rc circuits physics 102: lecture 7 exam i: monday february 18 practice exam is available-click on...
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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
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
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
RC CircuitsRC Circuits
RC Circuits
Charging Capacitors
Discharging Capacitors
Intermediate Behavior
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
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.
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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