Assignment # 9
Reading: Chapter 30, section 3, Chapter 31
Hirose and Longren, Chapter 9 and
10
WebAssign Set #9: Due Tuesday, Nov. 6
Problems covered on set #9 are: Ch. 30,
# 28, Ch. 31 # 34, 36, 54
Problem Set # 9: Due Thursday Nov. 8, 9 AM
Problems 5-10: Chapter 30 # 32; Ch. 31 # 37, 39, 46,
59, 62
273H extra problem:
Assignment # 8
Reading: Chapter 30
Hirose and Longren, Chapter 9
WebAssign Set #8: Due Tuesday, Oct. 30 9 AM
Problems covered on set #7 are: Ch. 30 #
26, 34, 36, 62
Problem Set # 8: Due Thursday Nov. 1, 9 AM
Problem 4: Wave
Reflection 2
SpreadsheetWaveReflection.doc
WaveReflection.xls
Problems 5-8 : Chapter 30
# 20, 24, 50, 52
273H extra problems:
1. Ch. 30 # 67
2. The circuit shown models the effects of “skin-depth” in the
response of conductors to high-frequency EM fields. Use the
approach shown in class to derive the expression for the second
derivative of the current with respect to position (x) in terms of
derivatives of the current with respect to time.
Assignment # 7
Reading: Chapter 30, sections 1 and 2
Review Gauss’s law, Ampere’s law,
Faraday’s law
Hirose and Longren, Chapter 9
WebAssign Set #7: Due Tuesday, Oct. 23 AM
Problems covered on set #7 are: Ch. 22
#39, 28 #34, 30 #16, 27 #119
Problem Set # 7: Due Thursday Oct. 25, 9 AM
Problem 1: A coaxial cable has an inner radius of 0.2mm and an
outer radius of 3 mm, and is filled with dielectric materials with ε =
2.0 ε
o. Find the speed of electromagnetic signal in
the cable, and the impedance of the cable, assuming μ = μ
o.
Problem 2: If a coaxial cable has impedance Z = 50Ω, μ = μ
o,
outer radius 2mm and electromagnetic signal speed 0.7c (c is the speed
of light in vacuum), what are the dielectric constant of the insulator
and the radius of the inner conductor?
Problem 3: From the dimensions for E and B, show that the
Poynting flux S has the dimension of watts per square meter.
(Hint: introduce a vector H = B/μ
o, which has
dimension of amperes/meter).
Problem 8: Consider a
long cylindrical resistor of length L, radius a and resistivity ρ,
carrying a current I.
a) Find the direction of the Poynting vector in terms of the
surface normal direction of the resistor.
b) Integrate the Poynting vector over the surface of the resistor
and relate the result to the resistive power I
2R dissipated
in the form of heat.
Problem 9: Ch. 30 # 17
273H extra problems:
1. Consider the
possibility of a standing electromagnetic wave, with E = E
m(sinωt)(sinκx)
and B = B
m(cosωt)(cosκx).
a) Find the relationship between E
m and B
m
for which these waveforms satisfy the equations of problem 4.
b) Find the instantaneous Poynting vector.
c) Show that the time average power flow across any area is zero.
2. Ch. 30 # 23
Assignment # 6
Reading: Chapter 15, section 4, Ch 16, sections 1, 4
Chapter 24, sections 2, 3, and 5 (review)
Chapter 27, sections 3 and 4
(review)
Chapter 28, section 7 (review)
Hirose and Longren, Chapter 9,
sections 1-2
WebAssign Set #6: Due Tuesday, Oct. 16 AM
Problems covered on set #6 are: Ch. 24 #50,
Ch. 27 #66, Ch. 28 #27, Ch. 29 #76
Problem Set # 6: Due Thursday Oct. 18, 9 AM
Problem 1: A wave on a tightly stretched string is observed
to have the form:
y(x,t) =
(0.05m)sin[2.5m-1x]cos[500s-1t].
a) What type of wave is this, and what are its wavelength and
frequency?
b) Write this wave as the sum of an incident and reflected wave,
giving numerical values for all the parameters for each (let phase
shifts be zero).
c) What is the kinetic energy, K(x,t) of this string if the mass
per unit length is 0.030kg/m?
Problem 2: The fourier transform (frequency spectrum) of a
waveform is shown in the graph. Estimate the frequencies and relative
intensities from the graph and write the waveform as a fourier series
(assume all components are cosine functions, and do not multiply
factors of 2π into numeric values for f – for instance write
2π∗9.8x105s-1 for the angular frequency of the center component in the
graph.)
Problem 3: A pulse of amplitude 1 cm is propagating along a
string toward a boundary where the string is connected to another
string having a mass density four times larger. Both strings are
subject to a common tension. a) Find the amplitudes
of the reflected and transmitted pulses. b) Find the
fraction of the energy reflected. c) Sketch qualitatively
reflected and transmitted pulses after the incident pulse reached the
boundary. d) Find the fraction of energy reflected for a
pulse traveling from the heavier string to the lighter string.
