The Biot-Savart Law and Ampère's Law, physics vol

7/28/2019 The Biot-Savart Law and Ampre's Law, physics vol

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"The word "e lec t romagnet ic" [has been] used to characteri ze the phen om ena produced bythe conduc ting wires of the vol taic pi le. I have determ ined to use the wo rd "electrodynamic"in order to un i te unde r a com mo n n am e a l l o f these phen om ena [o f the sor t tha t Oers teddiscovered], and part icularly to designate those which I have observed between voltaicconductors ."

- -And re Mar ie Ampere (1820)"These a tt rac t ions and repu ls ions be tween e lec t ric curren ts d i f fer fund am enta l ly f rom theef fects prod uced by electr ici ty in repose." - -And re Mar ie Ampere (1822)

C h a p t e r 1 1How E lec t r ic Cur ren t s MakeM a g n e t ic F ie ld s : T h e B i o t -Sa var t Law and Am p re 's LawC h a p t e r O v e r v i e wThere are three ma jor par ts to th is chapter : the Biot-Sav ar t Law , Am p~re 's Law , andappl icat ions o f these laws to supercon ducto rs and e lectroma gnets. Sect ion 1 1.1 g ivesa br ief in troduct ion to th is chapter and a br ie f h istory of the d iscovery of how elec-tr ic currents make magnet ic f ie lds. The f i rst par t begins wi th Sect ion 11.2, whichstates the Biot-Savar t law, g iving the magnet ic f ie ld due to any current-carryingcircui t (som ewh at l ike C oulom b's law for the e lectr ic f ie ld due to e lectr ic charge) .Sect ion 11.3 der ives the B iot-Sava r t law using Ampere 's equivalence. Sect ion 11.4shows ho w to use the Biot-Sav ar t la w. Sect ion 1 1.5 appl ies i t , using the pr incip le ofsuperposi t ion. Sect ion 11.6 f inds forces due to these magnet ic f ie lds der ived fromthe B io t -Savar t law . Th is leads to the de f in i t ion o f the a mp e r e in terms of the forcebetween two paral le l , current-carrying wires.The second p ar t begins wi th Sect ion 1 1.7, wh ich states Am pere 's law . Sect ion 1 1.8der ives Am pere 's law using Amp ere 's equivalence, w hich re lates the ci rcu lat ion arou ndan a rb i t ra ry c losed c i r cu it and w hatever cu r ren t m ay pass perpend icu la rly th rou ghi t. Su ch an arb i t rary clo sed ci rcu i t is ca l led an Amp~rian circui t. Th is re lat ionship issom ew hat l ike Gauss's law re lat ing e lectr ic flux to the charge enclosed by an arb i t raryclosed surface, called a Gaussian surface. Sect ion 1 1 .9 shows how Amp~ re 's law can beused to make a non invas ive measurem ent o f the c ur ren t pass ing th rough an Amper iancircui t. S ect ion 1 1.10 obta ins the ma gnet ic f ie lds of very sym metr ica l c urrent sourcesby app l ica t ion o f Am p~re 's law .The th i rd par t is opt ion al . Sect ion 1 1.1 1 d iscusses f ie ld expulsion by per fect d iamag-nets (which are supercon ductors) . Here, sur face currents make a magn et ic f ie ld that,w i th in the d iam agnet , cance ls any exte rna l ly app l ied m agnet ic f ie ld . Sec t ion 11 .12

4 6 0


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11.1 Introduction 46 1

discusses sof t ma gnets, and h ow th ey intensi fy appl ied ma gnet ic f ie lds with theirAm p~r ian currents. Sect ion 11.13 discusses tw o e xper iments establ ishing that theelect ric currents associated w ith perfect diamagnet ism f low over mac roscop ic paths,whereas those assoc ia ted wi th magnets f low over mic roscopic cur rent paths . In bothcases , the cur rents d o n ot de cay wi th t ime, so they m ust f low w i tho ut res is tance.Sect ion 11.14 discusses elect roma gnets, wh ich are used to p roduce very large mag-net ic f ie lds. Superconductors are also used to produce ve ry large ma gnet ic f ie lds, w

11oi I n t r o d u c t i o nThi s i s t he t h i rd a nd f i nal c ha p t e r on s t a ti c ma g ne t i c f ie lds. C ha p t e r 9 s t ud i e dma g ne t i c ma t e r i a l s a nd use d a n a na l ogy t o e l e c t ros t a t ic s t o s t udy t he p rope r t i e s o fm a g n e t s . C h a p t e r 1 0 s h o w e d h o w A m p e r e ' s e q u i v a l e n c e b e t w e e n m a g n e t s a n dc ur re n t l oops y i e l ds t he fo rc e a nd t he t o rque on c ur re n t -c a r ry i ng wi re s i n ma g-ne t i c f i e l ds . The p re se n t c ha p t e r d i sc usse s how e l e c t r i c c ur re n t s ma ke ma gne t i cf i e l ds : t he B i o t -Sa va r t l a w (pronounc e d bee-oh-suh-var ) a n d A m p 6 r e ' s l a w . A p -p l i c at i ons run f rom t he m i c rosc op i c ( t he m a gne t i c f ie l d p ro duc e d a t t he a t om i cnuc l e us , due t o e l e c t rons wi t h i n t he a t om) t o t he ma c rosc op i c ( t he e a r t h ' s ma g-ne t i c f i e l d , due t o e l e c t r i c c ur re n t s de e p wi t h i n t he e a r t h ) . More i mpor t a n t fo rda i ly l if e , t he se l a ws a l so ha ve be e n use d o n t he hu m a n sc al e t o de s i gn mo -t o rs , e l e c troma gn e t s , w r i t e -he a ds fo r ma g ne t i c me m ory , a nd a va s t a r ray o f o t he rde v ic e s. In t he l a te 1800s , e le c t ri c a l e ng i ne e r ing d e pa r t m e nt s we re fou nde d t ot e a c h s t ud e n t s h ow t o de ve l op a nd e xp l o i t suc h a pp l i c a ti ons . B e c a use t he re i s somuc h ma t e r i a l , we orga n i z e i t i n t o t h re e pa r t s .Some H is to ryAra go , whi l e t r ave l ing , l e a rne d of Oe rs t e d ' s w ork . O n re t u rn i ng t o F ra nc e, hed e s c r i b e d O e r s t e d ' s e x p e r i m e n t s o n t h e i n t e r a c t i o n b e t w e e n a m a g n e t a n d ac ur re n t -c a r ry i ng wi re t o t he Ac a de mi e de s Sc i e nc e s on Se p t e mbe r 11 , 1820 .From t h i s , Ampe re de duc e d t ha t t wo c ur re n t -c a r ry i ng wi re s woul d a l so i n t e ra c t .W i t h i n a we e k , h e ha d c o nf i rme d t h i s r e su l t , showi ng t ha t t wo pa ra ll e l wi re sc a r ry ing c ur re n t i n t he sa me (oppos i t e ) d i r e c t i on a t t r a c t ( r e pe l) O n O c t ob e r 30 ,B i o t a nd Sa va r t p re se n t e d t h e r e su l t s o f t he i r e xp e r i me n t s on t he i n t e ra c t i on o f am a g n e t a n d a l o n g s tr a ig h t w i re . T h e y s h o w e d t h a t t h e d i r e ct io n o f t h e m a g n e t i cf ie l d o f a l ong s t r a i gh t wi re i s a s g i ve n by wh a t w e ha ve c a l le d O e rs t e d ' s l a w, a ndt he y show e d t ha t t he f ie l d fa ll s o f f wi t h r a d ia l d i s t a nc e r f rom t he wi re a s r -1 .T h e y w e n t o n t o s t u d y t h e m a g n e t ic f i el d p r o d u c e d w h e n t h e w i r e w a s b e n t i n t otw o sem i inf ini te s t ra ight sec tions mak ing an angle of 2~ (2~ - ; r be ing a s t ra ightwi re ) . Am pe r e po i n t e d ou t t w o e r ro r s in t he i r i n i ti a l t he ore t i c a l a na lys is o f t he i re xpe r i me nt a l r e su l t s . Howe ve r , t he ma t he ma t i c a l phys i c i s t La p l a c e , us i ng t he i rda t a , de duc e d t h e c o r re c t fo rm o f t he l a w for t he f i e ld p rod uc e d by a ny se c t iono f t h e w i re ; t h is i s w h a t w e k n o w a s t h e B i o t - S a v a r t la w . A m p e r e ' s w o r k , b o t he xpe r i me n t a l a nd t he ore t i c a l, c ons i s t e d o f s t ud i es o f t he i n t e ra c t i on be t we e n t w oc ur re n t -c a r ry i ng w i re s , a nd l e d t o t he A mp 6re fo rc e la w of t he p re v i ous c ha p t e r .Wha t i s c a l l e d Ampe re ' s l a w a c t ua l l y ma y be due t o Ma xwe l l ( i n t he 1860s ) ,r a t h e r t h a n A m p e r e ( in t h e 1 8 2 0 s ).


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4 6 2 Cha pter l l " The Bio t -Savar t Law and Am pere ' s Law

Observer /Sourcer

I/Origin

Figure 11 .1 Geomet ry for theBiot-Savart law giving the mag neticfield B at F due to a current-carryingwire, with vector length eleme nts dY atF '. He re /~ = F - F ' is the vectorpoint ing from the source at F ' to theobserver at ~.

1 1 . 21 1 , 2 , 1

M a g n e t i c F i e l d o f a C u r r e n t - C a r r y i n g W i r eS t a te m e n t o f th e B i o t -S a v ar t L a wT h e B i o t - S a v a r t l a w t e ll s us h o w t o c o m p u t e t h e m a g n e t i c f ie ld B a t t h e o b s e r -va t i on po i n t F due t o an e l ec t r ic c i r cu i t - ca r r y i ng e l ec t r i c cu r re n t I ( i. e ., a cu r r en tl oop ) . F i r s t s om e de f i n i t ions . A s u s ua l , R = F - Y ' is t he vec t o r po i n t i ng f romt h e s o u r c e a t F ' t o t h e o b s e r v e r a t F . D e f i n e R - I/ ~1 s o t h a t / ~ - / ~ / R . T h e c u r r e n tl o o p c o n s is t s o f s e g m e n t s o f l e n g t h d s t h a t p o i n t a l o n g ~ d e f i n e d b y t h e l o ca l d i-r e c t i o n o f t h e e l e c t r i c c u r r e n t l . T h u s t h e v e c t o r l e n g t h e l e m e n t i s d s = ~ d s ,w h e r e d s = Ids' l > 0 . See F igure 11 .1 .T h e B i o t - S a v a r t l a w s t a t e s t h a t

- f : . - . _ d s . l d gB = d B , d B = k m I . . . . . . R 2 .. = le ,, , R3:::~:::k i n - 1 0 - T T- m i l i i ~A "

( T h e s e c o n d f o r m f o r d / 3 f o l l o w s f r o m / ~ / R 2 = / ~ / R 3 . ) E x c e p t f o r t h e v e c t o rc r o s s - p r o d u c t d~" x / ~ , ( 1 1 . 1 ) is n o t m u c h m o r e c o m p l i c a t e d t h a n t h e e x p r e s s i o nfo r t he e l ec t r i c f i e l d due t o e l ec t r i c cha rges d i s t r i bu t ed a l ong a l i ne . Never t he l e s s ,b e c a u s e o f t h e c r o s s - p r o d u c t , i t is w o r t h e x a m i n i n g i n m o r e d e t a il .

N u m e r i c a l A n a l y s i sI n p a r t b e c a u s e o f t h e v e c t o r c r o s s - p r o d u c t , s t u d e n t s o f t e n f in d th e B i o t -S avar t l aw t o be co m pl i ca t ed and d i f f i cu l t t o app l y . In t he s p i r i t o f "kno wy o u r e n e m y , " i t i s u s e f u l t o a n a l y z e t h e B i o t - S a v a r t l a w f r o m t h e v i e w p o i n to f a s p r e a d s h e e t a n al ys is , im a g i n i n g h o w y o u w o u l d c a l c u l a te B i f y o u h a ds o m e s o u r c e - m e a s u r i n g e lv e s w h o w o u l d a p p r o x i m a t e i t b y m a n y t i n y e le -m e n t s o f l e n gt h . T h e e lv e s w o u l d t h e n d e t e r m i n e t h e m i d p o i n t F ' o f e a c hs o u r c e e l e m e n t , a n d t h e v e c t o r d~" a l o n g w h i c h t h e c u r r e n t f l o w s f o r t h a t e l-e m e n t . I n a d d i t i o n , t h e y w o u l d m e a s u r e t h e o b s e r v a t i o n p o s i t i o n F . A l t e r -n a t i v e l y , t h e y c o u l d m e a s u r e t h e f o u r q u a n t i t i e s d s and ~ , and the n d~" = ~ d sc o u l d b e c o m p u t e d w i t h i n t h e s p r e a d s h e e t . H o w e v e r , w e h a v e c h o s e n t o m a k ei t eas i e r on t hem , s i nce d~" i nvo l ves on l y t h ree quan t i t i e s . To be s pec i f ic , l e t t hei n d e x i o n t h e e l e m e n t s g o f r o m 1 t o 2 8 . S e e F i g u r e 1 1 . 2 .


