PHY131 HOMEWORK PROBLEMS 2011 JULY 29 - AUG 4
UNIT 4
1. 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?
2. The figure for this problem is here. Switch S is closed
at time t=0 with no charge initially on the capacitor.
(a) What is the reading of each meter just after S is closed?
(b) What does each meter read long after S is closed?
(c) What is the maximum charge on the capacitor and when does
it occur? Here is a pdf version of the same figure.
3. 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.
4. 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?
5. 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. 26.23 (page 898), 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(t) = (dQ/dt). Show BY DIRECT SUBSTITUTION that
Q(t) = (Q_0)e^(-t/RC) is a solution of your differential
equation.
6. 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.
7. 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?
8. The figure for this problem is Figure 30.11 on page 1041 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?
9. 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.
10. The figure for this problem is here. The capacitor is
originally uncharged. The switch starts in the open position
and is then flipped to position 1 for 0.500 s. It is then
flipped to position 2 and left there. (a) If the resistance
r is very small, what is the upper limit for the amount of
charge the capacitor could receive? (b) Even if r is very
small, how much electrical energy will be dissipated in it?
(c) Sketch a graph showing the reading of the ammeter as a
function of time after the switch is in position 2, assuming
that r is very small and that current from left to right
through the ammeter is positive. (d) Sketch a graph showing
the charge on the right-hand plate of the capacitor as a
function of time after the switch is in position 2, assuming
that r is very small. Here is a pdf version of the
same figure.
11. An LC circuit has an inductance of 0.360 H and a
capacitance of 0.280 nF. During the current oscillations,
the maximum current in the inductor is 1.50 A. (a) What
is the maximum energy stored in the capacitor at any time
during the current oscillations? (b) How many times per
second does the capacitor contain the amount of energy
found in part A? (c) At an instant in time when the
current in the inductor is 0.75 A, what is the amount of
energy stored in the capacitor? (d) If the cycle begins
(t=0) when the maximum energy is stored in the capacitor,
when is the first instant in time that the energy stored
in the capacitor is equal to the energy stored in the
inductor? (e) What is the current in the inductor at the
instant referred to in part (d)?
12. 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?
13. 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?
14. An AC generator has EMF = (EMF_max)sin((w_d)t); it is
connected to an inductor of inductance L. (a) Write
Kirchhoff's Loop Rule for this circuit at a particular
instant of time. (b) Find the current as a function of
time. Hint: the result of (a) should be a simple
differential equation that can be solved easily by
integration. In this case, it is simplest to use
indefinite integration (antiderivatives); eliminate the
possible constant by an argument that the current should
be purely AC, with no DC contribution. Now assume
EMF_max = 25.0 V, w_d = 377 rad/s (w stands for omega),
and L=12.7 H. (c) What is the maximum value of the
current? (d) When the current is a maximum, what is the
EMF of the generator? (e) When the EMF of the generator
is -12.5 V and increasing in magnitude, what is the
current? (f) For the conditions of part (e), is the
generator supplying or taking energy from the circuit?
15. The AC generator of problem 14 is connected to a
capacitor of capacitance C. (a) Write Kirchhoff's Loop
Rule for this circuit at a particular instant of time.
Is your result a differential equation? (b) Find the
current as a function of time. Now assume
EMF_max = 25.0 V, w_d = 377 rad/s (w stands for omega),
and C=4.15 microF. (c) What is the maximum value of the
current? (d) When the current is a maximum, what is the
EMF of the generator? (e) When the EMF of the generator
is -12.5 V and increasing in magnitude, what is the
current? (f) For the conditions of part (e), is the
generator supplying or taking energy from the circuit?
16. A resistor of resitance R is connected to an AC
generator with EMF(t)=(EMF_max)sin((w_d)t). (a) Write
Kirchhoff's Loop Rule for this circuit at a particular
instant of time. Is your result a differential equation?
(b) Find the current as a function of time. Now assume
R=50 Ohm and EMF_max = 30.0 V. (c) What is the amplitude
of the resulting alternating current if the frequency of
the EMF is 1.00 kHz and (d) if the frequency of the EMF
is 8.00 kHz?
17. 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 the new values of (d) X_C, (e) Z,
and (f) I and state whether those values increase, decrease,
or remain the same.
18. In an L-R-C series circuit, the resistance is 380 ohms, the
inductance is 0.400 henrys, and the capacitance is 0.0120
microfarads. (a) What is the resonance angular frequency
of the circuit? (b) The capacitor can withstand a peak
voltage of 570 volts. If the voltage source operates at the
resonance frequency, what maximum voltage amplitude can the
source have if the maximum capacitor voltage is not exceeded?
