**Unformatted text preview: **Physics 212
Lecture 11
RC Circuits Physics 212 Lecture 11, Slide 1 Capacitor uncharged, switch is moved to position "a" a a Kirchoff's Voltage Rule RC Circuit Charging I
b b C C q -Vbattery + +IR = 0 C Initially (q = q0 = 0) -Vbattery + 0 + I0R = 0 Vbattery Vbattery R R Vbattery I0 = R Long Term (Ic =0) q -Vbattery + + 0 = 0 R C q = CVbattery
11 In general: q dq -Vbattery + + R = 0 C dt q(t ) = q (1 - e - t /RC ) I (t ) = I0e - t /RC
Physics 212 Lecture 11, Slide 2 A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Checkpoint 1a & Checkpoint 1b
A) V1 = V V2 = V B) V1 = 0 V2 = V C) V1 = 0 V2 = 0 D) V1 = V V2 = 0 Close S1, V1 = voltage across C immediately after V2 = voltage across C a long time after Immediately after the switch S1 is closed: Q = 0
13 After the switch S1 has been closed for a long time I = 0 VR = 0 V2 = V
Physics 212 Lecture 11, Slide 3 V = Q/C V1 = 0 R V C 2R Close S1 at t=0 (leave S2 open) S1 S2 R R I
V I=0
C V VC = V For t Physics 212 Lecture 11, Slide 4 C VC = Q/C = 0
15 At t = 0 RC Circuit (Discharging) Capacitor has q0 = CV, switch is moved to position "b" Kirchoff's Voltage Rule aa
C C q + + IR = 0 C Initially (q=q0) Vbattery + IR = 0 + Vbattery Vbattery b b I
R R -Vbattery I0 = R
q + 0 =0) R Long Term (Ic = 0 C q =0
19 In general: V q dq + + R=0 C dt q(t ) = q0e - t /RC I I(t ) = I0e - t /RC
Physics 212 Lecture 11, Slide 5 A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Checkpoint 1c
+ - IR After being closed a long time, switch 1 is opened and switch 2 is closed. What is the current through the right resistor immediately after switch 2 is closed? A A. B. IR = V/3R C. IR = V/2R D. IR = V/R B IR = 0 C D 22 Physics 212 Lecture 11, Slide 6 A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Checkpoint 1c
+ - IR After being closed a long time, switch 1 is opened and switch 2 is closed. What is the current through the right resistor immediately after switch 2 is closed? A A. B. IR = V/3R C. IR = V/2R D. IR = V/R B IR = 0 C D I
V C 2R V 22 Physics 212 Lecture 11, Slide 7 R V C 2R Open S1 at t=big and close S2 S1 S2
V I
V C I = V/2R 2R 23 Physics 212 Lecture 11, Slide 8 A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Checkpoint 1d Now suppose both switches are closed. What is the voltage across the capacitor after a A very long time? B V =0 A. C B. VC = V C. VC = 2V/3 C 26 Physics 212 Lecture 11, Slide 9 A circuit is wired up as shown below. The capacitor is initially uncharged and switches S1 and S2 are initially open. Checkpoint 1d Now suppose both switches are closed. What is the voltage across the capacitor after a A very long time? B V =0 A. C B. VC = V C. VC = 2V/3 C After both switches have been closed for a long time The current through the capacitor is zero Vcapacitor = V2R V2R = 2/3 V
26 The current through R = current through 2R Physics 212 Lecture 11, Slide 10 R V C 2 R C lo s e b o th S 1 a nd S 2 a nd wa it a lo ng tim e ... S1 S2 I
No c urre nt flo ws th ro u g h th e c a p a c ito r a fte r a lo ng tim e . This will always be the case if the sources of EMF don't change with time. R V C VC 2R I = V/(3R)
27 V2R = I(2R) = (2/3)V = VC VC = (2/3)V
Physics 212 Lecture 11, Slide 11 DEMO ACT 1
Bulb 2 S V Bulb 1 R R C What will happen after I close the switch?
A) B) C) D)
No initial charge on capacitor
30 Both bulbs come on and stay on. Both bulbs come on but then bulb 2 fades out. Both bulbs come on but then bulb 1 fades out. Both bulbs come on and then both fade out.
V(bulb 1) = V(bulb 2) = V V(bulb 2) = 0 Both bulbs light
Physics 212 Lecture 11, Slide 12 No final current through capacitor DEMO ACT 2
Bulb 2 S V Bulb 1 R R C Suppose the switch has been closed a long time. Now what will happen after open the switch?
A) B) C) D) Both bulbs come on and stay on. Both bulbs come on but then bulb 2 fades out. Both bulbs come on but then bulb 1 fades out. Both bulbs come on and then both fade out.
Capacitor discharges through both resistors
Physics 212 Lecture 11, Slide 13 Capacitor has charge (=CV)
32 R1 V S R2 C Calculation
R3 In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? Conceptual Analysis: Circuit behavior described by Kirchhoff's Rules: KVR: Vdrops = 0 KCR: Iin = Iout S closed and C charges to some voltage with some time constant Determine currents and voltages in circuit a long time after S closed Strategic Analysis 35 Physics 212 Lecture 11, Slide 14 Calculation
R1 V S R2 C R3 In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? Immediately after S is closed: what is I2, the current through C what is VC, the voltage across C?
