Electromagnetism Physics 15b

Transcription

1 lctromagntism Physics 15b Lctur #8 lctric Currnts Purcll Today s Goals Dfin lctric currnt I Rat of lctric charg flow Also dfin lctric currnt dnsity J Charg consrvation in a formula Ohm s Law vryon nows alrady right? I ll try to confus you by taling about rsistanc, rsistivity, conductivity... and conductanc as a bonus Watch out for th units! Gorg Ohm ( ) 1

2 Currnt Currnt = movmnt of somthing continuous Watr currnt in rivrs, ocans, fish tans Air currnt = wind W oftn masur currnts by th vlocity Thin wathr rport: Winds at 20 to 30 mph tc. Not always th bst way, though Watr flow Small v, larg A Larg v, small A Constant flow of watr through a pip may go fastr or slowr dpnding on th thicnss of th pip What s important in this cas is (mass of watr)/tim, which is givn by (dnsity of watr) x (vlocity) x (ara) lctric Currnt lctric chargs flow in a conductor lctric currnt = th amount of charg going through a cross sction of th conductor I Q t In cas th flow is changing with = tim, considr vry short priod dt [charg] su coulomb Unit: = in CGS, = ampr (A) in SI [tim] scond scond dq I = dt Gnraliz to arbitrary ara Thn imagin maing th ara vry small... 2

3 Currnt Dnsity Vry small ara a insid uniform flow of chargs Currnt is I = na u dnsity individual charg ara Dfin currnt dnsity by I = J a for infinitsimally small a Local dnsity of charg flow vlocity volum a u u a In x-y-z coordinats I x J = lim ˆx lim ŷ lim ẑ ax 0 a ay 0 x a az 0 y a z I y I z [currnt] su A Unit: = in CGS, in SI [ara] sc cm 2 m 2 Charg Consrvation I and J ar connctd by I = Apply this to a closd surfac I = th rat of charg flowing out of S Total charg insid S must b dcrasing J da S da S J I = dq nclosd dt = d dt V ρ dv Us th Divrgnc Thorm This is th local vrsion of charg consrvation div J = ρ t For a stady-stat systm (i.. no tim dpndnc) divj = 0 3

4 Currnt Carrirs In most mtals, fr lctrons mov around and caus lctric currnt Thy ar th currnt carrirs Sinc lctrons ar ngativly chargd, th dirction of th I currnt is opposit to th dirction of th lctron vlocity This is confusing it s all Bn s fault In gnral, currnt carrirs don t hav to b lctrons Salty watr conduct currnt with Na and Cl ions In smiconductors, positiv carrirs calld hols coxist with lctrons Carrir Dnsity & Vlocity For a singl carrir spcis, I = n a u = a J If mor than on typs of carrirs ar involvd J = n u = 1, 2, 3,... In rality, u is far from constant for ach carrir spcis J = n u Diffrnt typ may hav diffrnt dnsity n, charg, or vlocity u Sparat ach carrir into small groups moving with a common u J = j n j u j n = j n u j n u j n j = 1, 2, 3,... : subgroups of spcis Avrag vlocity of all typ- carrirs 4

5 Conductivity In most cass, carrirs mov bcaus of th fild Wait! Is thr supposd to b no fild in conductors? That was whn thr was no currnt Now thr is Carrir rcivs forc F = It would acclrat (as Nwton says) li a falling appl It dosn t bcaus it ps bumping into positiv ions It rachs a constant vlocity (= u) uicly mpirically, u is proportional to, and so is J: J = σ Natural for isotropic matrials at small σ is th conductivity of this matrial F Ohm s Law Considr a lngth of conductiv wir Potntial diffrnc V = φ 1 φ 2 = L φ Currnt 1 I = JA = σa V = L σ A I RI This is th familiar vrsion of Ohm s Law It s mor of a corollary to th ral Ohm s Law A J L J = σ φ 2 R is th rsistanc of this wir statvolt volt Unit: in CGS, = ohm (Ω) in SI su/sc = sc cm ampr 5

6 Confusd? Four uantitis rlatd to rsistanc Macroscopic Microscopic Quantity Dfinition CGS SI Rsistanc R = V/I sc/cm ohm (Ω) Conductanc G = I/V cm/sc simns (S) Rsistivity ρ = /J sc Ω m Conductivity σ = J/ 1/sc 1/(Ω m) Macro and micro vrsions ar connctd by intgrals ovr lngth and cross sction For a constant cross sction A, R = L σ A Lt s loo at a tricy xampl Cylindrical Rsistor A matrial of rsistivity ρ is filld btwn two highlyconductiv tubs of radii a and b (a < b), lngth Apply potntial diffrnc V btwn tubs and J ar both pointing out At radius r, J(r) = σ(r) = 1 ρ (r)ˆr Intgrat J ovr a cylindr at r I = J(r) da = 2πrLJ(r) (r) = ρj(r) = ρi 2πrL ˆr cylindr Intgrat from = a to b b V = (r) dr = ρi b dr a 2πL = ρi a r 2πL ln b a R = V I = ρ 2πL ln b a a r J b 6

7 Physics of Conduction Can w driv Ohm s law from th first principls? Just how dos th carrirs mov in a conductor? W ar laving 15b and ntring 143a 181 Considr a fr lctron moving in a mtal It movs randomly, bumping into positiv ions At vry collision, vlocity changs bump u(t) u(t τ ) uncorrlatd Avrag tim btwn collisions is τ sconds Also: vlocitis of diffrnt lctrons (u i and u j ) ar uncorrlatd Physics of Conduction Avraging ovr many lctrons, Now, apply fild ach lctron is acclratd by F = Tim t i aftr th last collision u i (t i ) = u i (0) m t i Avrag ovr lctrons u = 1 N = m 1 N = τ m i u i (0) m t i i t i u = 1 N i u i du i dt = 0 That s what random mans = m J = nu = n 2 τ m σ = n 2 τ m 7

8 Subtltis σ = 1 ρ = n 2 τ m W assumd τ was constant It should (at last) dpnd on th spd of th lctron changs u τ should chang as wll OK only if τ m u =0 fild is wa nough so that random motion du to thrmal nrgy dominats Quantum Mchanics says τ = in a prfct crystal lctrons don t bump into ions that ar lind up rgularly But ions vibrat (thrmal fluctuation) at tmpratur T > 0 Highr T mor vibration shortr τ highr ρ Most mtals xhibit ρ T in broad rangs of T Ohm s Law holds at constant T and for small Numbrs Pur coppr has σ = /sc at 273 K (Purcll p.133) If ach atom supplis on fr lctron n = 8.96 g/cm /mol 63.5 g/mol = /cm 3 Avrag thrmal vlocity of lctrons at 273 K is Avrag distanc btwn bumps is cm = 3 nanomtrs... or about 13 intr-atomic spacs 29 Cu Dnsity 8.96 g/cm 3 Atomic wight 63.5 g/mol τ = σm n = /sc g /cm 3 ( su) = sc u = 3T m = rg/k 273K g = cm/sc 8

9 Summary Dfin currnt I and currnt dnsity J I = J da S Charg consrvation divj = ρ or = 0 in stady currnt t Carrir dnsitis n, vlocitis u J = n u J = σ Ohm s Law (micro) (macro) For constant cross sction A, Physics of lctrical conduction V = RI R = L σ A Assuming that carrir (=lctron) motion is dominatd by thrmal fluctuation σ = n 2 τ m 9