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Physics 212 Lecture 11, Slide 1
Physics 212 Lecture 11
RC Circuits
Change in schedule
Exam 2 will be on Thursday, July 12 from 8 – 9:30 AM.
Physics 212 Lecture 11, Slide 2
RC Circuit Charging
• Kirchoff’s Voltage Rule
• Initially (q = q0 = 0)
• Long Term (Ic =0)
R Vbattery
C a
b battery
q-V + +IR = 0
C
00 0batteryV I R
battery
0
VI =
R
0 0battery
qV R
C
batteryq CV
In general:
0battery
q dqV R
C dt
/( ) (1 )t RCq t q e
/
0( ) t RCI t I e
11
a
R Vbattery
C
b
• Capacitor uncharged, switch is moved to position “a”
I
Physics 212 Lecture 11, Slide 3 13
Checkpoint 1a
Checkpoint 1b &
Close S1, V1 = voltage across C immediately after V2 = voltage across C a long time after
A) V1 = V V2 = V
B) V1 = 0 V2 = V
C) V1 = 0 V2 = 0
D) V1 = V V2 = 0
Immediately after the switch S1 is closed:
Q = 0 V1 = 0 V = Q/C
After the switch S1 has been closed for a long time
I = 0 V2 = V VR = 0
A circuit is wired up as shown below. The capacitor is initially uncharged and switches
S1 and S2 are initially open.
Physics 212 Lecture 11, Slide 4
V 2R C
R
S1 S2
Close S1 at t=0 (leave S2 open)
For t
V C
R
I=0
VC = V
At t = 0
V C
R
I
VC = Q/C = 0
15
Physics 212 Lecture 11, Slide 5
RC Circuit (Discharging)
• Kirchoff’s Voltage Rule
• Initially (q=q0)
• Long Term (Ic =0)
R Vbattery
C a
b 0q
IRC
00 0
qI R
C
0
0
qI
RC
0 0q
RC
0q
In general:
0q dq
RC dt
/
0( ) t RCq t q e
/0( ) t RCdq qI t e
dt RC
V
19
• Capacitor has q0 = CV, switch is moved to position “b”
R Vbattery
C a
b
I + -
-I
Physics 212 Lecture 11, Slide 6
A
B
C
D
22
Checkpoint 1c
+ -
IR
A circuit is wired up as shown below. The capacitor is initially uncharged and switches
S1 and S2 are initially open.
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. IR = 0 B. IR = V/3R C. IR = V/2R D. IR = V/R
Physics 212 Lecture 11, Slide 7
V
2R C V
I
A
B
C
D
22
Checkpoint 1c
+ -
IR
A circuit is wired up as shown below. The capacitor is initially uncharged and switches
S1 and S2 are initially open.
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. IR = 0 B. IR = V/3R C. IR = V/2R D. IR = V/R
Physics 212 Lecture 11, Slide 8
A
B
C
26
Checkpoint 1d
A circuit is wired up as shown below. The capacitor is initially uncharged and switches
S1 and S2 are initially open.
Now suppose both switches are closed. What is the voltage across the capacitor after a
very long time?
A. VC = 0 B. VC = V C. VC = 2V/3
Physics 212 Lecture 11, Slide 9
A
B
C
26
Checkpoint 1d
• After both switches have been closed for a long time
• The current through the capacitor is zero
• The current through R = current through 2R
• Vcapacitor = V2R
• V2R = 2/3 V
A circuit is wired up as shown below. The capacitor is initially uncharged and switches
S1 and S2 are initially open.
Now suppose both switches are closed. What is the voltage across the capacitor after a
very long time?
A. VC = 0 B. VC = V C. VC = 2V/3
Physics 212 Lecture 11, Slide 10
2R C
R
S1 S2
Close both S1 and S2 and wait a long time…
V
2R C
R
V
No current flows through the capacitor after a long time. This will always be the case if the sources of EMF don’t change with time.
VC
I
I = V/(3R)
27
V2R = I(2R) = (2/3)V = VC VC = (2/3)V
Physics 212 Lecture 11, Slide 11
DEMO – ACT 1
V C
S
Bulb 1
Bulb 2
What will happen after I close the switch? A) Both bulbs come on and stay on. B) Both bulbs come on but then bulb 2 fades out. C) Both bulbs come on but then bulb 1 fades out. D) Both bulbs come on and then both fade out.
