Unformatted text preview: Physics 212
Lecture 11
Today's Concept: RC Circuits
(Circuits with resistors & capacitors & batteries)
50 40 30 20 10 0 Confused Avg = 3.0 Confident Physics 212 Lecture 11, Slide 1 Music
Who is the Artist?
BB A) B) C) D) E) Tito Puente Buena Vista Social Club Louis Prima Freddie Omar con su banda Freddie su Los Hombres Calientes Los Calientes Cuban Jazz !! Thanks to Ry Cooder for bringing these guys Thanks Ry for to our attention !! to
A YouTube clip from the 1999 YouTube Wim Wenders movie of the same name. Wim Physics 212 Lecture 11, Slide 2 Physics 212
Lecture 11
Today's Concept: RC Circuits
(Circuits with resistors & capacitors & batteries)
50 40 30 20 10 0 Confused Avg = 3.0 Confident Physics 212 Lecture 11, Slide 3 Key Concepts:
1) Understanding the behavior of capacitors Understanding in circuits with resistors in 2) Understanding the RC time constant Today’s Plan:
1) Examples with switches Examples closing and opening closing  What changes?  What is constant? 2) Example problem 3) Exponentials
Physics 212 Lecture 11, Slide 4 07 RC Circuit (Charging)
• Capacitor uncharged, Switch is moved to position “a” a a • Kirchoff’s Voltage Rule C C −Vbattery + q + IR = 0 C Vbattery Vbattery b b R R • Short Term (q = q0 = 0) −Vbattery + 0 + I 0 R = 0
Vbattery I0 = R • Long Term (Ic =0)
q∞ −Vbattery + + 0 ⋅ R = 0 C q∞ = CVbattery
Intermediate q dq −Vbattery + + R = 0 C dt q(t ) = q∞ (1 − e −t / RC ) I (t ) = I 0e −t / RC
Physics 212 Lecture 11, Slide 5 11 Preflight 2 70 60 50 • Immediately after the switch S1 is closed • The charge on the capacitor is zero • The voltage across the capacitor is zero • Vcapacitor = Q/C = 0 40 30 20 10 0 13 Physics 212 Lecture 11, Slide 6 Preflight 4 70 60 • After the switch S1 has been closed for a long has
time time • The current through the capacitor is zero • The voltage across the capacitor is the The battery voltage battery • Vcapacitor = V
15 50 40 30 20 10 0 Physics 212 Lecture 11, Slide 7 R V C 2R Close S1 at t=0 (leave S2 open) S1 S2 R At t = 0 R At t = big I
V C V VC = Q/C Q/C =0 I=0
C VC = V 15 Physics 212 Lecture 11, Slide 8 RC Circuit (Discharging)
• Capacitor has q0 = CV, Switch is moved to position “b” aa • Kirchoff’s Voltage Rule C C q + + IR = 0 C • Short Term (q=q0) Vbattery Vbattery b b R R Vbattery + IR = 0 −Vbattery I0 = R
• Long Term (Ic =0) Intermediate V q∞ + 0⋅ R = 0 C q∞ = 0
19 q dq + + R=0 C dt q(t ) = q0e −t / RC I (t ) = I 0e −t / RC
Physics 212 Lecture 11, Slide 9 Preflight 6
BB A B C D
50 40 I
V C 2R V 30 20 10 0 22 Physics 212 Lecture 11, Slide 10 10 R V C 2R Open S1 at t=big and close S2 S1 S2 I
V C 2R I = V/2R V 23 Physics 212 Lecture 11, Slide 11 11 Preflight 8
BB A B C
50 • After both switches have been closed for a long After
time time • The current through the capacitor is zero • The current through R = current through 2R • Vcapacitor = V2R • V2R = 2/3 V
26 40 30 20 10 0 Physics 212 Lecture 11, Slide 12 12 R V C 2R Close both S1 and S2 and wait a long time… S1 I R V C S2
No current flows through No the capacitor after a long time. This will always be VC 2R the case in any static circuit!! circuit!! VC = 2V/3 27 Physics 212 Lecture 11, Slide 13 13 DEMO – ACT 1 DEMO
Bulb 2
BB S V Bulb 1 R R C What will happen after I close 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.
