UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 14 Prof. Steven Errede LECTURE NOTES 14

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1 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede LETURE NOTES 4 THE MAROSOPI MAGNETI FIELD ASSOIATED WITH THE RELATIVE MOTION OF AN ELETRIALLY-HARGED POINT-LIKE PARTILE An electically chaged pint-like paticle f chage + (> 0 ving in the lab fae with elative velcity v (with espect t a fixed cdinate syste, igin ϑ geneates an appaent slenidal agnetic field B in the lab efeence fae ( cf with paticle s wn efeence fae: B = 0 thee! F the nn-elativistic case (i.e. f v << c {c = speed f light in vacuu}, B f a ving pint-chaged paticle with electic chage is: the stength f ( B v E c ( ( = ( whee: E ( = ε = electstatic field f chaged paticle in its wn est fae! μ B = v = v (using c = ε ε 0c μ = = bs. pt sc. pt and: = = since: = = Then: ( whee: ( Side View: v = vz B B ϕ Face View: > 0: ( =+ ( x ϕ B B ϕ B = B ϕ = B ϕ F + > 0: ( =+ ( F < 0: ( ( ( ( v = vz, z ŷ ut f page The acscpic B -field assciated with a ving electically chaged paticle is a slenidal B = B ϕ! field, i.e. ( ( Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

2 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Take css-pducts e.g. v and v t deteine the diectin f B-field at bsevatin/field P and P(, as shwn in the figue belw: B ( ( = B y Lcal Oigin, ϑ Obsevatin x ẑ Pint P( v = ŷ = v Obsevatin v = vz B =+ B y pints ( Pint ( P ( ( μ μ ( = N = ( = Ap B v v Teslas This is the agnetic field bseved in the lab fae due t a pint-like paticle with electic chage ving with elative velcity v << c in the lab fae. By delibeate cnstuctin (hee, = and = ae (entaily t v In geneal: v = v sin Θ ϕ, whee Θ = angle between v and (= 90 hee. haged paticle s velcity vect is v = vz and vect lies in x-z plane (hee. The abve pix shws the situatin at the s-called distance f clsest appach f the pint-like P, then = = iniu, and chaged paticle t the bsevatin/field pint ( ( 0 0 Θ= 90, sinθ= sin 90 =. Let s say that this ccus at tie t = 0. μ v Then (hee at the distance f clsest appach: B (, 0 t = = ϕ At tiet = the chaged paticle is infinitely fa away, lcated at ( xyz,, = (0,0 taveling with v = vz. Then B (, t = = 0. Siilaly, at tiet = + the chaged paticle is als infinitely fa away, lcated at ( xyz,, = (0,0 + taveling with v = vz B, t =+ = 0. B -field vs. = : t vtz vt z ( = + = ( + in in. Then ( B -field vs. t: B t = (, 0 B ( t, 0 t = = z t = 0 = z in t = + = + z Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

3 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede = = = = At the distance f clsest appach (t = 0: in in = iniu. Lcal Oigin, ϑ ẑ x ŷ v = vz Suce pint S( = (lies in x-z plane {hee} bs suce pint pint Obsevatin/field pint P( (lies in x-z plane {hee} μ v μ v B (, 0 t= = ϕ = yat the distance f clsest appach (t = 0. Nte the siilaity between E and B -fields f a pint-like electically chaged paticle: E ( = = Bth E and B -fields decease as f pint ε ε μ v μ v B ( = = chage, due t flux law f vitual phtns!!! The Macscpic Magnetic Field B( due t a Steady uent I Flwing in an Infinitely Lng Filaentay Wie The pinciple f linea supepsitin tells us that we can view the acscpic agnetic field due t a steady cuent I flwing in a lng filaentay wie as the linea supepsitin f agnetic field cntibutins assciated with each f the individual electic chages flwing in the filaentay wie that icscpically cnstitute/ake up the acscpic steady cuent I: N N i ( t B I ( μ, t = B (, t = v i i= i= i ( t Whee we have assued (hee that all chage caies have the sae electic chage and ve alng the filaentay wie with the sae velcity v = vz. In the liit that the nube f electic chages that ae icscpically invlved in aking up the acscpic cuent I beces s lage that the spacing between adjacent chage caies beces eely sall, e.g. that f atic diensins, ~ O(0 Angsts = n, then if we ae nly inteested in the net/ttal acscpic agnetic field assciated with acscpic distance scales, e.g. ~μ(and lage, then we can safely eplace the suatin in the abve expessin by an integal ve a cntinuu f infinitesial electic chage cntibutins d all ving alng the filaentay wie with velcity v = vz. F steady acscpic cuents I fcn ( t the tie dependence dps ut/vanishes in the 4 N ~ Ο 0 ttal chage caies, then the fluctuatins ae icscpic aveaging pcess! If ( ( σ N = N ~ Ο 0, thus factinal fluctuatins σ N N N ~ ( 0 = Ο - negligible!!! Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