(A) Only I2 = 0 (B) Only VC = 0 (C) Both I2 and VC = 0 (D) Neither I2 nor VC = 0 (A) (B) (C) (D) Why?? We are told that C is initially uncharged (V = Q/C) I2 cannot be zero because charge must flow in order to charge C 37 Physics 212 Lecture 11, Slide 15 I1 R1 V Calculation
S R2 C R3 In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? Immediately after S is closed, what is I1, the current through R1 ?
V R R R1 + 2 3 R2 + R3 (A) (B) (C) (D) (E) V R1 V R1 + R3 V R1 + R2 + R3 R1 + R2 + R3 V R1 R2 + R2 R3 + R1 R3 Why?? Draw circuit just after S closed (knowing VC = 0) R1 S R2
VC = 0 R3 R1 is in series with the parallel combination V of R2 and R3 39 Physics 212 Lecture 11, Slide 16 Calculation
R1 V S R2 C R3 In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? After S has been closed "for a long time", what is IC, the current through C ? V R1 V R2 0
I (A) (B) (C) Why?? 41 After a long time in a static circuit, the current through any capacitor approaches 0 ! This means we Redraw circuit with open circuit in middle leg R1 V IC = 0 VC R3 Physics 212 Lecture 11, Slide 17 Calculation
R1 V S R2 C R3 In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. What is the voltage across the capacitor after a long time ? After S has been closed "for a long time", what is VC, the voltage across C ?
R3 V R1 + R3 R2 R2 RR V V R1 + 2 3 R1 + R2 R2 + R3 (A) (B) (C) (D) (E) I V VC = V3 = IR3 = (V/(R1+R3))R3 R1 V
VC 0
I R3 Why?? 43 Physics 212 Lecture 11, Slide 18 R1 V S R2 C In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. R3 Challenge What is c, the charging time constant? Strategy Write down KVR and KCR for the circuit when S is closed 2 loop equations and 1 node equation Use I2 = dQ2/dt to obtain one equation that looks like simple charging RC circuit ( (Q/"C") + "R"(dQ/dt) "V" = 0 ) Make correspondence: "R" = ?, and "C" = ?, then = "R" "C" We get: R1R3 C c = R2 + R1 + R3 Physics 212 Lecture 11, Slide 19 How do exponentials work?
1 Q ( t ) = Q0e - t RC Q( t) Q0
"Fraction of initial charge that remains" 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 "How many time constants worth of time that have elapsed"
45 t RC
Physics 212 Lecture 11, Slide 20 Q( t) Q0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Q ( t ) = Q0e - t RC RC = 2 Time constant: = RC The bigger is, the longer it takes to get the same change...
47 0 0 RC = 1
1 2 3 4 5 6 7 8 9 10 t Physics 212 Lecture 11, Slide 21 The two circuits shown below contain identical capacitors that hold the same charge at t = 0. Circuit 2 has twice as much resistance as circuit 1. Checkpoint 2a Which circuit has the largest time constant? A) B) C) Circuit 1 Circuit 2 Same 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 = RequivC
49 0.2 0.1 0 0 RC = 2 RC = 1
1 2 3 4 5 6 7 8 9 10 Physics 212 Lecture 11, Slide 22 The two circuits shown below contain identical capacitors that hold the same charge at t = 0. Circuit 2 has twice as much resistance as circuit 1. Checkpoint 2b Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0? A. Q1 < Q2 B. Q1 > Q2 C. Q1 = Q2 D. Q1 < Q2 at first, then Q1 > Q2 after long time E. Q1 > Q2 at first, then Q1 < Q2 after long time 50 Physics 212 Lecture 11, Slide 23 The two circuits shown below contain identical capacitors that hold the same charge at t = 0. Circuit 2 has twice as much resistance as circuit 1. Checkpoint 2b Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0? A. Q1 < Q2 B. Q1 > Q2 C. Q1 = Q2 D. Q1 < Q2 at first, then Q1 > Q2 after long time E. Q1 > Q2 at first, then Q1 < Q2 after long time 50 Physics 212 Lecture 11, Slide 24 The two circuits shown below contain identical capacitors that hold the same charge at t = 0. Circuit 2 has twice as much resistance as circuit 1. Checkpoint 2b Checkpoint 2b Which of the following statements best describes the charge remaining on each of the the two capacitors for any time after t = 0? A. Q1 < Q2 B. Q1 > Q2 C. Q1 = Q2 1 D. Q1 < Q2 at first, then Q1 > Q2 after long time E. Q1 > Q2 at first, then Q1 < Q2 after long time 0.9
0.8 0.7 Q = Q0e t/RC 0.6 0.5 0.4 0.3 Look at plot !!! 0.2 0.1 0 0 RC = 2 RC = 1
1 2 3 Physics 212 Lecture 11, Slide 25 4 5 6 7 8 9 10 ...

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