Chapter 23: Magnetic Field Shielding

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Meagan Lane
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1 ELECTROMAGNETIC COMPATIBILITY ANDBOOK 1 Chapt : Magntc Fld Shldng.1 Usng th Bt-Savat law, vfy th magntc fld xpssn X (pvdd by yu nstuct) gvn n th cunt dstbutns and th magntc flds tabl n ths chapt.. Usng th ntgal dfntn f th magntc vct ptntal, cmpltly stup (but d nt slv) th ntgal(s) t dtmn th magntc vct ptntal f th cunt dstbutn Y (pvdd by yu nstuct) gvn n th cunt dstbutn and th magntc flds tabl n ths chapt.. Dtmn th magntc vct ptntal usng th vct Pssn s quatn f th fllwng cunt dstbutns: a) nfnt slab f vlum cunt J = Jzaˆy f thcknss d paalll t and cntd abut th xy plan z b) nfnt slab f vlum cunt J = J aˆ y f thcknss d paalll t and cntd abut th xy plan c) nfntly lng cylnd f vlum cunt J = Jaˆz f adus a cntd abut th z axs J = J ρ a f adus a cntd d) nfntly lng cylnd f vlum cunt ( ) ˆz abut th z axs. Dtmn th magntc flux dnsty usng ths vct ptntal..4 Th cunt dstbutn f multpl cncntc cylndcal cnductng (nnmagntc) shlls fllws. Usng Ampè s law, cmpltly st up th ntgals t dtmn th magntc fld vywh. Evaluat all dt pducts. Assum, as n ths chapt, that th axs f th cylnds a alng th z axs and th shlls a nsulatd fm ach th. Th vaabls A, I, and δ a ndpndnt f pstn. 0 ρ < a I n th + z dctn a < ρ < b f spac a) b < ρ < c I n th z dctn c < ρ f spac ( ρ a ) δ 0 ρ < a J z = A b) a < ρ < b f spac b < ρ < c I n th z dctn c < ρ f spac Cpyght 00 by Knnth L. Kas, Vsn 10/05/05

2 ELECTROMAGNETIC COMPATIBILITY ANDBOOK c) d) 0 ρ < a f spac a < ρ < b I n th + z dctn b < ρ < c f spac ( ρ c) δ c < ρ < d J = z A d < ρ f spac 0 ρ < a I n th + z dctn ( b ρ ) δ a < ρ < b J = z A b < ρ < c f spac c < ρ < d I n th + z dctn d < ρ f spac.5 A flat ntfac btwn tw lna magntc matals xsts n th xz plan. A sufac cunt f K a ˆ xsts alng th ntfac. F y > 0 na th ntfac, x ˆ ˆ ˆ = 4ax + ay + az y > 0 wh µ = Dtmn bth and B f y < 0 na th ntfac wh µ =..6 A flat ntfac btwn tw lna magntc matals xsts n th yz plan. A sufac cunt f K aˆ xsts alng th ntfac. F x < 0 na th ntfac, y B ˆ ˆ ˆ 4 = 4ax ay + az x < 0 wh µ = 4 Dtmn bth 5 and B 5 f x > 0 na th ntfac wh µ = 5..7 A flat ntfac btwn tw dlctcs s dscbd by th sufac φ = π/4 n th cylndcal cdnat systm. A sufac cunt f K aˆ xsts alng th ntfac. F φ > π/4 na th ntfac, π B ˆ ˆ ˆ 4 = aρ + 4aφ + a z φ > wh µ = 4 4 Dtmn bth and B f φ < π/4 na th ntfac wh µ =..8 A cuvd ntfac btwn tw lna magntc matals s dscbd by th sufac ρ = n th cylndcal cdnat systm. A sufac cunt f xsts alng th ntfac. F ρ < na th ntfac, B ˆ ˆ ˆ = 4aρ aφ 5a z ρ < wh µ = z ˆ K a φ Cpyght 00 by Knnth L. Kas, Vsn 10/05/05

