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PHY481: Electromagnetism Current & Resistance Lecture 24 Carl Bromberg - Prof. of Physics Current and current densities No charge can accumulate in a uniform wire carrying current. Charge going in = charge going out. ∂ρc ρc (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) K(x) = nS qv(x) (2 D) Local current density J for a differential area element Surface current K K ê ⊥ d Current is charge through A per unit time I = dQ dt (1 D) Lecture 24 Current through dA dI = J(x) ⋅ dA (1 D) Current crossing d dI = K ⋅ ê ⊥ d (1 D) Carl Bromberg - Prof. of Physics 1 Continuity equation -- conservation of charge Continuity equation: Integrate over volume V, and use Gauss’s theorem ∇⋅J = − ∫V ∇ ⋅ J d 3 Remember that ρc and J are functions of 3D position ∂ρc ∂t x = −∫ V ρc (x) & J(x) ∂ρc ∂t 3 d x Rate of change of charge inside of V Continuity equation, integral form. Flux of J through surface S d 3 J ⋅ dA = − ρ d x ∫S dt ∫V c Rate of change of charge inside of V Boundary condition on J at a surface Discontinuity in normal ∂σ J 2n − J1n = − c Rate of change of component of J at a surface ∂t surface charge density ∂ρc ∂t = 0 and Lecture 24 ∂σ c ∂t =0 with constant currents Reminiscent of σc E2n − E1n = ε0 Carl Bromberg - Prof. of Physics 2 Good conductors & poor conductors (resistors) Macroscopic Resistance Microscopic - Local forms of Ohm’s law Resistivity ρ 1 J(x) = E(x) ρR 1 I= V R Conductivity σ σR = J(x) = σ R E(x) 1 ρR ρ and σ are an intrinsic property of a material “Wire” with area A, and resistivity ρR With constant resistivity ρR z 1 R = ∫ dR = ∫ ρ R ( z ′ )dz ′ A0 ρR z A Resistance linear in z (length) and ρ . R= What is still true about E ? E = −∇V field <--> potential relationship, e.g., E = E k̂; V = − E z z 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 ∇⋅E = ρ free ρc ⇒ ε0 ε Lecture 24 Charge density doesn’t change ∂ρc ∂t =0 but ρc ≠ 0 must specify “free charge density” and permittivity, ε . Carl Bromberg - Prof. of Physics 3 Free charge and resistivity What is free charge, ρ free ? Charge carriers move w.r.t. atomic background, but resistance causes charge to (instantaneously) collect locally and then remain constant. ρ free = 0 with constant resistivity (uniform material) No free charge in resistor volume Free charge on up- & down-stream surfaces, due to resistivity discontinuity ρ free = ε I ⎛ d ρ R ⎞ with changing resistivity over z ⎜ ⎟ A ⎝ dz ⎠ Free charge in resistor volume, and free charge on up- & down-stream surfaces Lecture 24 Carl Bromberg - Prof. of Physics 4 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) = σ RE(x) Gauss’s Law: I = σ R ∫ E(x) ⋅ n̂ dAI = σ R ∫ ∇ ⋅ E(x)d 3 x S V σR σR 3 I= ρ(x)d x = Q ε V∫ ε −Q2 (t) Q1 (t) Q = CV conductor 2 Gaussian surface ∇ ⋅ E(x) = σ R ,ε ρ(x) ε V σR = CV R ε RC = ε σR constants with dimensions of time dQ(t) σ R 1 = Q(t) = Q(t) dt ε RC Lecture 24 Q1 (t) = Q0 e −t / RC Exponential decay of the charge Carl Bromberg - Prof. of Physics 5 HW 7.3 Solid sphere radius a, uniformly distributed Q. Constant angular velocity. Calculate J(x) . ∫S J(r) ⋅ n̂ dA = dφ = ω dt d 3 ρ d x dt ∫V c 7.11 Find surface charge density at discontinuity in resistivity. σc E2n − E1n = ε0 L 2 Lecture 24 L 2 J = J k̂ ρ R1 ++ + + ρ R2 ρR2 > ρ R1 J = J k̂ ρ R1 – –– – ρ R2 ρR2 < ρ R1 Carl Bromberg - Prof. of Physics 6 HW 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 ? Use general solution for cylidrical symmetry twice V1 (r) = Aln r + B V2 (r) = C ln r + D Lecture 24 Carl Bromberg - Prof. of Physics 7