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Physics 2113 Jonathan Dowling Lecture 24: FRI 13 MAR DC circuits III Ch27.4-9 One Battery? Simplify! Resistors Key formula: V=iR In series: same current dQ/dt Req=∑Rj In parallel: same voltage 1/Req= ∑1/Rj P = iV = i2R = V2/R Capacitors Q=CV same charge Q 1/Ceq= ∑1/Cj same voltage Ceq=∑Cj U = QV/2 = Q2/2C = CV2 Many Batteries? Loop & Junction! One Battery: Simplify First Three Batteries: Straight to Loop & Junction RC Circuits: Charging a Capacitor In these circuits, current will change for a while, and then stay constant. We want to solve for current as a function of time i(t)=dq/dt. The charge on the capacitor will also be a function of time: q(t). The voltage across the resistor and the capacitor also change with time. To charge the capacitor, close the switch on a. E + VR (t) + VC (t) = 0 VC=Q/C VR=iR E - i(t)R - q(t) /C = 0 E - ( dq /dt ) R - q(t) /C = 0 A differential equation for q(t)! The solution is: Time constant: = RC q(t) = CE (1- e-t / RC ) Time i drops to 1/e. -t / RC ® i(t) º dq /dt = (E /R)e i(t) E/R CE t t RC Circuits: Discharging a Capacitor Assume the switch has been closed on a for a long time: the capacitor will be charged with Q=CE. +++ --- Then, close the switch on b: charges find their way across the circuit, establishing a current. V +V = 0 R C -i(t)R + q(t) /C = 0 ® ( dq /dt ) R + q(t) /C = 0 + -C -t / RC -t / RC q(t) = q(0)e = CEe Solution: i(t) = dq /dt = (q(0) /RC)e-t / RC = (E /R)e-t / RC i(t) E/R t t 1 1 1 = + ® R23par = 12W par R23 R2 R3 Too Many Batteries! • Fire Hazard: Filling gas can in pickup truck with plastic bed liner. • Safe Practice: Always place gas can on ground before refueling. • Touch can with gas dispenser nozzle before removing can lid. • Keep gas dispenser nozzle in contact with can inlet when filling.