
 1.
 A locomotive and a car approach an intersection with
perpendicular velocities as shown in the diagram. The locomotive and
the car travel with constant speeds of 30 m/s and 25 m/s,
respectively. At time , the car is 100 m to the south of the
intersection, and the locomotive is 50 m to the west of the
intersection. The locomotive blows its whistle. What is the Doppler
correction factor for the observer in the car (a) at
, (b) as the car crosses the intersection? (c) The sound detected
by the observer in the car at was actually emitted at an earlier
time, when the locomotive was some distance greater than 50 m from the
intersection. Hence, 50 m is not the source position you should have
used in part (a) to determine the source and observer
angles. Determine how much of a difference correcting for this makes
in your result for part (a). Is this correction important in this
situation?
 2.
 An object traveling faster than the speed of sound in air
generates a conical surface of wave fronts interfering
constructively, called a shock wave, illustrated in the
diagram. (a) Find an expression for , the angle the shock wave
surface makes with the source velocity , called the
(Mach angle). Hint: the velocities of the wave
fronts along the shock wave are perpendicular to its conical surface.
(b) The air speed record is 979 m/s. If a plane travels at this speed
at an altitude of 10 km, roughly how long after the plane passes
directly overhead do observers on the ground hear the sonic boom?
Ignore variation in the speed of sound with altitude, and assume the
STP sound velocity of 331 m/s.
Note: These results are not only approximate due to the
variation of the speed of sound with altitude. We also neglect the
fact that the large amplitude of the shock wave violates the
acoustic approximation.
 3.
 A 10.0 cm linear antenna connected to an oscilloscope is used to
pick up an oscillating voltage signal induced by the electric field of
a radio wave. Adjustment of the orientation of the antenna reveals
that a maximum signal of 10.0 mV is measured when the antenna is
oriented along .
(a) What are the amplitudes and of the electric and
magnetic fields of the wave?
(b) What can we say about the directions of and on
the basis of the above information?
(c) What is the intensity of the wave?
(d) The radio station has a 1 MW transmitter. Assuming the radio
station emits its signal isotropically, and ignoring the effects of
obstructions, how far from the station is the pickup antenna? (A real
transmitter with a linear antenna actually emits maximum power in the
plane perpendicular to it.)
 4.
 The average electromagnetic power flux from the sun incident on
the upper atmosphere of the earth is about 1.3 kW/m.
(a) If the average earthsun distance is
m, and the sun
emits a uniform angular distribution of electromagnetic radiation,
what is its electromagnetic power output?
(b) We have a 1000 kg spaceship which can unfurl a sail which
reflects about 50% of solar radiation and transmits the
rest. Reflection reverses the momentum of the incident light and hence
yields twice the light pressure. The ship is equipped with
conventional thrusters to ensure that its sail always faces the
sun. What area must the sail have in order to balance the
gravitational force exerted on it by the sun? (Canceling the sun's
gravitational force would give the ship a linear trajectory tangent to
its initial orbit around the sun.)
Hints: Both forces depend on the inverse square of the
distance from the sun, so the answer does not depend on position.
 5.
 A quarter wave plate is an optical element which
introduces a phase difference of (one quarter wave)
between perpendicular components of incident electromagnetic
waves. One way to accomplish this is to use a material which exhibits
double refraction, meaning that it has
different indices of refraction and for
light with its electric field oriented along two perpendicular
axes. Show that the thickness of a quarter wave plate made of such
a material must satisfy the equation
A linear polarizer and a quarter wave plate can be used to
produce elliptically polarized light.