ADDITIONS TO WOLFSON PROBLEMS AND PROBLEMS FROM OTHER TEXTS
JULY 30 - AUG 5
WOLFSON - CHAPTER 28
52. This problem is badly worded. An appropriate figure for
this problem is Figure 28-71, with a two-position switch
inserted at the upper junction. R1 is assumed to be very
small, so that the capacitor charges almost instantaneously
(i.e. in much less that 1/60th of a second) when the switch
in its charging position. In part (a), you are asked for
the effective resistance of the discharging circuit (R2 in
the figure). The 1.2 A is the initial current in the
discharging circuit immediately after the switch is thrown
to that side. The 35 V "supply" refers to the charged
capacitor.
FISHBANE PROBLEMS - CHAPTER 28
57. 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. 28-33, with an open switch
inserted into the circuit and 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 this 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.
(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 from the central
axis of the plate is the magnitude of the induced magnetic
field 3.00 mT?
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 32-11(a) with
EMF = 10.0 V, R = 6.70 Ohms, and L = 5.50 H. The switch
was moved to A 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. 32-11(a). After being at
position A for a long time, the switch is thrown to B. 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?
WOLFSON - CHAPTER 34
E6. This is the exercise accompanying Example 34-1 on page 886.
(P prefixes) HRW Problem Supplement #1 - CHAPTER 34
6. The figure for this problem is Figure 34-22(a) in Wolfson.
What is the wavelength of the electromagnetic wave emitted
by the oscillator-antenna system in the figure if
L = 0.253 microH and C = 25.0 pF?
Halliday and Resnick - 2nd Ed - CHAPTER 36
NOTE: You are to do parts (a)-(c) of problems 1 and 2 for Aug 4,
and then part (d) of both problems on Aug 5.
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?
EXTRA PROBLEMS FOR CHAPTER 33
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 page A9 of HRW.
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 inductance.
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).