Lecture 21: MON 12 OCT

1. Physics 2113 Jonathan Dowling Lecture 21: MON 12 OCT Circuits III

2. i(t) CE E/R 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. VC=Q/C VR=iR A differential equation for q(t)! The solution is: Time constant: τ = RC Time i drops to 1/e.

3. Solution: i(t) + C E/R - 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.

5. i(t) E/R Time constant: τ = RC Time i drops to 1/e≅1/2.

6. Example The three circuits below are connected to the same ideal battery with emf E. All resistors have resistance R, and all capacitors have capacitance C. • Which capacitor takes the longest in getting charged? • Which capacitor ends up with the largest charge? • What’s the final current delivered by each battery? • What happens when we disconnect the battery? Simplify R’s into into Req. Then apply charging formula with ReqC = τ If capacitors between resistors in parallel use loop rule.

7. Example In the figure, E = 1 kV, C = 10 µF, R1 = R2 = R3 = 1 MΩ. With C completely uncharged, switch S is suddenly closed (at t = 0). • What’s the current through each resistor at t=0? • What’s the current through each resistor after a long time? • How long is a long time? At t=0 replace capacitor with solid wire (open circuit) then use loop rule or Req series/parallel. At t>>0 replace capacitor with break in wire, i3=0, and use loop rule on R1 and R2 branch only. A long time is t>>𝜏.

8. i(t) CE E/R 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. VC=Q/C VR=iR A differential equation for q(t)! The solution is: Time constant: τ = RC Time i drops to 1/e.

9. Solution: i(t) + C E/R - 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.

10. 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.

13. One Battery? Simplify! ResistorsCapacitors Key formula: V=iRQ=CV In series: same current dQ/dtsame charge Q Req=∑Rj1/Ceq=∑1/Cj In parallel: same voltagesame voltage 1/Req=∑1/RjCeq=∑Cj P = iV = i2R = V2/R U = QV/2 = Q2/2C = CV2

14. Many Batteries? Loop & Junction! Three Batteries: Straight to Loop & Junction One Battery: Simplify First

17. Too Many Batteries!