Physics 102: Lecture 7, Slide 1 RC Circuits Textbook Section Physics 102: Lecture 7

Physics 102: Lecture 7, Slide 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 3 RC Circuits: Charging KVL: Just after S 1 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) = I 0 e -t/RC C  R S1S1 S2S I The switches are originally open and the capacitor is uncharged. Then switch S 1 is closed. t q RC 2RC 0 qq

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

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

Physics 102: Lecture 7, Slide 7 Preflight 7.1, 7.3 2R C  R S2S2 Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S 1 is closed? 1) I b = 02) I b =  /(3R) 3) I b =  /(2R)4) I b =  /R S1S1 IbIb 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?long time 1) I b = 0 2) I b = V/(3R) 3) I b = V/(2R) 4) I b = V/R

Physics 102: Lecture 7, Slide 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? 1) Q = 02) Q = C  /3 3) Q = C  /24) Q = C  R 2R C  S2S2 S1S1 IRIR

Physics 102: Lecture 7, Slide 9 ACT: RC Circuits Both switches are closed. What is the final charge on the capacitor after the switches have been closed a long time? 1) Q = 02) Q = C  / 3 3) Q = C  /24) Q = C  KVL (right loop): -Q/C+IR = 0 KVL (outside loop), -  + I(2R) + IR = 0 I =  /(3R) combine: -Q/C +  /(3R) R = 0 Q = C ε / 3 R 2R C  S2S2 S1S1 IRIR

Physics 102: Lecture 7, Slide 10 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: R C E S1S1 R=10  C=30 mF E =20 Volts -

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: R C E S1S1 R=10  C=30 mF E =20 Volts -  + I 0 R + q 0 /C = 0 I  + I 0 R + 0 = 0 I 0 =  /R After a long time current through capacitor is zero! -  + IR + q ∞ /C = 0 -  q ∞ /C = 0 q ∞ =  C

Physics 102: Lecture 7, Slide 12 Charging: Intermediate Times Calculate the charge on the capacitor 3  seconds after switch 1 is closed. q(t) = q  (1-e -t/RC )  C R1R1 S2S2 S1S1 IbIb R 1 = 20  R 2 = 40  ε = 50 Volts C = 100  F R2R2

Physics 102: Lecture 7, Slide 13 Charging: Intermediate Times  C R1R1 S2S2 S1S1 IbIb Calculate the charge on the capacitor 3  seconds after switch 1 is closed. q(t) = q  (1-e -t/R 2 C ) = q  (1-e - 3  /(40  100  10 -6) ) ) = q  (0.53) Recall q  = ε C = (50)(100x10 -6 ) (0.53) = 2.7 x10 -3 Coulombs R 1 = 20  R 2 = 40  ε = 50 Volts C = 100  F R2R2

Physics 102: Lecture 7, Slide 14 RC Circuits: Discharging KVL: ____________ Just after…: ________ –Capacitor is still fully charged Long time after: ____________ Intermediate (more complex) –q(t) = q 0 e -t/RC –I c (t) = I 0 e -t/RC C  R S1S I S2S2 q RC2RC t

Physics 102: Lecture 7, Slide 15 RC Circuits: Discharging KVL: - q(t) / C - I(t) R = 0 Just after…: q=q 0 –Capacitor is still fully charged –-q 0 / C - I 0 R = 0  I 0 = q 0 /(RC) Long time after: I c =0 –Capacitor is discharged –-q  / C = 0  q  = 0 Intermediate (more complex) –q(t) = q 0 e -t/RC –I c (t) = I 0 e -t/RC C  R S1S I S2S2 q RC2RC t

Physics 102: Lecture 7, Slide 16 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) I R = 02) I R =  /(3R) 3) I R =  /(2R)4) I R =  /R 2R C  R S2S2 S1S1 IRIR

Physics 102: Lecture 7, Slide 17 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) I R = 02) I R =  /(3R) 3) I R =  /(2R)4) I R =  /R KVL: -q 0 /C+IR = 0 Recall q is charge on capacitor after charging: q 0 =  C (since charged w/ switch 2 open!) -  + IR = 0  I =  /R 2R C  R S2S2 S1S1 IRIR

Physics 102: Lecture 7, Slide 18 ACT: RC Challenge 1) q 0 2) q 0 3) q 0 4) q 0 2R C E R S1S1 E = 24 Volts R = 4  C = 15 mF 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?

Physics 102: Lecture 7, Slide 19 ACT: RC Challenge 1) q 0 2) q 0 3) q 0 4) q 0 2R C E R S1S1 E = 24 Volts R = 4  C = 15 mF 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? q(t) = q 0 e -t/RC = q 0 (e /(4  (15  )) ) = q 0 (0.368)

Physics 102: Lecture 7, Slide 20 RC Summary ChargingDischarging q(t) = q  (1-e -t/RC )q(t) = q 0 e -t/RC V(t) = V  (1-e -t/RC )V(t) = V 0 e -t/RC I(t) = I 0 e -t/RC Short term: Charge doesn’t change (often zero or max) Long term: Current through capacitor is zero. Time Constant  = RC Large  means long time to charge/discharge

Physics 102: Lecture 7, Slide 21 See you next lecture! Read Textbook Sections