for the magnetic induction at the point P with coordinate x produced by an increment of current
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1 5. tatng wth th ffnta psson B fo th magntc nucton at th pont P wth coonat pouc by an ncmnt of cunt at, show pcty that fo a oop cayng a cunt th magntc nucton at P s B Ω wh Ω s th so ang subtn by th oop at th pont P. Ths cospons to a magntc scaa potnta,. Th sgn connton fo th so ang s that Ω s post f th pont P ws th nn s of th sufac spannng th oop, that s, f a unt noma n to th sufac s fn by th cton of cunt fow a th ght han, Ω s post f n ponts away fom th pont P, an ngat othws. Ths s th sam connton as n cton.6 fo th ctc po ay. ( B( c c a c a ( ( a ( a a a Ω Ω( B Ω( B( Ω( 5. ght ccua sono of fnt ngth L an aus a has N tuns p unt ngth an cas a cunt. how that th magntc nucton on th cyn as n th mt s N B ( θ θ wh th angs a fn n th fgu at pag 5.
2 B fo on oop B NL N N asn θ sn θ a θ sn θ θa N N sn θ N N B BN N θ sn θ θ θ θ θ θ θ [ ] 5.6 cynca conucto of aus a has a ho of aus b bo paa to, an cnt a stanc fom, th cyn as ba. Th cunt nsty s unfom thoughout th manng mta of th cyn an s paa to th as. Us mp s aw an pncp of na supposton to fn th magntu an th cton of th magntc fu nsty n th ho. Bcaty Bnoho Bho ϕ ϕ J J J J Bcaty ϕ ϕ ( ϕ ϕ Bcaty J J B caty ( a b ( a b 5. ccua oop of w cayng a cunt s ocat wth ts cnt at th ogn of coonats an th noma to ts pan hang sphca angs θ,ψ. Th s an app magntc f, B X B (βy an B y B (β. (a Cacuat th foc actng on th oop wthout mang any appomaton. Compa you sut wth th appomat sut (5.69. Commnt. (b Cacuat th toqu n owst o. Can you uc anythng about th hgh o contbutons? Do th ansh fo th ccua oop? What about fo oth shaps? (a
3 B B y B y ( β ( β n sn θ ϕ sn θsnϕy θ F B F B F F B C C C ( B ( B ( By C C B β ( β y y β ( β βy s F ( ( B βy n Bβ sn sn s y n B β θ ϕ s F Bβ sn θ sn ϕa F ( y B β n Bβ sn θ ϕa s F ( ( { F B By By By By } C C y B y By y ( β ( β B B y F m m F Bβ snθ sna Bβmy Qsnθ sna Fy Bβ snθ a Bβm fom 5.69 F ( m B F B βm B βm y y N m B m B m B y B m m ( y Eact toqu N J B J B B J y pojcton of oop aa { ( β ( β ( β ( β } J B y B y B y B y J f w can soat th owst o tm, a oths a hgh o contbutons. Not that 5.5 m J ( yj J y J J y J yyj m my m y