Problem 4: A steel rod is
joined to a copper rod at a smooth flat surface. Steel has a mass
density of 7800 kg/m3 and a Young’s modulus of 2.0x1011 N/m2, and
copper has a mass density of 8900 kg/m3 and a Young’s modulus of
1.1x1011N/m2. A sinusoidal sound wavetrain in the steel rod is
incident on the boundary. a) Find the mechanical impedances for
the steel and the copper rods for sound waves. b) Calculate the
fraction of reflected and transmitted wave energies relative to the
incident wave energy. c) Calculate the ratio of the
amplitudes of the reflected and trasnmitted waves relative to the
incident wave.
Problem 5: The conductor
in a cable will really have a finite resistance, which will modify the
ideal distributed LC line discussed in class as shown in the
figure.
a) Show that the Kirchoff’s law voltage relationship for one
element of this circuit is:
b) Find the corresponding current equation.
c) Combine the two equations as we did in class for the pure LC
case, to obtain the partial differential equation for V(x,t).
Problem 6: An LC transmission line has the following
parameters: L/Δx = 1.0x10-4 H/m, and C/ Δx = 20 x10-12F/m.
Find the velocity of the electromagnetic waves on the transmission
line. What is the impedance?
Problems 7-8: Ch. 24 # 68, Ch. 27 # 68
273H extra problem:
H&L Ch. 6, problem 5.
Assignment # 5
Reading: Chapter 15, sections 1, Ch 16, sections 1, 3-5
WebAssign Set #5: Due Tuesday, Sept. 25, 9 AM
Problems covered on set #5 are: Ch. 15 #109,
Ch. 16 # 48, 51, 79
Problem Set # 5: Due Thursday Oct. 4, 9 AM
Problem 1: When a spring of mass 0.100 kg and natural length 2.00
m is stretched by a force of 30.0 N, an elongation of 10.0 cm
results. Find the velocity of longitudinal waves along the
spring, using the equilibrium mass per length.
Problem 2: A series of equal masses are joined by springs,
with equilibrium spacing Δx = 100 μm. Under the action of a
longitudinal wave (not necessarily sinusoidal), the displacements of
three neighboring masses are:
ξ
1 = 0.0987 μm, ξ
2 = 0.1069 μm
and ξ
3 = 0.0796 μm. Find the approximate value of the
displacement curvature ( ) at the position of the second mass.
(Give units on curvature.)
Problem 3: Consider a longitudinal pulse propagating with a
speed c
w along a spring of mass density ρ
l
(measured in kg/m) and elastic modulus K (measured in N), with the wave
form:
ξ(x,t) =ξ
oexp{-(x-c
wt)
2/a
2)},
where a is a constant with units meters.
a) find the expressions for the instantaneous kinetic and
potential energies of the wave, KE(x,t) and U(x,t)
b) find the expression for the average total energy of the wave
Problem 4: Longitudinal earthquake waves typically have a
velocity of 5x10
3 m/s. Assuming the average earth
density is 1500 kg/m
3, estimate the elastic modulus of the
earth.
Problem 5: SpreadsheetAMwaves.doc,
AMwaves.xls
Problems 6-10: Ch. 15 # 29, 36; Ch. 16 # 50, 54,
68
273H extra problems: Ch. 15 # 48, H&L Ch. 4,
#13 (if you don’t have H&L, e-mail me to get a copy of the
problem)
Assignment 4 -
Sept. 20 - Sept. 27
Reading: Chapter 15, Section 1
Chapter 16, Sections 1, 3-5
WebAssign Set #4: Due Tuesday, Sept. 25, 9 AM
Problems covered on set #4 are: Ch. 15
#65, 87 and Ch. 16, # 22, 43
Problem Set # 4: Due Thursday Sept. 27, 9 AM
Problem 1: Determine as
quantitatively
as possible the periods present in the sum wave shown, and use the
estimates to find the frequencies, f1 and f2, of the two cosine waves
that were summed to produce the wave.
Problem 2:
SpreadsheetWaveSum1.doc,
and
SumWaves1.xls.
Problem 3: Use complex variables in polar form to derive the
trigonometric identity:
cos(A) + cos(B)
=2cos[(A+B)/2]cos[(A-B)/2].
Problem 4: A speaker radiates spherical sound waves with power 5
W. The radiation is limited to a cone with a 20
o
angle, as shown. Within the cone the radiation is uniform.
Show that the power per area at a distance 10 m from the speaker is
0.13 W/m
2 (not 0.12 W/m
2). At what distance
does the intensity become 10
-6W/m
2?
Problem 5 Show that the function z, defined below, satisfies the
two-dimensional form of the wave equation shown, with the relationship
for k, k
1, k
2 and
ω and c as given.
Problems 6-9: Ch.
16 # 65, 88, 90, 98
273H extra problems: Ch. 15 # 120, 16 # 35
Assignment 3 - Sept.
13 - Sept. 20
Reading: Chapter 15, Sections 1-3 and 5
WebAssign Set #3: Due Tuesday, Sept. 18, 9 AM
Problems covered on set #3 are: Ch. 15
#26, 44, 51, 56
Problem Set #3
Due: Thursday, Sept. 20 at the beginning of class
