Download Document

Your Comments
I am not feeling great about this midterm...some of this stuff is really confusing
still and I don't know if I can shove everything into my brain in time, especially
after spring break. Can you go over which topics will be on the exam
Wednesday?
This was a very confusing prelecture. Do you think you could go over
thoroughly how the LC circuits work qualitatively?
I may have missed something simple, but in
question 1 during the prelecture why does the
charge on the capacitor have to be 0 at t=0? I feel
like that bit of knowledge will help me with the test
wednesday
I remember you mentioning several weeks ago that there was one equation
you were going to add to the 2013 equation sheet... which formula was that?
Thanks!
Electricity & Magnetism Lecture 19, Slide 1
Some Exam Stuff
Exam Wed. night (March 27th) at 7:00



Covers material in Lectures 9 – 18
Bring your ID: Rooms determined by discussion section (see link)
Conflict exam at 5:15
Don’t forget:
•
•
Worked examples in homeworks (the optional questions)
Other old exams
For most people, taking old exams is most beneficial
Final Exam dates are now posted
The Big Ideas L9-18
Kirchoff’s Rules



Sum of voltages around a loop is zero
Sum of currents into a node is zero
Kirchoff’s rules with capacitors and inductors
• In RC and RL circuits: charge and current involve exponential functions
with time constant: “charging and discharging”
•

E.g.
IR 
Q
dQ Q
R
 V
C
dt C
Capacitors and inductors store energy
Magnetic fields




Generated by electric currents (no magnetic charges)
Magnetic forces only on charges in motion Fmag  qvxB
Easiest to calculate with Ampere’s Law
B  d  0 I enclosed
Changing magnetic fields can generate electric fields! FARADAY’S LAW

dmag
d
 E  d  EMF  V   dt  B  dA   dt
Physics 212 Lecture 19, Slide 3
Physics 212
Lecture 19
Today’s Concepts:
A) Oscillation Frequency
B) Energy
C) Damping
Electricity & Magnetism Lecture 19, Slide 4
LC Circuit


I
dI
VL  L
dt
L
C

Q
VC 
C
Q

Q
dI

L
0
Circuit Equation:
C
dt
dQ
I
dt
d 2Q
Q


dt 2
LC
d 2Q
2



Q
2
dt
where

1
LC
Electricity & Magnetism Lecture 19, Slide 5
CheckPoint 1a
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
The voltage across the capacitor is Qmax/C Kirchhoff's
Voltage Rule implies that must also be equal to the
voltage across the inductor
Pendulum.
Electricity & Magnetism Lecture 19, Slide 6
LC Circuits analogous to mass on spring
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 7
Time Dependence
I
L
C


Electricity & Magnetism Lecture 19, Slide 8
CheckPoint 1b
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
dI/dt is zero when current is maximum
Electricity & Magnetism Lecture 19, Slide 9
CheckPoint 1c
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
Total Energy is constant!
ULmax  ½ LImax2
UCmax  Qmax2/2C
I  max when Q  0
Electricity & Magnetism Lecture 19, Slide 10
CheckPoint 2a
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

1
LC
C
1
2L

1
3

10
(25 10 4 )( 4 10 3 )
Electricity & Magnetism Lecture 19, Slide 11
CheckPoint 2b
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
Energy proportional to I2  C cannot be negative
Current is changing  UL is not constant
Initial current is zero
Electricity & Magnetism Lecture 19, Slide 12
CheckPoint 2c
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 13
Add R: Damping
Just like LC circuit but energy but the oscillations get smaller because of R
Concept makes sense…
…but answer looks kind of complicated
Electricity & Magnetism Lecture 19, Slide 14
Physics Truth #1:
Even though the answer sometimes looks complicated…
Q(t )  Qo cos(t   )
the physics under the hood is still very simple!
d 2Q
2