4 5. sph of aus a cas a unfom sufac chag stbuton σ. Th sph s otat about a amt wth constant angua octy ω. Fn th cto potnta an magntc fu nsty both ns an outs th sph. Choos th aong th as K σ σ ω ω ω & asn θ ϕ asn θ sn ϕ j aθ j {( a ( a j} σ ω σ ω snθ snϕ ω snθ ϕ asnθ ϕ asnθ snϕ a θ K ( a * a K Y (, (, m θ ϕ Ym θ ϕ m a * Y m( θϕ, K( Ym( θ, ϕ Ω m σωa * Y (, {( sn sn ( sn m θϕ θ ϕ θ ϕ j} Ym ( θ, ϕ Ω m σω a 8 8 (, ( Y ( m θϕ Y Y Y Y j * Ym( θ, ϕ Ω m σω a 8 8 ( (, (, Y θϕ Y θϕ ( Y( θϕ, Y ( θϕ, j σω a sn { ( sn sn ( sn σωa θ θ ϕ θ ϕ j} Bcaus othogona, th ntgaton bcoms σ a ω σ a ω sn θ [ snϕ ϕy] sn θϕ σ aω n sn θϕ Bn n σaω ( mnn m m a, hnt : out B out m a σω
5 5.8 ccua oop of w hang a aus a an cayng a cunt s ocat n acuum wth ts cnt a stanc away fom a sm nfnt sab of pmabty. Fn th foc actng on th oop whn (a Th pan of th oop s paa to th fac of th sab, (b Th pan of th oop s ppncua to th fac of th sab. (c Dtmn th mtng fom of you answ to pats a an b whn a. Can you obtan ths mtng aus on som smp an ct way? (a & J δ a δ θ a a Fom (5.8 & (5.9 a!! B P ( θ!!! a a a Bθ P (! a F J( B( ( θ!! ê δ a δ θ P ( θ θ ê a a a!!! a a δ a δ θ P ( θ a (! a a a a ( a (!! a a a a P P! ( a a a a a a ( a upp : a ow : a ( b ( ( J δ ± a δ a Fom pobm ( b J a J B, B ( F J( B( J B J B 5 (, (
6 a a B J( a J B J a J a a J( a J J( a J ( F J B J B J B ( J B a ( ( ( J a J a ( ± ( a J ( ( ( a ± ± ( J B ( ( s ( a a ± ± ( J ( a J a co 5.9 magntcay ha mata s n th shap of a ght ccua cyn of ngth L an aus a. Th cyn has a pmannt magntaton, unfom thoughout ts oum an paa to ts as. (a Dtmn th magntc f H an magntc nucton B at a ponts on th as of th cyn, both ns an outs. (b Pot th atos on th as as functons of fo. (a Magntc ha mata, s 5.9(c, scaa potnta H( ΦM ( M( n M( Φ M ( a s M, on th top M( M M( & n M( M, on th bottom, oth n M( M a M a Φ M ( a s L L a a M L L L L
7 L ( ns M L L Φ M ( a a L L M H( Φ M L L a a L L M B ( H M L L a a L ( outs M L L Φ M ( a a L L L M H( Φ M ( L L a a L L M B ( H M H L L a a 5. (a tatng fom th foc quaton (5. an th fact that a magntaton M ns a oum boun by a sufac s quant to a oum cunt nsty an a sufac cunt nsty, show that n th absnc of macoscopc conucton cunts th tota magntc foc on th boy can b wttn F ( M B ( M n B a wh s th app magntc nucton (not ncung that of th boy n quston. Th foc s now pss n tms of th ffct chag nsts an. f th stbuton of magntaton s not scontnuous, th sufac can b at nfnty an th foc gn by just th oum ntga. (b sph of aus wth unfom magntaton ha ts cnt at th ogn of coonats an ts cton of magntaton mang sphca angs,. f th tna magntc f s th sam as n Pobm 5., us th psson of pat a to auat th componnts of th foc actng on th sph.
8 (b (a F J B K Ba Jm M K m M n F M B M n Ba B M M n Ba ( M B ( M B ( B M M ( B B ( M ( M B ( B M B ( M Q B ( M n Ba B ( M n a [ ( B n M ( B M n ] a F [( M B ( B M ( M B ] [ ( B n M ( B M ] us ( C D ( C D ( n C Da ( B M ( B M ( n B Ma ( n B ( M B ( M B ( n M Ba F [ ( M B ( M B ] [( n M B ( B ] ( M B ( n M Ba M M( sn θ ϕ,sn θsn ϕ,θ B B y B M n a Ma ( β, β, ( β snθsn ϕ, β snθ θ, ( θ ϕ θ ϕ θ n sn,sn sn, Thn F M n B a n a ( θθ sn θsn θ ( ϕ ϕ ( β sn θsn ϕ, β sn θ θ, MB Ω MB β sn θsn ϕ,sn θ ϕ, 5. how that n gna a ong, staght ba of unfom coss sctona aa wth unfom ngthws magntaton M, whn pac wth ts fat n aganst an nfnty pmab fat sufac, ahs wth a foc gn appomaty by F M ctostatc consatons n cton... at you scusson to th
9 Ths pobm s bst so by consng an mag magnt. Th nfnt pmabty of th at sufac nsus that th magntc f must b ppncua to th sufac. s a sut, ths s sma to th ctostatc cas of ctc f ns bng ppncua to th sufac of a pfct conucto. Fo magntostatcs, ths mans that w may us a magntc scaa potnta _M (snc th a no f cunts subjct to th conton _M at (tang th sufac to n th -y pan at. Th mag pobm s thn st up as foows W M B H t s mpotant to not that, wh w so ths pobm usng an mag magnt, th ony quantts that show up n ths ngy ntga a th actua soucs of magntaton ~M an th actua magntc nucton ~B. W pac th magnt at a stanc fom th sufac so that M a M, L, othws s a sut Z L L W M a B ( M B Z wh w ha appomat that th magntc nucton s oughy unfom acoss th fac of th magnt. Usng th mag magnt stup, th a two soucs of magntc nucton B B B a mag Usng ( w s that L Ba um a ( L a L L Bmag um a ( L a ( L H w ha shft th coonats such that th a magnt s btwn an L an th mag magnt s btwn --L an -. n pncp, w may nst ths pssons nto (5 to comput th magntostatc ngy. How, as a smpfcaton, w not that th ntga of ~M.Ba gs a poston npnnt ( npnnt sf ngy. Hnc ths w not contbut to th
10 foc. s a sut, w ony n to nst Bmag nto (5. Ths gs us Z L L W um Z ( a a ( L Z L um a ( a ( L Z um a ( L a a ( L Th foc s thn W L L F um a ( ( ( L a a L L L -u M a L a L - um wh n th ast n w us L a (a conton that w n anyway to nsu that B s nay unfom on th n caps. Not that w cou ha atnaty us th sut of Pobm 5. F ( M B ( M n B a s wh th app magntc nucton ~B s gn by Bmag n (6 wth. nc th magntaton s unfom, th foc ass nty fom th sufac tm F ( M n B a M B ( B ( L a s [ ] L L L [ (] um M B L B um a L a L a L What w ha on h s to cacuat th foc though th magntostatc ngy F W( wh ~ nots th poston of th ba magnt. Ths s th magntostatc quant of th foc scusson n cton., whch stats that \Focs actng btwn chag bos can b obtan by cacuatng th chang n th tota ctostatc ngy of th systm un sma
11 tua spacmnts." n fact, ths statmnt s tu n gna, po w us th compt (ctostatc pus magntostatc ngy of th systm. Cuousy, a conucto wth sufac-chag nsty _ fs an outwa foc of th fom σ F ε whch s oughy th ctostatc quant of F um 5. (a how that a sufac cunt nsty K / fowng n th aa cton on a ght ccua cynca sufac of aus poucs ns th cyn a unfom magntc nucton n a cton ppncua to th cyn as. how that th f outs s that of a two mnsona po. (b Cacuat th tota magnt statc f ngy p unt ngth. How s t ns an outs th cyn? (c What s th nuctanc p unt ngth of th systm, w as a ng ccut wth cunt fowng up on s of th cyn an bac th oth? (a J ( K( δ ( K( ( J sn y by.9 m m ( [ ( ] ( K ( m( ( [ ( ] ( K ( th ntga pc up m by othogonaty of ns. Usng sn sn an ( m ( m m m m
12 [ ] [ ] [ ] K K δ δ Q th asymptotc fom. appyng Γ Γ δ δ δ y B, sn, sn,, sn y y (b sn B B W υ
13 (c nc w ha ony on ccut. Compang to th abo w a off.
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