ADDITIONS TO Y&F PROBLEMS AND PROBLEMS FROM OTHER TEXTS JULY 27 - AUG 3 FISHBANE PROBLEMS - CHAPTER 28 57. Show how to use Kirchhoff's loop rule (write a clear and concise explanation for each term) to find a differential equation which can be solved for Q(t), the charge as a function of time, on a discharging capacitor. The figure for this problem is Fig. 26.22, with an initial charge of Q_0 on capacitor C when the switch is closed at t = 0. Let positive current represent a charging capacitor, i.e. i = (dQ/dt). Show by direct substitution that Q(t) = (Q_0)e^(-t/RC) is a solution of your differential equation. Halliday and Resnick - 2nd Ed - CHAPTER 29 22. An initially uncharged capacitor C is fully charged by a device of constant EMF in series with a resistor R. (a) Show that the final energy stored in the capacitor is half the energy supplied by the EMF device. (b) By direct integration of (i^2)R over the charging time, show that the thermal energy dissipated by the resistor is also half the energy supplied by the EMF device. Young and Freedman - 11th Ed - CHAPTER 30 68. When making the graph for part (c), you may use T for the period of the oscillations. Tomorrow you will learn how to calculate T in seconds for such a circuit. Young and Freedman - 11th Ed - CHAPTER 31 32. Please add the following parts: (d) Assume the circuit elements are arranged as shown in Figure 31.21. What are the readings from voltmeters V4 and V5? Remember that AC meters give RMS values, not peak values. (e) If the circuit elements are not changed but the angular frequency of the source is changed to 645 rad/s, what are the readings on the five voltmeters, V1-V5? (f) Repeat part e. for 1245 rad/s. (P prefixes) HRW Problem Supplement #1 - CHAPTER 32 28. Suppose that a parallel-plate capacitor has circular plates with radius R = 30 mm and a plate separation of 5.0 mm. Suppose also that sinusoidal potential difference with a maximum value of 150 V and a frequency of 60 Hz is applied across the plates; that is V = (150 V)sin[2pi(60 Hz)t]. (a) Find B_max(R), the maximum value of the induced magnetic field that occurs at r = R. (b) Plot B_max(r) for 0 < r < 10 cm. 38. A capacitor with parallel circular plates of radius R is discharging via a current of 12.0 A. Consider a loop of radius R/3 that is centered on the central axis between the plates. (a) How much displacement current is encircled by the loop? The maximum induced magnetic field has a magnitude of 12.0 mT. (b) At what radial distance, or distances, from the central axis of the plate is the magnitude of the induced magnetic field 3.00 mT? Young and Freedman - 11th Ed - CHAPTER 32 47. Hint for Problem 32.47: This is a Faraday's Law problem, but with a radiated B field that is varying sinusoidally (so you can write B(t) as Bmax*sin(omega*t) and differentiate to get the maximum value of dB/dt). Halliday and Resnick - 2nd Ed - CHAPTER 33 11. A coil with an inductance of 2.0 H and a resistance of 10 Ohms is suddenly connected to a resistanceless battery with EMF = 100 V. At 0.10 s after the connection is made, what are the rates at which (a) energy is being stored in the magnetic field, (b) thermal energy is appearing in the resistance, and (c) energy is being delivered by the battery? 12. The figure for this problem is Figure 30.11 with EMF = 10.0 V, R = 6.70 Ohms, and L = 5.50 H. With S2 open, S1 was closed at time t = 0. (a) How much energy is delivered by the battery during the first 2.00 s? (b) How much of this energy is stored in the magnetic field of the inductor? (c) How much of this energy is dissipated in the resistor? FISHBANE PROBLEMS - CHAPTER 33 46. The figure for this problem is Fig. 30.11. After being closed for a long time, S1 is suddenly opened while, at the same instant, S2 is suddenly closed. Show how to use Kirchhoff's loop rule to find the differential equation which can be solved for the resulting I(t), and show that I(t) = (EMF/R)e^(-tR/L) is a solution of that differential equation. (P prefixes) HRW Problem Supplement #1 - CHAPTER 33 5. The frequency of oscillation of a certain LC circuit is 200 kHz. At time t = 0, plate A of the capacitor has maximum positive charge. At what times t > 0 will (a) plate A again have maximum positive charge, (b) the other plate of the capacitor have maximum positive charge, and (c) the inductor have maximum magnetic field? 12. In an oscillating LC circuit in which C = 4.00 microF, the maximum potential difference across the capacitor during the oscillations is 1.50 V and the maximum current through the inductor is 50.0 mA. (a) What is the inductance L? (b) What is the frequency of the oscillations? (c) How much time is required for the charge on the capacitor to rise from zero to its maximum value? 