19. 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.)
20. The figure for this problem is here. The figure shows
voltage and current graphs for a series LRC circuit with a
generator. The voltage curve first crosses the time axis
at t=25 microseconds; the current curve first crosses the
time axis at t=41.67 microseconds. (a) What is the
resistance in the circuit? (b) If L=200 microH, what is
the capacitance in the circuit? (c) What is the resonant
angular frequency for the circuit? Here is a pdf
version of the same figure.
21. 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?
22. You enjoy listening to KONG-FM, which broadcasts at 94.1 MHz.
You destest listening to KRUD-FM, which broadcasts at 94.0 MHz.
You live the same distance from both stations and both
transmitters are equally powerful, so both radio signals
produce the same 1.0 V source voltage as measured at your
house. Your goal is to design an LRC radio circuit with
the following properties: i) It gives the maximum power
response to the signal from KONG-FM; ii) the average power
delivered to the resistor in response to KRUD-FM is 1.00%
of the average power in response to KONG-FM. This limits
the power received from the unwanted station, making it
inaudible. You are required to use an inductor with
L = 1.00 microH. Find the capacitance C and effective
resistance R that satisfy the design requirements. NOTE:
To get R, you will need to save C in your calculator and
use that saved answer; this is because R depends on the
small DIFFERENCE between X_C and X_L, which magnifies any
rounding error in X_C. GBA
23. An LRC series circuit with L=0.125 H, R=242 Ohms, and
C=7.26 microF carries an RMS current of 0.452 A with a
frequency of 402 Hz. (a) What is the phase constant?
(b) What is the power factor? (c) What is the
impedance of this circuit? (d) What is the RMS EMF
of the source? (e) What is the average power delivered
by the source? (f) What is the average rate at which
electrical energy is converted to thermal energy in
the resistor? (g) What is the average rate at which
electrical energy is converted to other forms in the
capacitor and in the inductor? (h) If the driving
frequency were changed to the resonant frequency of
this circuit, what would be the resulting RMS current
(instead of 0.452 A)? (i) If the driving frequency
were changed to the resonant frequency of this circuit,
what would be the resulting average power delivered by
the source (instead of your answer to part (e))?
24. A parallel-plate, air-filled capacitor has plates of
area 0.00260 square meters. The magnitude of charge
on each plate is 8.20 pC, and the potential difference
between the plates is 2.40 V. What is the electric
field energy density in the volume between the plates?
25. An air-filled toroidal solenoid has 600 turns and a mean
radius of 6.90 cm. Each winding has a cross-sectional area
of 0.0350 square cm. Assume that the magnetic field is
uniform over the cross section of the windings.
(a) When the current in the solenoid is 2.50 A, what is the
magnetic field within the solenoid? (b) Calculate the
energy density in the solenoid directly from the magnetic
field strength. (c) What is the total volume enclosed by
the windings? (d) Use the volume from part c and the
energy density from part b to calculate the total energy
stored in the solenoid. (e) Use the geometry of the
solenoid to calculate its inductance. (f) Use 0.5LI^2
to calculate the energy stored in the solenoid. Compare
your answer to what you obtained in part d.
26. A sinusoidal radio wave of frequency 1.0 MHz is traveling
in the negative z direction. The electric field of the
radio wave oscillates in the plus or minus y direction.
The maximum electric field strength is 1000 V/m. What are
(a) the maximum radiated magnetic field strength,
(b) the radiated magnetic field strength and direction
at a point where the radiated electric field is
500 V/m in the negative y direction, and
(c) the smallest distance between a point on the wave
having the magnetic field of part b and a point where
the magnetic field is at maximum strength?
27. A satellite in geostationary orbit is used to transmit
data via electromagnetic radiation. The satellite is
at a height of 35000 km above the surface of the Earth,
and we assume it has an isotropic power output of
4.00 kW (in practice, satellite antennas transmit
signals that are less powerful but more directional).
The satellite dish which will receive the signal sent
from the satellite is located on the surface of the
Earth directly below the satellite. (a) What is the
intensity of the signal from the satellite at the
location of the antenna? (b) What is the energy
density of the electromagnetic radiation from the
satellite which is being received by the antenna?
(c) The satellite dish detects the variation in the
radiated electric field from the satellite. What is
the amplitude of the radiated electric field vector
of the satellite broadcast at the location of the dish?
Young and Freedman - 12th Ed - CHAPTER 31
36. Please add the following parts:
(d) Assume the circuit elements are arranged as shown
in Figure 31.25 on page 1088. 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.
Young and Freedman - 12th Ed - CHAPTER 32
49. Hint for Problem 32.49: 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).