R
R
30
No initial charge on capacitor V(bulb 1) = V(bulb 2) = V Both bulbs light
No final current through capacitor
V(bulb 2) = 0
Physics 212 Lecture 11, Slide 12
DEMO – ACT 2
V C
S
Bulb 1
Bulb 2
Suppose the switch has been closed a long time. Now what will happen after open the switch? A) Both bulbs come on and stay on. B) Both bulbs come on but then bulb 2 fades out. C) Both bulbs come on but then bulb 1 fades out. D) Both bulbs come on and then both fade out.
R
R
32
Capacitor has charge (=CV) Capacitor discharges through both resistors
Physics 212 Lecture 11, Slide 13
Calculation
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 ?
– Circuit behavior described by Kirchhoff’s Rules:
• KVR: SVdrops = 0 • KCR: SIin = Siout
– S closed and C charges to some voltage with some time constant
– Determine currents and voltages in circuit a long time after S closed
V
R1 R2
C
R3
S
35
Physics 212 Lecture 11, Slide 14
Calculation
V
R1 R2
C
R3
S
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
• 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
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 ?
37
Physics 212 Lecture 11, Slide 15
Calculation
V
R1 R2
C
R3
S
• Why? – Draw circuit just after S closed
(knowing VC = 0)
• Immediately after S is closed, what is I1, the current through R1 ?
V
R1 R2
S
R3
VC = 0
I1
– R1 is in series with the parallel combination of R2 and 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 ?
39
(A) (B) (C) (D) (E)
1 3
V
R R1
V
R1 2 3
V
R R R 1 2 3
1 2 2 3 1 3
R R RV
R R R R R R
32
321
RR
RRR
V
Physics 212 Lecture 11, Slide 16
Calculation
V
R1 R2
C
R3
S
• 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
After S has been closed “for a long time”, what is IC, the current through C ?
(A) (B) (C)
1
V
R 0
V
R1 R3 IC = 0 VC
I
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 ?
2
V
R
41
Physics 212 Lecture 11, Slide 17
Calculation
V
R1 R2
C
R3
S
• Why??
After S has been closed “for a long time”, what is VC, the voltage across C ?
(A) (B) (C) (D) (E)
V3
1 3
RV
R R 0
2
2 31
2 3
RV
R RR
R R
2
1 2
RV
R R
– VC = V3 = IR3 = (V/(R1+R3))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 ?
V
R1 R3 VC
I I
43
Physics 212 Lecture 11, Slide 18
Challenge In this circuit, assume V, C, and Ri are known. C initially uncharged and then switch S is closed.
V
R1 R2
C
R3
S
• 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 t = “R” ”C”
C
What is tc, the charging time constant?
R2
C R3
2 3
2 3
1
2 ( )
R Rt C
R R
We get:
CRR
RRRc
31
312t
Physics 212 Lecture 11, Slide 19
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
0
Q t
Q
0
t
RCQ t Q e
t
RC
“Fraction of initial charge that remains”
“How many time constants worth of time that have elapsed”
45
How do exponentials work?
Physics 212 Lecture 11, Slide 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
0
Q t
Q
t
RC = 1
RC = 2
Time constant: t = RC The bigger t is, the longer it takes to get the same change…
0
t
RCQ t Q e
47
Physics 212 Lecture 11, Slide 21
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
RC = 1
RC = 2
Which circuit has the largest time constant?
A) Circuit 1
B) Circuit 2
C) Same
49
t = RequivC
Checkpoint 2a
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.
Physics 212 Lecture 11, Slide 22 50
Checkpoint 2b
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.
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
Physics 212 Lecture 11, Slide 23 50
Checkpoint 2b
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.
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
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
Physics 212 Lecture 11, Slide 24
Checkpoint 2b
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
RC = 1
RC = 2
Q = Q0e-t/RC
Look at plot !!!
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
Physics 212 Lecture 8, Slide 25
Charge capacitor – store a logical “1”
Discharge capacitor – store a logical “0”
Capacitor discharges through resistance
between plates.
Only holds Q for < 1 msec.
“Dynamic” random access memory
Charge Q must be “refreshed” constantly,
So memory is called dynamic.
Physics 212 Lecture 11, Slide 26
1. Capacitor is an open circuit for dc (direct current).
R
+Q
V C
IC
-Q
VC = Q/C and IC = dQ/dt. If IC is flowing then Q is changing, so VC is changing.
But in a dc circuit, nothing changes with time, so we must have IC = 0.
For t
V C
R
IC = 0
VC = V
Situation once things stop changing.