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11.2 M agnet ic Field of a Current-Carrying Wire 463

I n w h a t f o ll o w s , r e f e r t o T a b l e 1 1 . 1 , w h i c h s h o w s t h e f i rs t t w o r o w s o f as p r e a d s h e e t c a l cu l a t io n . T h e f ir st c o l u m n ( u n l a b e l e d ) w o u l d c o n t a i n t h e e n t r ie s

17 16 15 14-xs"v'e--l'w-l~ So ur ceO bserver 1 ~ " "" ~ 8 /\ 19~" Rv = r~ - t : 'v ~- / .T ~ = 2 ~ ' ' _ j ~ ~ 6 7 = 7 " a s .~ ds7

/ 21__~ ~ r' 7 Tsu r i g i n . . . . -

Figure 1 1 . 2 Disc re t ized ve r s ion of F igurel l . l , w i th c i r cu i t b roken up in to 28 e lem ents .

1 t o 2 8 , o n e f o r e a c h s o u r c e e l e -m e n t . C o l u m n s A t o C c o n ta i n th et h r e e c o m p o n e n t s o f t h e s o u r c ep o s i t i o n v e c t o rs ~ '. C o l u m n s D t oF c o n t a in t h e t h r e e c o m p o n e n t s o ft h e s o u r c e e l e m e n t v e c to r d s As e p a r a te p a r t o f t h e s p r e a d s h e e ts t o r e s t h e t h r e e c o m p o n e n t s o ft h e o b s e r v e r v e c t o r ~ , a s w e l l a s kma n d l . O n c e t h e i n p u t e n t r ie s h a v eb e e n m a d e , t h e c o m p u t a t i o n sb e g i n .

C o l u m n s G to I c o n t a in th e c o m p o n e n t s o f t h e v e c t o r / ~ - ~ - ~ ' f r o m th es o u r c e p o i n t t o t h e o b s e r v a t i o n p o i n t . F o r e x a m p l e , R x - r x - r ' x . C o l u m n Jc o n t a i n s t h e l e n g t h R - I/~ l, o b t a i n e d b y t h e P y t h a g o r e a n t h e o r e m w i t h t h e e n -t r ie s in c o l u m n s G t o I. C o l u m n s K t o M c o n t a i n t h e c o m p o n e n t s o f d s x / ~ .F i n a l l y , c o l u m n s N t o P c o n t a i n t h e c o m p o n e n t s o f d B - k m I (d s x R ) / R 3.T h u s , s u m m i n g c o l u m n N y i el ds B x; s i m i l a r l y , c o l u m n O y i e l d s B y, a n d c o l u m nP y ie lds B~ .

F ro m a co m p u t a t i o n a l p o i n t o f v i e w , i t is p re f e ra b l e t o wo r k w i t h d / ~ = k r n l ( d ~ x R ) / R 3 ,r a t h e r th a n d / ~ = k m l (d _~ x / ~ ) / R 2 , s i n ce t h e n w e d o n o t n e e d t o h a ve se p a ra t e co l u m n sf o r / ~ a n d d _~ x / ~ . Ho we ve r , f o r a n a l y t i c wo r k t h e i n ve rse sq u a re fo rm is o f t e n p re f e ra b l e .No te tha t ~ ' ~ 'i + 1 - r i - t- d .~ - conn ects cons ecut i ve en t r ies o f F ' , so the re la t ionsh ip d F i + 1 -r i + l - r i = d -~i ho lds .

~ Calculation of dB- " ( 7 , 2 , 0 ) m ,n the p reced ing c i r cu i t , l e t I 4 A, 7 =_- ( - 0 . 5 , 2 .2 ,0 ) m , r 7 =

a n d d s -~ ( - 0 . 2 , 0 . 4 , 0 ) m . D e t e r m i n e dB T .Solut ion: R7 = F - r 7 = ( - 7 . 5 , 0 . 2 , 0 ) m , f r o m w h i c h [ R7 ] = 7 .5 0 3 m a n dds x/~7 - (0,0,2 .96) m 2. He nce

kmlds 0_9d/37 = = 2.80 x 1 Tie. (11.2 )Colu mn s N to P in the s eventh row success ively r ece ive the three en t r ies cor re -sponding to (dBx, dBy, dBz) - (0, 0, 2.80 x 10 -9) T.

Table 1 1 . 1 Spreadshee t ca lcu la tion o f d/3i i i ! i i i ! i i i i i i i ! i l i ! g i i i i i i i ! i i i i i i il i ! ! i ! i i l i il i i i i l l i i i l i i i i i i ! i i i i i !i:iiiiiiiiiii:i:i:: il:iiiiiiiiii~ii~iiiiiiii~iii~ii~iii~i!ii~iiiiii~!iii!i!i~!i~iii::i~i~!::i!i:iii!::~i~i;i~iii~ii:~iii!iiiiiiiiiiiiiii~!iiiiiiiii~iiii~iii~iiiii~iiiiiiiiiii!iiiiiiii~iii~iiiii~!i~iiiiiiiiiii~iiiiiJiiiii!i~iiiiiiiiiiiiiii

1 rx ry r" dsx dsy dsz Rx Ry Rz I / ~ 1 ( d s ( d s ( d s d Bx d By d Bz


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4 6 4 Chapter 11 ~ The Biot-Savart Law and Amp~re's Law

T h i s p r o c e d u r e is c o m p l i c a t e d b u t c o m p l e t e l y w e l l d e f in e d , a n d i t w o r k s e v enw h e n t h e c i r c u it is s o c o m p l e x t h a t a n a ly t ic m e t h o d s f ai l. O n c e t h e s o u r c e e n t ri e sa r e made in co lumns A to F , and the obs e rve r en t r i e s a r e made e l s ew here , a l lt h e o t h e r r e s u l ts c a n b e o b t a i n e d c o m p l e t e l y a u t o m a t i c a l l y b y t h e s p r e a d s h e et .I f t he o bs e rva t ion po in t i s changed , on ly the o bs e rve r pos i t ion need be c hanged .S ec t ions 11 .4 and 11 .5 p r es en t s ome s i tua t ions tha t do no t r equ i r e numer ica lme thods . F i rs t, how ever , w e de r ive the B io t -S av ar t law us ing A m pS re ' s equ iva -lence.

1 1 . 3 D e r i v a t i o n o f B i o t - S a v a r t L a w : F i el do f C u r r e n t -C a r r y i n g W i r eTo f ind the f i e ld B a t the po s i t ion F , d u e to a cu r r en t loop I, w e f i r s t u s e thede f in i t ion o f B in t e rms o f the fo r ce F q m o n a m a g n e t i c p o l e q m placed a t F . SeeFigure 11 .3 .

T h u sFqr,, = qmB. (11.3)

N ow no te tha t , by ac t ion and r eac t ion , the fo r ce F I on a c i r cu i t due to them o n o p o l e q m is e q u a l a n d o p p o s i t e Fq,:Fq

C o m b i n i n g ( 1 1 . 3 ) a n d ( 1 1 . 4 ) y i e l d s(11.4)

~ - F q m - '- - - - F I . (1 1.5)qm qmH ence , f rom the fo r ce F I on the cu r r e n t loop , w e can deduc e B . F or exam ple ,i f qm - 2 A - m a n d t F I I - 0 . 0 8 N , t h e n 1 / 3 1 - 0 . 0 4 T . N o w c o n s i d e r t h e g e n e ra lcase.F rom A mp~ re ' s f o r ce l aw , (10 .20 ) , t he fo r ce F I on the cu r r en t - ca r ry ing loopin the f ie ld B qm d u e t o t h e m o n o p o l e i s

FI - f d .F - l f d s ' x.Bqm. (11.6)

N o t e t h a t th e v e c t o r - / ~ - - ( ~ - F ' ) p o in t s fr o m F t o ~ ' . T h e n ( 9 .4 ) g iv e s t h e

qm ~ SourceN = ! ; - - -;; / ? ds; T

Origin

Figure 11.3 Geom etry for proof of theBiot-Savart law. Here a long magnet,with nearby pole q,z, is acted on by thecurrent loop, and vice versa.


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11.4 Applications of the Biot-Savart Law 465

f ield Bqm a t pos i t i on F, d u e t o qm a t F ', as

B qm k m q m ( - [ ~ )R 2 ( R - F - F ' ) (11.7)P l a c ing ( 11 .7 ) i n to ( 11 .6 ) y i e lds

-~ f km qm (-[-{ )Fi - I ds x R2 , (11.8)Fina ll y , us ing ( 11 .8 ) i n ( 11 .5 ) g ive s t he B io t - Sa v a r t l a w f o r t he f i e ld B a t F duet o t h e c u r r e n t l o o p l "

f3 - k m I f d~ 'x R f d s x R -. , )R2 = k m I R-----T-. (R - ~ - F (11.9)For su r f a c e c ur r e n t s , we use f ( d A r a t h e r t h a n I d s ( wh e r e /7 ( i nc lude s t hed i r e c t i o n o f t h e s u r fa c e c u r r e n t K ) , a n d w e i n t e g r a t e o v e r t h e e l e m e n t o f a r e ad A . Sim i l a r ly , f o r vo lum e c ur r e n t s , we m us t use J d V r a t h e r t h a n Ida , a n d w em u s t i n t e g ra t e o v e r th e e l e m e n t o f v o l u m e d V .

11! 1 , 4 ~ !

A p p l ic a t i o n s o f t h e B i o t - S a v a r t L a wC i r c u i t i n P l a n e, O b s e r v e r i n P l a n eF i g u r e s 1 1 . 1 t h r o u g h 1 1 . 3 d e p i c t a c i r c u i t i n a p l a n e a n d a n o b s e r v e r i n t h a tsa m e p l a ne . A t t he obse r ve r , t he f i e ld i s pe r pe nd ic u l a r t o t he p l a ne . T h i s f o l l owsi m m e d i a t e l y b e c a u s e , w h e n d ~ a n d / ~ a r e i n t h e s a m e p la n e , t h e i r c r o s s - p r o d u c td s x / ~ i s p e r p e n d i c u l a r t o t h e p l a n e . I n o u r e a r l i e r n u m e r i c a l c a l c u l a t i o n o f( 11 .2 ) , t h e f i e ld wa s a long ~ . T h i s s im pl i f ie s t he p r ob l e m of f i nd ing B be c a u sew e o n l y h a v e t o a d d u p t h e d B z c o l u m n ( P ) w i t h n o n z e r o e n t r i e s . T h e B x a n dB y c o l u m n s s u m t o z e r o b e c a u s e e a c h d B x a n d d B y is zero.F i e l d at C e n t e r o f C i rc u l ar C u r r e n t L o o p

Y

Observer

gFigure 11.4 Ge om e t r y f o rmagn etic f ield at the centerof a circular curren t loop ofradius a.

C i r c u l a r c u r r e n t l o o p s a r e a n i m p o r t a n t a n dc o m m o n l y e n c o u n t e r e d g e o m e t r y . C o n s i d e r aloop o f r a d ius a i n t he y z - p l a n e , c e n t e r e d a t t h eor ig in , w i th i t s nor m a l a long ) ? , whic h we t a ket o b e o u t o f t h e p a g e. L e t t h e o b s e r v e r b e a tt he c e n t e r o f t he l oop ; t h i s i s a spe c i fi c e xa m pleo f t h e p r e v i o u s g e n e r a l c as e o f t h e o b s e r v e rin t he p l a ne o f t he c i r c u it . Se e F igur e 11 .4 . W i tht h e c u r r e n t c o u n t e r c l o c k w i s e a s v i e w e d f r o mthe pos i t i ve x - a x i s , ds x / ~ po in t s a long ) ? , a ndI d~ x / ~ l - [ d~ ll / ~ l] s i n90 ~ - ( ds ) ( 1 ) ( 1 ) - d s .


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4 6 6 C h a p t e r 11 m Th e B i o t-S a v a r t La w a n d A m p ~ re ' s La w

B y ( 1 1 . 1 ) w i t h R - a , a l l s o u rc e d s p r o d u c e d B x = k m l d s / a 2. W i t h f d s -2 t e a , t h i s i n t e g r a t e s t o2rr km IB~ = ~ . ( fi el d a t c e n t e r o f c i rc u la r l o op ) ( 11 .1 0)

I f I - 2 A a n d a - 2 c m - . 0 2 m , t h e n (1 1 . 10 ) g iv es B x - 6 .2 8 x 1 0 - s T ,a p p r o x i m a t e l y t h e s t r e n g t h o f t h e e a r t h ' s f ie ld . T e n tu r n s o f w i r e ( al l w o u n di n t h e s a m e d i r e c t i o n ) w o u l d g i v e t e n t i m e s t h i s f ie l d.

