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TRANSCRIPT
Today’s Concepts:
A) LC circuits and Oscillation Frequency
B) Energy
C) RLC circuits and Damping
Physics 2112
Unit 19
Electricity & Magnetism Lecture 19, Slide 1
Your Comments
Electricity & Magnetism Lecture 19, Slide 2
differential equations killing me.
Differential equations have taken over my life. I'm not sure I We've
been doing spring stuff in Diff Eq, so this is lining up nicely!
I was never good with springs during 2111 so that pre-lecture just
was rugged. The pre-lecture said without a battery an LC circuit
with a full capacitor would just oscillate back and forth since there is
no energy that can be added, but isn't there heat energy being
released through the wires that will also dampen the oscillation?
For capacitors you told us that huge capacitors were used in car
stereo systems. A similar example for how capacitors and inductors
are used would be nice.
Can you go over the differential equations again?
I find it very convenient that this is pretty much exactly like last
semesters spring problems. I'll need practice, but it seems very
straight forward!
Can we relate flux to snow?
C
I
L Q
LC Circuit
Electricity & Magnetism Lecture 19, Slide 3
+
-
dt
dILVL
+
-
C
QVC
Circuit Equation: 0+dt
dIL
C
Q
dt
dQI
LC
Q
dt
Qd-
2
2
where
Qdt
Qd 2
2
2
-LC
1
)cos()( max + tQtQSolution to this DE:
CL
k
x
m
F = -kx
a
Same thing if we notice that and
Electricity & Magnetism Lecture 19, Slide 4
Qdt
Qd 2
2
2
-LC
1
m
kx
dt
xd 2
2
2
-
Ck
1
Lm
Lm
Just like in 2111
k
x
m
F = -kx
a
Electricity & Magnetism Lecture 19, Slide 5
Also just like in 2111
22
2
1
2
1kxmv
Total Energy constant (when no friction present)
CL
2
2
2
1
2
1
C
QLI
Total Energy constant (when no resistor present)
Unit 19, Slide 6
Prelecture Question
What happens to the frequency ω of the oscillations
when the switch is closed?
A. Increases (faster oscillations)
B. Remains the same
C.Decreases (slower oscillations)
The switch is initially open
and the circuit is
oscillating with a
frequency ω =1/sqrt(LC).
CL
I+ +
--
Q and I Time Dependence
Electricity & Magnetism Lecture 19, Slide 7
Charge and Current “90o out of
phase”
What is the potential difference across the inductor at t = 0?
A) VL = 0
B) VL = Qmax/C
C) VL = Qmax/2C
At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C
The voltage across the capacitor is Qmax/C Kirchhoff's Voltage Rule implies that must also be equal to the voltage across the inductor
since VL VC
CheckPoint 1(A)
Electricity & Magnetism Lecture 19, Slide 8
dI/dt is zero when current is maximum
CheckPoint 1(B)
Electricity & Magnetism Lecture 19, Slide 9
At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0.
L C
What is the potential difference across the inductor at when the current is maximum?
A) VL = 0
B) VL = Qmax/C
C) VL = Qmax/2C
How much energy is stored in the capacitor when the current is a maximum ?
A) U Qmax2/(2C)
B) U Qmax2/(4C)
C) U 0
Total Energy is constant!ULmax ½ LImax
2
UCmax Qmax2/2C
I max when Q 0
CheckPoint 1(C)
Electricity & Magnetism Lecture 19, Slide 10
At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C
The switch is closed for a long time, resulting in a steady current Vb/R through the inductor.
A. Q(t) = Qmaxcos(ωt) B. Q(t) = Qmaxcos(ωt + π/4) C. Q(t) = Qmaxcos(ωt + π/2) D. Q(t) = Qmaxcos(ωt + π)
Prelecture Question
Electricity & Magnetism Lecture 19, Slide 11
At time t = 0, the switch is opened, leaving a simple LC circuit. Which formula best describes the charge on the capacitor as a function of time?
L=0.82H
Example 19.1 (Charged LC circuit)
Unit 19, Slide 12
1 2After being left in position 1 for a long time, at t=0, the switch is flipped to position 2.
All circuit elements are “ideal”.
C=0.01FV=12V
What is the equation for the charge on the upper plate of the capacitor at any given time t?
How long does it take the lower plate of the capacitor to fully discharge and recharge?
R=10W
Example 19.1 (Charged LC circuit)
Unit 19, Slide 13
A) Find the current through the inductor when the switch is in position 1. Energy will be conserved.
B) Find the rate of energy lost in the resistor when the switch is position 1. Energy will be conserved.
C) Find the voltage drop across the inductor when the switch is in position 1. The inductor and the capacitor will be in parallel.
To figure out the equation for the charge on the upper plate of the capacitor at any given time t, I’m going fill in the number in this equation:
How will I figure out Qmax?
)cos()( max + tQtQ
L=0.82H
Example 19.1 (Charged LC circuit)
Unit 19, Slide 14
1 2After being left in position 1 for a long time, at t=0, the switch is flipped to position 2.
