SLIDE 1
Physics 2102 Jonathan Dowling
Lecture 29: WED 25 MAR 09 Lecture 29: WED 25 MAR 09
- Ch. 31.1
- Ch. 31.1
- Ch. 31.1
- Ch. 31.1–
Lecture 29: WED 25 MAR 09 Lecture 29: WED 25 MAR 09 Ch. 31.1 Ch. - - PowerPoint PPT Presentation
Lecture 29: WED 25 MAR 09 Lecture 29: WED 25 MAR 09 Ch. 31.1 Ch. - - PowerPoint PPT Presentation
Physics 2102 Jonathan Dowling Lecture 29: WED 25 MAR 09 Lecture 29: WED 25 MAR 09 Ch. 31.1 Ch. 31.1 4: Ele 4: Ele lectric ical l Oscilla illatio ions, LC Ch. 31.1 Ch. 31.1 lectric ical l Oscilla illatio ions, LC Cir ircuit
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SLIDE 3
What are we going to learn? hat are we going to learn? What are we going to learn? hat are we going to learn? A road map A road map A road map A road map
- Electric charge
Electric force on other electric charges Electric field, and electric potential
- Moving electric charges : current
- Electronic circuit components: batteries, resistors, capacitors
- Electric currents Magnetic field
Magnetic force on moving charges
- Time-varying magnetic field Electric Field
- More circuit components: inductors.
- Electromagnetic waves light waves
- Geometrical Optics (light rays).
- Physical optics (light waves)
SLIDE 4
Oscillators are very useful in practical applications, for instance, to keep time, or to focus energy in a system. All oscillators can store energy in more than one way and exchange it back and forth between the different storage possibilities. For instance, in pendulums (and swings)
- ne exchanges energy
xchanges energy between kinetic and potential kinetic and potential form.
Oscillators in Physics Oscillators in Physics Oscillators in Physics Oscillators in Physics
We have studied that inductors and capacitors nductors and capacitors are devices that can store electromagnetic energy lectromagnetic energy electromagnetic energy lectromagnetic energy. In the inductor it is stored in a B field, in the capacitor in an E field.
SLIDE 5
Utot = Ukin +U pot = const
Utot = 1 2 mv2 + 1 2 k x2
dUtot dt = 0 = 1 2 m 2v dv dt
- + 1
2 k 2x dx dt
- v =
x (t) a = v (t) = x (t)
m dv dt + k x = 0
) cos( ) ( : Solution
- +
= t x t x phase : frequency : amplitude :
- x
m k =
- PHYS2101: A Mechanical Oscillator
PHYS2101: A Mechanical Oscillator PHYS2101: A Mechanical Oscillator PHYS2101: A Mechanical Oscillator
2 2
= + x k dt x d m
Newton’s law
F=ma!
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The magnetic field on the coil starts to collapse, which will start to recharge the capacitor. Finally, we reach the same state we started with (with
- pposite polarity) and the cycle restarts.
PHYS210 101 A 1 An E n Ele lectr trom
- magne
gnetic tic LC LC O Osc scilla illator tor PHYS210 101 A 1 An E n Ele lectr trom
- magne
gnetic tic LC LC O Osc scilla illator tor
Capacitor discharges completely, yet current keeps going. Energy is all in the inductor. Capacitor initially charged. Initially, current is zero, energy is all stored in the capacitor. A current gets going, energy gets split between the capacitor and the inductor. EnergyConservation:Utot = UB +UE
Utot = 1 2 Li2 + 1 2 q C
2
UB = 1 2 Li2UE = 1 2 q C
2
SLIDE 7
Utot = UB +UE
Utot = 1 2 Li2 + 1 2 q C
2
dUtot dt = 0 = 1 2 L 2i di dt
- + 1
2C 2q dq dt
- VL + VC = 0 = L di
dt
- + 1
C q
( )
i = q (t)
- i (t) =
q (t)
C q dt q d L + =
2 2
1 LC
q = q0 cos( t + 0)
Ele lectric ric Oscilla illators rs: the Math Ele lectric ric Oscilla illators rs: the Math
Or loop rule!
