Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits

Chapter 30Inductance, Electromagnetic Oscillations, and AC Circuits

Copyright © 2009 Pearson Education, Inc.

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.

30-8 LRC Series AC Circuit


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:


0

0 0 0

0 0 0

0 0 0

cos

cos ;

cos 90 ;

cos 90 ;

R

L L

C C

I t I t

V t V t V I R

V t V t V I X

V t V t V I X


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

The factor cos φ is called the power factor of the circuit.

or



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.


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.

30-9 Resonance in AC Circuits


30-10 Impedance MatchingWhen 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


• 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

Units of Chapter 15


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,

15-1 Characteristics of Wave Motion


The motion of particles in a wave can be either perpendicular to the wave direction (transverse) or parallel to it (longitudinal).

15-2 Types of Waves: Transverse and Longitudinal


Sound waves are longitudinal waves:




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.



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.

38900Cu kg m



The velocity of a longitudinal wave depends on the elastic restoring force of the medium (numerator) and on the mass density (denominator):

or



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?


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.


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 15.2ConcepTest 15.2 The WaveThe 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.

ConcepTest 15.2ConcepTest 15.2 The WaveThe Wave

Follow-up:Follow-up: What type of wave occurs when you toss a pebble in a pond? What type of wave occurs when you toss a pebble in a pond?

ConcepTest 15.3aConcepTest 15.3a Wave Motion IWave Motion I

Consider a wave on a string moving to the right, as shown below.

What is the direction of the velocity of a particle at the point labeled A ?

1)

2)

3)

4)

5) zero

A

ConcepTest 15.3aConcepTest 15.3a Wave Motion IWave Motion I

Consider a wave on a string moving to the right, as shown below.

What is the direction of the velocity of a particle at the point labeled A ?

1)

2)

3)

4)

5) zero

The velocity of an

oscillating particle

is (momentarilymomentarily) zerozero

at its maximum

displacement.

A

Follow-up:Follow-up: What is the acceleration of the particle at point A? What is the acceleration of the particle at point A?

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 15.6cConcepTest 15.6c Wave Speed IIIWave 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.

ConcepTest 15.6cConcepTest 15.6c Wave Speed IIIWave Speed III


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 three-dimensionally 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


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:



After a time t, the wave crest has traveled a distance vt, so we write:

0 0

2, sin sin

Note: this is NOT2with and 2 .

the spring constant

Choice of where 0 and when 0 arbitrary, so

, sin sin

sin

D x t A x vt A kx t

kk f

x t

D x t A kx t A k x x t t

A kx t

0 0; kx t



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 0t


15-5 The Wave Equation

Look at a segment of string under tension:

2

2 1 22 1

2 2

2

22 1

2

2 22 2

so any wave with is a

solution can propagate

sin sin

1In calculus l

1

imit: so

y T T T

T

D

D D DF F F F x

x x t

D D D

x x x x

D

x

v

F

D

t

T

v t


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 SuperpositionConceptual 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.”)

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Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits