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RC Circuits

Physics 102: Lecture 7. RC Circuits. Textbook Section 18.10-11. Combine R+C Circuits. Gives time dependence Current is not constant I(t) Charge is not constant q(t) Used for timing Pacemaker Intermittent windshield wipers Models nervous system include R, C

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RC Circuits

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  1. Physics 102:Lecture 7 RC Circuits • Textbook Section 18.10-11 Physics 102: Lecture 7, Slide 1

  2. Combine R+C Circuits • Gives time dependence • Current is not constant I(t) • Charge is not constant q(t) • Used for timing • Pacemaker • Intermittent windshield wipers • Models nervous system include R, C • Sports Trivia: How soon after starting gun can you run w/o getting a False Start? Physics 102: Lecture 7, Slide 2

  3. RC 2RC q q 0 t RC Circuits: Charging The switches are originally open and the capacitor is uncharged. Then switch S1 is closed. + • KVL: • Just after S1 is closed: ______ • Capacitor is uncharged (no time has • passed so charge hasn’t changed yet) • Long time after: __________ • Capacitor is fully charged • Intermediate (more complex) q(t) = q(1-e-t/RC) I(t) = I0e-t/RC R + e - I - C + - S2 S1 Physics 102: Lecture 7, Slide 3

  4. Capacitor “Rules of Thumb” • Initially uncharged capacitor: • acts like a wire (short circuit) at t = 0 • acts like an open circuit (broken wire) as t   • Initially charged capacitor: • acts like a battery at t = 0 Physics 102: Lecture 7, Slide 4

  5. 1 2 t = 2RC Example Time Constant Demo Each circuit has a 1 F capacitor charged to 100 Volts. When the switch is closed: • Which system will be brightest? • Which lights will stay on longest? • Which lights consumes more energy? t = RC/2 Physics 102: Lecture 7, Slide 5

  6. 1 2 t = 2RC Example Time Constant Demo Each circuit has a 1 F capacitor charged to 100 Volts. When the switch is closed: • Which system will be brightest? • Which lights will stay on longest? • Which lights consumes more energy? 2 I=2V/R 1 Same U=1/2 CV2 t = RC/2 Physics 102: Lecture 7, Slide 6

  7. Preflight 7.1, 7.3 Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S1 is closed? 2R + - 1) Ib = 0 2) Ib = e/(3R) 3) Ib = e/(2R) 4) Ib = e/R Ib + + C e R - - S2 S1 Both switches are initially open, and the capacitor is uncharged. What is the current through the battery after switch 1 has been closed a long time? 1) Ib = 0 2) Ib = V/(3R) 3) Ib = V/(2R) 4) Ib = V/R Physics 102: Lecture 7, Slide 7

  8. ACT: RC Circuits • Both switches are closed. What is the final charge on the capacitor after the switches have been closed a long time? 2R + - IR + + 1) Q = 0 2) Q = C e /3 3) Q = C e/2 4) Q = C e + C e - R - - S1 S2 Physics 102: Lecture 7, Slide 8

  9. ACT: RC Circuits 2R + - • Both switches are closed. What is the final charge on the capacitor after the switches have been closed a long time? IR 1) Q = 0 2) Q = C e / 3 3) Q = C e/2 4) Q = C e + + + C e - R - - KVL (right loop): -Q/C+IR = 0 KVL (outside loop), - e + I(2R) + IR = 0 I = e/(3R) combine: -Q/C + e/(3R) R = 0 Q = C ε / 3 S1 S2 Physics 102: Lecture 7, Slide 9

  10. R C E S1 R=10W C=30 mF E =20 Volts Example Practice! Calculate current immediately after switch is closed: - Calculate current after switch has been closed for 0.5 seconds: Calculate current after switch has been closed for a long time: Calculate charge on capacitor after switch has been closed for a long time: Physics 102: Lecture 7, Slide 10

  11. R I C E S1 R=10W C=30 mF E =20 Volts Example Practice! + - Calculate current immediately after switch is closed: -e + I0R + q0/C = 0 + + - -e + I0R + 0 = 0 - I0 = e/R Calculate current after switch has been closed for 0.5 seconds: Calculate current after switch has been closed for a long time: After a long time current through capacitor is zero! Calculate charge on capacitor after switch has been closed for a long time: -e + IR + q∞/C = 0 -e + 0 + q∞/C= 0 q∞ = eC

