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Physics 212
Lecture 11
Today's Concept:
RC Circuits
(Circuits with resistors & capacitors & batteries)
Physics 212 Lecture 11, Slide 1
Music
Who is the Artist?
A)
B)
C)
D)
E)
Professor Longhair
John Cleary
Allen Toussaint
Fats Domino
Tuts Washington
Salute to
Mardi Gras (today!)
“Piano Players Rarely Ever Play
Together”
Your Comments
“thought this section would be easy, this made it hard”
“This was the first time I was virtually lost on the checkpoints. I am hoping that a
clearer explanation of things in class will help.”
“Everything is a little jumbled. Please go over what changes and what stays the
same in the long and short term of these systems”
Most of our time on checkpoints
and examples
“Urrrg, there were a lot of crazy equations today, and I feel like they went through
the derivations kinda fast... Yup, im confused!”
“Differential Equation!! I dont know what is going on there!!!!”
“When did differential equations become a prerequisite? ”
First, remember this is just Kirchoff.
We have to learn the solutions of two simple d.e.s
“This stuff really has me charged up. I can’t resist!”
“Trying to conceptually join resistors and capacitors is currently demanding all my
power . . .”
“If this is on the test we R going to get a C”
05
Physics 212 Lecture 11, Slide 3
Key Concepts:
1) Understanding the behavior of capacitors
in circuits with resistors
2) Understanding the RC time constant
Today’s Plan:
1) Examples with switches
closing and opening
- What changes?
- What is constant?
2) Example problem
3) Exponentials
07
Physics 212 Lecture 11, Slide 4
The 212 Differential Equations
• We describe the world (electrical circuits, problems in heat
transfer, control systems, etc., etc.) using differential
equations
• You only need to know the solutions of two basic
differential equations
dq 1
 q0
dt 
2
d q
2
 q  0
2
dt
q  qconst e
 t /
q  qconst sin t   
Physics 212 Lecture 11, Slide 5
RC Circuit (Charging)
• Capacitor uncharged, Switch is moved to position “a”
aa
• Kirchoff’s Voltage Rule
C
C
q
-Vbattery + +IR = 0
C
bb
V
Vbattery
battery
RR
• Short Term (q = q0 = 0)
Vbattery  0  I0R  0
Vbattery
I0 =
R
• Long Term (Ic =0)
11
q
Vbattery 
 0R  0
C
q  CVbattery
Intermediate
Vbattery
q dq
  R0
C dt
q(t )  q (1  e  t /RC )
I(t )  I0e  t /RC
Physics 212 Lecture 11, Slide 6
Solving the Differential Equation
a
C
• What do we do with
q dq
 R  Vbattery ??
b
V
R
C dt
• First consider (without Vb – a constant)
dq q

0
dt RC
• Guess the solution (in 212 there are only 2 choices!)
q
dq
q  qae t /RC:
  a e t /RC
d const 
dt
RC
 0q  qae t /RC  const
dt
dq q Vb
• Is this it?

 
dt RC R
A) Add another exponential
qa t /RC qa t /RC const Vb
B) Add a constant

e

e


RC
RC
RC
R
C) This IS it
and q t  0  0  qa  VbC  qa  VbC
batter
y
11
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 1a
&
Checkpoint 1b
A) V1 = V
Close S1,
V1 = voltage across C immediately after
V2 = voltage across C a long time after
13
V2 = V
B) V1 = 0
C) V1 = 0
V2 = V
V2 = 0
D) V1 = V
V2 = 0
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 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 9
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
At t = big
Physics 212 Lecture 11, Slide 10
RC Circuit (Discharging)
• Capacitor has q0 = CVbattery, Switch is moved to position “b”
aa
• Kirchoff’s Voltage Rule
CC
q
  IR  0
C
• Short Term (q=q0)
+ -
I
bb
VVbattery
battery
RR
Vbattery  IR  0
Vbattery
I0 
R
• Long Term (Ic =0)
q
 0R  0
C
q  0
19
Intermediate
q dq
  R0
C dt
q(t )  q0e  t /RC
V
-I
I(t )  I0e  t /RC
Physics 212 Lecture 11, Slide 11
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
“The loop is no longer closed so there will be no voltage”
“I = V / total resistance which is 3R.”
“The capacitor initially acts as a battery with voltage V, so Ohm’s law gives a current of V/2R”
“The capacitor has potential difference V across its plates. When that charge is given a path from
one plate to another, it will travel through the resistor until the plates are at equal potential”
22
Physics 212 Lecture 11, Slide 12
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 13
Open S1 at t=big
and close S2
R
C
V
S1
2R
S2
V
I
V
C
2R
I = V/2R
23
Physics 212 Lecture 11, Slide 14
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
“Everything eventually goes to zero.”
“Fully charged with large time.”
“the resistance only allows for so much of the potential to be stored in the cap”
26
Physics 212 Lecture 11, Slide 15
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 16
Close both S1 and S2 and
wait a long time…
R
C
V
S1
2R
S2
I
R
V
C
I = V/(3R)
VC = V2R
27
VC
2R
No current flows
through the capacitor
after a long time. This
will always be the case
in any static circuit!!
VC = I(2R)
VC = (2/3)V
Physics 212 Lecture 11, Slide 17
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 18
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 19
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 ?
• Conceptual Analysis:
–
–
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
• Strategic Analysis
–
35
Determine currents and voltages in circuit a long time after S closed
Physics 212 Lecture 11, Slide 20
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 21
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  R2  R3
V
R1 R2  R2 R3  R1 R3
(E)
S
R2
R3
VC = 0
Physics 212 Lecture 11, Slide 22
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 23
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 24
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 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 25
How do exponentials work?
Q t 
Q0
Q  t   Q0 e
1

t
RC
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Time constant:
 = RC
RC = 2
0.2
0.1
RC = 1
0
0
The bigger  is,
the longer it takes to get
the same change…
47
1
2
3
4
5
6
7
8
9
10
Physics 212 Lecture 11, Slide 27
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
 = 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 28
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
“C1 will discharge faster than C2 because of the resistance”
“The smaller time constant means the charge will dissipate slower”
“Charge after time doesn’t depend on resistance. Only V and C which are the same”
“The charge decreases exponentially so at first 2 will be greater then eventually 1 will be greater. ”
“C1 charges faster, but C2 has a higher max charge”
50
Physics 212 Lecture 11, Slide 29
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 30
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 31
10