~ Field on axis of c ircular current loopC o n s i d e r t h e s a m e l o o p a s i n F i g u re 1 1 .4 , b u t l e t t h e o b s e rv e r b e a n y w h e reo n t h e a x is ( x ) o f t h e l o o p . S e e F i g u re 1 1 .5. Th e c u r r e n t i s c o u n t e r c l o c k -

Z~ y ~ I Observer- {a L l x . . . . . .

N ear side p ~ i ~ ) dB

Figure 11 .5 Ge om et ry fo r magn et ic f i eldon the ax is o f a c i rcu la r cu r ren t loop o f

w i s e a s v i e w e d f ro m t h e p o s i -t i v e x - a x i s . A t p o i n t P o n t h ec u r r e n t l o o p , d s p o i n t s a l o n g ~ ,/~ l ies in the x z - p lane , a n d t h ea n g l e b e t w e e n d~" a n d / ~ i s 9 0 ~(a) For the o bserver , ind ica te thed i r e c t i o n o f t h e t o t a l f i e ld B .( b ) F o r t h e d / 3 p r o d u c e d b y I d sa t P, i n d ic a t e w h i c h c o m p o n e n t sa r e n o n z e ro . ( c ) C o m p u t e t h et o t a l f i e l d B e x p re s s e d i n t e rm sra d iu s a . o f t h e c u r r e n t . ( d ) C h e c k y o u r

r e s u l t a t x = O . ( e ) C o m p u t e t h e t o t a l f i e ld B e x p re s s e d i n t e rm s o f t h e m a g -n e t ic m o m e n t # . ( f ) C h e c k y o u r r e s u l t f o r l a rg e x / a . (g ) Ev a l u a t e B x fo rI - 1 A a n d a - 4 c m = 0 . 0 4 m , a t x - 3 c m = 0 . 0 3 m .So lu t i on : (a ) By ro ta t iona l sym me t ry abou t z, By an d Bz are zero. (Thinking ofth i s as a magnet , by A mp~ re ' s equ iva lence the f ie ld po in t s a long the ax is o f theloop.) (b) As indicated in Figure 11.5 , d B has nonzero componen ts a long x andz. In general , al l three components are nonzero. (c) Because only Bx is nonzero,we need on ly cons ider dB x = ]d/~ l cos0 , w here cos0 = a / R = a / ( x 2 + a2) 89 By( 1 1 . 1 ) , I d B I - k m l ]d~ x [~ I /R2. Also,

Ida* x/~1 = Id~'tl /~ll sin90 ~ = ds (1 )(1 ) - ds,w h e r e d s - Ids'l . Thus

dB x = I dB I c osO - ~ d S - R =k m I ( d s ) a

R 3Since R is a constant , the in tegral y ields

B x k m la f k m la 2 r r k m la 2= R 3 d s = R3 2 r r a = ( x 2 + a 2 ) ~ . (field on axis of loop)(1 ] . 1 1 )

(d) For x = O, so R = a, (11.11) gives B x = 2 r r k m l /a , in ag reemen t wi th (11 .10) .


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1 1 . 4 A p p l i c a t io n s o f t h e B i o t - S a v a r t L a w 46 7

( e) U s i n g / , = I A = I zra 2, (11 .11) t akes the fo rmB x = 2 k i n # (f ield on axis of a circular loop) (11.12 )(x2 + a2) V

(f) For large x , so R - + x , (11 .11) and (11 .12) go to B x = 2 k m b t / x 3 . This isas expec ted fo r a d ipo le , by Ampere ' s equ iva lence ; see (9 .8 ) . Fo r compar i son ,fa r a long the ax i s o f an e lec t r ic d ipo le o f mom en t p , Ex = 2 k p / x 3 ; see (3 .12).(g ) For ou r cu rren t loop , / , = I A = 0 .503 A- m 2 . At a d i s tance x = 3 cm = 0 .03 m,so ~/x 2 + a 2 = 0 .05 m , ( 11.12 ) gives Bx = 8 x 10 -6 T, abo ut a f if th oft he earth 'smagn et ic f ie ld . T en tu rns o f wi re wou nd in the same d i rec t ion wi ll increase th e / ,and the B by a fac to r o f t en .

T h i s e x a m p l e h a s s o m e i m p o r t a n t g e n e r a li z a ti o n s . ( 1 ) B y t a k i n g t w o i d e n t ic a lc o - a x i a l c o il s, w i t h e q u a l c u r r e n t s , a n d b y s p a c i n g t h e m j u s t r i g h t , i t is p o s s i b l et o m a k e a f i el d t h a t i s v e r y u n i f o r m i n t h e v i c i n i t y o f t h e i r m i d p o i n t . T h i s s i m p l em e t h o d t o p r o d u c e a n e a r l y u n i f o r m f ie ld a lo n g t h e x - a x i s is d u e t o H e l m h o l t z .( 2 ) B y t a k i n g t w o i d e n t i c a l c o - a x i a l c oi ls , w i t h e q u a l a n d o p p o s i t e c u r r e n t s , a n db y s p a c i n g t h e m j u s t r i g h t , i t i s p o s s i b l e t o m a k e a f i el d t h a t i s z e r o a t t h e m i d p o i n ta n d h a s a v e r y u n i f o r m s l o pe a l o n g t h e x - a x is i n t h e v i c i n it y o f t h e i r m i d p o i n t .M R I o f t e n e m p l o y s m a g n e t i c f ie ld s w i t h v e r y u n i f o r m s lo p es . ( 3 ) B y w i n d i n g aw i r e a r o u n d a l o n g c y l in d e r , w e c a n s i m u l a t e a l o n g s e t o f e q u a l l y s p a c e d c o - a x i a lc o il s, w i t h e q u a l c u r r e n t s . A s i n d i c a t e d i n C h a p t e r 1 0 , t h i s is k n o w n a s a s o l e n o i d .I t is u s e f u l b e c a u s e i t p r o d u c e s a u n i f o r m f ie ld e v e r y w h e r e w i t h i n t h e s o le n o i d .( I n f a c t, t o p r o d u c e a u n i f o r m f ie l d, t h e c r o s s - s e c t i o n n e e d n o t b e c i r c u l a r s o l o n ga s i t is t h e s a m e a ll a l o n g t h e a x is o f t h e t u b e . )

~ Field on axis of disk-shaped ma gnetC o n s i d e r a m a g n e t i c d i s k o f t h i c k n e s s l = 2 m m , r a d i u s a = 2 c m , a n d w i t hM ~ 1 .0 x 1 06 A / m ( t h e r e m a n e n t m a g n e t i z a t i o n M r o f t h e p e r m a n e n tm a g n e t i c m a t e r i a l N EO ) . S e e F i g u re 1 1 .6 . F i n d t h e f i e ld a d i s t a n c e x = 3 c malong i ts axis .S o l u t i o n : By Am p~re ' s cu r ren t loop dec omp os i t ion (Sec t ion 10 .2 ) , a f in it e cu r-ren t loop can be decom posed in to man y t iny cu rre n t loops . By Am p~re ' s equ iv -a lence , a t a d i stance we ma y cons ider each t iny cu rre n t loop to be a t iny magnet .These m agnets add up to a th in magn et ic d i sk wi th the same per im eter and th ick -ness as the cur rent loop of Figure 11.5 . Since the disk real ly is th in (l


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4 6 8 C h a p t e r l l m T h e B i o t - S a v a r t L a w a n d A m p ~ r e ' s L a w

"~ Ads = dxiI - - x

O bserver - r ~

Figure 1 1 . 7 G e o m e t r y f o r m a g n e t i c f i el d d u e t o a ni n f i n it e ly l o n g c u r r e n t - c a r r y i n g w i r e .

~ Field due to long current-carrying w ireL o n g w i re s ar e a n i m p o r t a n t a n d c o m m o n l y e n c o u n t e r e d g e o m e t r y . L e t a w i r eb e a l o n g t h e x - a x i s , c a r r y i n g c u r r e n t a l o n g + ) ~, s o d s = )~ d x . L e t t h e o b s e r v e rb e a d is t a n c e r b e l o w t h e o r i g in . S e e F i g u r e 1 1 . 7 . ( a) F i n d t h e d i r e c t i o n a n dm a g n i t u d e o f t h e f i e ld d u e t o t h e w i r e . ( b ) E v a l u a t e t h i s f o r I = 1 0 A a n dr = 1 c m = 0 . 01 m .Solution: F i g u r e 1 1 . 7 s h o w s t h a t t h e c i r c u i t a n d o b s e r v e r a r e i n t h e s a m e p l a n e .B y S e c t i o n 1 1 . 4 . 1 , t h e f i e l d i s p e r p e n d i c u l a r t o t h i s p l a n e . L e t 0 b e t h e a n g l eb e t w e e n )~ a n d / ~ ( t o t h e o b s e r v e r ) . F r o m F i g u r e 1 1 . 7 , i f x ~ - o e , t h e n 0 - * 0 ,a n d if x - * + e c , t h e n 0 ~ z r. A l s o f r o m F i g u r e 1 1 .7 , d s x / ~ p o i n t s a l o n g - ~ , a n dd s x [ ~ = - ~ d x s i n O , w h e r e w e h a v e d s = d x . T o e v a l u a t e t h e i n t e g r a l in ( 1 1 . 1 ) ,u s e 0 a s t h e v a r i a b l e s o t h a t x a n d R m u s t b e e x p r e s s e d i n t e r m s o f 0 a n d r . S i n c ec o t ( z r - O ) = - c o t O ,

d ( c o t O)dO = r csc 20dO,= r co t (zr - 0 ) = - r co t 0 , d x = - r - f f ~and R 2 = x 2 + r 2 = r 2 c sc 2 0 . T he n (11 . 1) bec om es

B z - k m l / ( d s x = - k m l f _o c d x s inO 0

= - k m l Z ~ r c s c 2 Od O s i n O - - k m f o ~ d ~ si n Oc s c 2 0 r - ~ ( 1 1 1 3 ) .S i n c e ~ p o i n t s o u t o f t h e p a p e r , t h e f i el d p o i n t s i n t o t h e p a p e r , i n a g r e e m e n t w i t hO e r s t e d ' s r i g h t - h a n d r u le . T h i s p r o c e d u r e a ls o a p p l ie s t o f i n it e l e n g t h s o f w i r e ,w i t h a p p r o p r i a t e c h a n g e s i n t h e l i m i t s o f i n t e g r a t io n .

More gene ra l ly , a t a d i s t ance r f rom a long cur ren t -ca r ry ing wi re , the f i e ld hasm a g n i t u d e

~ B = 2 k = , ( fi el r o d r l o n g l ~ . . . . . .... i l l i ~ i

I t s d i r e c t i o n i s g i v e n b y O e r s t e d ' s r i g h t - h a n d r u le .

11.5 Ap pl ica tions o f the Pr inc ip le o f S uperpos i t ionT h e m a g n e t i c f ie ld s fo r m a n y c o n f ig u r a ti o n s c a n b e d e t e r m i n e d b y s u p e r i m p o s i n gs o m e o f t h e p r e c e d i n g r e su l ts .~ Field along plane midw ay between two long wires

T h i s e x a m p l e , a r e p r i s e o f t h e c o n f i g u r a t i o n o f F i g u r e l O . 2 ( c) , i s r e l a t e d t ot h e m a g n e t i c f i el d p r o d u c e d b y p o w e r l in e s o r o r d in a r y l i ne c o r d , a l o n g th e i r


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11.5 Appl icat ions of the Principle of Superposit ion 4 6 9

Figure 1 1 . 8 U s e o f s u p e r p o s i t i o n t o f in dt h e f i e l d o n t h e p e r p e n d i c u l a r b i s e c t o rb e t w e e n t w o l o n g w ir e s c a rr y i ng e q u a lc u r r e n t s i n o p p o s i t e d i r e c ti o n s .

p e r p e n d i c u l a r b i s e c t o r . L e t e a c hw i r e c a r r y c u r r e n t I , o n e i n t oa n d o n e o u t o f t h e p a p e r .S e e F i g u r e 1 1 .8 , w h e r e w e u s eO e r s t e d ' s r i g h t - h a n d r u l e f o rt h e i r d i r e c t i o n s . ( a ) F i n d t h et o t a l f i e l d d u e t o t h e s e w i r e s .( b ) F i n d t h e e q u i v a l e n t m a g -n e t . ( c ) E v a l u a t e t h e f i e l d d u et o p o w e r l i n es c a r r y i n g I =2 0 0 0 A , w i t h w i r e s e p a r a t i o na = 0 . 8 m , a t a d i s t a n c e l yl =9 m . ( d ) E v a l u a t e t h e f i e l d d u et o a n e l e c t r i c b l a n k e t c a r r y i n gI = 2 A w i t h w i r e s e p a r a t i o na = 2 m m a t a d i s t a n c e l yl =2 c m .

