Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits
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. 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. Copyright © 2009 Pearson Education, Inc. 30-8 LRC Series AC Circuit Some time t later, the phasors have rotated. Copyright © 2009 Pearson Education, Inc. 30-8 LRC Series AC Circuit 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 Copyright © 2009 Pearson Education, Inc. 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. Again, the current is the same throughout the circuit. Copyright © 2009 Pearson Education, Inc. 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 BUT – only an actual resistance dissipates energy. The inductor and capacitor store it then release it. Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. 30-9 Resonance in AC Circuits The rms current in an ac circuit is Clearly, Irms depends on the frequency. Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. Summary of Chapter 30 • LR circuit: . . • Inductive reactance: • Capacitive reactance: Copyright © 2009 Pearson Education, Inc. Summary of Chapter 30 • LRC series circuit: . • Resonance in LRC series circuit: Copyright © 2009 Pearson Education, Inc. Chapter 15 Wave Motion Copyright © 2009 Pearson Education, Inc. 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 Copyright © 2009 Pearson Education, Inc. Units of Chapter 15 • Interference • Standing Waves; Resonance • Refraction • Diffraction Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. 15-1 Characteristics of Wave Motion Wave characteristics: • Amplitude, A • Wavelength, λ • Frequency, f and period, T • Wave velocity, Copyright © 2009 Pearson Education, Inc. 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). Copyright © 2009 Pearson Education, Inc. 15-2 Types of Waves: Transverse and Longitudinal Sound waves are longitudinal waves: Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. 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 0.750 seconds after it landed. Determine the tension in the wire. Cu 8900 kg m3 Copyright © 2009 Pearson Education, Inc. 15-2 Types of Waves: Transverse 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 Copyright © 2009 Pearson Education, Inc. 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? Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. 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: Copyright © 2009 Pearson Education, Inc. 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? Copyright © 2009 Pearson Education, Inc. 15-4 Mathematical Representation of a Traveling Wave Suppose the shape of a wave is given by: Copyright © 2009 Pearson Education, Inc. 15-4 Mathematical Representation of 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 Copyright © 2009 Pearson Education, Inc. 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. at t 0 Copyright © 2009 Pearson Education, Inc. 15-5 The Wave Equation Look at a segment of string under tension: D D 2D Fy FT sin 2 FT sin1 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 Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. 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. Copyright © 2009 Pearson Education, Inc. 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.”) Copyright © 2009 Pearson Education, Inc.
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