4 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Thus f a acscpic, steady electic cuent I flwing in a lng (ne-diensinal db due t a filaentay wie (f infinitesial thickness, the infinitesial cntibutin ( acscpic cuent I = λ feev= dv/ d flwing in an infinitesial length d f cuentcaying filaentay wie with assciated infinitesial suce chage inceent d = Id / v is: μ db ( = I d ( whee = nnectin: Suce Pint ( S( I = v Id = dv Id = Id (since I d d = vdt d d = λd = λvdt λ line chage (ulbs/ = I = d dt = λv B( Id = λd v = dv I, v, d ( Lcal Oigin, ϑ P(, Obsevatin/field Pint d P n.b. figue dawn f ( cntibutin clsest t bsevatin/field pint ( b μ bd( μ bd ( B = db = I = I a a n.b. assued a Then: ( ( t be cnstant eveywhee n.b. If I cnstant eveywhee, I ( d= I( d I ( I (, d ( Suce Pint, S( b I ( = then I ust eain inside the integal! a B( ẑ P( Obsevatin/field pint x ϑ ŷ In geneal this integal can ften be difficult t pef analytically, but it can be easy t d n cpute, e.g. using nueical integatin techniues! 4 Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

5 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede μ b ( b I μ I( d I ( d= I( d B( = d = a a Suce pint, μ b d( If I( = cnstant eveywhee, this expessin siplifies t: B( = I a Giffiths Exaple The Macscpic Magnetic Field Due t a Steady uent Flwing in an Infinitely Lng Wie: Find the agnetic field B( a pependicula distance away f an infinitely lng staight filaentay wie caying a steady / cnstant / unif cuent I. μ d ( d ( + μ + x B( = I = I Obsevatin / P( Field Pint (in x-z plane θ = = + = = = + Lcal igin ϑ ( ŷ ut f page = F this pble, which way des ( β α d z, I = Iz ( Suce Pint S( d pint? Nte that: d( = d ( z and that lies in the x-z plane. By the ight-hand ule, the css pduct: d = dz ( ( ( xx+ zz = xd z x + zd z z = xdy = y = 0 Thus, d( = xdy pints in the ŷ diectin (i.e. ut f the page hee, because the field pint P( (as dawn abve lies in the x-z plane. Hweve as entined ealie, d( actually pints in the ϕ -diectin (n.b. ϕ = y when ϕ = 0. What is the agnitude f d( d d α? F the definitin f this css pduct, d and. ( sin whee α = pening angle between ( Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved. 5

6 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede π π Geetically: α = π β and als: π = θ + β +, hence β = θ π π α = π β = π + θ = + θ, thus: = 0 π π π sinα = sin + θ = sin csθ + cs sin cs θ = θ d = d sinαϕ = d csθϕ F the abve figue, nte that: csθ = ( euivalently: = csθ ( Then: ( d μ B + = I with: = + and: ( d = dcsθϕ = d ϕ μ + d I Then: B( I μ μ I = ϕ = ( ϕ = ϕ + π μ I B( = ϕ f an infinitely lng filaentay wie caying a steady cuent I. π An altenative deivatin f this esult: Since = csθ, we can euivalently expess d in tes f an integal ve θ : dsinα = d csθ but nte that: = tanθ ( euivalently: tanθ = csθ Nte als that: = cs θ = d = dθ is the Jacbian f the tansfatin f d dθ cs θ π π Nw when =+, θ = and when =, θ = Then: dcsθ π μ d ( + μ + cs θ B( = I = I π csθdθϕ cs θ π π μ + cs θdθ μ I + = I cs d π ϕ π θ θϕ = μ I π π μ I μ I = sin sin ϕ [ ] ϕ ϕ = = B( μ I = ϕ f an infinitely lng filaentay wie caying steady cuent I. π falls ff as whee = distance f the infinitely lng wie. Nte that B( 6 Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

