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 1
RC Circuit Charging
• Capacitor uncharged, switch is moved to position “a”
aa
• Kirchoff’s Voltage Rule
C
C
I
q
-Vbattery + +IR = 0
C
VVbattery
battery
bb
RR
• Initially (q = q0 = 0)
Vbattery  0  I0R  0
Vbattery
I0 =
R
• Long Term (Ic =0)
11
q
Vbattery   0  R  0
C
q  CVbattery
In general:
Vbattery
q dq
  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
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
V = Q/C
V1 = 0
V2 = V
B) V1 = 0
C) V1 = 0
V2 = V
V2 = 0
D) V1 = V
V2 = 0
After the switch S1 has
been closed for a long time
I=0
VR = 0
V2 = V
Physics 212 Lecture 11, Slide 3
Close S1 at t=0
(leave S2 open)
R
C
V
S1
2R
S2
R
I
V
15
R
I=0
C
V
C
VC = Q/C
=0
VC = V
At t = 0
For t 
Physics 212 Lecture 11, Slide 4
RC Circuit (Discharging)
• Capacitor has q0 = CV, switch is moved to position “b”
aa
• Kirchoff’s Voltage Rule
CC
q
  IR  0
C
• Initially (q=q0)
q0
 I0R  0
C
q0
I0 
RC
• Long Term (Ic =0)
q
 0R  0
C
q  0
19
+ -
I
bb
VVbattery
battery
RR
In general:
V
q dq
  R0
C dt
q(t )  q0e  t /RC
-I
dq
q0 t / RC
I(t ) 

e
dt
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. IR = 0
B. IR = V/3R
C. IR = V/2R
D. IR = V/R
B
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. IR = 0
B. IR = V/3R
C. IR = V/2R
D. IR = V/R
B
C
D
I
V
22
C
V
2R
Physics 212 Lecture 11, Slide 7
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
A. VC = 0
B. VC = V
C. VC = 2V/3
C
26
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
A. VC = 0
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
• The current through R = current through 2R
• Vcapacitor = V2R
• V2R = 2/3 V
26
Physics 212 Lecture 11, Slide 9
Close both S1 and S2 and
wait a long time…
R
C
V
S1
2R
S2
I
No current flows
through the capacitor
after a long time. This
R
V
C
I = V/(3R)
27
VC
2R
will always be the case
if the sources of EMF
don’t change with time.
V2R = I(2R) = (2/3)V = VC
VC = (2/3)V
Physics 212 Lecture 11, Slide 10
DEMO – ACT 1
Bulb 2
S
V
Bulb 1
R
R
C
What will happen after I close the switch?
A)
B)
C)
D)
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.
No initial charge
on capacitor
V(bulb 1) = V(bulb 2) = V
No final current
through capacitor
V(bulb 2) = 0
Both bulbs light
Physics 212 Lecture 11, Slide 11
DEMO – ACT 2
Bulb 2
R
S
V
Bulb 1
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 has charge (=CV)
32
Capacitor discharges through both resistors
Physics 212 Lecture 11, Slide 12
Calculation
S
R1
R2
C
V
–
35
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 ?
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
Physics 212 Lecture 11, Slide 13
Calculation
S
R1
R2
C
V
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
• Why?
–
–
37
(B) Only VC = 0 (C) Both I2 and VC = 0 (D) Neither I2 nor VC = 0
We are told that C is initially uncharged (V = Q/C)
I2 cannot be zero because charge must flow in order to charge C
Physics 212 Lecture 11, Slide 14
I1
R1
Calculation
S
R2
C
V
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
R1
V
R1  R3
V
R1  R2  R3
(A)
(B)
(C)
• Why?
39
–
Draw circuit just after S closed
(knowing VC = 0)
–
R1 is in series with the parallel
combination of R2 and R3
V
R R
R1  2 3
R2  R3
(D)
R1
V
R1  R 2  R3
V
R1 R 2  R 2 R3  R1 R3
(E)
S
R2
R3
VC = 0
Physics 212 Lecture 11, Slide 15
Calculation
S
R1
R2
C
V
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
(A)
(B)
(C)
• Why?
–
–
41
I
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
Physics 212 Lecture 11, Slide 16
Calculation
S
R1
R2
C
V
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
(A)
V
R2
R1  R2
V
(B)
(C)
• Why??
–
VC = V3 = IR3 = (V/(R1+R3))R3
V
R1
R2
RR
R1  2 3
R2  R3
(D)
I
0
(E)
I
VC
R3
V
43
Physics 212 Lecture 11, Slide 17
Challenge
In this circuit, assume V, C, and Ri are known.
C initially uncharged and then switch S is closed.
S
R1
R2
C
V
R3
What is tc, 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 t = “R” ”C”
We get:

R1 R3 

C
t c   R2 
R1  R3 

Physics 212 Lecture 11, Slide 18
How do exponentials work?
Q  t   Q0e
1
Q t 
Q0

t
RC
0.9
0.8
0.7
0.6
“Fraction of initial
charge that remains”
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
“How many time constants worth
of time that have elapsed”
45
4
5
6
7
8
9
10
t
RC
Physics 212 Lecture 11, Slide 19
Q t 
Q0
1
Q  t   Q0e
0.9
0.8

t
RC
0.7
0.6
0.5
0.4
0.3
RC = 2
0.2
0.1
Time constant:
t = RC
RC = 1
0
0
The bigger t is,
the longer it takes to get
the same change…
47
1
2
3
4
5
6
7
8
9
10
t
Physics 212 Lecture 11, Slide 20
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) Circuit 1
B)
C)
Circuit 2
Same
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
t = RequivC
RC = 2
0.2
0.1
RC = 1
0
0
49
1
2
3
4
5
6
7
8
9
10
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 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 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
Checkpoint2b
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 = Q0
e-t/RC
0.6
0.5
0.4
0.3
Look at plot !!!
RC = 2
0.2
0.1
RC = 1
0
0
1
Physics
Lecture
11,
Slide
2
3 212
4
5
6
7
8
9 24
10
“Dynamic” random access memory
Charge capacitor – store a logical “1”
Discharge capacitor – store a logical “0”
Capacitor discharges through resistance
between plates.
Only holds Q for < 1 msec.
Charge Q must be “refreshed” constantly,
So memory is called dynamic.
Physics 212 Lecture 8, Slide 25
1. Capacitor is an open circuit for dc (direct current).
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.
R
V
IC
R
IC = 0
+Q
C
-Q
V
C
VC = V
For t 
Situation once things stop changing.
Physics 212 Lecture 11, Slide 26