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Physics 102: Lecture 7
RC Circuits
Exam I: Monday February 18
PRACTICE EXAM IS AVAILABLE-CLICK ON
“COURSE INFO” ON WEB PAGE
Exam I
Review, Sunday Feb. 17
3 PM, 141 Loomis
•
•
I will go over a past exam
On Monday, you get to vote for which exam
a)
b)
c)
d)
e)
Fall 2007
Spring 2007
Fall 2006
Spring 2006
Fall 2005
Recall ….
•
First we covered circuits with batteries
and capacitors
–
•
Then we covered circuits with batteries
and resistors
–
–
–
•
series, parallel
series, parallel
Kirchhoff’s Loop Law
Kirchhoff’s Junction Law
Today: circuits with batteries, resistors,
and capacitors
RC Circuits

RC Circuits

Charging Capacitors

Discharging Capacitors

Intermediate Behavior
RC Circuits

Circuits that have both resistors and capacitors:
R2
S

R1
C
S
V
With resistance in the circuits, capacitors do not
charge and discharge instantaneously – it takes time
(even if only fractions of a second).
Capacitors


Charge on Capacitors cannot change instantly.
Short term behavior of Capacitor:
If the capacitor starts with no charge, it has no potential difference
across it and acts as a wire.
If the capacitor starts with charge, it has a potential difference across it
and acts as a battery.


Current through a Capacitor eventually goes to zero.
Long term behavior of Capacitor:
If the capacitor is charging, when fully charged no current flows and
capacitor acts as an open circuit.
If capacitor is discharging, potential difference goes to zero and no
current flows.
Charging Capacitors

Example: an RC circuit with a switch

When switch is first closed …
(short term behavior):
No charge on the capacitor since charge can not change
instantly – the capacitor acts as a wire or short circuit

After switch has been closed for a while…
(long term behavior):
No current flows through the capacitor – the capacitor
acts as a break in the circuit or open circuit
C
R
S

Charging Capacitors
Capacitor is initially uncharged and switch
is open. Switch is then closed. What is
current in circuit immediately thereafter?
 What is current in circuit a long time later?

C
R
S

ACT/Preflight 7.1
Both switches are initially open, and the capacitor is uncharged.
What is the current through the battery just after switch S1 is
closed?
+ 2R -
34%1) Ib = 0
6% 2) Ib = E /(3R)
44%3) Ib = E /(2R)
17% 4) Ib = E /R
Ib

+
C
-
KVL: -E + I(2R) + q/C = 0
q=0
+
-
R
S2
S1
 I = E /(2R)
20
ACT/Preflight 7.3
Both switches are initially open, and the capacitor is uncharged.
What is the current through the battery after switch 1 has been
closed a long time?
+ 2R -
56% 1) Ib = 0
5% 2) Ib = E/(3R)
28% 3) Ib = E/(2R) 11% 4) Ib = E/R
Ib

+
C
-
+
-
R
S2
S1
• Long time  current through capacitor is zero
• Ib=0 because the battery and capacitor are in series.
• KVL: - E + 0 + q/C = 0  q = E C
24
Discharging Capacitors


Example: a charged RC circuit with a switch
When switch is first closed…
(short term behavior):
Full charge is on the capacitor since charge can not
change instantly – the capacitor acts as a battery

After switch has been closed for a while…
(long term behavior):
No current flows through the capacitor – the capacitor
is fully discharged
R
C
S
ACT/Preflight 7.5
After switch 1 has been closed for a long time, it is opened and
switch 2 is closed. What is the current through the right resistor
just after switch 2 is closed?
+ 2R -
32% 1) IR = 0
16% 3) IR = V/(2R)
15% 2) IR = V/(3R)
37% 4) IR = V/R
KVL: -q0/C+IR = 0
IR
+
+
C
-
+
-
S1
R
-
S2
Recall q is charge on capacitor after charging:
q0=VC (since charged w/ switch 2 open!)
-V + IR = 0
 I = V/R
42
ACT: RC Circuits
Both switches are closed. What is the final charge on the
capacitor after the switches have been closed a long
time?
+ 2R -
1) Q = 0
3) Q = C E /2
2) Q = C E /3
4) Q = C E
IR

+
-
+
+
C
-
R
-
KVL (right loop): -Q/C+IR = 0
KVL (outside loop), -E + I(2R) + IR = 0
I = E /(3R)
KVL: -Q/C + E /(3R) R = 0
Q = C E /3
S2
27
RC Circuits: Charging
The switches are originally open and the capacitor is uncharged. Then
switch S1 is closed.
• KVL: -  + I(t)R + q(t) / C = 0
• Just after…: q =q0
+
R