Solu t ion: (a ) By sym m et r y , IB~I = IB21. N ow add the f i e lds vec tor ia l ly . . I f thewi res a re a t a long , the x -ax i s, th en each i s a d i s t ance R = v /y 2 + a 2 away .( /31)y i s sm a l le r th an IB 11 by a facto r of co s ~ = a~ R , a n d s i m i l a r l y fo r w i r e 2 . T h ex - c o m p o n e n t s c a n c e l . T h u s , u s i n g ( 1 1 . 1 4 ) f o r t h e i r m a g n i t u d e s , w i t h r -+ R,

~ = ~ + ~2 _ 25,1~]cosff = 2~9(2kRI ) a 4 k m l a- ~ = 3 ) y 2 + a 2 " ( 1 1 . 1 5 )

( b ) T h e c i r c u i t o f F i g u r e 1 1 . 8 c a n b e c o n v e r t e d t o a n in f i n it e m a g n e t i c s l abg e o m e t r y o n u s i n g ( 1 1 . 1 5 ) , w i t h I - + M l . S e e F i g u r e 1 1 . 9 , w h e r e A m p ~ r i a ns u r fa c e cu r r e n t s K = I / l = M m u s t c i r c u la t e a r o u n d t h e o u t si d e o f t h e m a g n e t .M a g n e t s o f th i s s h a p e ( b u t n o t m a g n e t i z e d n o r m a l t o t h e i r p l a n e) a r e u s e d t o s e alre f r ige ra to r doors .

(c ) F or the po we r l ines , (11 . 15) g ives 1/31 = 7 . 84 x 10 -6 T . (d ) F or the e lec t r i cb l a n k e t , ( 1 1 . 1 5 ) g i v es I / ~ 1 - 3 . 9 6 x 1 0 - 6 T . T h e s e f ie l ds a re a b o u t a t e n t h o f t h ee a r t h ' s m a g n e t i c f ie l d. B y tw i s t i n g a p a i r o f w i r e s c a r r y i n g e q u a l a n d o p p o s i t ec u r r e n t , t h e n e t m a g n e t i c f i e ld c a n b e d e c r e a s e d e v e n m o r e . T h i s is t h e o r i g i n o ft h e t e r m twis ted pair , o f t e n u s e d to d e s c r i be w i r e s e m p l o y e d f o r c o m m u n i c a t i o n sp u r p o s e s .

Figure 1 1 . 9 I n f i n it e m a g n e t i c sl a b m a g n e t i z e d n o r m a lt o t h e s la b. I t is e q u i v a l e n t t o t h e t w o w i r e s o fF igure 11 . 8 .


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47 0 C h a p t e r 11 m T h e B i o t- S a v a r t L a w a n d A m p ~ r e ' s L a w

I ~ bserver/ Figure 11.10 C u r r e n t - c a r r y i n g h a i r p in . I t c a n b ed e c o m p o s e d i n t o t w o s e m i - i n f i n i t e c u r r e n t - c a r r y i n gwi res and a cu r r en t - car ry i ng semi c i r c l e .

~ Field at center of current-carrying hairpinI n F i g u r e 1 1 . 1 O , f i n d t h e f i e ld a t t h e o b s e r v e r ' s p o s i t i o n .S o l u t i o n : W e d e c o m p o s e t h e h a i r p i n i n t o t w o s e m i - i n f in i te w i r e s a n d a h a lf - lo o pof rad i us a . See F i gu re 1 1 .10 . Th e f i e ld due t o t he h a l f - l oop i s ha l f t ha t du e t ot he fu l l l oop , g i ven by (11 .10 ) . I t s d i r ec t i on i s i n t o t he paper . In add i t i on , t he t w osem i - i n fi n i t e wi res each p r od uce ha l f t he e f f ec t o f a fu l l w i r e , t oge t h er y i e l d i ngt he sam e ef f ec t , ( 11 .1 4 ) , as a fu l l w i r e . The i r f ie l d al so po i n t s i n t o t he paper . Th ust h e n e t f i e ld is in t o t h e p a g e, w i t h m a g n i t u d e

1 2 J r k m I 2 k m I k m l ( z r + 2)B - + = . ( 11 .16 )2 a a a

~ ~ ~ ~ Field due to current sheetL e t a c u r r e n t s h e e t l ie i n t h e x z - p l a n e , w i t h c u r r e n t i n t o th e p a p e r ( - ~ ) ,a n d c u r r e n t p e r u n i t l e n g t h K m e a s u r e d a l o n g x. S e e F i g u r e 1 1 . 1 1 . F i n d t h em a g n e t i c f ie ld a b o v e a n d b e l o w t h e c u r r e n t s h e e t .S o l u t i o n : A t h i c k n e s s d x o f t h e s h e e t i s l ik e a s u b w i r e w i t h c u r r e n t I -+ d I =K d x d i r e c t e d i n t o t h e p a p e r . W e w i ll a d d u p t h e e f f e c t s o f e a c h o f t h e s e s u b -wi res . Oer s t ed ' s r i gh t -han d ru l e g i ves t he d i r ec t i on o f t he f i e ld d B o f t h e s u b -wi re , as sho wn i n F i gu re 11 .11 . Fo r an observer_a l ong t he y - ax i s , t he d i s t ancet o t h e s u b w i r e , o f t h ic k n e s s d x , i s R = v /y 2 + x 2 . By (11 .14) , I dBI d u e t o t h es u b w i r e i s

IdBI = ~.2kmKdx (11 .17)R

Figure 11.11 Cross - sec t i on o f i n f i n i t e cu r r en ts h e e t c a r r y i n g c u r r e n t p e r u n i t l e n g t h K , w h e r et h e l e n g t h i s m e a s u r e d a l o n g t h e x - d i r e c t io n .


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1 1 . 6 F o r ce s o n M a g n e t s a n d C u r r e n t -C a r r y i n g W i r e s 471

Two var iables appear : x and ~) . Let us work with ~) , using the fact thaty is a con stan t. Th en x - y tan ~), so d x - y sec 2 ~)d~) an d R 2 = y2 -b-x 2 =y2 + y2 ta n 2 0 -- y2 se c 2 ~9. T he ny 2 k m K y d x _ 2 k i nK y d x 2 km K y 2 sec 2 {)d0 = 2km Kd~)

dB~ = IdBI ~ = R2 - y2 + x 2 = y2 sec 2 0This in tegra te s on ~) f rom -z t /2 to + z r /2 to y ie ld (wi th Bx --> B)

By symm etry , th is is the only nonze ro com pon ent of the f ie ld . By Oers ted ' s r igh t -han d rule , /~ is a long ~ for y > O, and a long - ~ for y < O. In each of these regions,/3 i s un i form~tha t i s , i t does no t weaken as one moves away f rom the cur ren tsheet .

U n i f o r m i t y i s o f t e n d e s i r e a b l e . B y c o n n e c t i n g a s i n g le r i b b o n o f c o n d u c t o r a si n F i g u r e 1 1 . 1 2 ( a ) , a r e g i o n o f u n i f o r m f i el d c a n b e p r o d u c e d . L e t u s n e g l e c t e d g ee f fe c ts a n d s u p e r i m p o s e t h e f i el ds o f t h e t o p a n d b o t t o m o f t h e r i b b o n . T h e n , u s -i n g (1 1 . 1 8 ) , t h e t o t a l f ie ld w i t h i n t h e r i b b o n h a s m a g n i t u d e B = 4 J r k m K . T o p r o -d u c e B = 2 1 0 - 4 T t h e n r e q u i r e s t h a t K = 1 . 5 9 2 x 1 02 A / m = 1 . 5 9 2 A / c m .T h e d i r e c t io n o f t h e f i el d w i t h i n t h e c o n d u c t i n g r i b b o n is i n a g r e e m e n t w i t hw h a t w e w o u l d e x p e c t u s in g A m p e r e ' s r i g h t - h a n d ru le .

W e w i l l s h o r t l y s h o w t h a t a u n i f o r m f ie ld c a n a ls o b e p r o d u c e d b y t h e c y li n -d r i c a l c u r r e n t s h e e t o f F i g u r e 1 1 . 1 2 ( b ) , w h i c h a ls o gi v es B = 4 Jr k i n K . A s i n d i -c a t e d e a r l i e r , t h i s s o l e n o i d a l g e o m e t r y o f t e n i s r e a l i z e d b y a w i r e w o u n d a s i nF i g u re 1 1 . 1 2 ( c ). T h i s c a n b e t h o u g h t o f as a s u p e r p o s i t i o n o f a c u r r e n t s h e e t ( asi n F i g u r e 1 1 . 1 2 b ) t h a t p r o d u c e s a f i e l d a l o n g t h e a x i s , a n d o f a l o n g w i r e t h a tp r o d u c e s a f ie l d t h a t c i r c u l a t e s a b o u t t h e a x is . A s a c o n s e q u e n c e , t h e f i e ld li n e sd o n o t c l o s e o n t h e m s e l v e s , b u t s p i r a l o u t t o i n fi n it y . ( I n F i g u r e 1 1 . 1 2 c w e d on o t d r a w t h e c o m p l e x s p ir a l f o r B .)

! 1,6 F orc es o n M a g n e t s a n d C u r r e n t -C a r r y i n g W i r e sU p t o t h is p o i n t , t h e d i s cu s s i on h a s b e e n o n l y o f t h e m a g n e t i c f ie ld s p r o d u c e db y e l e c t r i c c u r r e n t s . W e n o w c o n s i d e r t h e f o r c e s t h a t s u c h f i e l d s p r o d u c e o nm a g n e t i c p o l e s a n d c u r r e n t - ca r r y i n g - w i r e s.

(a) Co) (c)F igure 11 .12 Sequence of re la ted geometr ie s wi th the same cur ren t pe r un i t length Ktha t p roduce the same uni form f ie ld wi th in them: ( a ) bent -a round r ibbon,(b) cy l indr ica l shee t , ( c ) so lenoid wi th n tu rns pe r un i t length and cu r ren t I , w h e r eK = n I .


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472 Cha pter 11 s The Bio t -Savar t Law and Am pere ' s Law

The Bio t -Savar t law or ig ina l ly was deve loped to f ind the ma gnet ic f ie ld o f a long w i re ,in o rder to exp la in the to rque on a pe rmane nt ma gnet (a compass need le ) . Ampere 'swork was more ambi t ious : to de te rmine the fo rce be tween two cur ren t -car ry ingc i rcu i ts . By theore t ica l reason ing , he found a fo rm fo r B, wh ich w e do no t p resen t ,tha t en fo rced ac t ion and reac t ion be twee n e very pa i r o f cu r ren t e lements in the twocircuits . Al thou gh th is form correct ly g ives the net fo rce be tween any cur ren t e lementand a c losed cur ren t -car ry ing c i rcu i t, i t is no t accep ted today . Al tho ugh the A mpe refo rce law and the Bio t -Savar t law do no t en fo rce ac t ion and reac t ion be tween everypa i r o f cu r ren t e lements in the tw o c i rcu its , overa l l mo me ntum is conserved wh en themomentum o f the e lec t romagnet ic f ie ld (Sec t ion 15 .10) is inc luded.

T o o b t a i n t h e f o r c e o n c i r c u i t # l d u e t o c i r c u i t # 2 , w e u s e t h e B i o t - S a v a r t l a wt o f in d B 2 e v e r y w h e r e o n c i r c u i t # 1 d u e t o c i r c u i t # 2 , a n d t h e n u s e t h e A m p e r ef o r c e l a w i n t h e f o r m , d P l - I l d s 'l x /~ 2 , t o f i nd t he ne t fo rce on c i r cu i t #1 .W e n o w a p p l y t h e B i o t - S a v a r t l a w t o fi n d b o t h t h e f o r c e o n a m a g n e t d u e t o ac u r r e n t l o o p a n d t h e f o r c e b e t w e e n t w o l o n g w i r e s .

1 1 . 6 . 1 F o r ce o n a M o n o p o l e a lo n g A x i s o f C u r r e n t L o o pC o n s i d e r t h e f o r c e o n a p o l e qm o f a l o n g n a r r o w m a g n e t , a t th e o b s e r v e r ' s p o s i t i o ni n F i gu re 11 .5 . Th en (11 .3 ) fo r Fq~ and ( 11 .12 ) fo r B x g i ve t ha t t he fo rce i s a l ongt h e x - a x i s , w i t h m a g n i t u d e

2km q ml,tF q m - - q m B x = (x 2 -[- a2)3 /2 . (11 .19)

T h i s h a s t h e s a m e m a g n i t u d e a s, b u t i s o p p o s i t e t o , t h e f o r c e o n t h e c u r r e n t l o o pd u e t o t h e m a g n e t i c p o l e , d i s c u s se d i n t h e p r e v i o u s c h a p t e r a s a n e x a m p l e o f af l a r i ng m agne t i c f i e l d .

1 1 . 6 . 2 F o r c e b e t w e e n T w o P a r a ll e l Wi r esC o n s i d e r a p a i r o f o v e r h e a d w i r e s t h a t c a r r y c u r r e n t t o a n d f r o m a p o w e r s t a -t io n . S e e F i g u r e 1 1 . 1 3 ( a ) . W h a t i s t h e m a g n e t i c f o r c e b e t w e e n t h e s e w i r e s, i f

F ig u r e 1 1 .1 3 (a) Two paral lel wires carrying curren t in oppo si tedirections. Their interaction is repulsive. (b) Two closed-circuitversions o f the p aral lel wires, show ing their ma gnetic m omen ts,which lead to repulsion.


