1. Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits
Chapter 30 Opener. A spark plug in a car receives a high voltage, which produces a high enough electric field in the air across its gap to pull electrons off the atoms in the air–gasoline mixture and form a spark. The high voltage is produced, from the basic 12 V of the car battery, by an induction coil which is basically a transformer or mutual inductance. Any coil of wire has a self-inductance, and a changing current in it causes an emf to be induced. Such inductors are useful in many circuits.

2. 30 – Circuits with Resistors, Capacitors, Inductors, and V Source

3. 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 add them simply. Furthermore, the reactances depend on the frequency.
Figure An LRC circuit.

4. Kirchoff’s Loop Rule Treat each term as the x-component of a vector
(called a PHASOR) that is rotating counterclock-
wise with angular frequency .

5. 30-8 LRC Series AC Circuit
Here, at t = 0, the current and voltage are both at a maximum. As time goes on, the phasors will rotate counterclockwise.
Figure 30-20a. Phasor diagram for a series LRC circuit at t=0.

6. 30-8 LRC Series AC Circuit
Some time t later, the phasors have rotated.
VL0, VR0, and VC0 are constant BUT the projections on the x-axis change with time
Figure 30-20b. Phasor diagram for a series LRC circuit at a time t later.

7. 30-8 LRC Series AC Circuit
The voltage across each device is given by the x-component of each, and the current by its x-component. The current is the same throughout the circuit.
Figure 30-20c. Phasor diagram for a series LRC circuit. Projections on x axis reflect Eqs. 30–20, 30–22a and 30–24a.

8. 30-8 LRC Series AC Circuit
We find from the ratio of voltage to current that the “effective resistance,” called the impedance, of the circuit is given by

9. 30-8 LRC Series AC Circuit
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:

10. 30-8 LRC Series AC 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.
Solution: a. XL = 2πfL = 94.2 Ω; XC = 1/(2πfC) = 26.5 Ω; so Z = 72.2 Ω and I = 1.25 A.
b. The voltages are the currents multiplied by the reactances (or resistance). VL = 118 V; VC = 33.1 V; VR = 31.2 V.
c. cos φ = 0.346, so φ = 69.7°.
d. P = IV cos φ = 39.0 W.

11. 30-9 Resonance in AC Circuits
The rms current in an ac circuit is
Clearly, Irms depends on the frequency.

12. 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.
Why are FM stations always 200 kHz apart?
Figure Current in LRC circuit as a function of angular frequency, showing resonance peak at ω = ω0 = (1/LC)1/2.

13. 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 minimum when
dP/dR2 = 0
This occurs when R1 = R2.
Figure Output of the circuit on the left is input to the circuit on the right.

14. Summary of Chapter 30 Mutual inductance: Self-inductance:
Energy density stored in magnetic field:

15. Summary of Chapter 30 LR circuit: Inductive reactance:. .
Inductive reactance:
Capacitive reactance:

16. Summary of Chapter 30 LRC series circuit:.
Resonance in LRC series circuit:

17. Chapter 15
Wave Motion
Chapter opener. Caption: Waves—such as these water waves—spread outward from a source. The source in this case is a small spot of water oscillating up and down briefly where a rock was thrown in (left photo). Other kinds of waves include waves on a cord or string, which also are produced by a vibration. Waves move away from their source, but we also study waves that seem to stand still (“standing waves”). Waves reflect, and they can interfere with each other when they pass through any point at the same time.

18. 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

19. Units of Chapter 15 Interference Standing Waves; Resonance Refraction
Diffraction

20. 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.
Figure Motion of a wave pulse to the right. Arrows indicate velocity of cord particles.

21. 15-1 Characteristics of Wave Motion
Wave characteristics:
Amplitude, A
Wavelength, λ
Frequency, f and period, T
Wave velocity,
Figure Characteristics of a single-frequency continuous wave moving through space.

22. 15-2 Types of Waves: Transverse and Longitudinal
Figure (a) Transverse wave; (b) longitudinal wave.
The motion of particles in a wave can be either perpendicular to the wave direction (transverse) or parallel to it (longitudinal).

23. 15-2 Types of Waves: Transverse and Longitudinal
Sound waves are longitudinal waves:
Figure Production of a sound wave, which is longitudinal, shown at two moments in time about a half period (1/2 T) apart.

24. ConcepTest 15.3a Wave Motion I1) 2) 3) 4)
5) zero
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 ? A
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25. ConcepTest 15.3a Wave Motion I1) 2) 3) 4)
5) zero
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 ? A
The velocity of an
oscillating particle
is (momentarily) zero
at its maximum
displacement.
Follow-up: What is the acceleration of the particle at point A?

26. ConcepTest 15.3b Wave Motion II1) 2) 3) 4)
5) zero
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 B ? B
Click to add notes

27. ConcepTest 15.3b Wave Motion II1) 2) 3) 4)
5) zero
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 B ?
The wave is moving
to the right, so the
particle at B has to
start moving upward
in the next instant of
time. B
Follow-up: What is the acceleration of the particle at point B?