3 ELECTROMAGNETIC COMPATIBILITY ANDBOOK Dtmn bth 4 and B 4 f ρ > na th ntfac wh µ = 4..9 A cuvd ntfac btwn tw lna magntc matals s dscbd by th sufac = n th sphcal cdnat systm. A sufac cunt f xsts alng th ntfac. F > na th ntfac, ˆ ˆ ˆ = 4a aθ 5aφ > wh µ = ˆ K a θ Dtmn bth 4 and B 4 f < na th ntfac wh µ = A cuvd ntfac btwn tw lna magntc matals s dscbd by th sufac θ = π/ n th sphcal cdnat systm. A sufac cunt f xsts alng th ntfac. F θ < π/ na th ntfac, π ˆ ˆ ˆ = a + 4aθ 5aφ θ < wh µ = Dtmn bth and B f θ > π/ na th ntfac wh µ =..11 A flat ntfac btwn tw lna magntc matals xsts n th xy plan dscbd by th quatn y = x +. N sufac cunt xsts alng ths cuvd ntfac. F y > x + na th ntfac, ˆ ˆ ˆ = 4ax + ay az y > x + wh µ = ˆ K a φ Dtmn bth 5 and 5 B f y < x + na th ntfac wh µ = 5..1S Pw s nductvly tansfd t a vhcl by usng a pa f paalll cnducts, cayng qual but ppstly dctd cunts, whch a bud slghtly blw th sufac. Assum that th ampltud f th cunt n ach cnduct angs fm 50 t 100 A ms, and th cnt-t-cnt dstanc btwn th cnducts s 0.15 m. If th fquncy s aund 0 t 5 kz, stmat th magntud f th magntc fld dctly btwn th cnducts alng th sufac. Is ths fld lvl saf? Pvd a cpy f th spcfc fnc usd t dtmn ths magntc fld xpsu lmt..1c As n ths chapt f th th-phas cllna systm, plt th ampltud f th x and y cmpnnts f th th-phas symmtcal ln dstbutn shwn n Fgu 1 at a dstanc f 4d n all dctns aund th gn. Lt d = 4.1 mm and I = 10 A. Thn, als as n ths chapt, plt th maxmum and mnmum pssbl magntuds f th flux dnsty n mg alng th x and y axs. Cmpa ths sults t th cllna th phas systm. Assum that th th cnducts a paalll t th z axs and lng cmpad t th spacng. Th cnducts a cntd abut th gn and qually spacd. Cpyght 00 by Knnth L. Kas, Vsn 10/05/05

4 4 ELECTROMAGNETIC COMPATIBILITY ANDBOOK y I 0 d d x I 10 d I 10 Fgu 1.14 If a hgh-fquncy plan wav hts (.., ncdnt t) a human bdy, ds th lctc magntc fld pntat th bdy and ptntally ntact wth any gans?.15 Can stay magntc flds fm a mt pass thugh th mt husng? Explan usng th apppat quatns n ths chapt..16c Slct a magntc matal f a bdy sut t ptct a csmtlgst aganst magntc flds fm a hady f 0.1 µt ms f fld s accptabl. Assum th xtnal magntc fld na th hady s 10 µt ms at 60 z. Dtmn th thcknss f th bdy sut. Th bdy sut matal shuld nt satuat..17 Th magntc fld nsd th cavty f a magntc sphcal shll f nn adus b, ut adus a, and latv pmablty µ whn placd n a unfm magntc fld f stngth s 9µ = b 9µ ( µ 1) 1 a Dtmn an appxmatn f ths quatn whn th thcknss s small (but nt z) and th latv pmablty s nt ncssaly lag..18 Th magntc fld nsd th cavty f a magntc cylndcal shll f nn adus b, ut adus a, and latv pmablty µ whn placd n a unfm magntc fld f stngth at ght angl t t s 4µ = b 4µ ( µ 1) 1 a Dtmn an appxmatn f ths quatn whn th thcknss s small (but nt z) and th latv pmablty s nt ncssaly lag..19 An appxmatn smtms sn f th magntc cylndcal shll quatn gvn n Pblm.18 s Cpyght 00 by Knnth L. Kas, Vsn 10/05/05

