Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits

Transcription

Chapter 30
Inductance, Electromagnetic
Oscillations, and AC Circuits
30-8 LRC Series AC Circuit
Analyzing the LRC series AC circuit is
complicated, as the voltages are not in phase
– this means we cannot simply add them.
Furthermore, the reactances depend on the
frequency.
Copyright © 2009 Pearson Education, Inc.
We calculate the voltage (and current) using
phasors – these are vectors representing the
individual voltages.
NOTE: we reference the
voltages across the
components to the
current as it is the same
everywhere in the circuit.
Here, at t = 0, the current
and voltage are both at a
maximum. As time goes
on, the phasors will rotate
counterclockwise.
Some time t later,
the phasors have
rotated.
Some time t later,
the phasors have
rotated.
Recall:
I  t   I 0 cos t
VR  t   V0 cos t ; V0  I 0 R
VL  t   V0 cos  t  90  ; V0  I 0 X L
VC  t   V0 cos  t  90  ; V0  I 0 X C
The voltage across
each device is given
by the x-component of
each, and the current
by its x-component.
Again, the current is
the same throughout
the circuit.
We find from the ratio of voltage to
current that the “effective resistance,”
called the impedance, of the circuit is
given by
BUT – only an actual resistance
dissipates energy. The inductor and
capacitor store it then release it.
The phase angle between the voltage and
the current is given by
or
The factor cos φ is called the
power factor of the circuit.
Example 30-11: LRC circuit.
Suppose R = 25.0 Ω, L = 30.0 mH, and
C = 12.0 μF, and they are connected in
series to a 90.0-V ac (rms) 500-Hz
source. Calculate (a) the current in the
circuit, (b) the voltmeter readings
(rms) across each element, (c) the
phase angle , and (d) the power
dissipated in the circuit.

30-9 Resonance in AC Circuits
The rms current in an ac circuit is
Clearly, Irms depends on the frequency.
30-9 Resonance in AC Circuits
We see that Irms will be a maximum when XC
= XL; the frequency at which this occurs is
f0 = ω0/2π is called the
resonant frequency.
30-10 Impedance Matching
When one electrical circuit is connected to
another, maximum power is transmitted when
the output impedance of the first equals the
input impedance of the second.
The power
delivered to the
circuit will be a
maximum when
dP/dR2 = 0;
this occurs
when R1 = R2.

Summary of Chapter 30
• LR circuit:
.
.
• Inductive reactance:
• Capacitive reactance:
Summary of Chapter 30
• LRC series circuit:
.
• Resonance in LRC series circuit:
Chapter 15
Wave Motion
Units of Chapter 15
• Characteristics of Wave Motion
• Types of Waves: Transverse and Longitudinal
• Energy Transported by Waves
• Mathematical Representation of a Traveling
Wave
• The Wave Equation
• The Principle of Superposition
• Reflection and Transmission
Units of Chapter 15
• Interference
• Standing Waves; Resonance
• Refraction
• Diffraction
15-1 Characteristics of Wave Motion
All types of traveling waves transport energy.
Study of a single wave
pulse shows that it is
begun with a vibration
and is transmitted
through internal forces in
the medium.
Continuous waves start
with vibrations, too. If the
vibration is SHM, then the
wave will be sinusoidal.
15-1 Characteristics of Wave Motion
Wave characteristics:
• Amplitude, A
• Wavelength, λ
• Frequency, f and period, T
• Wave velocity,
15-2 Types of Waves: Transverse
and Longitudinal
The motion of particles in a wave can be either
perpendicular to the wave direction (transverse)
or parallel to it (longitudinal).
and Longitudinal
Sound waves are longitudinal waves:
and Longitudinal
The velocity of a transverse wave on a
cord is given by:
As expected, the
velocity increases
when the tension
increases, and
decreases when
the mass
increases.
and Longitudinal
Example 15-2: Pulse on a wire.
An 80.0-m-long, 2.10-mm-diameter
copper wire is stretched between two
poles. A bird lands at the center point of
the wire, sending a small wave pulse out
in both directions. The pulses reflect at
the ends and arrive back at the bird’s
location 0.750 seconds after it landed.
Determine the tension in the wire.
Cu  8900 kg m3

and Longitudinal
The velocity of a longitudinal wave depends
on the elastic restoring force of the medium
(numerator) and on the mass density
(denominator):
or
and Longitudinal
Example 15-3: Echolocation.
Echolocation is a form of sensory perception
used by animals such as bats, toothed
whales, and dolphins. The animal emits a
pulse of sound (a longitudinal wave) which,
after reflection from objects, returns and is
detected by the animal. Echolocation waves
can have frequencies of about 100,000 Hz.
(a) Estimate the wavelength of a sea animal’s
echolocation wave. (b) If an obstacle is 100
m from the animal, how long after the animal
emits a wave is its reflection detected?