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11 .6 Forces on Ma gnets and Current-Carrying W ires 4 7 3

the y car ry equa l cur re nts 11 = 12 = I in oppo s i te d i rect ions? Le t us f i r s t obta inqua l i t a t ive in fo rmat ion abou t th i s in t e r ac t ion . F igu re 11 .13 (b ) i l l u s t r a t e s tw oad jacen t r ec tan gu la r cu r r en t loops ca r ry ing l c i r cu la t ing in the s a m e d i r ec t ion .The i r nea res t a rms , l ike the w i r es in F igu re 11 .13 (a ) , ca r ry cu r r en t in oppos i t ed i r e c t i o n s , a n d d o m i n a t e t h e i n t e r a c t i o n b e t w e e n t h e t w o c u r r e n t l o o p s . U s i n gA m p e r e ' s e q u i v a l e n c e , t h e s e c i r c u i t s a r e e q u i v a l e n t t o t w o m a g n e t s o r i e n t e d i nthe s ame d i r ec t ion s o tha t the fo r ce w i l l be r epu l s ive . Thus an t ipa ra l l e l cu r r en t sr epe l ; co r r es pond ing ly , pa r a l l e l cu r r en t s add .W e n o w d e t e r m i n e t h e f o r c e o n t h e w i r e t o t h e r i g h t d u e t o t h e f i e l d o fthe w i r e to the l e f t . Bo th w i r es a r e no rmal to the paper , one pas s ing th rough theo r ig i n w i t h c u r r e n t i n t o t h e p a p e r , a n d t h e o t h e r c o m i n g o u t o f t h e R a p e r a t x = r .By ( l l . 14 ), t he f ie ld due to w i r e #1 on the l e f t has mag n i tud e IB I I = 2 k m I 1 / ra n d p o i n t s d o w n t h e p a g e ( a l o n g - 3 ) ) . N o w a p p l y t h e A m p e r e f o r c e l a w , i nthe fo rm d / 52 = 12d~2 x B1 , to w i r e #2 on the r igh t , w i th d s p o i n t i n g a l o n gthe c u r r en t ( ou t o f the paper , a long ~ ). S ince ds and B1 a re pe rpend icu la r ,the i r c ro s s -p rod uc t d s 2 x B1 has m agn i tude Ids s in90~ = ( d s 2 ) ( B 1 ) . T h u sd F - ] d / ~ 2 ] - I 2 (d s2 )B 1 . M o r e o ve r , d F 2 - I 2 d & x B1 p o i n t s t o t h e r i g h t , b y t h evec to r c ro s s -p rodu c t r igh t -han d ru le o r by ~ x ( -3 ) ) - x . Th e fo r ce is r epu l s ive .F ina lly , t he fo r ce pe r un i t l eng th on c i r cu i t #2 i s g iven by

d F 2kin l l 2km 11I2d s = 1 2B 1 - 1 2 ~ r = ~ ' r (11.20)For 11 = 12 = 100 A and r = 1 cm , (1 1.2 0) gives d F / d s = 0 . 2 N / m .

i1~6~3 D e f i n i n g t h e A m p e r e : S I U n i t sI f t he tw o w i r es ca r ry cu r r e n t in the s ame d i r ec t ion , th e fo r ce pe r un i t l eng th hasthe s ame m agn i tud e a s (11 .20 ) , b u t in s t ead o f be ing r epu l s ive , i t is a t t r ac tive .Th e in te rna t iona l s t a ndard de f in ing the am pere a r is es f rom th i s exp res s ion fo r thefo rce . P lac ing tw o long w i r es a m e te r apa r t , and ad jus t ing the cu r r e n t I i n eachun t i l t he fo r ce pe r u n i t l en g th i s 2 x 10 -7 N / m , g ives by de f in i t ion a cu r r en t o f1 ampere . Th i s e l ec t rom agne t i c de f in i t ion is the bas is o f the S I un i t o f cu r r en t ,and thus th e bas i s o f the S I un i t o f cha rge, f o r w h ich k m us t be m eas u red .N u m e r o u s o t h e r a p p r o a c h e s h a v e b e e n t a k e n t o d e f in e t h e u n i t o f c h a rg e . I nelect ro s ta t ic-cgs uni ts , k = 1 serves to def ine the un i t of e lect r ic charge, an d k~m u s t b e m e a s u r e d . I n m a g n e t o s t a t i c - c g s u n it s, km = 1 s e rves to de f ine the un i to f m a g n e t i c p o l e s t r e n g th , a n d k m u s t b e m e a s u r e d . W e w i ll w o r k o n l y w i t h S Iun i ts , bu t be aw are o f the ex i s t ence o f thes e o the r , equa l ly va l id , s e t s o f un i t s.

11~G4[.J ~ lm ~ a M a gn e t i c P r e s s u r eA cu r r en t - ca r ry ing c i r cu i t i s s ub jec t to s e l f - s t r e s s es tha t t end to expand i t . Con-s ide r, fo r exam ple , the r ib bon o f con duc to r in F igu re 11 .12 (a ) . N eg lec t ing edgee f f ec t s , t he f i e ld on the low er s u r f ace p roduced by the upper s u r f ace i s g iven by(11.18) . In d /~ = I d ~ x / 3 , w i t h I d ~ r e p l a c e d b y [ ( d A , t he fo r ce on an a r ea d Aof the low er s u r face becom es d / ~ = f ( d A x Bz. A s fo r tw o w i r es ca r ry ing cu r r en tin opp os i te d i rect ions , th is force is repuls ive . S ince K an d B are perpe ndicu lar ,


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4 7 4 Chap ter 11 m The B iot-Savar t Law and Am pere ' s Law

the fo r ce pe r un i t a r ea i s g iven by d F / d A = K B l . This f i e ld B l is ha l f o f th e to t a lf ield B - 4 z r k m I ( . H e n c e d F / d A - 8 9 B - B 2 / 8 z r k m . T h i s c a n b e i n t e r p r e t e da s a m a g n e t i c p r e s s u r eB 2Pmag = 8Jr k in " (ma gne t i c p r es s u re ) (11 .21 )

F or B - 1 T , ( 11 .21 ) g ives Pm~g ~ 4 X 10 ~ N / m 2, o r a b o u t f o u r a t m o s p h e r e s .A l t h o u g h i t w a s d e r i v e d o n l y f o r t h e c a s e o f a r i b b o n c o n d u c t o r , ( 1 1 . 2 1 ) i st rue m ore genera l ly . N o te tha t f o r the r ib bon conduc to r , t he f i eld l ines a r e pa ra l l e lt o t h e r i b b o n . T h e y c a n b e t h o u g h t o f a s e x e r t i n g p r e s s u re o n o n e a n o t h e r. I n( 1 1 . 2 1 ) , l e t t in g k ~ - ~ k a n d B ~ E g i v es t h e c o r r e s p o n d i n g p r e s s u r e b e t w e e ne lec t r i c f i e ld l ines , i n ag reemen t w i th S ec t ion 6 .10 on e l ec t r i c f lux tubes .Ex te rna l m agn e t i c f i e ld s t en d to o r i en t a c i r cu i t s o tha t the f i e ld o f the c ir -cu i t po in t s a long the app l i ed f i eld ( as fo r a cu r r en t loop , tho ug h t o f as a mag -ne t i c s hee t ) . I n add i t ion , s e l f - f i e ld s t end to make a c i r cu i t expand ( i . e . , p r es -s u r e) , t o e n c lo s e th e m a x i m u m a m o u n t o f m a g n e t i c f lu x . ( T h i n k o f a s q u a r ec i r cu i t , w he re o ppo s i t e s ides r epe l . ) H ence , a f l ex ib le c i r cu i t o f f ixed leng th ina un i fo rm f i e ld B w i l l f o rm a c i r c l e w hos e enc los ed a r ea has i t s no rmal a l ignedw i t h B .Y o u ' v e n o w f i n is h e d t h e d i s c us s io n o f t h e B i o t - S a v a r t l a w , w h i c h i s p a r t o n eo f t h e c h a p t e r. T h i s i s a g o o d p l a c e t o t a k e a b r e a k . T h e n w e w i l l c o n t i n u e w i t hA m p ~ r e ' s l a w ( o f m a g n e t i c c i r c u l a t io n ) , w h i c h is p a r t t w o o f t h e c h a p t e r.

11.7

1 1 . 7 , 1

S t a t e m e n t o f A m p ~ r e ' s L awA m p ~ r e ' s l a w r e l a t e s t h e m a g n e t i c c i r c u l a t i o n a r o u n d a c l o s e d p a t h , c a l l e d a nA m p ~ r i a n c i r c u i t , t o w h a t e v e r c u r r e n t p a s s e s t h r o u g h t h e a r e a d e f i n e d b y t h a tc lo s ed pa th . A n y c lo s e d p a t h c a n b e u s e d w i t h A m p ~ r e ' s l a w , j u s t a s a n y c l o s eds u r fa c e c a n b e u s e d w i t h G a u s s ' s l aw . A mp~ r ian c i r cu i t s a r e pu re ly imag ina ry .T h e y c a n b e m a d e t o c o r r e s p o n d t o r e a l e l e c t r i c c i r c u i t s , b u t s u c h A m p ~ r i a nc i r cu i t s a r e r a r e ly u s e fu l . I n S ec t ion 11 .7 .1 , w e d i s cus s how A mp~ r ian c i r cu i t scan be app l i ed .Magnetic C i rculation F8M agne t i c c i r cu la t ion F B , fo r a g iven c lo s ed pa th de f ined by a s e t o f d i r ec ted l ineele m en ts d~" = ~ d s , i s a meas u re o f the " s w i rl ines s" o f the m agn e t i c f i e ld a rou ndt h a t p a t h . H e r e d s = ]ds > 0. Specifically, FB is de fine d as

H e r e d e s / d s - [~ 9~ , t h e c o m p o n e n t o f B a l o n g t h e p a t h d i r e c t i o n ~, i s t h e m a g -n e t i c c ir c u l a t i o n p e r u n i t l e n g th . I t h a s u n it s o f T, w h e r e a s t h e m a g n e t i c c i r c u l a ti o nhas un i t s o f T-m.


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11.7 S ta tement o f Amp~ re 's Law 4 7 5

Figure 11.14 (a) An Am p~rian circuit that encloses anactual, current-carrying circuit. (b) The circuit-normalright-hand rule relating the direction of circulation dsaround, and the normal ~ to, an Amperian circuit.

Fo r a g iven Amp~r i an c i rcu i t , t he magne t i c f i e ld due to a magne t i c po l eq m has ze ro ne t mag ne t i c c i rcu l a t i on . Th i s fo l l ows by ana logy to e l ec tros t a ti c s ,where by (5 .35 ) t he e l ec t r i c f i e ld due to a cha rge q has ze ro c i rcu l a t i on fo rany c i rcu i t . However , i f e l ec t r i c cu r ren t i n an ac tua l phys i ca l c i rcu i t passesth rou gh tha t same Am p~r i an c i rcu it , t he ma gne t i c f i eld B due to an e l ec -t r i c cu r ren t can have nonzero magne t i c c i rcu l a t i on . See F igu re 11 .14 (a ) , wherethe c losed pa th we re fe r t o i s t he Amp~r i an c i rcu i t . Amp~re ' s l aw impl i e s t ha ta magne t i c f i e ld B d ue to e l ec t r ic cu r ren t s d i ffe rs f rom a ma gne t i c f i e ld Bqmd u eto m agne t i c cha rges.For t he cu r ren t i n F igu re 11 .14 (a ) , by Oers t ed ' s r i gh t -hand ru l e t he mag-ne t i c f i e ld c i rcu l a t e s coun te rc lockwise . Le t t he magne t i c c i rcu l a t i on be ca l cu -l a t ed fo r t he Amp~r i an c i rcu i t o f F igu re 11 .14 (a ) , wh ich enc loses t h i s cu r ren t .Wi th ds coun te rc lockwise (c lockwise ) , by (11 .22 ) t he c i rcu l a t i on i s pos i t i ve(negat ive) .

11~,7~2 C u r r e n t E n c l o s e d lencF r o m C h a p t e r 7 , t h e c u r r e n t [enc pass ing th rough the Amp~r i an c i rcu i t de f inedby d~" i s g iven by an in tegral ov er the associa ted c ross-sect ion:

T h e r e l a t i o n s h i p b e t w e e n t h e n o r m a l ~ i n ( 1 1 . 2 3 ) a n d t h e A m p ~ r i a n c i r c u i te l em en t d s i n (11 .22 ) i s g iven by the c i rcu i t -no rm al r i gh t -hand ru l e .