i = q (t) = q0 sin( t + 0)
- i (t) =
q (t) = 2q0 cos( t + 0)
Energy Cons. Both give Diffy-Q: Solution to Diffy-Q: LC Frequency In Radians/Sec
SLIDE 8
UB = 1 2 L i
[ ]
2 = 1
2 L q0 cos( t + 0)
[ ]
2
VL = L i (t) = 2q0 sin( t + 0)
- 2
q = q0 cos( t + 0)
Ele lectric ric Oscilla illators rs: the Math Ele lectric ric Oscilla illators rs: the Math
i = q (t) = q0 sin( t + 0)
- i (t) =
q (t) = 2q0 cos( t + 0)
UE = 1 2 q
[ ]
C
2
= 1 2C q0 cos( t + 0)
[ ]
2
Energy as Function of Time Voltage as Function of Time
VC = 1 C q(t)
[ ] = 1
C q0 cos( t + 0)
[ ]
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2 2
= + x k dt x d m
Analogy Between Electrical And Mechanical Oscillations
q x 1/ C k i v L m
LC 1 =
- )
cos( ) (
- +
= t x t x
m k =
- C
q dt q d L + =
2 2
q = q0 cos( t + 0) i = q (t) = q0 sin( t + 0)
- i (t) =
q (t) = 2q0 cos( t + 0) v = x (t) = x0 sin( t + 0) a = x (t) = 2x0 cos( t + 0) Charqe q -> Position x Current i=q’ -> Velocity v=x’ D-Current i’=q’’-> Acceleration a=v’=x’’
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- 1.5
- 1
- 0.5
0.5 1 1.5 Time Charge Current
) cos(
- +
= t q q
) sin(
- +
- =
= t q dt dq i
UB = 1 2 Li2 = 1 2 L 2q0
2 sin2( t + 0)
0.2 0.4 0.6 0.8 1 1.2 Time Energy in capacitor Energy in coil
UE = 1 2 q C
2
= 1 2C q0
2 cos2( t + 0)
LC x x 1 and , 1 sin cos that, g rememberin And
2 2
= = +
- Utot = UB +UE = 1
2C q0
2
The energy is constant and equal to what we started with.
LC Circ ircuit it: Conserv rvatio ion of Energ rgy LC Circ ircuit it: Conserv rvatio ion of Energ rgy
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Example le 1 : Tunin ing a Radio io Receiv iver Example le 1 : Tunin ing a Radio io Receiv iver
The inductor and capacitor in my car radio are usually set at L = 1 mH & C = 3.18
- pF. Which is my favorite FM
station? (a) KLSU 91.1 (b) WRKF 89.3 (c) Eagle 98.1 WDGL FM radio stations: frequency is in MHz. = 1 LC = 1 1106 3.18 1012 rad/s = 5.61108rad/s
f = 2 = 8.93107Hz = 89.3MHz
SLIDE 13
Example Example 2 2
- In an LC circuit,
L = 40 mH; C = 4 µF
- At t = 0, the current is
a maximum;
- When will the capacitor
be fully charged for the first time?
= 1 LC = 1 16x108 rad/s
- ω = 2500 rad/s
- T = period of one
complete cycle
- T = 2π/ω = 2.5 ms
- Capacitor will be
charged after T=1/4 cycle i.e at
- t = T/4 = 0.6 ms
- 1.5
- 1
- 0.5
0.5 1 1.5 Time Charge Current
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Example le 3 Example le 3
- In the circuit shown, the
switch is in position “a” for a long time. It is then thrown to position “b.”
- Calculate the amplitude ωq0
- f the resulting oscillating
current.
- Switch in position “a”: q=CV = (1 µF)(10 V) = 10 µC
- Switch in position “b”: maximum charge on C = q0 = 10 µC
- So, amplitude of oscillating current =
q0 = 1 (1mH)(1µF) (10µC) =
0.316 A
) sin(
- +
- =
t q i
b a
E=10 V 1 mH
1 µF
SLIDE 15
Example 4 Example 4
In an LC circuit, the maximum current is 1.0 A. If L = 1mH, C = 10 µF what is the maximum charge q0 on the capacitor during a cycle of oscillation?
) cos(
- +
= t q q
) sin(
- +
- =