– Capacitor is uncharged
+
- I
-
+
-
– -  + I0R = 0  I0 =  / R
• Long time after: Ic= 0
C
S2
S1
– Capacitor is fully charged
– -  + q/C =0  q =  C
q
1
RC
2RC
q(t) = q(1-e-t/RC)
I(t) = I0e-t/RC
Q
• Intermediate (more complex)
q
f( x ) 0.5
00
0
1
t
2
3
4
1
RC Circuits: Discharging
• KVL: - q(t) / C - I(t) R = 0
+
• Just after…: q=q0
– Capacitor is still fully charged
– -q0 / C - I0 R = 0  I0 = -q0 / (RC)
R

+
-
I
-
C
• Long time after: Ic=0
– Capacitor is discharged (like a wire)
S1
– -q / C = 0  q = 0
• Intermediate (more complex)
q(t) = q0 e-t/RC
Ic(t) = I0 e-t/RC
+
-
1
RC
1
S2
2RC
q
f( x ) 0.5
0.0183156
0
0
1
t
2
x
3
4
404
What is the time constant?

The time constant  = RC.

Given a capacitor starting with no charge, the time
constant is the amount of time an RC circuit takes to
charge a capacitor to about 63.2% of its final value.

The time constant is the amount of time an RC
circuit takes to discharge a capacitor by about 63.2%
of its original value.
Time Constant Demo
Each circuit has a 1 F capacitor charged to 100 Volts.
When the switch is closed:
• Which system will be brightest?
• Which lights will stay on longest?
• Which lights consumes more energy?
1
 = 2RC
2 I=2V/R
1
Same
U=1/2 CV2
2
 = RC/2
50
Summary of Concepts

Charging Capacitors
Short Term: capacitor has no charge, no potential difference, acts
as a wire
Long Term: capacitor fully charged, no current flows through
capacitor, acts as an open circuit

Discharging Capacitors
Short Term: capacitor is fully charged, potential difference is a
maximum, acts as a battery
Long Term: capacitor is fully discharged, potential difference is
zero, no current flows

Intermediate Behavior
Charge & Current exponentially approach their long-term values
 = RC
Charging: Intermediate Times
Calculate the charge on the capacitor 310-3 seconds
after switch 1 is closed.
R = 10 W
q(t) = q(1-e-t/RC)
=
V = 50 Volts
C = 100mF
-3 /(2010010-6))
-310
q(1-e
)
+ 2R -
= q (0.78)
Recall q =  C
Ib

+
C
-
R
S2
= (50)(100x10-6) (0.78)
= 3.9 x10-3 Coulombs
+
-
S1
38
Practice!
+
Calculate current immediately after switch is closed:
-E + I0R + q0/C = 0
-E + I0R + 0 = 0
I0 = E/R
I  I 0e


R
e
0.5
RC
-
I
+
-
Calculate current after switch has been closed for 0.5 seconds:
t
RC
R
0.5
20 100.5
20
 e 0.03  e100.03  0.38 Amps
10
10
C
E
+
-
S1
R=10W
C=30 mF
E =20 Volts
Calculate current after switch has been closed for a long time:
After a long time current through capacitor
is zero!
Calculate charge on capacitor after switch has been closed for a long time:
-E + IR + q∞/C = 0
-E + 0 + q∞ /C = 0
q∞ = E C
32
ACT: RC Challenge
E = 24 Volts
R=2W
After being closed for a long time, the
switch is opened. What is the charge
Q on the capacitor 0.06 seconds after
the switch is opened?
1) 0.368 q0
2) 0.632 q0
3) 0.135 q0
4) 0.865 q0
C = 15 mF
R
C
E
2R
S1
q(t) = q0 e-t/RC
= q0
-3))
-0.06
/(4(1510
(e
)
= q0 (0.368)
45
RC Summary
Charging
q(t) = q(1-e-t/RC)
V(t) = V(1-e-t/RC)
I(t) = I0e-t/RC
Discharging
q(t) = q0e-t/RC
V(t) = V0e-t/RC
I(t) = I0e-t/RC
Time Constant  = RC
Large  means long time to charge/discharge
Short term: Charge doesn’t change (often zero or
max)
Long term: Current through capacitor is zero.
45