44. An AC generator with EMF_max = 220 V and operating at 400 Hz causes oscillations is a series RLC circuit having R = 220 Ohms, L = 150 mH, and C = 24.0 microF. Find (a) the capacitive reactance X_C, (b) the impedance Z, and (c) the current amplitude I. A second capacitor of the same capacitance is then connected in series with the other components. Determine whether the values of (d) X_C, (e) Z, and (f) I increase, decrease, or remain the same. 50. An AC voltmeter with large impedance is connected in turn across the inductor, the capacitor, and the resistor in a series circuit having an alternating EMF of 100 V (rms); it gives the same reading in volts in each case. What is the reading? (To get any credit for your answer to this problem, you must very carefully explain the logic that you used to obtain that answer.) WOLFSON - CHAPTER 34 6. An electric field points into the page and occupies a circular region of radius 1.0 m. There is a magnetic field forming circular closed loops centered on the circular region and pointing clockwise. The magnetic field strength 50 cm from the center of the region is 2.0 microT. (a) What is the rate of change of the electric field? (b) Is the electric field increasing or decreasing? Halliday and Resnick - 2nd Ed - CHAPTER 36 NOTE: You are to do parts (a)-(c) of problems 1 and 2 for July 31, and then part (d) of both problems on Aug 1. 1. An AC generator has EMF = (EMF_max)sin((w_d)t), with EMF_max = 25.0 V and w_d = 377 rad/s (w stands for omega). It is connected to a 12.7 H inductor. (a) What is the maximum value of the current? (b) When the current is a maximum, what is the EMF of the generator? (c) When the EMF of the generator is -12.5 V and increasing in magnitude, what is the current? (d) For the conditions of part (c), is the generator supplying or taking energy from the circuit? 2. The AC generator of problem 1 is connected to a 4.15 microF capacitor. (a) What is the maximum value of the current? (b) When the current is a maximum, what is the EMF of the generator? (c) When the EMF of the generator is -12.5 V and increasing in magnitude, what is the current? (d) For the conditions of part (c), is the generator supplying or taking energy from the circuit? 4. A 50 Ohm resistor is connected to an AC generator with EMF_max = 30.0 V. What is the amplitude of the resulting alternating current if the frequency of the EMF is (a) 1.00 kHz and (b) 8.00 kHz? 17. A typical "light dimmer" used to dim the stage lights in a theater consists of a variable inductor L (the inductance of which is adjustable between zero and L_{max}) connected in series with a lightbulb (a lightbulb is a non-ohmic resistor, but assume it is ohmic for the purpose of this problem). The electrical supply for this series circuit is 120 V (rms) at 60.0 Hz; the lightbulb is rated as "120 V, 1000 W". (a) What L_{max} is required if the rate of energy dissipation in the lightbulb is to be varied by a factor of five from its upper limit of 1000 W? (b) Could one use a variable resistor (adjustable between zero and R_{max}) instead of an inductor? If so, what R_{max} is required? Why isn't this done? G1 and G2: The differential equation for an undriven LRC circuit is (q/C) + R(dq/dt) + L(d^2q/dt^2) = 0 Eq (1) Your task is to show that q(t) = (Q_0)e^(-(alpha)t)cos((omega)t) is a solution of this differential equation, where Q_0, alpha, and omega are constants. G1. Substitution of q(t) in Eq (1) gives terms in sin((omega)t) and terms in cos((omega)t). Show that the sin((omega)t) terms sum to zero only if alpha = R/2L. G2. Show that the cos((omega)t) terms sum to zero only if omega^2 = omega0^2 - alpha^2 where omega0^2 = 1/LC. G3 and G4: The differential equation for a driven LRC circuit is (q/C) + R(dq/dt) + L(d^2q/dt^2) = (EMFm)sin((omegaD)t) Eq (2) where (EMFm)sin((omegaD)t) is the driving EMF as a function of time, EMFm is the amplitude of that driving EMF, and omegaD is the driving angular frequency. Your task is to show that q(t) = -(Q_0)cos((omegaD)t - phi) is a solution of this differential equation, where Q_0 and phi are constants. G3. Substitution of q(t) in Eq (2) gives terms in sin((omegaD)t) and terms in cos((omegaD)t). HINT: Use the trig identities for sin(alpha+-beta) and cos(alpha+-beta) as found on Y&F page A3. Show that the sum of the cos((omegaD)t) terms = 0 only if tan(phi) = (1/R)(X_L-X_C) where X_L and X_C are the inductive and capacitive reactances. G4. (a) Show that the sum of the sin((omegaD)t) terms = (EMFm)sin((omegaD)t) only if EMFm/(omegaD)Q_0 = R/cos(phi) and therefore that Z = R/cos(phi), where Z is the impedance. HINT: Use sin^2(phi) + cos^2(phi) = 1. (b) From (a), show that Z = sqrt((X_L-X_C)^2+R^2). HINT: sec^2(phi) = 1 + tan^2(phi).