5 ELECTROMAGNETIC COMPATIBILITY ANDBOOK 5 a µ Dv ths xpssn and dtmn ts ang f usfulnss..0 F magntc cylndcal shlls, f Q = ωl R, wll th shldng ffctvnss ncas dcas as th Q f th shll ncass? Explan fm a ccut s pspctv..1 Th gnal latnshp btwn th ntnal and xtnal magntc flds paalll t th axs f a magntc thn-walld cylnd f latv pmablty µ, ppagatn cnstant γ, thcknss, and adus a (much gat than ) s ( ω ) 1 ( ) = ω csh ( ) γ a γ + snh ( γ ) Statng wth ths xpssn, vfy th lw-fquncy and hgh-fquncy xpssns gvn n ths chapt. [Cly]. Th gnal latnshp btwn th ntnal and xtnal magntc flds tansvs t th axs f a magntc thn-walld cylnd f latv pmablty µ, ppagatn cnstant γ, thcknss, and adus a (much gat than ) s ( ω ) ( ) µ 1 = ω 1γ a µ csh µ γ a ( γ ) + + snh( γ ) Statng wth ths xpssn, vfy th lw-fquncy and hgh-fquncy xpssns gvn n ths chapt. [asslgn]. Dtmn th pmablty, cnductvty, and thcknss f bdy sut aund a ndvdual usng an lctc dll f 0.1 mt f fld s accptabl. Assum th xtnal magntc fld na th dll s.5 mt at 60 z..4 Wuld yu xpct th magntc flux dnsty na a ha dy t ncas, dcas, man th sam whn th hgh-hat buttn s dpssd? Explan..5 Th dnsty qumnts f a ccut dctats that a slnd must b cls t a snstv cmpnnt. What lw-cst mthd s avalabl t duc th magntc fld cuplng btwn th slnd and cmpnnt?.6c Th avag 60 z magntc fld fm a fgat s mt s.6 mg at 10.5". At a dstanc f 4" th fld dps t 1.1 mg. At a dstanc f 48" th fld dps t 0.4 mg. At what at s ths fld dcasng? Explan ths bhav..7c Th magntc flux dnsty alng th axs f a cylndcal magnt unfmly magntzd alng th axs f th magnt s Cpyght 00 by Knnth L. Kas, Vsn 10/05/05

6 6 ELECTROMAGNETIC COMPATIBILITY ANDBOOK B z L L z + z Bs = + L z + z + + L m m wh B s s th magntud f th satuatn nductn, L s th ttal lngth f th magnt, and m s th adus f th magnt. Shw that th fld vas as 1/z fa fm th magnt. Th magnt s alng th z axs cntd abut z = 0. Why ds th fld nt vay as 1/z fa fm th magnt? F a 1" lng magnt wth a damt f /16", magntzd t a satuatn lvl f 1.5 T, plt n th sam st f axs th flux dnsty f 0 < z < 10L usng bth pvus xpssn and th as 1/z appxmatn and cmpa th sults. Th pvus quatn cmpas wll wth masumnts at dstancs gat than L fm th fac f th magnt and f lag lngth-t-damt ats. [Watsn; La].8 Is a gd magntc fld shld usually a gd lctc fld shld? Is th cnvs tu?.9 Th pctu tub f a TV s bng ntfd wth by naby spak magnts. Pv n n cst slutn and n lw-cst slutn t mdy ths pblm..0 w ffctv s th wat suundng a slf-cntand, batty-patd a pump n a fsh tank n ducng th magntc and lctc flds gnatd by th pump? Can th wat actually ncas th magntc flds? Cpyght 00 by Knnth L. Kas, Vsn 10/05/05

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