Q
2
dt
Electricity & Magnetism Lecture 19, Slide 15
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!
Electricity & Magnetism Lecture 19, Slide 16
Calculation
The switch in the circuit shown has
been closed for a long time. At t  0,
the switch is opened.
What is QMAX, the maximum charge on
the capacitor?
V
L
C
R
Conceptual Analysis
Once switch is opened, we have an LC circuit
Current will oscillate with natural frequency 0
Strategic Analysis
Determine initial current
Determine oscillation frequency 0
Find maximum charge on capacitor
Electricity & Magnetism Lecture 19, Slide 18
Calculation
The switch in the circuit shown has
been closed for a long time. At t = 0,
the switch is opened.
IL
V
L
C
R
What is IL, the current in the inductor, immediately after the switch is opened?
Take positive direction as shown.
A) IL < 0
B) IL  0
C) IL > 0
Current through inductor immediately after switch is opened
is the same as
the current through inductor immediately before switch is opened
before switch is opened:
all current goes through inductor in direction shown
Electricity & Magnetism Lecture 19, Slide 19
Calculation
The switch in the circuit shown has
been closed for a long time. At t = 0,
the switch is opened.
IL
V
L
VC  0
C
R
IL (t  0 ) > 0
The energy stored in the capacitor immediately after the switch is opened is zero.
A) TRUE
B) FALSE
before switch is opened:
after switch is opened:
dIL/dt ~ 0  VL  0
VC cannot change abruptly
VC  0
BUT: VL  VC
since they are in parallel
VC  0
UC  ½ CVC2  0 !
IMPORTANT: NOTE DIFFERENT CONSTRAINTS AFTER SWITCH OPENED
CURRENT through INDUCTOR cannot change abruptly
VOLTAGE across CAPACITOR cannot change abruptly
Electricity & Magnetism Lecture 19, Slide 20
Calculation
The switch in the circuit shown has
been closed for a long time. At t = 0,
the switch is opened.
IL
V
C
L
R
IL (t  0 ) > 0
VC (t  0 )  0
What is the direction of the current immediately after the switch is opened?
A) clockwise
B) counterclockwise
Current through inductor immediately after switch is opened
is the same as
the current through inductor immediately before switch is opened
Before switch is opened: Current moves down through L
After switch is opened: Current continues to move down through L
Electricity & Magnetism Lecture 19, Slide 21
Calculation
The switch in the circuit shown has
been closed for a long time. At t = 0,
the switch is opened.
V
C
L
R
VC (t  0 )  0
IL (t  0 ) > 0
What is the magnitude of the current right after the switch is opened?
A) I o  V C
L
B) I o  V2
R
V
C) I o 
R
L
C
D) I o 
V
2R
Current through inductor immediately after switch is opened
is the same as
the current through inductor immediately before switch is opened
Before switch is opened:
IL
V
IL
L
IL
R
VL  0
C
VL  0
V  ILR
Electricity & Magnetism Lecture 19, Slide 22
Calculation
The switch in the circuit shown has
been closed for a long time. At t = 0,
the switch is opened.
IL
V
L
C
IL (t  0 )  V/R
VC (t  0 )  0
R
Hint: Energy is conserved
What is Qmax, the maximum charge on the capacitor during the oscillations?
A) Qmax
V

LC
R
Imax
Qmax
C
L
When I is max
(and Q is 0)
1
U  LI 2
2
1
Q

CV
B) max
2
C
L
When Q is max
(and I is 0)
2
1 Qmax
U
2 C
C) Qmax  CV
D) Qmax 
V
R LC
2
1 2 1 Qmax
LI 
2
2 C
Qmax  I max LC 
V
LC
R
Electricity & Magnetism Lecture 19, Slide 23
Follow-Up
The switch in the circuit shown has
been closed for a long time. At t = 0,
the switch is opened.
IL
V
R
Is it possible for the maximum voltage on the
capacitor to be greater than V ?
A) YES
Qmax 
V
LC
R
Imax  V/R
B) NO
Vmax
V L

R C
C
L
Vmax can be greater than V IF:
Qmax 
V
LC
R
L
>R
C
We can rewrite this condition in terms of the resonant frequency:
0 L > R
OR
1
>R
0 C
We will see these forms again when we study AC circuits!
Electricity & Magnetism Lecture 19, Slide 24