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

Physics 102: Lecture 7, Slide 2 Recall …. First we covered circuits with batteries and capacitors –series, parallel Then we covered circuits with batteries and resistors –series, parallel –Kirchhoff’s Loop and Junction Relations Today: circuits with batteries, resistors, and capacitors

Physics 102: Lecture 7, Slide 3 RC Circuits Charging Capacitors Discharging Capacitors Intermediate Behavior

Physics 102: Lecture 7, Slide 4 RC Circuits Circuits that have both resistors and capacitors: With resistance in the circuits, capacitors do not charge and discharge instantaneously – it takes time (even if only fractions of a second). RKRK C S R Na R Cl εKεK ε Na ε Cl

Physics 102: Lecture 7, Slide 5 Capacitors Charge (and therefore voltage) on Capacitors cannot change instantly: remember V C = Q/C Short term behavior of Capacitor: –If the capacitor starts with no charge, it has no potential difference across it and acts as a wire –If the capacitor starts with charge, it has a potential difference across it and acts as a battery. Long term behavior of Capacitor: Current through a Capacitor eventually goes to zero. –If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit –If capacitor is discharging, potential difference goes to zero and no current flows

Physics 102: Lecture 7, Slide 6 Charging Capacitors Capacitor is initially uncharged and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter? What is current I  in circuit a long time later?  R C S

Physics 102: Lecture 7, Slide 7 Charging Capacitors: t=0 Capacitor is initially uncharged and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter? –Capacitor initially uncharged –Therefore V C is initially 0 –Therefore C behaves as a wire (short circuit) –KLR:  – I 0 R = 0 I 0 =  /R  R C S R 

Physics 102: Lecture 7, Slide 8 Charging Capacitors: t>0 –I 0 =  /R –Positive charge flows Onto bottom plate (+Q) Away from top plate (-Q) As charge builds up, V C rises (V C =Q/C) Loop:  – V C – I R = 0 –I = (  -V C )/R –Therefore I falls as Q rises –When t is very large (  ) I  = 0: no current flow into/out of capacitor for t large V C =   R C + - R  Demo

Physics 102: Lecture 7, Slide 9 ACT/Preflight 7.1 Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S 1 is closed? 2R C  R S2S2 1) I b = 02) I b = E /(3R) 3) I b = E /(2R)4) I b = E /R S1S1 IbIb

Physics 102: Lecture 7, Slide 10 ACT/Preflight 7.3 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 = 02) I b = E /(3R) 3) I b = E /(2R)4) I b = E /R 2R C  R S2S2 S1S1 IbIb

Physics 102: Lecture 7, Slide 11 Discharging Capacitors Capacitor is initially charged (Q) and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter? What is current I  in circuit a long time later? R C S

Physics 102: Lecture 7, Slide 12 Discharging Capacitors Capacitor is initially charged (Q) and switch is open. Switch is then closed. What is current I 0 in circuit immediately thereafter? –KLR: Q/C – I 0 R = 0 –So, I 0 = Q/RC What is current I  in circuit a long time later? –I  = 0 R C + - Demo

Physics 102: Lecture 7, Slide 13 ACT/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 14 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 E /3 3) Q = C E /24) Q = C E R 2R C  S2S2 S1S1 IRIR

Physics 102: Lecture 7, Slide 15 RC Circuits: Charging Loop:  – I(t)R – q(t) / C = 0 Just after…: q =q 0 –Capacitor is uncharged –  – I 0 R = 0  I 0 =  / R Long time after: I c = 0 –Capacitor is fully charged –  – q  /C =0  q  =  C 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 16 RC Circuits: Discharging Loop: 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 (like a wire) –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 17 What is the time constant? The time constant  = RC. Given a capacitor starting with no charge, the time constant is the amount of time an RC circuit takes to charge a capacitor to about 63.2% of its final value. The time constant is the amount of time an RC circuit takes to discharge a capacitor by about 63.2% of its original value.

Physics 102: Lecture 7, Slide 18 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

Physics 102: Lecture 7, Slide 19 Summary of Concepts Charge (and therefore voltage) on Capacitors cannot change instantly: remember V C = Q/C Short term behavior of Capacitor: –If the capacitor starts with no charge, it has no potential difference across it and acts as a wire –If the capacitor starts with charge, it has a potential difference across it and acts as a battery. Long term behavior of Capacitor: Current through a Capacitor eventually goes to zero. –If the capacitor is charging, when fully charged no current flows and capacitor acts as an open circuit. –If capacitor is discharging, potential difference goes to zero and no current flows. Intermediate behavior: Charge and current exponentially approach their long-term values  = RC

Physics 102: Lecture 7, Slide 20 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 ε S1S1 R=10  C=30 mF ε =20 Volts E – I 0 R – q 0 /C = 0 I E – I 0 R – 0 = 0 I 0 = E /R After a long time current through capacitor is zero! E – IR – q ∞ /C = 0 E – 0 – q ∞ /C = 0 q ∞ = E C

Physics 102: Lecture 7, Slide 21 ACT: RC Challenge 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? 1) q 0 2) q 0 3) q 0 4) q 0 R C ε 2R S1S1 ε = 24 Volts R = 2  C = 15 mF

Physics 102: Lecture 7, Slide 22 Charging: Intermediate Times  2R C R S2S2 S1S1 IbIb Calculate the charge on the capacitor 3  seconds after switch 1 is closed. q(t) = q  (1-e -t/RC ) = q  (1-e - 3  /(20  100  10 -6) ) ) = q  (0.78) Recall q  =  C = (50)(100x10 -6 ) (0.78) = 3.9 x10 -3 Coulombs R = 10  V = 50 Volts C = 100  F

Physics 102: Lecture 7, Slide 23 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