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Ohm’s Law: V = IR Power dissipated (Joule heating) = P = I2R = IV Kirchoff’s Loop equation: Kirchoff’s junction rule: Σ closed loop Σ Vi = 0 entering / leaving junction Ii = 0 Resistors in series: Req = R1 + R2 + … Resistors in parallel: Req-1 = R1-1 + R2-1 +… Capacitors in series: Ceq-1 = C1-1 + C2-1 + … Capacitors in parallel: Ceq = C1 + C2 + … Group Problems 1. R combo – all 100 Ω R1 25V R2 R3 a) Find I from the battery R5 R4 b) Find I through each R 2. a) Find equivalent R (in terms of R) R b) Find the battery current if R = 100 Ω R R 10V R Simple RC Series circuit Norm alized Q or I • Discharging: Q(t) = Qoexp(-t/RC) and I(t) = Ioexp(-t/RC) 1 0.8 0.6 0.4 0.2 0 0 10 20 30 tim e (m s) When t = RC = 1 time constant, then Q = Qoexp(-1) = 0.37Qo 40 50 60 Microscopic Picture of Electric Current • E field in wire produces forces on free electrons leading to a net drift velocity: vdrift = aτ, where a is the acceleration (a = F/m = eE/m = e(V/L)/m where V is the applied voltage across the wire of length L) and τ is the meanfree time between collisions • Solving for the charge drifting by allows us to compute the current I: I=[(ne2τ/m)(A/L)]V = [1/R]V = GV, where n is the free charge density, e is the electron charge, A is the wire cross-sectional area, R is the wire’s resistance and G is its conductance • We can also introduce intrinsic parameters: conductivity, σ = ne2τ/m and resistivity, ρ = 1/σ to write R = ρL/A outside IC IN a IK GNa IL GK GL C V Na VK inside VL dendrites axon Myelin sheath Cell body Terminal endings retinal “starburst” cell (red) found in visual processing network, Courtesy of Thomas Euler, Max Planck Institute for Medical Research, Heidelberg I n s id e I i( x ) r iΔ x I i( x + Δ x ) Im (x ) gΔ x cΔ x V M ( x ,t) cΔ x V Io(x ) roΔ x x r iΔ x I i( x + 2 Δ x ) Im (x + Δ x ) V M ( x + Δ x ,t) gΔ x cΔ x V Io(x + Δ x ) x+Δx O u t s id e roΔ x Im (x + 2 Δ x ) gΔ x V Io(x + 2 Δ x ) x+2Δx I (pA) 0 1 Time (ms) V (mV) I (pA) Time (ms)