~ i ~ ~ ! i ~ ~ ~ ~ ~ i i i i iii;iiiii !ii iiiiii iiiiiiiii! iiiiiiiiiiil !!;iiii i iiiiiiiii i !iii!i liiii iiiiiiiiii!!i


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4 7 6 Chap ter 11 w T he Biot-Savart Law and Amphre's Law

1 1 . 7 . 3

11.7.4

A mp~r e ' s La wWith a l l t h i s de f ined , we now s t a t e Ampere ' s l aw:

: r . =4:,-,-k,.I~, o r ~ ~ =wi th d~ and h re la t ed by the c i rcu i t -no rm al r i gh t -hand - ru l e o f F igu re 11 .14 (b ) .Typ ica lly , i n u s ing Am p6re ' s l aw , we se l ec t an a rea t h rou gh wh ich cu r ren t f l ows ,and the pe r ime te r de f ines t he A mp er i an c i rcu i t. Thus , t he ac tua l phys ica l c ircu it ,w h i c h p r o v id e s I~c, t yp i ca l ly passes t h rough the Amper i an c i rcu i t .U s es o f A m p ~r e 's L a wReca ll t ha t G auss ' s l aw , wh ich em ploys G auss i an su rfaces , has th ree p r im aryuses : (1 ) non invas ive me asu rem en t o f the cha rge Qenc wi th in a c lo sed su r face ;( 2 ) r e la t io n s h i p b e t w e e n s u r fa c e c h a rg e d e n s i ty a s a n d t h e n o r m a l c o m p o n e n tE out" i t of t he e l ec t r i c f i e ld j u s t ou t s ide a conduc to r i n equ i l i b r ium ( fo r wh ich# i n - - 6 i n s ide ) ; (3 ) de t e rm ina t ion o f t he e l ec t r i c f ie ld /~ w hen the cha rge d i s tr i -bu t ion i s so symm et r i ca l t ha t i t p roduc es a un i fo rm e l ec tr i c f l ux dens ity .S imil ar ly , Amp6 re ' s l aw , wh ich e mploys A mp er i an c i rcu it s, has t h ree p r i -mary uses : (1 ) non invas ive m easu rem en t o f t he cu r ren t Ienc t h rou gh a c losedc i rcu i t (Sec t ion 11 .9 ) ; (2 ) re l a t i onsh ip be tween su r face cu r ren t dens i ty K andt h e t r a n s v e r s e c o m p o n e n t f i x B o u t of t he magne t i c f i e ld j u s t ou t s ide a pe r fec td i amagne t , fo r wh ich B in - - 6 i n s ide (Sec t ion 11 .11 ) ; (3 ) de t e rm ina t ion o f t hemagn e t i c f ie ld B wh en th e c u r ren t d i s t r i bu t ion i s so sym met r i ca l t ha t i t p roducesa un i fo rm magn e t i c c i rcu la t i on pe r un i t l eng th (Sec t ion 11 .10 ) .Bu t be fo re us ing Am pere ' s l aw , f ir s t we de r ive i t u s ing Amp ere ' s equ iva lence .

11o8

11,8,1

D e r i v a t i o n o f A m p ~ r e ' s L a w f o r M a g n e t i cC i r c u l a ti o n : T h e M a g n e t i c S h e l lThe derivat ion i s performed in s teps. F i rs t we consider a t r iv ia l case . Next wein t roduce the c oncep t o f t he m agne t i c she ll . Th en w e ou t l i ne t he s t ra t egy o f t heproof, using a numerical example . F inal ly , we give the proof i t se l f , fo l lowed bysome add i t iona l app l i ca t i ons o f the mag ne t i c she ll concep t .C i r c u i t N o t E n c lo s i n g C u r r e n tCons ide r an Amp~r i an c i rcu i t fo r wh ich Ienc= 0 , as in F igure 11.15(a) . I fAm pere ' s l aw ho lds , t hen b y (11 .24 ) , FB = 0 . Le t u s de rive t h is re su l t.By Am p6re ' s equ iva lence , a t a d i s tance f rom the cu r re n t l oop, t he f ie ld Bprod uce d by the cu r ren t l oop i s equ iva l en t t o t he f i eld Bm p r o d u c e d b y th e m a g -net ic poles qm of a d i s t r i bu t ion o f equ iva l en t magne t s . Because t he co r respond ingelect r ic f ie ld produced by e lect r ic d ipoles has zero c i rcula t ion for th is and anyother c i rcui t , the magnet ic c i rcula t ion for th is c i rcui t i s a l so zero . That i s ,

FB - f B . d s f B m . d s (d is tant sources) (11 .25)


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1 1 .8 Th e M a g n e t i c S h e l l 4 7 7

Figure 11.15 (a) An Am perian circuit that does not enclose an actual,current-carrying circuit . (b) A n Am perian circuit that encloses an actual,current-carrying circuit. (c) An A mperian circuit that pierces the magneticshell that is equivalent to an actual, current-carrying circuit.

11~..,8,2 A mp~re's Magnetic S hellN o w c o n s id e r t h e m a g n e t i c c i r c u l a ti o n f o r an A m p e r i a n c i r c u it w i t h Ie nc ~ O, asin F igu re 11 .15 (b ) . T h i s can be ob ta in ed us ing ano the r idea o f A m pere , ca l l ed am a g n e t i c s h e l l . I t has pos i t ive po les on one s ide and nega t ive po les on the o the r ,a n d i s a g e n e ra l i z a ti o n o f t h e m a g n e t i c d i s k o f S e c t i on 1 1 . 4. 4 . T h e a r g u m e n tbea r s r epea t ing .F i r s t , decompos e the ac tua l cu r r en t loop , w h ich can be qu i t e i r r egu la r , i n to as e t o f t iny equ iva len t c i r cu i ts , a s d i s cus s ed in S ec t ion 10 .2 . (Th i s deco mp os i t ioni s no t un ique , s o w e a r e f ree to em ploy a u s e fu l one . )S econd , u s e the equ iva lence , a t a d i s t ance , be tw een each t iny cu r r en t loopand a t iny magn e t . T hen add up the t in y magne t s , w h ich , p l aced s ide by side ,s imu la te th e ac tua l c i r cu i t o f F igu re 11 .15 (a ) o r F igu re 11 .15 (b ) . Th i s y ie ld s them a g n e t i c s h e l l of F igu re 11 .15 (c ) , o f sma l l and u ns pec i f i ed th icknes s l ; sho r tly ,w e w i l l t ake the l imi t l -+ O . L ike the cu r r en t loop , the magne t i c s he l l can bei r regu la r. S ee F igu re 11 .15 (c ) , w h ere the to p o f the ma gne t i c s he ll is pos i t ivea n d t h e b o t t o m is n e g a ti v e, b y A m p e r e ' s r i g h t - h a n d r u l e.W e n o w f in d t h e m a g n e t ic m o m e n t t h a t t h e m a g n e t ic s h el l m u s t h a v eto p ro duce the s ame f i e ld a s the ac tua l cu r r e n t loop . F rom # = I A o fC h a p t e r 1 0 a n d # = q m l = Crm A l o f C h a p t e r 9 , t h e m a g n e t i c m o m e n t p e r u n i tarea is

pt = crml - - I , (11.26)Ajus t a s in (10 .7 ) . Thus crm l - - I , for all l, no mat t e r how s mal l . Equ iva len t ly ,f rom Ch ap t e r 10 K = M = am, s o K l = I = crm l . T h u s t h e c u r r e n t l o o p h a s b e e nr e p l a c e d b y t h e m a g n e t i c s h e ll .

S trategy fo r P roving A mp~re's L awWe now s how tha t , i n F igu re 11 .15 (c ) , i n t eg ra t ion o f d F B m o n l y t h r o u g h t h es he l l f r om the b o t t om to the top equa ls , i n F igu re 11 .15 (b ) , t he in t eg ra l o f dF Bfo r a l l t h e A m p e r i a n c i r c u i t .


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4 7 8 Cha pter 11 t The Biot-Savart Law and Am pere 's Law

11.8,4

1 The i n t e r i o r c i r c u l a t i on pint fo r t he ma gne t i c she l l ( t a ke n f rom - - O " m to -1-om9 B mwi t h i n t he she l l ) i s de t e rmi ne d e a s i l y us i ng a n a na l ogy t o our o l d f r i e ndt he pa ra l l e l -p l a t e c a pa c i t o r . Le t ' s s a y i t t a ke s on t he nume r i c a l va l ue p i n t _ _B m m- 5 0 T - m , t h e m i n u s s i g n b e c a u s e i n F i g u r e 1 1 . 1 5 ( c ) B m , i n t w i t h i n t h e m a g -ne t i c she l l oppose s ds2 For t he ma g ne t i c she ll , t he e x t e r i o r c ir c u l a t i on p e x t ( f r o m - ]- o"m t o - - o " m out s i de9 B mt he she ll ) mu s t be t he n e ga t i ve o f t he i n t e r i o r c i r c u l a ti on be c a use t h e t o t a lc i rcula t ion for the m agn et ic she l l is zero. Hence , for the m agn et ic she l l , F ext - -m m50 Tom.

- . ~ _+3. B m , e x t , t he f ie ld e x t e r i o r t o t he ma g ne t i c she l l, e qua l s B e x t , the f i e ld exte r io r tot he c ur re n t l oop . Thus t he e x t e r i o r c i r c u l a t i on pext for t h e m a g n e t i c s h e l lB m( s e e F i gure 11 .15c ) a nd t he e x t e r i o r c i r c u l a t i on p~xt for t h e c u r r e n tl oop ( se e F i gure 11 15b) mus t be t he sa me . He nc e , fo r our e xa mpl e , pe x t _9 B - -50 T-m.4 . For t he c ur re n t l oop , t he e x t e r i o r c i r c u l a t ion p~xt is e sse n t ia l ly a l l t he ma g ne t i c

c i rc u l a t i on FB ; i t ne g l e c t s on l y t e rms propor t i ona l t o l, whi c h go t o z e ro a sl -+ 0 . He nc e , fo r our e xa mpl e , F~ - - 50 T -m.W o r k i n g b a c k w a r d f r o m 4 t o 1 , t h is a r g u m e n t i s e q u i v a l e n t t o t h e s e q u e n c eof e qua l i t ie s

P B - - F ~ t - - p e x t _ _ H i n tB m - - B i n" (11.27)P r o o f o f A m p ~r e 's L a wL e t ' s n o w o b t a i n H int a lgebra ica l ly . We must integra te o v e r Bm, ntf o r th e m a g n e t i cB mshe l l, t aking th e integra l f ro m --Om to +Om. This i s like taking the integra l o f thee l e c t r i c f i e l d E wi t h i n a c a pa c i t o r f rom -c ~ t o + ~ . The re E i n t = -- ] E i n t ] - 4rrkcr,a nd t he vo l t a ge d i f fe re nc e (e l e c t r ic c i r c u l a t ion) fo r p l a t e s e pa ra t i on d i s E i n t d -4zrkod. H e r e B m , i n t - 4 r r k m o - m wi t h i n t he ma gne t i c she l l . More ove r , a s no t e da bove , s inc e fo r our pa t h d~"o p p o s e s #m,int,

H i n t - - B m , i n tl - - 4 z r k m c r m t - -4 z rkmlm D (11.28)He nc e , a pp l y i ng (11 .28) t o (11 .27) y i e l ds

FB = 4zrkml. (11.29)Thi s is Am p~re ' s la w, whi c h , wr i t t e n o u t i n e ve n more de t a i l t ha n (11 .24) , i s

J B . d s 4rckmlenc- 4Jrkm f J. fidA. (11.30)As usua l , t he c i r c u i t -norma l r i gh t -ha nd ru l e , g i ve n i n F i gure 11 .14 , r e l a t e s dsand ft .W e h a v e o n l y p r o v e d A m p e r e ' s l aw f o r a s in g le c u r r e n t - c a rr y i n g w i re . H o w -e ver , by t he p r i nc i p l e o f supe rpo s i t i on , i f we a dd o t he r w i re s , the i r c i r c u l a t iona dds t o t he l e f t -ha nd s i de o f (11 .30) , a nd t he i r Ienca dds t o t he r i gh t -ha nd s i de .