and Longitudinal
Earthquakes produce both longitudinal and
transverse waves. Both types can travel through
solid material, but only longitudinal waves can
propagate through a fluid—in the transverse
direction, a fluid has no restoring force.
Surface waves are waves that travel along the
boundary between two media.
ConcepTest 15.2 The Wave
At a football game, the “wave”
might circulate through the stands
and move around the stadium. In
this wave motion, people stand up
and sit down as the wave passes.
What type of wave would this be
characterized as?
1) polarized wave
2) longitudinal wave
3) lateral wave
4) transverse wave
5) soliton wave
ConcepTest 15.2 The Wave
At a football game, the “wave”
might circulate through the stands
and move around the stadium. In
this wave motion, people stand up
and sit down as the wave passes.
What type of wave would this be
characterized as?
1) polarized wave
2) longitudinal wave
3) lateral wave
4) transverse wave
5) soliton wave
The people are moving up and down, and the wave is
traveling around the stadium. Thus, the motion of the
wave is perpendicular to the oscillation direction of the
people, and so this is a transverse wave.
Follow-up: What type of wave occurs when you toss a pebble in a pond?
ConcepTest 15.3a Wave Motion I
1)
Consider a wave on a string moving
to the right, as shown below.
2)
What is the direction of the velocity
of a particle at the point labeled A ?
3)
4)
5)
A
zero
ConcepTest 15.3a Wave Motion I
1)
Consider a wave on a string moving
to the right, as shown below.
2)
What is the direction of the velocity
of a particle at the point labeled A ?
3)
4)
5)
The velocity of an
zero
A
oscillating particle
is (momentarily) zero
at its maximum
displacement.
Follow-up: What is the acceleration of the particle at point A?
ConcepTest 15.6c Wave Speed III
A length of rope L and mass M hangs
from a ceiling. If the bottom of the
rope is jerked sharply, a wave pulse
will travel up the rope. As the wave
travels upward, what happens to its
speed? Keep in mind that the rope is
not massless.
1) speed increases
2) speed does not change
3) speed decreases
ConcepTest 15.6c Wave Speed III
A length of rope L and mass M hangs
from a ceiling. If the bottom of the
rope is jerked sharply, a wave pulse
will travel up the rope. As the wave
travels upward, what happens to its
speed? Keep in mind that the rope is
not massless.
1) speed increases
2) speed does not change
3) speed decreases
The tension in the rope is not constant in the case of a
massive rope! The tension increases as you move up
higher along the rope, because that part of the rope has
to support all of the mass below it! Because the
tension increases as you go up, so does the wave
speed.
15-3 Energy Transported by Waves
By looking at the
energy of a particle of
matter in the medium
of a wave, we find:
Then, assuming the entire medium has the same
density, we find:
Therefore, the intensity is proportional to the
square of the frequency and to the square of the
amplitude.

15-3 Energy Transported by Waves
If a wave is able to spread out threedimensionally from its source, and the medium is
uniform, the wave is spherical.
Just from geometrical
considerations, as long as
the power output is
constant, we see:
15-3 Energy Transported by Waves.
Example 15-4: Earthquake intensity.
The intensity of an earthquake P wave
traveling through the Earth and detected
100 km from the source is 1.0 x 106 W/m2.
What is the intensity of that wave if
detected 400 km from the source?

15-4 Mathematical Representation of
a Traveling Wave
Suppose the shape of a wave is given
by:
a Traveling Wave
After a time t, the wave crest has traveled a
distance vt, so we write:
 2

D  x , t   A sin   x  vt    A sin  kx   t 


 Note: this k is NOT 
2
with k 
and   2 f .



 the spring constant 
Choice of where x  0 and when t  0 arbitrary, so


D  x , t   A sin  kx   t   A sin k  x  x0    t  t 0 
 A sin  kx   t    ;    kx0   t0
a Traveling Wave
Example 15-5: A traveling wave.
The left-hand end of a long horizontal stretched cord
oscillates transversely in SHM with frequency f = 250 Hz
and amplitude 2.6 cm. The cord is under a tension of
140 N and has a linear density μ = 0.12 kg/m. At t = 0,
the end of the cord has an upward displacement of 1.6
cm and is falling. Determine (a) the wavelength of
waves produced and (b) the equation for the traveling
wave.
at t  0

15-5 The Wave Equation
Look at a segment of string under tension:
 D
D 
2D
 Fy  FT sin 2  FT sin1  FT  x  x    x t 2
2
1

1  D
D 
2D
In calculus limit:


so


2
 x  x 2 x 1 
x
2D  2D 1 2D

 2 2
2
2
x
FT t
v t
so any wave with v   T is a
solution  can propagate
15-5 The Wave Equation
This is the one-dimensional wave
equation; it is a linear second-order
partial differential equation in x and t.
Its solutions are all sinusoidal waves
satisfying v=/T.
15-6 The Principle of Superposition
Superposition: The
displacement at any
point is the vector sum
of the displacements of
all waves passing
through that point at that
instant.
Fourier’s theorem: Any
complex periodic wave
can be written as the
sum of sinusoidal waves
of different amplitudes,
frequencies, and phases.
15-6 The Principle of Superposition
Conceptual Example 15-7:
Making a square wave.
At t = 0, three waves are
given by D1 = A cos kx, D2 =
-1/3A cos 3kx, and D3 = 1/5A
cos 5kx, where A = 1.0 m
and k = 10 m-1. Plot the
sum of the three waves
from x = -0.4 m to +0.4 m.
(These three waves are
the first three Fourier
components of a “square
wave.”)

Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits

Transcription

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