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Physics 2112
Unit 19
Today’s Concepts:
A) LC circuits and Oscillation Frequency
B) Energy
C) RLC circuits and Damping
Electricity & Magnetism Lecture 19, Slide 1
LC Circuit
-
+
I
dI
VL  L
dt
L
C
Q
VC 
C
Q
+
-
Q
dI
+
L
0
Circuit Equation:
C
dt
dQ
I
dt
Solution to this DE:
d 2Q
Q
2
dt
LC
d 2Q
2


Q
2
dt
where

1
LC
Q(t )  Qmax cos(t +  )
Electricity & Magnetism Lecture 19, Slide 2
Just like in 2111
mL
d 2Q
2


Q
2
dt
2
d x
2


x
2
dt

1
LC
L
k
k

m
C
F = -kx
a
m
x
Same thing if we notice that
1
k
C
and
mL
Electricity & Magnetism Lecture 19, Slide 3
Also just like in 2111
1 2
1 2
mv  kx
2
2
k
F = -kx
a
m
Total Energy constant (when no friction present)
x
2
1Q
1 2
 LI
2C
2
L
C
Total Energy constant (when no resistor present)
Electricity & Magnetism Lecture 19, Slide 4
Q and I Time Dependence
I
L
++
C--
Charge and Current “90o out of
phase”
Electricity & Magnetism Lecture 19, Slide 5
CheckPoint 1
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 t = 0?
A) VL = 0
B) VL = Qmax/C
C) VL = Qmax/2C
since VL  VC
Electricity & Magnetism Lecture 19, Slide 6
CheckPoint 2
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
Electricity & Magnetism Lecture 19, Slide 7
CheckPoint 3
At time t = 0 the capacitor is fully
charged with Qmax and the current
through the circuit is 0.
L
C
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
Electricity & Magnetism Lecture 19, Slide 8
Example 19.1 (Charged LC circuit)
After being left in
position 1 for a long
time, at t=0, the
switch is flipped to
V=12V
position 2.
L=0.82H
All circuit elements
are “ideal”.
R=10W
1
2
C=0.01F
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?
Unit 19, Slide 9
Example 19.1 (Charged LC circuit)
•
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?
 Conceptual Analysis
Fill in all the terms in
Q(t )  Qmax cos(t +  )
 Strategic Analysis
Find initial current
Use energy conservation to find Qmax
Use L and C to find 
Use initial conditions to final 
Unit 19, Slide 10
Prelecture Question
The switch is closed for a long
time, resulting in a steady
current Vb/R through the
inductor.
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?
A.
B.
C.
D.
Q(t) = Qmaxcos(ωt)
Q(t) = Qmaxcos(ωt + π/4)
Q(t) = Qmaxcos(ωt + π/2)
Q(t) = Qmaxcos(ωt + π)
Electricity & Magnetism Lecture 19, Slide 11
CheckPoint 4
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.
L
++
C - -
L = 4 x 10-3 H
 = 500 rad/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
Electricity & Magnetism Lecture 19, Slide 12
CheckPoint 5
closed at t = 0
L
C
+Q0
-Q0
Which plot best represents the energy
in the inductor as a function of time
starting just after the switch is closed?
1 2
U L  LI
2
Electricity & Magnetism Lecture 19, Slide 13
CheckPoint 6
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)
B)
C)
D)
E)
closed at t = 0
L
C
+Q0
-Q0
+Q0
+Q0 /2
0
-Q0/2
-Q0
Q is maximum when current goes to zero
dQ
dt
Current goes to zero twice during one cycle
I
Electricity & Magnetism Lecture 19, Slide 14
Add Resistance
1
2
Switch is flipped to
position 2.
R
d 2Q
dQ 1
-L 2 -R
- Q0
dt
dt C
Use “characteristic
equation”
at + bt + c  0
2
Unit 19, Slide 15
Damped Harmonic Motin
Q  QMax e
- t
cos( ' t +  )
Damping
factor
R

2L
Natural
oscillation
frequency
o 
1
LC
Damped
oscillation
frequency
 '2  o2 -  2
Unit 19, Slide 16
Remember from 2111?
Overdamped
Critically damped
Under damped
Which one do you want for your shocks
on your car?
Mechanics Lecture 21, Slide 17
Example 19.2 (Charged LC circuit)
1
After being left in
position 1 for a long
time, at t=0, the switch
is flipped to position 2.
RL=7W
V=12V
The inductor now nonideal has some internal
resistance.
2
L=0.82H
C=0.01F
R=10W
What is the equation for the charge on the capacitor at
any given time t?
How long does it take the lower plate of the capacitor to
fully discharge and recharge?
Unit 19, Slide 18
V(t) across each elements
The elements of a circuit are very simple:
dI
VL  L
dt
VC 
V  VL + VC + VR
I
Q
C
dQ
dt
VR  IR
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 19
How would we actually do this?
Start with some initial V, I, Q, VL
Now take a tiny time step dt
(1 ms)
VL
d I  dt
L
dQ  Idt
Q
VC 
C
Repeat…
VR  IR
VL  V - VR - VC
What would this look like?
Electricity & Magnetism Lecture 19, Slide 20
Example 19.3
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. Q(t) is defined to be positive if V(a) – V(b) is positive.
What is the charge on the bottom plate of the capacitor at time t = 2.82 ms?
 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
Electricity & Magnetism Lecture 19, Slide 21
Example 19.4
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. Q(t) is defined to be positive if V(a) – V(b) is positive.
What is the energy stored in the inductors at time t = 2.82 ms?
 Conceptual Analysis
Use conservation of energy
 Strategic Analysis
Find energy stored on capacitor using information from Ex 19.3
Subtract of the total energy calculated in the clicker question
Electricity & Magnetism Lecture 19, Slide 22