7 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede The Macscpic Magnetic Field due t a Steady uent Flwing in a Finite Length Wie: What is the B -field assciated with a steady cuent I flwing dwn a finite length filaentay wie (f length? Sae cnfiguatin/geety as befe/abve f infinite length wie. ( μ I d ϕ μ I θ Bfinite = = cs d wie θ ( + μ I B ( ( sin sin finite = θ θ ϕ wie Whee: sin θ =, sinθ = + + θ θϕ P( θ θ I ϑ = ẑ Fall 06 P45 Student Michael Wicze ade plts f the agnitude f the B-field alng finite length wie f 5 diffeent chices f cuent: M. Wicze s cntu plt f agnitude f B-field f steady cuent I flwing dwn finite-length wie: Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved. 7

8 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Giffiths Exaple The Macscpic Magnetic Field Due t icula Steady uent On the Syety Axis f a uent Lp: A cicula filaentay cuent lp f adius R lies in x-y plane, with a steady cuent I ciculating anti-clckwise (viewed f abve, as shwn in the figue belw: I ẑ Obsevatin/Field Pint P = P z ( ( z = R Lcal Oigin,ϑ ϕ I ŷ x μ d ( Blp ( = I : ( d ( Suce Pint S( (lies in x-y plane Blp = I μ d ( What is ( d f this paticula pble? By the ac-length fula S Rθ, and: d Rd x y ( = ( ϕ sinϕ( + csϕ = Rdϕ ( sin ϕx + csϕy d = d = Rdϕ = the infinitesial segent f cuent lp ( ( and: = z since = z = zz = z = zz R csϕx+ sinϕ y ( ϑ R dϕ R d ( can see this easily e.g. when ϕ = 0 and and = R( csϕx+ sinϕ y π ϕ = 8 Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

9 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Thus: ( d( = d( ( = d( ( z = d( zz R( csϕx+ sinϕy = Rdϕ( sinϕx+ csϕy zz R ( csϕx+ sinϕy ( ( Nw: x y =+ z y x = z x x = 0 y z =+ x z y = x y y = 0 z x =+ y x z= y z z= 0 Vey useful table Thus: Finally: ( d( = Rdϕ zsinϕ( x z + zcsϕ( y z = y =+ x ( + Rsinϕcsϕ x x = 0 ( R cs ϕ y x = z + Rsin ϕ( x y Rsinϕcsϕ( y y =+ z = 0 ( d( = Rdϕ { + zsinϕy+ zcsϕx+ Rcs ϕz+ Rsin ϕz} { sin cs cs sin } = Rdϕ + z ϕy+ z ϕx+ R ϕ+ ϕ z { sin cs } = Rdϕ + z ϕy + z ϕx + Rz d = Rd z x+ y + Rz ( ( ϕ ( csϕ sinϕ { } Nw: = = = z + R and = + z R and als = ( z + R Then: μ ( ( cs sin d μ Rd z x y Rz π ϕ ϕ ϕ Blp ( = I = I 0 + { + + } ( z R μ Rz π π μ IR π = I ϕdϕx ϕdϕy dϕz cs sin ( z R π + ( z + R n.b each f these tes will individually vanish / cancel when integated ve all ϕ - i.e. f 0 ϕ π!! μ 0 Rz = = 0 π lp ( = + sinϕ 0 x cs B I + ( z R IR ϕ y π μ + π z + R ( 0 z Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved. 9

10 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Finally: μ R B ( lp = I z ( z + R The B -field n syety axis f a steady cuent-caying lp f adius R pints in ẑ diectin Nte als that B ( lp = zz falls ff as z just as a agnetic diple field shuld!! i.e. B ( lp = zz is the n-axis B -field f a physical (i.e. spatially-ended agnetic diple! A lp caying a cuent I geneates a agnetic diple field!!! B ( The B -field due t a cuent lp, n the syety ( ẑ -axis f the lp: -D Side View: ẑ B, na, An at I z B ( plane f page A I R at plane f page uent I cing ut f page hee A = π R = css-sectinal aea f cuent lp uent I ging int page hee Nte that at the bsevatin pint = zz, the nal cpnents f B (i.e. thse t the ẑ -axis cancel, wheeas the paallel cpnents f B (i.e. thse t the ẑ -axis add! The vect aea f the cuent-caying lp is A= An (whee n is defined by ight-hand ule assciated with diectin f cuent ciculatin. The Magnetic Diple Ment f a uent-aying Lp f ss-sectinal Aea A The agnetic diple ent is defined as: IA ( = π R Iz hee (SI units: Apee-etes The vect aea f the lp is: A R = π n ( n = z hee, the scala aea f the lp: = π R lp The Magnetic Field n the Syety Axis f an N-Tun uent-aying Lp: Instead f a single cuent-caying lp, what wuld be the B -field assciated with N tuns (f vey fine wie f the plana cuent-caying lp? Using the pinciple f linea supepsitin: ITOT = NI. Alp 0 Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

