Magnetostatics Bar Magnet. Magnetostatics Oersted s Experiment

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1 Mgneosics Br Mgne As fr bck s 4500 yers go, he Chinese discovered h cerin ypes of iron ore could rc ech oher nd cerin mels. Iron filings "mp" of br mgne s field Crefully suspended slivers of his mel were found o lwys poin in he sme direcion, nd s such could be used s compsses for nvigion. The firs compss is hough o hve been used by he Chinese in bou 76 B.C Greeks found his iron ore ner Mgnesi, in wh is presen dy Turkey. I conined mgneie (Fe O 4 ), nd cme o be known s mgneic lodesone. Mgneic field in br mgne Mgneosics Oersed s Experimen A compss is n exremely simple device. A mgneic compss consiss of smll, lighweigh mgne blnced on nerly fricionless pivo poin. Mgneic Compss The mgne is generlly clled needle. One end of he needle is ofen mrked "N," for norh, or colored in some wy o indice h i poins owrd norh. On he surfce, h's ll here is o compss In 1600, Willim Gilber of Englnd posuled h mgneic lodesones, or compsses, work becuse he erh is one big mgne. The mgneic field is genered by he spin of he molen inner core. The norh end of compss needle poins o he geogrphic norh pole, which corresponds o he erh s souh mgneic pole. Mgneic Compss The Erh cn be hough of gignic br mgne buried inside. In order for he norh end of he compss o poin owrd he Norh Pole, you hve o ssume h he buried br mgne hs is souh end he Norh Pole. 1

2 Mgneosics Oersed s Experimen In 180, Hns Chrisin Oersed ( ) used compss o show h curren produces mgneic fields h loop round he conducor Oersed s experimen Mgneism nd elecriciy were considered disinc phenomen unil 180 when Hns Chrisin Oersed conduced n experimen h showed compss needle deflecing when in proximiy o curren crrying wire. I ws observed h moving wy from he source of curren, he field grows weker. Oersed s discovery relesed flood of sudy h culmined in Mxwell s equions in Mgneosics Bio-Svr s Lw Shorly following Oersed s discovery h currens produce mgneic fields, Jen Bpise Bio ( ) nd Felix Svr ( ) rrived mhemicl relion beween he field nd curren. The Lw of Bio-Svr is dh IdL (A/m) To ge he ol field resuling from curren, you cn sum he conribuions from ech segmen by inegring IdL H Noe: The Bio-Svr lw is nlogous o he Coulomb s lw equion for he elecric field resuling from differenil chrge de dq 1 1 ε 1.. (A/m)

3 Mgneosics An Infinie Line curren Exmple.: Consider n infinie lengh line long he -xis conducing curren I in he + direcion. We wn o find he mgneic field everywhere. We firs inspec he symmery nd see h he field will be independen of nd φ nd only dependen on ρ. So we consider poin disnce r from he line long he ρ xis. The Bio-Svr Lw IdL H An infinie lengh line of curren IdL is simply Id, nd he vecor from he source o he es poin is +ρ ρ The Bio-Svr Lw becomes H ( + ρ ρ ) + ρ Id. Mgneosics An Infinie Line curren Pulling he consns o he lef of he inegrl nd reliing h x 0 nd x ρ φ, we hve I ρφ d H + ρ The inegrl cn be evlued using he formul given in Appendix D dx x + x + I + ρ ρ ρ ρ + + ρ ρ + When he limi d I ρ + ρ ρ ρ ρ + 1 ρ + 1 x d I ρ ρ ρ ρ ρ ρ ρ + ρ ρ +

4 Mgneosics An Infinie Line curren I ρφ d H + ρ d Using I ρ + ρ An infinie lengh line of curren We find he mgneic field inensiy resuling from n infinie lengh line of curren is I φ H πρ H 1 ρ φ Direcion: The direcion of he mgneic field cn be found using he righ hnd rule. Mgniude: The mgniude of he mgneic field is inversely proporionl o rdil disnce. Mgneosics A ing of Curren Exmple.: Le us now consider ring of curren wih rdius lying in he x-y plne wih curren I in he + direcion. The objecive is o find n expression for he field n rbirry poin heigh h on he -xis. The Bio-Svr Lw IdL H H The differenil segmen dl dφ φ The vecor drwn from he source o he es poin is h Mgniude: ρ h + Uni Vecor: ( h ) ρ The bio-svr Lw cn be wrien s ( ) ( ) Idφ h I dφ h π π φ ρ φ ρ φ 0 φ 0 h + h + 4

5 Mgneosics A ing of Curren We cn furher simplify his expression by considering he symmery of he problem ( ) ( ) π π I dφ h I dφ h + φ ρ ρ H 4 φ 0 h π + φ 0 h + A priculr differenil curren elemen will give field wih n ρ componen (from φ x ) nd n componen (from φ x ρ ). Tking he field from differenil curren elemen on he opposie side of he ring, i is ppren h he rdil componens cncel while he componens dd. componens dd ρ Componens Cncel I H π 0 ( h + ) I H dφ H I ( h + ) A h 0, he cener of he loop, his equion reduces o H Mgneosics A Solenoid Solenoids re mny urns of insuled wire coiled in he shpe of cylinder. Suppose he solenoid hs lengh h, rdius, nd is mde up of N urns of curren crrying wire. For igh wrpping, we cn consider he solenoid o be mde up of N loops of curren. A solenoid To find he mgneic field inensiy from single loop poin P long he xis of he solenoid, from we hve The differenil moun of field resuling from differenil moun of curren is given by dh P di ' ( + ) The differenil moun of curren cn be considered funcion of he number of loops nd he lengh of he solenoid s N di Id ' h 5

6 Mgneosics A Solenoid Fixing he poin P where he field is desired, will rnge from o h-, or h NI d ' H h ' + h NI d ' h ' ( + ). This inegrl is found from Appendix D, leding o he soluion NI h H + h h + + A he very cener of he solenoid ( h/), wih he ssumpion h he lengh is considerbly bigger hn he loop rdius (h >> ), he equion reduces o NI H h 6