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RC Circuits
Physics 102: Lecture 7
Exam I: Monday February 18PRACTICE EXAM IS AVAILABLE-CLICK ON “COURSE INFO” ON WEB PAGE
Exam I
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Review, Sunday Feb. 17 3 PM, 141 Loomis
• I will go over a past exam
• On Monday, you get to vote for which exama) Fall 2007
b) Spring 2007
c) Fall 2006
d) Spring 2006
e) Fall 2005
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Recall ….
• First we covered circuits with batteries and capacitors
– series, parallel
• Then we covered circuits with batteries and resistors
– series, parallel– Kirchhoff’s Loop Law– Kirchhoff’s Junction Law
• Today: circuits with batteries, resistors, and capacitors
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RC CircuitsRC Circuits
RC Circuits
Charging Capacitors
Discharging Capacitors
Intermediate Behavior
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RC CircuitsRC Circuits
Circuits that have both resistors and capacitors:
With resistance in the circuits, capacitors do not charge and discharge instantaneously – it takes time (even if only fractions of a second).
V
R2
R1C
SS
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CapacitorsCapacitors
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.
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Charging CapacitorsCharging 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
RC
S
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Charging CapacitorsCharging 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?
RC
S
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ACT/Preflight 7.1
2R
C R
S2
Both switches are initially open, and the capacitor is uncharged. What is the current through the battery just after switch S1 is closed?
1) Ib = 0 2) Ib = E /(3R)
3) Ib = E /(2R) 4) Ib = E /R
S1
KVL: -E + I(2R) + q/C = 0
q = 0
I = E /(2R)
Ib
+
-
+
+
-
-
34%
44%
6%
17%
20
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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?
1) Ib = 0 2) Ib = E/(3R)
3) Ib = E/(2R) 4) Ib = E/R
2R
C R
S2
S1
Ib
+
-
+
+
-
-
• 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
56%
28%
5%
11%
24
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Discharging CapacitorsDischarging 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
RC
S
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ACT/Preflight 7.5After 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?
1) IR = 0 2) IR = V/(3R)
3) IR = V/(2R) 4) IR = V/R
KVL: -q0/C+IR = 0
Recall q is charge on capacitor after charging:q0=VC (since charged w/ switch 2 open!)-V + IR = 0 I = V/R
2R
C R
S2S1
IR
+
-
+
+
-
-
+
-
32%
16%
15%
37%
42
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ACT: RC CircuitsBoth switches are closed. What is the final charge on
the capacitor after the switches have been closed a long time?
1) Q = 0 2) Q = C E /3
3) Q = C E /2 4) Q = C E
KVL (right loop): -Q/C+IR = 0
KVL (outside loop), -E + I(2R) + IR = 0I = E /(3R)
KVL: -Q/C + E /(3R) R = 0Q = C E /3
R
2R
C
S2S1
IR
+
-
+
+
-
-
+
-
27
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RC Circuits: Charging
• KVL: - + I(t)R + q(t) / C = 0
• Just after…: q =q0– Capacitor is uncharged
– - + I0R = 0 I0 = / R
• Long time after: Ic= 0– Capacitor is fully charged
– - + q/C =0 q = C
• Intermediate (more complex)q(t) = q(1-e-t/RC)
I(t) = I0e-t/RC
C
R
S1
S2
+
+
+
I-
-
-
The switches are originally open and the capacitor is uncharged. Then switch S1 is closed.
t
q
RC 2RC
0
q
17
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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)
• Long time after: Ic=0– Capacitor is discharged (like a wire)
– -q / C = 0 q = 0
• Intermediate (more complex)q(t) = q0 e-t/RC
Ic(t) = I0 e-t/RC
C
R
S1
-+
+
I-
+
-
S2
q
RC 2RC
t40
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What is the time constant?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.
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Time Constant Demo
• Which system will be brightest?• Which lights will stay on longest?• Which lights consumes more energy?
2
Each circuit has a 1 F capacitor charged to 100 Volts.When the switch is closed:
1
= 2RC = RC/2
2 I=2V/R1
Same U=1/2 CV2
50
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Summary of ConceptsSummary 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
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Charging: Intermediate Times
2R
C R
S2
S1
Ib
+
-
+
+
-
-
Calculate the charge on the capacitor 310-3
seconds after switch 1 is closed.
q(t) = q(1-e-t/RC)
= q(1-e-310-3 /(2010010-6)))
= q (0.78)
Recall q = C
= (50)(100x10-6) (0.78)
= 3.9 x10-3 Coulombs
R = 10
V = 50 Volts
C = 100F
38
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Practice!
Calculate current immediately after switch is closed:
Calculate current after switch has been closed for 0.5 seconds:
Calculate current after switch has been closed for a long time:
Calculate charge on capacitor after switch has been closed for a long time:
R
CE
S1
R=10C=30 mF
E =20 Volts
-E + I0R + q0/C = 0I
+
+
+
-
-
--E + I0R + 0 = 0I0 = E/R
After a long time current through capacitor is zero!
-E + IR + q∞/C = 0
-E + 0 + q∞ /C = 0q∞ = E C 32
0
t
RCI I e
0.5
RCeR
0.5
10 0.0320
10e
0.5
10 0.0320
10e
0.38 Amps
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ACT: RC Challenge
1) 0.368 q0 2) 0.632 q0
3) 0.135 q0 4) 0.865 q0
R
CE 2R
S1
E = 24 Volts
R = 2
C = 15 mFAfter 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?
q(t) = q0 e-t/RC
= q0 (e-0.06 /(4(1510-3)))
= q0 (0.368)45
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RC Summary
Charging Discharging
q(t) = q(1-e-t/RC)
q(t) = q0e-t/RC
V(t) = V(1-e-t/RC)
V(t) = V0e-t/RC
I(t) = I0e-t/RCI(t) = I0e-t/RC
Short term: Charge doesn’t change (often zero or max)
Long term: Current through capacitor is zero.
Time Constant = RC Large means long time to charge/discharge
45