11 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Thus n the syety axis f an N-tun plana lp f adius R caying a steady cuent I: R R B tun lp ( -tun lp ( N z = NB z = N μi z = μ NI z f adius R f adius R ( z R + ( z + R The supepsitin pinciple als wks f / is valid f agnetic phenena since it is intiately cnnected t electic phenena via cn / sae icscpic physics! The Macscpic Magnetic Fields Assciated with Line, Suface and Vlue uents: F line, suface and vlue cuents, we suaize thei cespnding agnetic fields belw: μ ( ( Id μ I Line uents: Bline ( = = d 4 4 cuent π π μ ( K Suface uents: Bsuface ( = da 4 S cuent π whee =, and = = μ ( J Vlue uents: Bvlue ( = dτ 4 v cuent π and = = n.b. The pied vaiables dentes integatin ve the elevant suce cuent distibutins. Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

12 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede The Bit-Savat Law: The alculatin f Fces n uent-aying nducts Due t Othe uent-aying nducts We have peviusly deived (in P45 Lectue Ntes that the net acscpic agnetic fce F n a cuent-caying wie iesed in an enal agnetic field B ( was: If I = I( F df = = I d B iff I = cnstant/steady/unif cuent! ( ( ( is nt cnstant/steady/unif in space, then e geneally: F = I B d ( ( Nw let us cnside what F wuld be if B ( is due t a nd cuent-caying lp. F siplicity s sake, we will assue that all cuents invlved ae steady cuents. Tw Apeian uent Lps: Lp #: Lp #: df ( = I d ( B ( I d ( Lcal Oigin,ϑ I = I d ( I The infinitesial fce df ( acting n a line segent d ( acscpic agnetic field B ( at the pint flwing in lp #. n lp # is due t the net that is ceated by the steady cuent I Thus: df ( = I d ( B( = infinitesial fce acting n line segent d ( f lp # due t net B -field f secnd cuent lp # Then the net fce n the iginal steady cuent-caying lp (lp # is: F df ( I d ( B ( = = = agnetic field at suce pint (n lp # at d due t nd cuent I flwing in lp # ( Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

13 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede What is the net, acscpic B -field at the pint aising f the steady cuent I flwing in the secnd cuent lp #? It is: μ d ( B ( = I whee ( and = Then: F = ( ( ( df = Id B μ d ( = Id ( I μ d ( = II d ( O: F μ = I I d ( d ( ( Nw, Newtn s d Law f Mtin hlds hee, i.e. that: F = F μ ( ( ( d d Thus: F = I I = whee ( ( Bit-Savat Law = = = and This elatin can be / is easily btained by peuting indices and Explicit pf - Use the vect tiple css-pduct identity: A B = B( Ai ( Ai B Expand ut the integal f F : d ( ( d ( d ( d ( i d id = Why des d d i ( ( ( = 0??? ( ( ( ( = 0!!! Why? d ( ( d ( d ( ( d ( i i ( d ( i = = d ( n.b. Switched de f integatin This integal is caied ut nly ve clsed cntu f lp # Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

14 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Nw: i i = ( d ( d ( ( since = But what is d ( i?? d ( i = dcsθ whee θ = pening between d ( d ( i = d csθ = d ( csθ d ( = d ( d d csθ and θ d ( d d = ( ( i Then: d i d i d ( ( ( ( = = d But we knw that 0 aund an (abitay clsed lp/cntu!!! e.g. Δ d d V = E ( d = i i 0 ε 4 πε = Thus finally: μ ( ( ( d id F = I I But: ( = ( And: = = = F = F And: AB i = BA i μ ( ( ( d id Thus: F = I I Q.E.D. f a pint electic chage!!! 4 Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