V(bulb 1) = V(bulb 2) = V 1) V(bulb V(bulb 2) = 0 Both bulbs light No initial charge No on capacitor on
30 No final current No through capacitor through Physics 212 Lecture 11, Slide 14 14 DEMO – ACT 2 DEMO
Bulb 2
BB 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 15 15 Capacitor has charge (=CV)
32 Calculation
S R1 V 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 ? long • Conceptual Analysis:
– – Circuit behavior described by Kirchhoff’s Rules: • KVR: ΣVdrops = 0 • KCR: ΣIin = ΣIout S closed and C charges to some voltage with some time constant • Strategic Analysis
– Determine currents and voltages in circuit a long time after S closed 35 Physics 212 Lecture 11, Slide 16 16 Calculation
S R1 V 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 ? long Immediately after S is closed: what is I2, the current through C what is VC, the voltage across C?
(A) Only I2 = 0 (A) BB (B) Only VC = 0 (C) Both I2 and VC = 0 (D) Neither I2 nor VC = 0 (B) (C) • 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 17 17 I1 S R1 V R2 C Calculation
In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. R3 What is the voltage across the capacitor after a long time ? long • Immediately after S is closed, what is I1, the current through R1 ?
V R1 V R1 + R3 V R1 + R2 + R3
V 1 1 R1 + + R2 R3 −1 V R1 + R2 + R3 R1 R2 + R2 R3 + R1 R3 BB (A) (A) (B) (C) (D) (D) S (E) • Why??
– – Draw circuit just after S closed (knowing VC = 0) R1 is in series with the parallel combination of R2 and R3 R1 V R2
VC = 0 R3 39 Physics 212 Lecture 11, Slide 18 18 Calculation
S R1 V 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 ? long After S has been closed “for a long time”, what is IC, the current through C ? V R1
(A) V R2
(B) 0
(C) I BB • Why??
– – 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 IC = 0 VC R3 V 41 Physics 212 Lecture 11, Slide 19 19 Calculation
S R1 V 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 ? long After S has been closed “for a long time”, what is VC, the voltage across C ?
V R3 R1 + R3
R2 V R1 + R2 V V (A) (A) (B) (C) R2 RR R1 + 2 3 R2 + R3 BB
0 (D) (D) (E) I I • Why??
– VC = V3 = IR3 = (V/(R1+R3))R3 R1
VC R3 V 43 Physics 212 Lecture 11, Slide 20 20 Challenge
S R1 V R2 C R3 What is τc, the charging time constant? In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed. • 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 τ = “R”∏ ”C” I got: τ c = R2 + R1 R3 C ( R1 + R3 ) Physics 212 Lecture 11, Slide 21 21 How do exponentials work?
1 Q ( t ) = Q0e − t RC Q (t ) Q0 0.9 0.8 0.7 0.6 “Fraction of initial 0.5 Fraction charge that remains” 0.4 charge
0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 “How many time constants worth How of time that have elapsed” of
45 t RC
Physics 212 Lecture 11, Slide 22 22 Q (t ) Q0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Q ( t ) = Q0e − t RC RC = 2
0.2 0.1 Time constant: τ = RC RC RC = 1
0 0 1 2 3 4 5 6 7 8 9 10 The bigger τ is, The the longer it takes to get the same change…
47 t Physics 212 Lecture 11, Slide 23 23 Preflight 10 Which circuit has the largest time constant? A) Circuit 1 B) Circuit 2 C) Same
80 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 60 τ = Requiv/C
49 40
0.2 RC = 2 RC = 1
0 1 2 3 4 5 6 7 8 9 10 20 0.1 0 0 Physics 212 Lecture 11, Slide 24 24 Preflight 12 BB 35 30 25 20 15 10 5 0 50 Physics 212 Lecture 11, Slide 25 25 Preflight 12 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Q= Q0et/RC RC = 2 RC = 1
0 1 Look at plot !!! 0.1 0 Physics 212 Lecture 711, Slide 26 2 3 4 5 6 8 9 26 10 ...
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 Spring '10
 JohnThaler
 Physics, mechanics, RC Circuits, RC circuit, RC time constant

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