ECE 37: Electricit n Mgnetism Spring nstructor: J.D. Willims, Assistnt Professor Electricl n Computer Engineering Universit of Alm in untsville 6 Optics Builing, untsville, Al 35899 Phone: (56) 8-898, emil: john.willims@uh.eu Course mteril poste on UA Angel course mngement wesite Textook: M.N.O. Siku, Elements of Electromgnetics 5 th e. Oxfor Universit Press, 9. Optionl eing:.m. She, Div Gr Curl n ll tht: n informl text on vector clculus, th e. Norton Press, 5. All figures tken from primr textook unless otherwise cite.
Topics Covere Chpter 7: Mgnetosttic Fiels Biot-Svrt s w Ampere s Circuit w Applictions of Ampere s w Mgnetic Flux Densit Mxwell s Eqns. For Sclr Fiels Mgnetic Sclr n Vector Potentils Derivtion of Biot-Svrt s w n Ampere s w omework: 8/7/ All figures tken from primr textook unless otherwise cite.
ntrouction to Mgnetic Fiels For the next two n hlf chpters we focus our ttention on mgnetic fiels n electrosttics we stuie ivergent potentil fiels n mgnetosttics, we will exmine solenoi rottionl fiels Also wheres, electrosttic fiels re generte sttic chrges, mgnetosttic fiels re generte sttic currents (chrges tht move with constnt velocit in prticulr irection) There re severl similrities etween electrosttic n mgnetosttic fiels For exmple, s we h E n D for electrosttics, we now use B n to exmine mgnetic sstems Our stu of these fiels llows us to evlute n solve for tremenous numer of electric n electromechnicl evices. Furthermore this stu, will provie the sis for formulting n universl theor of Electromgnetic Fiels tht is utilie in lmost ever spect of electricl engineering 8/7/ 3
Anlog Between Electric n Mgnetic Fiels Bsic ws Force w Source Element Fiel intensit Flux ensit eltionship Between Fiels Potentils Flux Energ Densit Poisson s Eqn. Electric Mgnetic v E r V E D w t V C CV Q S D r V V E E D m C S D m V l V E Q QE F Q S D r Q Q F / ) / ( J A B w t S B A J V B m W S B m A l Qu B Qu F B E m r ),( / ) / (
Biot-Svrt s w The ifferentil mgnetic fiel intensit,, prouce t point P, the ifferentil current element,, is proportionl to the prouct n the sine of the ngle etween the element n the line joining P to the element n is inversel proportionl to the squre of the istnce,, etween P n the element 3 sin 5
Current Densit One efines ifferentil current se on the geometr of the current element eing investigte Evlution of mgnetic fiel intensit,, using these three current ifferentils is 6 Jv KS v S Jv KS
letting cos cos sin csc csc csc cot 3 Fiel From Strit Current Crring Filment The fiel is etermine for strit filment of current in mnner ver similr to tht of the electric fiel etermine from line chrge 7 ine from = to 3/ 3 ine from = - to
Fiel From ing of Current Crring Filment () Agin, the fiel is etermine for ring filment of current in mnner ver similr to tht of the electric fiel etermine from circulr line chrge 8 h x h,,,, 3 Dir not present in ue to smmetr h h h h 3/
3/ 3/ 3/ h h h letting Fiel From ing of Current Crring Filment () Agin, the fiel is etermine for ring filment of current in mnner ver similr to tht of the electric fiel etermine from circulr line chrge 9 h h 3/ ine from = - to B smmetr, the terms sum to ero
Fiel From Solenoi A solenoi is coil or wire pssing current cross it with uniform rius n numer of loops, N. One cn etermine the fiel within solenoi summing ech of the respective mgnetic fiels ue to ech loop. Thus the totl fiel nwhere in the solenoi m e foun s: n where n if N / l l n cos cos N l ine from = - to Derivtion: tn n n 3/ csc sin n N l sin n sin cos cos n 3/ 3/
Ampere s Circuit w Ampere s lw: The line integrl of roun close pth is the sme s the net current,, lose the pth Similr to Guss lw since Ampere s lw is esil use to etermine when the current istriution is smmetricl Ampere s lw AWAYS hols, even if the current istriution is NOT smmetricl, however the eqution is tpicll use for smmetric cses ike Guss n Coulom s ws, Ampere s lw is specil cse of the Biot-Svrt lw n cn e erive irectl from it. Appling Stokes s theorem provies lterntive solution methos S J S J S S Definition of Current provie in Chpter 5 Mxwell s 3 r Eqn.
