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PHY481: Electromagnetism Constant currents Conductors & insulators Lecture 22 Carl Bromberg - Prof. of Physics Exam 2 Lecture 22 Carl Bromberg - Prof. of Physics 1 Current and current densities No charge can accumulate in a uniform wire carrying current. Charge going in = charge going out. !"c !c (x) = charge density !t Volume current density Current density J is position dependent =0 Beware: ρ is also used for resistivity of a material. J(x) = nV qv(x) (3 D) (2 D) Local current density J for K(x) = nS qv(x) a differential area element Surface current K K ê ! d! Current is charge through A per unit time I = dQ dt (1 D) Lecture 22 Current through dA dI = J(x) ! dA (1 D) Current crossing d! dI = K ! ê " d! (1 D) Carl Bromberg - Prof. of Physics 2 Continuity equation -- conservation of charge Continuity equation: !"J = # Remember that ρc and J are functions of 3D position $%c $t !c (x) & J(x) Integrate over volume %&c 3 3 V, and use Gauss’s ! " J d x = $# d x # V V %t theorem Continuity equation, integral form. Flux of J through surface S "S J ! dA = # d 3 $ d x " c V dt Rate of change of charge inside of V Rate of change of charge inside of V Boundary condition on J at a surface Discontinuity in normal "# c Rate of change of J ! J = ! component of J at a surface 2n 1n "t surface charge density with constant = 0 currents = 0 and !t !t !"c Lecture 22 !" c Reminiscent of "c E2n ! E1n = #0 Carl Bromberg - Prof. of Physics 3 Good conductors & poor conductors (resistors) Macroscopic Resistance 1 I= V R Microscopic - Local forms of Ohm’s law Resistivity ρ Conductivity σ 1 1 ! = J(x) = ! R E(x) J(x) = E(x) R "R !R ρ and σ are an intrinsic property of a material “Wire” with area A, and resistivity ρR z 1 R = ! dR = ! " R ( z # )dz # A0 With constant resistivity ρR !R z A Resistance linear in z (length) and ρ . R= What is still true about E ? E = !"V field <--> potential relationship, e.g., E = Ez k̂; V = ! Ez z !"E=0 violation requires changing magnetic fields In good conductors, e.g., Cu, Ag, etc., the charge density is negligible In resistors carrying constant current !"c Charge density doesn’t change = 0 but !c " 0 # # !t free !"E = c % $0 $ must specify “free charge density” and permittivity, ε . Lecture 22 Carl Bromberg - Prof. of Physics 4 Free charge and resistivity What is free charge, ! free ? Though currents have moving charge, resistance will allow charge to accumulate locally. ! free = 0 with constant resistivity (uniform material) No free charge in resistor volume Free charge, ! free on up- & down-stream surfaces " I # d ! R & with resistivity changing over z % ( A $ dz ' Free charge in resistor volume, and ! free = free charge on up- & down-stream surfaces Lecture 22 Carl Bromberg - Prof. of Physics 5 Voltage and charge decay Two conductors embedded in a resistive matrix Charge each with a battery and disconnect. E Relate Total current I to material Total current: I = " J(x) ! n̂ dA S conductor 1 J(x) = ! RE(x) Gauss’s Law: I = ! E(x) " n̂ dAI = ! $ " E(x)d 3 x R# R# S V !R !R 3 I= #(x)d x = Q " V$ " !Q2 (t) Q1 (t) Q = CV conductor 2 Gaussian surface ! " E(x) = " R ,# #(x) $ V !R = CV R " RC = ! "R constants with dimensions of time dQ(t) ! R 1 = Q(t) = Q(t) dt " RC Lecture 22 Q1 (t) = Q0 e !t / RC Exponential decay of the charge Carl Bromberg - Prof. of Physics 6 HW-7 7.3 Solid sphere radius a, uniformly distributed Q. Constant angular velocity. Calculate J(x) . "S J(r) ! n̂ dA = d! = " dt d 3 # d x dt "V c 7.11 Find surface charge density at discontinuity in resistivity. "c E2n ! E1n = #0 L 2 Lecture 22 L 2 J = J k̂ ! R1 ++ + + ! R2 !R2 > ! R1 J = J k̂ ! R1 – –– – ! R2 !R2 < ! R1 Carl Bromberg - Prof. of Physics 7 HW-7 7.9 Cylinders length L , resistive media between radius a and 3a . What is the resistance? What is the charge density on the interface at 2a ? General solutions Boundary Conditions V1 (r) = Aln r + B V1 (a) = V0 V2 (r) = C ln r + D V2 (3a) = 0 V1 (2a) = V2 (2a) J1 (2a) = J 2 (2a) Lecture 22 Carl Bromberg - Prof. of Physics 8