Problems
1201
A gas is slowly leaked into the cylinder until a pressure of
1 atm is reached. If N bright fringes pass on the screen
when light of wavelength & is used, what is the index of
refraction of the gas?
Additional Problems
44.
In the What If? section of Example 37.2, it was claimed
that overlapping fringes in a two-slit interference pattern
for two different wavelengths obey the following relation-
ship even for large values of the angle !:
(a) Prove this assertion. (b) Using the data in Example
37.2, find the value of y on the screen at which the fringes
from the two wavelengths first coincide.
45. One radio transmitter A operating at 60.0 MHz is 10.0 m
from another similar transmitter B that is 180° out of
phase with A. How far must an observer move from A
toward B along the line connecting A and B to reach the
nearest point where the two beams are in phase?
46.
Review problem. This problem extends the result of
Problem 12 in Chapter 18. Figure P37.46 shows two
adjacent vibrating balls dipping into a tank of water. At
distant points they produce an interference pattern of
water waves, as shown in Figure 37.3. Let & represent the
wavelength of the ripples. Show that the two sources
produce a standing wave along the line segment, of length
d, between them. In terms of & and d, find the number of
nodes and the number of antinodes in the standing wave.
Find the number of zones of constructive and of destruc-
tive interference in the interference pattern far away from
the sources. Each line of destructive interference springs
from a node in the standing wave and each line of
constructive interference springs from an antinode.
&
&+
#
m+
m
of constructive interference. (b) To make the angles in
the interference pattern easy to measure with a plastic
protractor, you should use an electromagnetic wave with
frequency of what order of magnitude? How is this wave
classified on the electromagnetic spectrum?
48.
In a Young’s double-slit experiment using light of wave-
length &, a thin piece of Plexiglas having index of refrac-
tion n covers one of the slits. If the center point on the
screen is a dark spot instead of a bright spot, what is the
minimum thickness of the Plexiglas?
49.
Review problem. A flat piece of glass is held stationary and
horizontal above the flat top end of a 10.0-cm-long vertical
metal rod that has its lower end rigidly fixed. The thin film
of air between the rod and glass is observed to be bright by
reflected light when it is illuminated by light of wavelength
500 nm. As the temperature is slowly increased by 25.0°C,
the film changes from bright to dark and back to bright
200 times. What is the coefficient of linear expansion of
the metal?
50.
A certain crude oil has an index of refraction of 1.25. A
ship dumps 1.00 m
3
of this oil into the ocean, and the oil
spreads into a thin uniform slick. If the film produces a
first-order maximum of light of wavelength 500 nm
normally incident on it, how much surface area of the
ocean does the oil slick cover? Assume that the index of
refraction of the ocean water is 1.34.
Astronomers observe a 60.0-MHz radio source both
directly and by reflection from the sea. If the receiving
dish is 20.0 m above sea level, what is the angle of the
radio source above the horizon at first maximum?
52.
Interference effects are produced at point P on a screen as
a result of direct rays from a 500-nm source and reflected
rays from the mirror, as shown in Figure P37.52. Assume
the source is 100 m to the left of the screen and 1.00 cm
above the mirror. Find the distance y to the first dark band
above the mirror.
51.
Figure P37.46
Courtesy of Central Scientific Company
Figure P37.52
O
Source
P
Viewing screen
Mirror
!
y
47.
Raise your hand and hold it flat. Think of the space
between your index finger and your middle finger as one
slit, and think of the space between middle finger and ring
finger as a second slit. (a) Consider the interference
resulting from sending coherent visible light perpendicu-
larly through this pair of openings. Compute an order-of-
magnitude estimate for the angle between adjacent zones
53.
The waves from a radio station can reach a home receiver
by two paths. One is a straight-line path from transmitter
to home, a distance of 30.0 km. The second path is by
reflection from the ionosphere (a layer of ionized air
molecules high in the atmosphere). Assume this reflection
takes place at a point midway between receiver and
transmitter and that the wavelength broadcast by the radio
station is 350 m. Find the minimum height of the
ionospheric layer that could produce destructive interfer-
ence between the direct and reflected beams. (Assume
that no phase change occurs on reflection.)