All circuit elements are “ideal”.
C=0.01FV=12V
What is the equation for the charge on the upper plate of the capacitor at any given time t?
How long does it take the lower plate of the capacitor to fully discharge and recharge?
R=10W
The capacitor is charged such that the top plate has a charge +Q0 and the bottom plate -Q0. At time t 0, the switch is closed and the circuit oscillates with frequency 500
radians/s.
What is the value of the capacitor C?
A) C = 1 x 10-3 F
B) C = 2 x 10-3 F
C) C = 4 x 10-3 F
L C
L = 4 x 10-3 H
= 500 rad/s+ +
--
CheckPoint 2(A)
Electricity & Magnetism Lecture 19, Slide 15
LC
1
3
34210
)104)(1025(
11 -
-
LC
Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed?
Current is changing UL is not constant
Initial current is zero
L C
closed at t = 0
+Q0
-Q0
CheckPoint 2(B)
Electricity & Magnetism Lecture 19, Slide 16
Energy proportional to I2 C cannot be negative
2
2
1LIUL
When the energy stored in the capacitor reaches its maximum again for the first time after t 0, how much charge is stored on the top plate of the capacitor?
A) +Q0
B) +Q0 /2
C) 0
D) -Q0/2
E) -Q0
CheckPoint 2(C)
Electricity & Magnetism Lecture 19, Slide 17
L C
closed at t = 0
+Q0
-Q0
Current goes to zero twice during one cycle
Q is maximum when current goes to zero
dt
dQI
Add Resistance
Unit 19, Slide 18
1 2
Switch is flipped to position 2.
R
01
2
2
--- QCdt
dQR
dt
QdL
02 ++ cbtatUse “characteristic
equation”
Damped Harmonic Motin
Unit 19, Slide 19
)'cos( + - teQQ t
Max
L
R
2
222' - o
Damping
factor
Damped
oscillation
frequency
LCo
1
Natural
oscillation
frequency
Remember from 2111?
Mechanics Lecture 21, Slide 20
Overdamped
Critically damped
Under damped
Which one do you want for your shocks
on your car?
L=0.82H
Example 19.2 (Charged LC circuit)
Unit 19, Slide 21
1 2After being left in position 1 for a long time, at t=0, the switch is flipped to position 2.
The inductor now non-ideal has some internal resistance.
What is the equation for the charge on the capacitor at any given time t?
C=0.01FV=12V
RL=7W
R=10W
How long does it take the lower plate of the capacitor to
fully discharge and recharge?
Example 19.2 (Charged LC circuit)
Unit 19, Slide 22
When we did this problem before without any resistance, we found that it takes 0.57 sec for the lower plate of the capacitor to fully discharge and recharge.
How long will it take now that I am accounting for the resistance of the inductor?
A. more than 0.57secB. less than 0.57secC. the discharge time is not effected by the
resistance
The elements of a circuit are very simple:
This is all we need to know to solve for anything.
But these all depend on each other and vary with time!!
Electricity & Magnetism Lecture 19, Slide 23
dt
dILVL
C
QVC
dt
dQI
IRVR
RCL VVVV ++
V(t) across each elements
Start with some initial V, I, Q, VL
Now take a tiny time step dt (1 ms)
Repeat…
How would we actually do this?
Electricity & Magnetism Lecture 19, Slide 24
C
QVC
IRVR
dtL
VId L
IdtdQ
CRL VVVV --
What would this look like?
Unit 19, Slide 25
VL
VR
VC
I
In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.
(The positive terminal of the battery is indicated with a + sign. All elements are considered ideal.)
Question
Electricity & Magnetism Lecture 19, Slide 26
The switch has been closed for a long time. What is the energy stored on the capacitor?
A. 0 JB. 0.0015JC. 0.0034JD. 0.0069JE. 1.2J
In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.
(The positive terminal of the battery is indicated with a + sign. All elements are considered ideal.)
Question
Electricity & Magnetism Lecture 19, Slide 27
The switch has been closed for a long time. What is the energy stored in the inductors?
A. 0 J
B. 0.0015J
C. 0.0034J
D. 0.0069JE. 1.2J
In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.
Example 19.3
Electricity & Magnetism Lecture 19, Slide 28
The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive. The switch has been closed for a long time.
What is the charge on the bottom plate of the capacitor 2.82 seconds after switch is open? Conceptual Analysis
Find the equation of Q as a function of time Strategic Analysis
Determine initial current
Determine oscillation frequency 0
Find maximum charge on capacitor
Determine phase angle
In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.
Example 19.4
Electricity & Magnetism Lecture 19, Slide 29
The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive.
What is the energy stored in the inductors 2.82 seconds after switch is opened?
Conceptual AnalysisUse conservation of energy
Strategic AnalysisFind energy stored on capacitor using information from Ex 19.3
Subtract of the total energy calculated in the clicker question