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11.8 The Magn etic Shell 479

1 1 , 8 . 5

1 1 ~ 8 , 5

Figure 11.16 (a) Long wire and fini te circui t that has a comm on arm w ith thatwire, showing the field and the equivalent magnetic m om ent fi of the circuit .(b) Field outside the mag netic sheet that is equivalent to the finite circuit ofpar t (a ). Imagine tha t the sheet thickness l is infinitesimal.F ie ld o f a L ong, Thin C urren t -C arry ing Wi reTh e m a gne t i c she ll c a n be use d t o f i nd the a l r e a dy know n B f ie l d o f a n in f i n i te l yl ong c ur re n t -c a r ry i ng w i re l . C on s i de r t h i s wi re to be p a r t o f a l a rge re c t a ngul a rc i rc u i t wi t h one s i de f ixe d , who se o t he r s ide s c a n be c hose n a t a n a rb i t r a ry a ng le .For s impl ic i ty , l e t the she l l l i e in the xy-plane for pos i t ive y , so t he e qu i va l e n tm a g n e t i c m o m e n t p o i n t s a l o ng th e + z - ax i s. S e e F i g u re 1 1 . 1 6 ( a ) , w h i c h s h o w s af i n i t e r e c t a ngul a r c i r c u i t . B y Oe rs t e d ' s r i gh t -ha nd ru l e , t he ma gne t i c f i e l d mus tc i rc u l a t e a bout t he a x is o f t he w i re, t a ke n t o be y . W e wi l l now f i nd t he m a gne t i cf ie l d due t o t he wi re by f i nd i ng t he e x t e r i o r m a gne t i c f i el d o f t he ma gne t i c she l l.W e d e t e r m i n e f B 9d~" fo r t wo pa t hs s t a r ti ng on t he pos i t i ve s i de o f t he ma g-ne t i c she l l , a nd e nd i ng on t he ne ga t i ve s i de . Se e F i gure 11 .16(b) , whe re t heshe l l thickness i s f ini te for c la r i ty . One i s an exte r ior pa th of radius r , chosensymme t r i c a l l y , so t ha t t he c i r c u l a t i on pe r un i t l e ng t h , d F B / d s - B ext" s , is uni-fo rm a l ong t he pa t h , i f t he i n f i n it e s ima l t h i c kne ss l o f t he she ll is ne g l e c t e d .The n f B . ds g i ve s B e x t ( 2 ; r r ) . T h e o t h e r p a t h g o es th r o u g h t h e s h e ll fr o m p o s -i t ive to nega t ive , and i s the nega t ive of (11.28) , or 4 z r k m I . W e t h u s c o n c l u d et h a t Bex t (27rr) = 47rkmI , o r Bext-- 2 k m I / r . W i t h Bext --~ B , thi s i s the same as(11 .14) .F r ing ing F i e l d o f a Capac i to rTh e e qu i va l e nc e o f t he ma g ne t i c she l l a nd a l ong wi re c an be t u rne d i n t o t hee lec t r ica l pr ob lem of the f r inging f ie ld of a para l le l -p la te capac i tor . For di s tancesm uc h l a rge r t ha n t he p l a t e s e pa ra t ion , a pa ra l l e l -p l a te c a pa c i t o r l ooks ve ry mu c hl ike the e lec t r ica l equiv a len t of a ma gn et ic she l l . See Figure 11 .17 (a ) .F rom Am pe re ' s e qu i va l e nc e, t he c ur r e n t I o f F i gure 11 .16(a ) goe s t oK l = M I = crml o f F i g u re 1 1 . 1 6 ( b ) . T h u s (B , km, 1 ) - -~ (B , km, O'ml ) . M a k i n g t h eana logy to e lec t r ic i ty , we le t a ~ O'm, SO (B , km, Crml) - -~ (E, k , crl) . T h e n f r o mB = 2 k m I / r fo r t he l ong w i re , t he a na l ogy y ie l ds

2 k ~ lEext -- (11.31)for the f r inging f ie ld of a c lose ly space d capac i tor . See Figure 11.17 (b) . N ote tha t


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480 Ch apte r 11 t The B io t -Savar t Law and Am p~re ' s Law

Figure1 1 . 1 7 (a) Paralle l-pla te capacitor , sho wing fr inging f ie ld.(b) Side view of paralle l-pla te capacitor , showing fr inging f ie ld that isana logous to tha t o f F igure 11 .16(b) .w i t h i n s u c h a c a p a c it o r E i n - 4 z r k o , s o ( 1 1 . 3 1 ) b e c o m e s

E i n l A V- = . ( 1 1 . 3 2 )E ext - 2 zr r 2 sr rI f A V - - 1 0 0 V a n d l - 0 . 0 0 1 m , f o r r - 0 . 0 2 m ( 1 1 . 3 2 ) g i v es E e x t - 7 9 6 V / m ,c o m p a r e d w i t h E i n t - I A V I / I - - 1 0 s V / m . P. H e l l e r h a s v e r i f i e d ( 1 1 . 3 2 ) e x p e r i -m e n t a l l y .

11,9 A m p ~ r e 's L a w I m p l ie s T h a t C i r c u la t io nY i e l d s C u r r e n tI f w e c a n c o m p u t e o r m e a s u r e t h e c i r c u la t i o n [ t h e l e f t- h a n d s i d e o f A m p ~ r e ' sl aw , ( 1 1 . 2 4 ) ] a r o u n d s o m e A m p ~ r i a n c i rc u it , t h e n w e c a n d e d u c e [ b y t h e r i g h t-h a n d s id e o f ( 1 1 . 2 4 ) ] t h e c u r r e n t I ~ c p a s s i n g t h r o u g h t h a t c i r c u i t , i n c l u d i n g i t sd i r e c t i o n o f f lo w . T h i s i s t h e b a s is o f t h e R o g o w s k i c o i l a n d t h e c l i p - o n a m m e t e r .

~ Using he circulationC o n s id e r a n i rr e g u l a r l y s h a p e d A m p ~ r i a n c i r c u i t o f l e n g th L - 1 2 c m ( e. g. , ac i r c u i t s h a p e d l i ke t h e A m p ~ r i a n c i r c u i t i n F ig u r e 1 1 . 1 5 b ) . F o r d~" c i r c u l a ti n gc o u n t e r c lo c k w i s e , l e t F B - - 0 . 0 4 2 T - c m. F in d ( a ) h o w B c i r c u la t e s ( o n a v e r -a g e ); ( b ) t h e a v e r a g e c o m p o n e n t o f / ~ a lo n g t h e c i r c u i t; ( c ) t h e d i r e c t i o n o fIenc; and (d) the m agn i tud e o f I ~ c .Solution: (a) Since the c ircula tion is posit ive , B and d~* m ust c ircula te in th esame d i rec t ion , on ave rage. Thu s bo th d # and B c i r cu la te counte rc lockwise . (b)T h e c o m p o n e n t o f / ~ a lon g t h e c i r c u it is B . ~ = d F B / d s . Hence , the ave rageva lue of B . ~ i s the ave rage va lue of d F B / d s , which equa ls F B d iv ided b y i tslength L , or (0 .042 T-cm ) / (12 cm) - 0 .0035 T . ( c ) S ince B c i r cu la te s coun-te rclockwise , O ers ted ' s r igh t -hand ru le te ll s us tha t I enc poin ts o u t of the page,as in Figure l l .15b. (d) Fin ally, use o f Amp ~re 's law as FB = 4 z r k m l e n c givesl en c = F B / 4 J r k m = 3 .34 102 A. Thus we have de te rm ined I in F igure 11 .15 . I fthe s ign of the c i r cu la tion w ere nega t ive , the cur ren t w ould reve r se in d i r ec t ion .


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1 1 .9 Amp O re ' s La w Imp l i es Th a t C i rcu la t i o n Y i e ld s Cu rren t 4 8 1

(0,b) (a,b)

(0, (a,0)

F i g u re 1 1 .1 8 A m p ~ r i a n c i r c u i t f o r t h e c o m p u t a t i o n o ft h e m a g n e t i c c i r cu l a ti o n , a n d t h e s u b s e q u e n tc a l c u l a ti o n o f t h e c u r r e n t Ienc t h a t p a s s e s t h r o u g h i t.

Com puting the circulationC o n s i d e r a m a g n e t i c f ie l d

/ 3 - ( 2 - x 2 ) ] , ( 1 1 . 3 3 )

w h e r e x i n c m g i v e s / ~ i n 1 0 - 4 T . T a k e a s t h e A m p ~ r i a n c i r c u i t a r e c t a n g l ei n t h e z = 0 p l a n e , w i t h c o r n e r s (0 , 0 ) , ( a , 0 ) , ( a ,b ) , (O ,b ) . L e t a = 7 c m a n db = 4 c m , a n d c o n s i d e r c l o c k w i s e c i r c u l a ti o n . S e e F i g u r e 1 1 . 1 8 . D e t e r m i n et h e c i r c u l a ti o n f o r t h i s A m p ~ r i a n c i r c u it , a n d d e t e r m i n e t h e a m o u n t a n dd i r e c t i o n o f c u r r e n t f l o w t h r o u g h i t.S o l u t i o n : T o f in d t h e t o t a l c i r c u l a t io n , c o n s i d e r e a c h a r m s e p a r a t e l y .1 . T h e f i rs t a r m g o e s f r o m ( 0 , 0 ) t o (O,b) = ( 0 , 4 c m ) , s o d s j d y , w i t h d s -

d y > 0 . T h e n/ 3 . d s = ( 2 - x 2 ) x 1 0 - 4 d y = 2 x l O-4 d y ,

s i n c e x - 0 f o r t h a t a r m . I t gi v e s a c o n t r i b u t i o n o fb2 x 1 0 - 4 j 0 d y = ( 2 x 1 0 - 4 T ) ( 4 c m ) = 0 . 0 8 x 1 0 - 4 T - m

t o t h e t o t a l c i r c u l a t i o n .2 . T h e s e c o n d a r m g o e s f r o m ( 0 , b ) = ( 0 , 4 c m ) t o (a ,b ) = ( 7 c m , 4 c m ) s o t h a t

d s ~ d x , w i t h d s - d x > 0 . T h i s d~ is p e r p e n d i c u l a r t o / 3 , s o i t g i ve s z e r oc o n t r i b u t i o n t o t h e t o t a l c i r c u l a t i o n .

3 . T h e t h i r d a r m g o es f r o m ( a , b ) = ( 7 c m , 4 c m ) t o ( a , O ) - ( 7 c m , 0 ) s o t h a td-~ - - ]d s, w h e r e d s = - d y > 0 , f r o m t h e l i m i t s o f i n t e g r a t i o n . T h u s / 3 9d~ =(2 - x 2 ) d y = - 4 7 x 1 0 - 4d y , s i n c e x = a = 7 c m f o r t h a t a r m . I t g i v e s a c o n -t r i b u t i o n o f 1 0- 4 7 x 1 0 - 4 ) d y = ( - 4 7 x 1 0 . 4 T ) ( - 4 c m ) - 1 . 8 8 x 1 0 - 4 T - mt o t h e t o t a l c i r c u l a t i o n .

4 . T h e f o u r t h a r m g o e s f r o m ( a , 0 ) = ( 7 c m , 0 ) t o ( 0 , 0 ) so t h a t d ~ = - ~ d s , w h e r ed s - - d x > 0 , f r o m t h e l i m i t s o f i n t e g r a t i o n . T h i s d ~ i s p e r p e n d i c u l a r t o B ,s o i t g i v e s z e r o c o n t r i b u t i o n t o t h e t o t a l c i r c u l a t i o n .S u m m i n g a ll f o u r t e r m s g i v e s a t o t a l c i r c u l a t i o n F B = 1 . 9 6 x 1 0 -4 T - m . B y

( 1 1 . 2 4 ) , i t m u s t e q u a l 4 zrk mlen c , w h e r e Ienc i s p o s i t i v e i n t o t h e p a p e r f o r a


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4 8 2 Chapter l 1 ~ The Bio t -Savar t Law and Ampere ' s Law

posi t ive clockwise ci rculat ion. Thus lanc= Fs /4zrkm = 156 A; we have non-invasively dete rm ined Ianc, both in mag ni tude and in i t s sense relat ive to thepage.

Me asuring the circulation" coils and clip-onogowskiammetersD e v i c e s k n o w n a s Rogowski coi ls s u r r o u n d a c u r r e n t - c a r r y i n g w i r e a n d p e r m i tt h e c u r r e n t t o b e r e a d non invas ive ly ( i . e . , w i thou t b reak ing the c i rcu i t t oi n c l u d e a n a mm e t e r ) . A l t h o u g h t h e y h a v e a s ma l l g ap , t h e y m a y b e t h o u g h t o fas c losed Amper ian c i rcu i t s . (E lec t r i c cu r ren t can on ly c i rcu la t e a round a rea lc i r c u i t t h a t is c lo s e d , b u t a ma g n e t i c f i e ld c a n c i r c u l a te a r o u n d a n A m p e r i a nc i rcu i t t ha t i s near ly c losed . ) Rogowsk i co i l s do the equ iva len t o f measur ingt h e c i r c u l a ti o n Fs . T h e y a r e c a li b r a te d , b y A m p e r e ' s l aw , ( 1 1 . 3 0 ) , t o d i s p la yt h e c u r r e n t lane. Fo r e x a mp l e , a Ro g o w s k i co i l c o u l d s u r r o u n d t h e c u r r e n t l o o pi n F ig u r e 1 1 . 1 5 ( b ) , l ik e t h e A mp e r i a n c i r c u i t i n t h a t f ig u re . A m o r e c o m m o nb u t l e s s p r e c i s e d e v i c e , w h i c h me a s u r e s a c h a r a c t e r i s t i c ma g n e t i c f i e l d d u eto the wi re , i s t he cl ip-on ammeter: i t c l ips on (p roper ly , a round) a cu r ren t -car ry ing wi re . The Hal l vo l t age y ie lds Ianc fo r dc c l i p - o n a mme t e r s , a n d t h ee m f f r o m F a r a d a y 's l a w - - t o b e d i s c us s e d i n t h e n e x t c h a p t e r - - y i e l d s Ia nc f o rac c l i p - o n a mm e t e r s . P . M u r g a t r o y d h a s d e v e l o p e d Ro g o w s k i c o il d e v ic e s t h a tme a s u r e l o ca l c u r r e n t d e n s i ti e s.

