Videos
SP212
Ch. 31 - E&M Oscillations
Tank Circuit http://youtu.be/8vu9WDjBAho
Foxhole Radio http://youtu.be/oEsejXyGyVM
Maj Jeremy Best USMC
Physics Department, U.S. Naval Academy
March 30, 2016
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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Why Study EM ?
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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LC Circuits
We’ve looked at combining resistors (R) with inductors
(L) and capacitors (C): RC circuits, LR circuits. Those
systems exponentially decayed (or built up). We will find
that LC circuits oscillate with a period T and angular
frequency ω. Sound familiar?
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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Energy in LC Circuits
Energy and Charge in LC circuit
We have expressions for the energy stored in a capacitor
and in an inductor.
q2
UE =
2C
Li 2
UB =
2
In an ideal LC circuit, the total energy is conserved, it
just transfers back and forth between these two forms.
What is T, and why does it take 2 wavelengths?
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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Energy in LC Circuits
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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The Electrical-Mechanical Analogy
During SP211, you looked at another system where
energy was exchanged between two forms, a mass
attached to a spring.
There, energy continually shuttled between kinetic and
potential energy, but the total energy remained
constant(in an ideal frictionless world).
Unsurprisingly, these two systems share a great deal of
underlying mathematical structure.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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The Electrical-Mechanical Analogy
Circuit to Spring Analogy
Many quantities in the two systems map directly onto
one another:
q→x
i →v
(1/C ) → k
L→m
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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Circuit to Spring Analogy2
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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Circuit to Spring Analogy3
March 30, 2016
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Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Mechanical Circuit Analog
Hooke’s Law
The basis for looking at a mass on a spring was Hooke’s
Law:
F = −kx
ma + kx = 0
d 2x
m 2 + kx = 0
dt
⇒ x(t) = X cos(ωt + φ)
Capital letters mean maximum values, also known as
amplitude.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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The Electrical-Mechanical Analogy
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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LC Circuits
The total energy in an ideal
circuit must be conserved.
You know (or should know) that a mass
p on a spring
oscillates with angular frequency ω = (k/m). You can
therefore immediately derive the corresponding
expression for the oscillations of an LC circuit
1
ω=√
LC
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
VC
−
March 30, 2016
15 / 25
+
+
EL
−
q2
Li 2
U = UB + UE =
+
2
2
2C
2
dU
d Li
q
=
+
=0
dt
dt
2
2C
di
q dq
Li +
=0
dt C dt
d 2q
1
L 2 + q=0
dt
C
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
16 / 25
Charge of LC oscillator
Current of LC oscillator
The solution to the differential equation is the same form
as the spring block oscillator equation since they are
mathematically identical.
q = Q cos (ωt + φ)
i=
Where Q is the Amplitude of the charge variations, ω is
the angular frequency of the electromagnetic oscillations,
and φ is the phase constant.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
If we take the first derivative of this equation with
respect to time, we can get the current of the LC
oscillator:
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RLC Circuits
dq
= −ωQ sin (ωt + φ)
dt
And we can define the maximum current, I = ωQ.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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RLC Circuit
If we place a resistor in the circuit, it dissipates energy,
and therefore the system is no longer perfectly
conservative. This is exactly like the damped harmonic
oscillator you did in SP211!
From Chapter 26:
dU
P = i 2R = −
dt
Which is the loss in energy due to heat in the resistor .
We add this to the equation of the LC oscillator and get:
d 2q
dq
1
L 2 +R
+ q=0
dt
dt
C
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
19 / 25
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
RLC charge per time
LC with an R: Damped Oscillator
q(t) = Qe −Rt/(2L) cos(ω 0 t + φ)
where
p
ω 0 = ω 2 − (R/2L)2
Why does Omega change?
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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Real World LRC circuits
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Frequency
Cell Phones, Ship radios, FM Radio; all have voltage
sources in the circuit to keep the wave from
damping out.
Our modern world is absolutely filled with
electromagnetic waves of all different frequencies,
amplitudes, modulations, and energies.
How does your cell phone not pick up the radio
station? How does the radio station not interfere
with the TV station?
Different oscillators pick up different signals.
Variable Capacitance or inductance.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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Secure Communications:
Chaotic Oscillators.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Wiley Plus Homework
Chapter 31: Questions 1, 2. Problems: 5, 10, 18, 25.
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
March 30, 2016
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