MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
Physics 8.033
September 11, 2003
Problem Set 2
Due: Thursday, September 18 (by 4:30 pm)
Reading: Chapter 2 of Resnick & Halliday. Parallel reading: Chapters 3 & 4 in French.
Problem 1
Consider a general wave equation of the form:
∂2y
1 ∂2y
= 2 2
∂x2
V ∂t
.
(a) Show that any function of the quantity (x ± V t) will satisfy this equation.
(b) Make an argument to show why V represents the propagation speed of waves or other
disturbances described by this equation.
Problem 2
Maxwell’s wave equation for the z­component of an electric field propagating in the x­
direction is:
∂ 2 Ez
1 ∂ 2 Ez
=
.
∂x2
c2 ∂t2
Show that this equation is not invariant under a Galilean transformation to a reference frame
moving with relative speed v in the x direction. (See problem #16, chapter 1, page 35, in
Resnick & Halliday.)
Optional: Show that any function Ez (x− V t) is a solution to the transformed wave equation,
where V = v ± c.
Problem 3
“Michelson­Morley Experiment With a Real Wind”
Resnick & Halliday, problem #19, chapter 1, page 35.
Problem 4
“Not Much of a Contraction”
Resnick & Halliday, problem #21, chapter 1, page 35.
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Problem 5
In Special Relativity we will often want to evaluate the relativistic factor γ = (1 −
v /c2 )−1/2 for (1) small values of speed, v, and (2) for values of v close to the speed of
light. While this can be done with a calculator, it is often more instructive to derive an
approximate analytic result explicitly in terms of powers of (v/c)2 for case (1), and powers
of 1/γ 2 for case (2).
a) Expand γ in a Taylor series in powers of (v/c)2 , keeping the first 3 terms (constant
plus next two terms). (If you have not yet learned about Taylor series, simply adopt the
expression given on page 334 of Resnick & Halliday.) Check your answers with those given at
the bottom of page 17 in Resnick & Halliday. Evaluate your expression for γ with values of
v/c = 0.1, 0.001, and 10−5 . Compare to the values you get by evaluating the full expression
for γ with a calculator.
b) Show that for large γ, 1 − β � 1/(2γ 2 ). Use this expression to find the values of β
when γ = 2, 10, and 103 . Compare the approximate results with those obtained from the
full expression using a calculator.
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Problem 6
“Aberration and Relativity”
Resnick & Halliday, problem #27, chapter 1, page 36.
Problem 7
“The Speed of Light Really is the Same in All Frames”
Resnick & Halliday, problem #6, chapter 2, page 82.
(Start with equation 2.4 and apply the Lorentz­Einstein transformations to yield equation
2.5)
Problem 8
“A Moving Clock”
Resnick & Halliday (2nd edition), problem #9, chapter 2, page 82.
Problem 9
“A Moving Rod”
Resnick & Halliday (2nd edition), problem #10, chapter 2, page 82.
Problem 10
“Hidden Symmetry in the Lorentz Equations”
Resnick & Halliday (2nd edition), problem #11, chapter 2, page 82.
Show that the variables x and w are completely interchangeable.
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Optional Problem A
“Michelson­Morley, Generalized”
Resnick & Halliday (2nd edition), problem #18, chapter 1, page 35.
Optional Problem B
“A Fascinating “Thought Experiment”
Resnick & Halliday (2nd edition), problem #25, chapter 1, page 36.
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