~i"i~'~ii:!i~:il:

L e t u s s e e h o w t h is w o r k s f o r a f e w e x a m p l e s .


23/48

35 8 C h ap t e r 8 ~ C o m p l ex C i r cu i t s

ER

- Q [QICFigure 8.1 5 A n R Cci r cu i t w i th an idea le m f g .

C o n s i d e r a n R C c i r c u it w i t h a r e si s to r R , a c a p a c i t o rC , a n d a n i d e a l b a t t e r y g w i t h z e r o i n t e r n a l r e s i s t a n c e .See F i g u r e 8 .1 5 .

T h e o n l y e m f a c t in g o n t h e r e s is t o r is t h e v o l t a g eA V R = V a - 8 9 a c r o s s i ts e n d s, w h i c h b y t h e p a t h i n d e -p e n d e n c e o f t h e v o l t a g e is t h e s u m o f t h e v o l t a g e s a c r o sst h e b a t t e r y a n d a c r o s s t h e c a p a c i t o r . B e c a u s e t h e b a t -t e r y i s t a k e n t o h a v e z e r o r e s i s ta n c e , i t s t e r m i n a l v o l t a g eA V r is g , t e n d i n g t o d r i v e t h e c u r r e n t o n e w a y , a n d t h ec a p a c i t o r v o l t a g e

Qzx vc- V c- V a- ( 8 . 3 0 )Ct e n d s t o d r i v e t h e c u r r e n t t h e o t h e r w a y . T h u s ( 8 . 1 5 a ) a n d ( 8 . 3 0 ) y i e ld

d Q = A VR = V a - Vb = ( V a - V c ) + ( V c - V b)I d t R R R- A v c + z x v r - Q / C + c= = . ( 8 . 3 1 )R R

I n it ia l ly , t h e c a p a c i t o r i s u n c h a r g e d , s o Q 0 - O, b u t i t h a s a n i n i ti a l c u r r e n t ,g i v e n b y Io - g~ R. A f t e r a l o n g t i m e , t h e c a p a c i t o r h a s c h a r

physics book about batteries, kirchhoff's rules, and complex circuits

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