A simple ppliction of Ampere s lw cn e use to esil erive the mgnetic fiel intensit from n infinite line current Applictions of Ampere s Circuit w
Consier n infinite sheet of current in the = plne with uniform current ensit, K=K K x x 3 3,, Ampere s Circuit w: nfinite Sheet of Current 3,, K K K from x x o Ampere s lw n integrl summtion n K Thus, for n infinite sheet Appl Ampere s lw
Consier infinite sheets of current in the = n = plnes with uniform current ensit, K=- x A/m n K= x A/m respectivel At point etween the two prllel pltes, P (,,) where < ( = ) < At point outsie of the pltes, P(,-3,) where ( = ) > > Ampere s Circuit w: nfinite Prllel Plte Cpcitor m A m A K m A K x n x n / / 5 / 5 m A m A K m A K x n x n / / 5 / 5
Ampere s Circuit w: nfinitel ong Coxil Cle One cn use Ampere s lw to irectl show the shieling of mgnetic fiels using coxil wires 5
One cn use Ampere s lw to irectl show the shieling of mgnetic fiels using coxil wires S J S J S J Ampere s Circuit w: nfinitel ong Coxil Cle 6
Ampere s Circuit w: nfinitel ong Coxil Cle One cn use Ampere s lw to irectl show the shieling of mgnetic fiels using coxil wires 7
One cn use Ampere s lw to irectl show the shieling of mgnetic fiels using coxil wires t t t t t t t J S J t Ampere s Circuit w: nfinitel ong Coxil Cle 8
One cn use Ampere s lw to irectl show the shieling of mgnetic fiels using coxil wires t t t t,,,, Ampere s Circuit w: nfinitel ong Coxil Cle 9
Ampere s Circuit w: Toroi A toroi is solenoi turne in on itself like onut N N, o o pprox N o N l
Mgnetic Flux Densit Mgnetic Flux ensit, B, is the mgnetic equivlent of the electric flux ensit, D. As such, one cn efine Similrl, Ampere s w is An the Mgnetic flux through surfce is B The mgnetic flux through n lose sstem is B S B S where S B B v S 7 BS Definition of solenoi fiel n Mxwell s th eqn. / m S S
Mgnetic Flux Densit Unlike electrosttic flux however, mgnetic flux lws follows close pth n fol in on themselves. This simple sttement hs profoun conseques. n electrosttics, we cn esil efine point chrge in which electric fiels emnte to infinit. owever, the solenoi nture of the mgnetic fiel requires mgnetic flux to trvel from positive (north) to negtive (south) pole n it is not possile to hve single mgnetic pole t n time. There re NO mgnetic monopoles, stipulting tht n isolte mgnetic chrge DOES NOT EXST The minimum fiel requirement for mgnetics is ipole.
Mxwell s Eqns. for Sttic Fiels Differentil Form ntegrl Form emrks D B E v J DS S S B S E Guss s w Nonexiste of the Mgnetic Monopole Conservtive nture of the Electric Fiel Ampere s w S 8/7/ 3 v v J S
Mgnetic Sclr & Vector Potentil n Chpter -6, we iscusse severl electrosttic prolems tht were more esil solve using the electric potentil to efine the electric fiel intensit, E. The sme pproches lso reuce the ifficult in exmining mgnetic fiel prolems s well s couple fiel prolems tht will e iscusse in the n course on electromgnetic fiels. eclling from chpter three tht solenoi fiel cn e escrie its sclr n vector potentils, we cn efine mgnetic fiel using the following requirements. E V V A Just s, we cn efine mgnetic sclr potentil V m relte to when the current ensit is ero s Vm, J J V V, J m 8/7/ m
Mgnetic Sclr & Vector Potentil The requirement for solenoi fiel (n Mxwell s th lw of electrosttics) stipultes B An we cn therefore efine mgnetic vector potentil, A, s B A Just s we efine the Electric Potentil s We cn efine the Mgnetic Vector Potentil s A A A Q V r for ine Current for Surfce Current for Volume Current 8/7/ 5 KS Jv
Mgnetic Sclr & Vector Potentil One cn lso erive these expressions irectl from the mgnetic fiel ' B 3 Where is the istnce vector from the line element t the source to the fiel point (x,,) r r' x x' ' ' 3 Yieling owever, Del opertes on(x,,) n is function of (x,, ) thus B ' Appling _ the _ ientit ff f F f F ' ' ' ' ' ' ' B A 8/7/ 6
Mgnetic Sclr & Vector Potentil Appling Stokes s Theorem provies some rther useful prcticl reltions, incluing ut not limite to the totl mgnetic flux through n re, S, lose contour,. B S A S A s A s 8/7/ 7
ecll the sic vector ientit Derivtion of Biot-Svrt s w 8/7/ 8 B x x r r B B A A A 3 ' ' ' ' ' ' ' ' '
Derivtion of Ampere s w Derivtion of the Vector Poisson s Eqn. for mgnetic fiels n current ensit Derivtion of Ampere s w B A A A A A S S J 8/7/ 9 S A A J J S S S
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x x x x 5 3 5 3 5 5 3 5 5 6 9 3 5 3 5 5 5 3 cos cos cos cos 5 3 6 6 5 3/ cos cos cos cos Fiel From n Shpe Current Crring Filment The fiel is etermine for strit filment of current in mnner ver similr to tht of the electric fiel etermine from line chrge 3 3/ 3