Physics 264L: Assignment 2
Made available: Monday, August 31, 2015
Due:
Monday, September 7, 2015, before class.
Problems
1. Problem 2.1 on pages 58-59 of French.
2. Problem 2.6 on page 60 of French.
3. Problem 2.8 on page 60 of French
4. Problem 2.9 on page 61 of French.
5. From Maxwell’s equations for electromagnetic phenomena, one can show that, in vacuum, each component Ei (x, y, z, t) of the electric field E(x, y, z, t) and each component Bi (t, x, y, z) of the magnetic
field B(t, x, y, z) satisfies an important partial differential equation known as the wave equation:
∂2f
∂2f
∂2f
1 ∂2f
+ 2 + 2 − 2 2 = 0,
2
∂x
∂y
∂z
c ∂t
(1)
where f = f (x, y, z, t) is a function that depends on space and time.
(a) Show that a function f of the special form f (x, y, z, t) = g(x − ct), where g(x) is a function of a
single variable, satisfies Eq. (1). The expression g(x − ct) describes a curve of shape g(x) moving
rigidly with speed c in the positive x direction, which is one of the simplest kinds of waves, so it
is not so crazy to call Eq. (1) a “wave equation”. (The waves you usually see in an intro physics
course are sinusoidal waves with g(x) = sin(x) but waves are much more general than this.)
Hint: use the chain rule of calculus to evaluate the various derivatives in Eq. (1). For example,
you can label the argument x − ct of the wave by the variable u = x − ct so f (x, y, z, t) = g(u),
then
∂[g(u)]
dg
∂u
∂f
=
=
×
= g ′ (u) × 1 = g ′ (u),
(2)
∂x
∂x
du ∂x
and similarly
( )
∂ ∂f
∂ ( ′ )
d [ ′ ] ∂u
∂2f
=
=
g (u) =
g (u) ×
= g ′′ (u),
(3)
2
∂x
∂x ∂x
∂x
du
∂x
and so on.
(b) Show that Eq. (1) is not Galilean invariant under the Galilean transformation Eq. (3-4) on page 68
of the French text. This means that, when you transform variables (x, y, z, t) → (x′ , y ′ , z ′ , t′ ) using
the Galilean transformation Eq. (3-4), the new partial differential equation does not have the same
mathematical form as Eq. (1).
Hint: to change variables in a differential equation, use the chain rule again. For example, the
function f (x, y, z, t) can also be considered a function f (x′ , y ′ , z ′ , t′ ) of the new variables since x =
x(x′ , y ′ , z ′ , t′ ), t = t(x′ , y ′ , z ′ , t′ ), etc. So you can start to transform each of the derivatives in
Eq. (1) like this
∂f
∂f ∂x′
∂f ∂y ′
∂f ∂z ′
∂f ∂t′
=
+ ′
+ ′
+ ′
,
(4)
′
∂x
∂x ∂x
∂y ∂x
∂z ∂x
∂t ∂x
1
where each of the chain-rule factors such as ∂x′ /∂x can be easily calculated from the Galilean
transformation. A second derivative like ∂ 2 f /∂x2 is similarly expressed in terms of derivatives
with respect to the new variables by using the chain rule Eq. (4) a second time like this
[
( ) [
( )]
( )]
∂ ∂f
∂
∂f
∂x′
∂
∂f
∂y ′
∂2f
+
+ ··· .
(5)
=
=
2
′
′
∂x
∂x ∂x
∂x ∂x
∂x
∂y ∂x
∂x
(c) Show that Eq. (1) is invariant under the Lorentz transformation Eq. (3-16) on page 80 of French,
the differential equation has the same form in the new coordinates (x′ , y ′ , z ′ , t′ ). This is consistent
with the claim that Maxwell’s equations are already compatible with special relativity.
6. Problem 3.5 on pages 86-87 of French.
7. Problem 3.6 on page 87 of French.
8. Problem 3.7 on page 87 of French.
9. To the nearest integer, please give the time in hours that it took you to complete this assignment,
including reading in the text.
Also, if you got help with this assignment, please give the names of the people from whom you got
help (classmates, the TA, myself, etc).
10. Optional Challenge Problems
(a) Challenge problem 2.1: An indestructible sphere of mass 100 kg is placed in the interstellar space
of the Milky Way galaxy, a few light years from any star. What will its speed be (as measured
from Earth) after a sufficiently long time?
Some comments: you can assume that the probability for the sphere to fall into a star is zero. (This
is true even for stars, which are so tiny (2 light-seconds) compared to their average separation (a
light year) that they essentially never physically collide, even when two galaxies collide with one
another.) The mass of a typical star is of order 1030 kg, the typical relative speed of one star with
respect to another is of order 10 km/s, and there are about 1010 stars in a typical galaxy.
(b) Challenge problem 2.2: Consider a vertical bar of length L (say about a meter), with a loud speaker
attached to its bottom that emits a sound wave with constant frequency f , and with a microphone
attached to its top that can record the frequency of the emitted sound. The bar is then dropped
from a motionless helicopter from a large height above the Earth’s surface. Determine whether
the microphone detects a shift in frequency and, if so, calculate how the frequency changes with
time. (For this problem, ignore air friction so the bar undergoes constant acceleration.)
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