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).