28. 15-2 Types of Waves: Transverse 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.
Figure Diagram of simple wave pulse on a cord for derivation of Eq. 15–2.The vector shown in (b) as the resultant of FT + Fy has to be directed along the cord because the cord is flexible. (Diagram is not to scale: we assume v’ << v; the upward angle of the cord is exaggerated for visibility.)

29. 15-2 Types of Waves: Transverse 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 seconds after it landed. Determine the tension in the wire.
Solution: We need to find the mass per unit length of the wire; this comes from the cross-sectional area and the density of copper (8900 kg/m3). Then the tension is 353 N.

30. 15-2 Types of Waves: Transverse and Longitudinal
The velocity of a longitudinal wave depends on the elastic restoring force of the medium and on the mass density. or

31. 15-2 Types of Waves: Transverse 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?
Solution: a. The speed of sound in sea water is 1400 m/s; therefore the wavelength is 14 mm.
b. The round trip time is 0.14 s.

32. 15-2 Types of Waves: Transverse 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.
Figure 15-9: A water wave is an example of a surface wave, which is a combination of transverse and longitudinal wave motions.
Figure 15-10: How a wave breaks. The green arrows represent the local velocity of water molecules.

33. ConcepTest 15.4 Out to Sea t t + Dt 1) 1 second
2) 2 seconds
3) 4 seconds
4) 8 seconds
5) 16 seconds
A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 m and the speed of the waves is 5 m/s, how long does it take the boat to go from the top of a crest to the bottom of a trough ? t
t + Dt
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34. ConcepTest 15.4 Out to Sea t t + Dt 1) 1 second
2) 2 seconds
3) 4 seconds
4) 8 seconds
5) 16 seconds
A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 m and the speed of the waves is 5 m/s, how long does it take the boat to go from the top of a crest to the bottom of a trough ?
We know that v = f l = l / T, hence T = l / v. If l = 20 m and v = 5 m/s, then T = 4 secs.
The time to go from a crest to a trough is only T/2 (half a period), so it takes 2 secs !! t
t + Dt

35. ConcepTest 15.6b Wave Speed II
A wave pulse is sent down a rope of a certain thickness and a certain tension. A second rope made of the same material is twice as thick, but is held at the same tension. How will the wave speed in the second rope compare to that of the first?
1) speed increases
2) speed does not change
3) speed decreases
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36. ConcepTest 15.6b Wave Speed II
A wave pulse is sent down a rope of a certain thickness and a certain tension. A second rope made of the same material is twice as thick, but is held at the same tension. How will the wave speed in the second rope compare to that of the first?
1) speed increases
2) speed does not change
3) speed decreases
The wave speed goes inversely as the square root of the mass per unit length, which is a measure of the inertia of the rope. So in a thicker (more massive) rope at the same tension, the wave speed will decrease.

37. 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:
Figure Calculating the energy carried by a wave moving with velocity v.
Therefore, the intensity is proportional to the square of the frequency and to the square of the amplitude.

38. 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:
Figure Wave traveling outward from a point source has spherical shape. Two different crests (or compressions) are shown, of radius r1 and r2.

39. 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?
Solution: Assume the wave is spherical; the intensity decreases as the square of the distance, so it is 6.3 x 104 W/m2.

40. 15-4 Mathematical Representation of a Traveling Wave
Suppose the shape of a wave is given by:
Figure In time t, the wave moves a distance vt.

41. 15-4 Mathematical Representation of a Traveling Wave
After a time t, the wave crest has traveled a distance vt, so we write:
Or:
with ,

42. 15-4 Mathematical Representation of 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.
Figure Example The wave at t = 0 (the hand is falling). Not to scale.
Solution: a. The wave velocity is 34 m/s, so the wavelength is 14 cm.
b. From the amplitude and the displacement at t = 0, we find that the phase angle is 0.66 rad. Then D = sin(45x – 1570t ).

43. 15-5 The Wave Equation Look at a segment of string under tension:
Newton’s second law gives:
Figure Deriving the wave equation from Newton’s second law: a segment of string under tension FT .

44. 15-5 The Wave Equation
Assuming small angles, and taking the limit Δx → 0, gives (after some manipulation):
This is the one-dimensional wave equation; it is a linear second-order partial differential equation in x and t. Its solutions are sinusoidal waves.

45. 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.
Figure The superposition principle for one-dimensional waves. Composite wave formed from three sinusoidal waves of different amplitudes and frequencies (f0, 2f0, 3f0) at a certain instant in time. The amplitude of the composite wave at each point in space, at any time, is the algebraic sum of the amplitudes of the component waves. Amplitudes are shown exaggerated; for the superposition principle to hold, they must be small compared to the wavelengths.

46. 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.”)
Solution is shown in the figure.