1 1 . 1 0 A p p l i c a t i o n s o f A m p ~ r e ' s L a w a n d S y m m e t r yG a u s s ' s l a w c a n b e u s e d t o d e t e r m i n e E w h e r e t h e e l e c t r i c c h a r g e d i s t r i b u t i o nh a s a h i g h d e g r e e o f s y m m e t r y ( s p h e r ic a l , c y l i n d ri c a l, o r p l a n a r ) a n d i s k n o w n .S i m il ar ly , A m p e r e ' s l a w c an b e u s e d t o d e t e r m i n e B w h e r e t h e e l e c t ri c c u r r e n td i s t r i b u t i o n h a s a h i g h d e g r e e o f s y m m e t r y a n d i s k n o w n .T o s e e th i s, n o t e t h a t f o r a c i r c u i t o f l e n g t h s , t h e a v e r a g e c ir c u l a t i o n p e r u n i tl e n g t h i s g i v e n b y

-0d F B = ~ . ~ _ ~ B . d-d _ - 4 zr kml en c . (11.34)d s s s

W h e n t h e c i r c u l a t io n p e r u n i t l e n g t h i s u n i f o r m a l o n g t h e c i r c u i t , o r p a r t o f ac i r c u it , t h e a v e r a g e c i r c u l a t io n p e r u n i t l e n g t h is t h e s a m e a s t h e l o c a l c ir c u l a t i o np e r u n i t l e n g t h a n d b o t h e q u a l ]B I. B o t h t h e n a r e g iv e n b y ( 1 1 . 3 4 ) .

~ Field inside a wire with uniform current densityCo n s i d e r a lo n g , s t r a ig h t w i r e o f r a d i u s a c a r r y i n g a u n i f o r m c u r r e n t d e n s i t yi n t o t h e p a p e r , w i t h J - [ J [ . See Figure 11 .19 . I t ca r r i es to t a l cu r ren tl = J ( z r a 2 ) , s o J - l / ( z r a 2 ) . U s e A m p h r e ' s l a w a n d s y m m e t r y t o f i n d t h emagnet i c f i e ld ins ide the wi re ( r < a ) .Solution: By Oers ted 's r ight-hand rule, for r > a the magnet ic f ield /3 wi l l ci r-culate clockwise aroun d the cen ter of the c urren t dis tr ibut ion. Moreover, by


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I 1 . 1 0 A p p l i c a t io n s o f A m p b r e ' s L a w a n d S y m m e t r y 48 3

U n i f o r m c u r r e n t d e n s i ty J A m p 6 r i a n c i r c u it

O u t e r s u r fa c e o f w i r eFigure 11.19 Cross - sec t i on a nd f i e ld /~ o f al o n g w i r e o f r a d iu s a w i t h u n i f o r m c u r r e n t p e run i t a r ea . The cu r r en t i s i n t o t he page and a reai s m e a s u r e d n o r m a l t o t h e p a g e .

t h e c i r c u l a r s y m m e t r y o f-+t h e c u r r e n t d i s t r i b u t i o n , IB Is h o u l d d e p e n d o n l y u p o n r .Th i s a l so shou l d be t rue fo rr < a . L e t 0 r e p r e s e n t t h ec l ockwi se t angen t i a l d i r ec-t io n . T h e n ~ = 0 a n d / 3 = B O ,wh ere B = I /3 t . Because t hep r o b l e m h a s s o m u c h s y m -m e t r y , w e c a n c h o o s e a nA m p 6 r i a n c i r c u i t i n F i g u r e11 .19 fo r wh i ch t he c i r cu l a -t i o n p e r u n i t le n g t h d r ' B / d s =[ 3 . 2 ~ = B O . O = B has t hesame va l ue a l l a l ong t heA m p 6 r i a n c i r c u i t . S u c hAmp6r i an c i r cu i t s a r e c i r c l esc o n c e n t r i c w i t h t h e c u r r e n td i s t ri b u t io n . ( A n o n c o n c e n -

t r i c c i r c l e w o u l d n ' t g i v e a u n i f o r m c i r c u l a t i o n p e r u n i t l e n g t h , n o r w o u l da n o n c i r c u l a r c i r c u i t . ) T h e t o t a l c ir c u l a ti o n , w i t h p a t h l e n g t h 2 z r r, is t h u sPB = ( d F B / d s ) ( 2 r r r ) = B(2J r r ) .

T h e c o r r e s p o n d i n g e n c l o s e d c u r r e n t i s I e n ~ - J (~r r2 ) . Hence Ampere ' s l aw,(11 .24 ) , y i e l d s

B ( 2 z r r ) - 4 J r k m [ e n c = 4 J r k m J ( z r r 2 ) , ( 1 1 . 3 5 )s o

2 k m l r ( f ie l d wi t h i n wi re o f f i n it e r ad i us ) (11 .36 )= 2 r r k m J r = a - -- T -.T h i s r e s u l t c o u l d h a v e b e e n o b t a i n e d f r o m t h e B i o t - S a v a r t la w o n l y w i t h g r e a t e f-fo r t . A l t e rna t i ve l y (11 .34 ) , w i t h l enc - - J ( r r r 2 ) and s - 2 J r r , r ep roduc es (11 .36 ) .N o t e th a t , f o r r - a , w h e r e I e n c - J ( z ra2 ) , (11 .36 ) g i ves B - ( 2 r c k m ) ( I / r t a 2 ) a -2 k m I / a , w h i c h m a t c h e s ( 1 1 . 1 4 ).

~ Field o ut si d e curren t-carrying ire of f inite radiusongC o n s i d e r a c lo c k w i s e - c i r c u l a t in g A m p ~ r i a n c i r c u i t w i t h r a d i u s r > a . S e eF i g u r e 1 1 . 20 . U s e A m p h r e ' s l a w a n d s y m m e t r y t o fi n d t h e m a g n e t i c f i e ldo u t s i d e t h e w i r e ( r > a ) .S o l u t i o n : T h e c i r c u l a t io n p e r u n i t le n g t h d V ~ / d s = / ~ 9~ fo r t h i s symmet r i c c i r -cu i t t akes on t he un i fo rm va l ue B , and t h e t o t a l c i r cu l a t i on i s B (2J r r ) . By (11 .24 ) ,

w i r eo ng ~ 1 ~ ' ~ . . . . . N / A m p ~ r i a n c ir cu it

Figure 11.20 Fi e l d /~ o f a l ong , t h i n wi rec a r r y in g c u r r e n t i n t o t h e p a g e .


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48 4 C h a p t e r 1 1 ~. T h e B i o t - S a v a r t L a w a n d A m p ~ r e ' s L a w

t h i s m u s t e q u a l 4zrkmlenc , and he re I enc - - I . He nce B(2zr r ) - 4 r c k m I , o r

B _ _ .2kml

t"( fi el d o u t s id e w i r e o f f in i te r a d i u s ) C i i ~

i n a g r e e m e n t w i t h ( 1 1 . 1 4 ) . F o r r = a , t h i s m a t c h e s t h e v a l u e o b t a i n e d i n t h ep r e v i o u s p r o b l e m . A l t e r n a ti v e ly , ( 1 1 . 3 7 ) f o l lo w s f r o m ( 1 1 . 3 4 ) w i t h I e n c = I a n ds = 2 z r r . E q u a t i o n ( 1 1 . 3 7 ) i s m u c h m o r e g e n e r a l t h a n ( 1 1 . 1 4 ) , w h i c h w a s o n l yder i ved fo r an i n f i n i t es i mal l y t h i n wi re .

~ Field inside toroidal coilC o n s i d e r a f in i te s o l e n o i d ( c o m p a r e F i g u r e 1 1 . 1 2 . c ) w i t h N t u r n s o f w i r eb e n t i n t o a c i r c l e . S e e F i g u r e 1 1 . 2 1 . T h i s i s c a l l e d a t o r o i d a l c o i l , o r t o r o i d .( a) U s e A m p 6 r e ' s l a w a n d s y m m e t r y t o f in d t h e m a g n e t i c f ie ld w i t h in t h ec o il , o f i n n e r r a d i u s a a n d o u t e r r a d i u s b, a n d h e i g h t w n o r m a l t o t h e p a g e .( b ) F o r a = 3 c m a n d b = 4 c m , a n d 2 5 0 t u r n s , e v a l u a te t h e f i el d a t t h e m e a nrad i us .Solution: ( a ) By Am p~ re ' s r i gh t -ha nd ru l e , fo r t he c u r r en t i n F i gu re 11 .21 ,c i r c u l a t e s c o u n t e r c l o c k w i s e . ( N e g l e c t t h e n o n u n i f o r m i t i e s d u e t o t h e t u r n s n o t

be i ng i n f i n i t es i mal l y c l o seToro i d w i t h N t u rn s Am l :~6 r ian c i rcu i t(cu r ren t I pe r t u rn )\

rB

Figure 11 . 21 T o r o i d w i t h i n n e r r a d i u s a a n do u t e r r a d iu s b, wi t h N t u rns , each car ry i ngcur ren t I . The f i e l d B i s con f i ned t o t he i n t e r i o ro f t h e t o r o id .

t o o n e a n o t h e r , a n d c o n s i d e ron l y t he average f i e l d . ) Takea n A m p 6 r i a n c i r c u i t t h a ti s a ci rcle of radius r , con-c e n t r i c w i t h t h e t o r o i d ,w i t h d ~ c i r c u l a t i n g c o u n -t e r c l ockwi se , so ~ po i n t sa l ong / ] . He re B = [B[i s u n i f o r m b e c a u s e t h eA m p 6 r i a n c i r c u i t i s c o n c e n -t r i c w i t h t h e t o r o i d . T h e nd F 1 3 / d s - B . ~ = B i s uni -f o r m . H e n c e F /3 = f ( d F / 3 /

d s ) d s = B f d s = B ( 2 J r r ) . S i n c e t h e t o r o i d h a s N t u rn s , e a c h c a r r y in g c u r r e n t I ,w e h a v e Ienc - - N I . Thus , by (11 .24 ) , B (2zr r ) = 4 z rk m l e nc = 4 z r k m ( N I ) , so

B .==.2/~NX ~ ! ~i~i~,!84 !i!il/ii~:~~i! ~i!!~7:~i!~!!~i!!!!!!?!!!~i( fi el d w i t h i n t o r o i d a l c o i l ) ( 1 ~ I i ~8 )

Al t e rna t i ve l y , (11 .34 ) wi t h I e n c - N I a n d s = 2 Jr r r e p r o d u c e s ( 1 1 . 3 8 ) . T h i sr e s u l t c o u l d n o t h a v e b e e n o b t a i n e d f r o m t h e B i o t - S a v a r t l a w w i t h o u t a g r e a tdea l o f e ffo r t . A l t ho ugh t he t o ro i d o f F igu re 11 .21 has a smal l c i rcu l a r c ross -sec t i on , (11 .38 ) ho l ds fo r m ore ge nera l c ross - sec t ions . (b ) Fo r ou r param et er s , a tt he m ean r ad i us o f 3 .5 cm, (11 .38 ) y i e l d s B = 2 .86 x 10 -3 T .

~ Field outside current sheetC o n s i d e r a c u r r e n t s h e e t w i t h c u r r e n t p e r u n i t le n g t h K t h a t p o i n t s in t ot h e p a p e r , w i t h t h e s h e e t c e n t e r e d a b o u t t h e x - ax i s. S e e F ig u r e 1 1 . 2 2 . U s eA m p ~ r e ' s l a w a n d s y m m e t r y t o f in d th e m a g n e t i c f ie ld .


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11 .10 App l i ca t ions o f AmpOre 's Law a nd S ymm e t ry 48 5

Figure 11 .22 F i n i t e cu r r e n t shee t car ry i ngc u r r e n t p e r u n i t l e n g t h K i n t o th e p a g e , w i t hl e n g t h m e a s u r e d a l o n g t h e x - d i r e c t i o n .Am p~r i an c i r cu i t A1 con t a i ns no cu r r en t , andt hus has zero c i r cu l a t i on , wh i ch i mp l i es t ha tt he f i e l d i s t he same on i t s t op and bo t t omarms . Amp~r i an c i r cu i t A2 con t a i ns a f i n i t ec u r r e n t , a n d t h u s h a s n o n z e r o c i r c u la t i o n .

S o l u t i o n : B y O e r s t e d ' s r i g h t - h a n d r u l e , t h e f i e l d a b o v e t h e s h e e t w i l l b e t o t h er i gh t , and be l ow t he shee t i t w i l l be t o t he l e f t . Cons i der Amp~r i an c i r cu i t Aa ,

where d~ c i r cu l a t es c l ock -wise. I t has t e

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The Biot-Savart Law and Ampère's Law, physics vol