  12. Example Charging: Intermediate Times Calculate the charge on the capacitor 310-3 seconds after switch 1 is closed. R1 = 20 W R2 = 40 W ε = 50 Volts C = 100mF • q(t) = q(1-e-t/RC) R2 + - Ib + + e C R1 - - S2 S1 Physics 102: Lecture 7, Slide 12

  13. Example Charging: Intermediate Times Calculate the charge on the capacitor 310-3 seconds after switch 1 is closed. R1 = 20 W R2 = 40 W ε = 50 Volts C = 100mF • q(t) = q(1-e-t/R2C) = q(1-e-310-3/(4010010-6))) = q (0.53) Recall q = εC = (50)(100x10-6) (0.53) = 2.7 x10-3 Coulombs R2 + - Ib + + e C R1 - - S2 S1 Physics 102: Lecture 7, Slide 13

  14. RC 2RC q t RC Circuits: Discharging + • KVL: ____________ • Just after…: ________ • Capacitor is still fully charged • Long time after: ____________ • Intermediate (more complex) • q(t) = q0 e-t/RC • Ic(t) = I0 e-t/RC R + e - I - C + - S1 S2 Physics 102: Lecture 7, Slide 14

  15. RC 2RC q t RC Circuits: Discharging + • KVL: - q(t) / C - I(t) R = 0 • Just after…: q=q0 • Capacitor is still fully charged • -q0/ C - I0 R = 0  I0 = q0/(RC) • Long time after: Ic=0 • Capacitor is discharged • -q / C = 0  q = 0 • Intermediate (more complex) • q(t) = q0 e-t/RC • Ic(t) = I0 e-t/RC R + e - I - C + - S1 S2 Physics 102: Lecture 7, Slide 15

  16. 2R + - IR + + + e C R - - - S1 S2 Preflight 7.5 • After switch 1 has been closed for a long time, it is opened and switch 2 is closed. What is the current through the right resistor just after switch 2 is closed? 1) IR = 0 2) IR = e /(3R) 3) IR = e /(2R) 4) IR = e /R Physics 102: Lecture 7, Slide 16

  17. 2R + - IR + + + e C R - - - S1 S2 Preflight 7.5 • After switch 1 has been closed for a long time, it is opened and switch 2 is closed. What is the current through the right resistor just after switch 2 is closed? 1) IR = 0 2) IR = e /(3R) 3) IR = e /(2R) 4) IR = e /R KVL: -q0/C+IR = 0 Recall q is charge on capacitor after charging: q0= e C (since charged w/ switch 2 open!) - e + IR = 0  I = e /R Physics 102: Lecture 7, Slide 17

  18. ACT: RC Challenge E= 24 Volts R = 4W C = 15 mF 2R After being closed for a long time, the switch is opened. What is the charge Q on the capacitor 0.06 seconds after the switch is opened? C R E 1) 0.368 q0 2) 0.632 q0 3) 0.135 q0 4) 0.865 q0 S1 Physics 102: Lecture 7, Slide 18

  19. ACT: RC Challenge E= 24 Volts R = 4W C = 15 mF 2R After being closed for a long time, the switch is opened. What is the charge Q on the capacitor 0.06 seconds after the switch is opened? C R E 1) 0.368 q0 2) 0.632 q0 3) 0.135 q0 4) 0.865 q0 S1 • q(t) = q0 e-t/RC = q0 (e-0.06/(4(1510-3))) = q0 (0.368) Physics 102: Lecture 7, Slide 19

  20. RC Summary Charging Discharging q(t) = q(1-e-t/RC) q(t) = q0e-t/RC V(t) = V(1-e-t/RC) V(t) = V0e-t/RC I(t) = I0e-t/RC I(t) = I0e-t/RC Time Constant t = RC Large t means long time to charge/discharge Short term: Charge doesn’t change (often zero or max) Long term: Current through capacitor is zero. Physics 102: Lecture 7, Slide 20

  21. See you next lecture! • Read Textbook Sections 19.1-4 Physics 102: Lecture 7, Slide 21

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