15 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Exaples f the Use f the Bit-Savat Fce Law: The Net Fce Between Tw Infinitely Lng, Paallel Wies aying Steady uents I and I Refeing t the figue belw, let s calculate the net fce n wie # (caying cuent I = Iz due t the enal agnetic field pduced by the (paallel wie # (caying cuent I = Iz lcated a pependicula distance = d away f wie #. Bth wies ae infinitely lng. Slenidal B -field I = Iz = Iz ẑ f wie # at f wie # hee wie # pints d d x ŷ pints int page ut f page d d wie # wie # Slenidal B -field The agnetic field Bwie # ( = d aising f a steady cuent I = Iz flwing in wie #, at a pependicula distance = d away f wie # is: μ Id μ Id μ I See pages 5-6 f B # ( wie = d = = = ϕ π d these lectue ntes The Bit-Savat Law: The net agnetic fce n cuent-caying wie # due t B-field f cuent-caying wie # is: F = ( ( ( df = I d Bwie# = d # wie = I d B = d = I d B = d But hee: ( ( ( ( ( wie# wie# d = dzz μ I μ II μ II F = I dzz z dz z dz ϕ = ϕ = ϕ # wie π d π d π d ( ( Which way des ( ẑ ϕ pint? If -wies ae in the y-z plane and sepaated by distance d: Wie # Wie # ẑ = Iz = Iz d Bwie # ( y = d y = d y = 0 ŷ pints in the ϕ x ϑ = Lcal Oigin ϕ -diectin Nte that ϕ at wie # pints in the x -diectin ( ϕ f the page. x at wie # i.e. ϕ and x bth pint ut Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved. 5

16 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede z = z x =+ y (efeing t useful table n p. 8 f these lectue ntes. μ II Then: F ( = dy = y dz # wie π d Then: ( ϕ But if cuent-caying wie # is infinitely lng, then: dz = dz = + = The net agnetic fce F ( n an infinitely lng wie caying cuent I = Iz due t wie# anthe infinitely lng, paallel wie caying cuent I = Iz a distance away is infinite!!!!!! L dz Hweve, nte that the agnetic fce pe unit length is finite: = L 0 μ II Define fce pe unit length as: f ( ( = dy F = d L = y wie# wie# π d Then by Newtn s d Law, the agnetic fce pe unit length acting n wie # due t the B - field a pependicula distance d away f wie # is: μ II μ II f ( 0 ( ( = F =+ d L = y = y wie# wie# π d π d Thus we see that: F = F : f = f as they ust by Newtn s d Law wie# wie# wie# wie# Nte that: f = ( + y and f = ( y ~~~ wie# wie# ~~~ F evey actin thee is an eual and ppsite eactin i.e. paallel wies caying cuents in the sae diectin attact each the! paallel wies caying cuents in ppsite diectins epel each the! 6 Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.

17 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Line I = λv Macscpic Magnetic Fces and Tues n Suface uent Densities K Vlue J. Mving pint chage lcated at F = v( B( τ = F = v( B( In an Extenal Magnetic Field B ( in a enally-applied agnetic field B ( :. Filaentay line cuent caying cnduct in enally-applied agnetic field B ( F I ( B ( d Id ( = = B ( τ = df ( = Id ( B ( :. Suface/sheet cuent densities in enally-applied agnetic field B ( F = K( B( da S τ = df( = K( B ( da S S 4. Vlue cuent densities in enally applied agnetic field B ( F = J( B ( dτ v τ = df ( = J( B ( dτ v v : : If B ( is e.g. due t a nd cuent-caying lp (lp #, with cuent I / K / J : μ Id ( i.e. B ( = whee ( and = then plug this expessin f B ( int any f the abve elatins t btain Bit-Savat fulae f F, τ, etc. Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved. 7

18 UIU Physics 45 EM Fields & Suces I Fall Seeste, 007 Lectue Ntes 4 Pf. Steven Eede Macscpic Magnetic Field Intensities B( Pduced by a Mving Pint hage Line/Suface/Vlue uent Density n.b. The pied vaiables in the fulae belw dente integatin ve the elevant suce cuent distibutins, and, thus: = = and = =. cf t paallel expessin f E-field:. B -field due t a ving pint electic chage ( v c: ( μ μ B ( = v( v( fee = μ μ ( = v ( = v (. B -field due t a line cuent I ( (Aps: μ d Id B I ( = = μ Id Id = = ( μ ( ( ( μ ( (. B -field due t a suface/sheet cuent K( (Aps/: μ ( ( ( K μ K B K ( = da da = S S μ K ( μ K ( ( = da = da S S 4. B -field due t a vlue cuent J( (Aps/ : μ ( ( ( J μ J B J ( = dτ dτ = v v μ J ( μ J ( ( = dτ = dτ v v E = ε ( λ ( Eλ = d ε ( σ ( Eσ = da ε S ( ρ ( Eρ = dτ ε v ( 8 Pfess Steven Eede, Depatent f Physics, Univesity f Illinis at Ubana-hapaign, Illinis All Rights Reseved.