27.6Limits of Resolution: The Rayleigh Criterion
57.The 300-m-diameter Arecibo radio telescope pictured inFigure
27.28detects radio waves with a 4.00 cm average wavelength.
(a) What is the angle between two just-resolvable point sources for this
(b) How close together could these point sources be at the 2 million
light year distance of the Andromeda galaxy?
58.Assuming the angular resolution found for the Hubble Telescope in
Example 27.5, what is the smallest detail that could be observed on the
59.Diffraction spreading for a flashlight is insignificant compared with
other limitations in its optics, such as spherical aberrations in its mirror.
To show this, calculate the minimum angular spreading of a flashlight
beam that is originally 5.00 cm in diameter with an average wavelength
of 600 nm.
60.(a) What is the minimum angular spread of a 633-nm wavelength
He-Ne laser beam that is originally 1.00 mm in diameter?
(b) If this laser is aimed at a mountain cliff 15.0 km away, how big will
the illuminated spot be?
(c) How big a spot would be illuminated on the Moon, neglecting
atmospheric effects? (This might be done to hit a corner reflector to
measure the round-trip time and, hence, distance.) Explicitly show how
you follow the steps inProblem-Solving Strategies for Wave Optics.
61.A telescope can be used to enlarge the diameter of a laser beam
and limit diffraction spreading. The laser beam is sent through the
telescope in opposite the normal direction and can then be projected
onto a satellite or the Moon.
(a) If this is done with the Mount Wilson telescope, producing a 2.54-m-
diameter beam of 633-nm light, what is the minimum angular spread of
(b) Neglecting atmospheric effects, what is the size of the spot this
beam would make on the Moon, assuming a lunar distance of
62.The limit to the eye’s acuity is actually related to diffraction by the
(a) What is the angle between two just-resolvable points of light for a
3.00-mm-diameter pupil, assuming an average wavelength of 550 nm?
(b) Take your result to be the practical limit for the eye. What is the
greatest possible distance a car can be from you if you can resolve its
two headlights, given they are 1.30 m apart?
(c) What is the distance between two just-resolvable points held at an
arm’s length (0.800 m) from your eye?
(d) How does your answer to (c) compare to details you normally
observe in everyday circumstances?
63.What is the minimum diameter mirror on a telescope that would
allow you to see details as small as 5.00 km on the Moon some
384,000 km away? Assume an average wavelength of 550 nm for the
64.You are told not to shoot until you see the whites of their eyes. If the
eyes are separated by 6.5 cm and the diameter of your pupil is 5.0 mm,
at what distance can you resolve the two eyes using light of wavelength
65.(a) The planet Pluto and its Moon Charon are separated by 19,600
km. Neglecting atmospheric effects, should the 5.08-m-diameter Mount
Palomar telescope be able to resolve these bodies when they are
from Earth? Assume an average wavelength of 550
(b) In actuality, it is just barely possible to discern that Pluto and Charon
are separate bodies using an Earth-based telescope. What are the
reasons for this?
66.The headlights of a car are 1.3 m apart. What is the maximum
distance at which the eye can resolve these two headlights? Take the
pupil diameter to be 0.40 cm.
67.When dots are placed on a page from a laser printer, they must be
close enough so that you do not see the individual dots of ink. To do
this, the separation of the dots must be less than Raleigh’s criterion.
Take the pupil of the eye to be 3.0 mm and the distance from the paper
to the eye of 35 cm; find the minimum separation of two dots such that
they cannot be resolved. How many dots per inch (dpi) does this
An amateur astronomer wants to build a telescope with a diffraction
limit that will allow him to see if there are people on the moons of
(a) What diameter mirror is needed to be able to see 1.00 m detail on a
Jovian Moon at a distance of
from Earth? The
wavelength of light averages 600 nm.
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
69.Construct Your Own Problem
Consider diffraction limits for an electromagnetic wave interacting with a
circular object. Construct a problem in which you calculate the limit of
angular resolution with a device, using this circular object (such as a
lens, mirror, or antenna) to make observations. Also calculate the limit
to spatial resolution (such as the size of features observable on the
Moon) for observations at a specific distance from the device. Among
the things to be considered are the wavelength of electromagnetic
radiation used, the size of the circular object, and the distance to the
system or phenomenon being observed.
27.7Thin Film Interference
70.A soap bubble is 100 nm thick and illuminated by white light incident
perpendicular to its surface. What wavelength and color of visible light
is most constructively reflected, assuming the same index of refraction
71.An oil slick on water is 120 nm thick and illuminated by white light
incident perpendicular to its surface. What color does the oil appear
(what is the most constructively reflected wavelength), given its index of
refraction is 1.40?
72.Calculate the minimum thickness of an oil slick on water that
appears blue when illuminated by white light perpendicular to its
surface. Take the blue wavelength to be 470 nm and the index of
refraction of oil to be 1.40.
73.Find the minimum thickness of a soap bubble that appears red
when illuminated by white light perpendicular to its surface. Take the
wavelength to be 680 nm, and assume the same index of refraction as
74.A film of soapy water (
) on top of a plastic cutting board
has a thickness of 233 nm. What color is most strongly reflected if it is
illuminated perpendicular to its surface?
75.What are the three smallest non-zero thicknesses of soapy water (
) on Plexiglas if it appears green (constructively reflecting
520-nm light) when illuminated perpendicularly by white light? Explicitly
show how you follow the steps inProblem Solving Strategies for
76.Suppose you have a lens system that is to be used primarily for
700-nm red light. What is the second thinnest coating of fluorite
(magnesium fluoride) that would be non-reflective for this wavelength?
77.(a) As a soap bubble thins it becomes dark, because the path
length difference becomes small compared with the wavelength of light
and there is a phase shift at the top surface. If it becomes dark when
the path length difference is less than one-fourth the wavelength, what
is the thickest the bubble can be and appear dark at all visible
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wavelengths? Assume the same index of refraction as water. (b)
Discuss the fragility of the film considering the thickness found.
78.A film of oil on water will appear dark when it is very thin, because
the path length difference becomes small compared with the
wavelength of light and there is a phase shift at the top surface. If it
becomes dark when the path length difference is less than one-fourth
the wavelength, what is the thickest the oil can be and appear dark at
all visible wavelengths? Oil has an index of refraction of 1.40.
79.Figure 27.34shows two glass slides illuminated by pure-
wavelength light incident perpendicularly. The top slide touches the
bottom slide at one end and rests on a 0.100-mm-diameter hair at the
other end, forming a wedge of air. (a) How far apart are the dark bands,
if the slides are 7.50 cm long and 589-nm light is used? (b) Is there any
difference if the slides are made from crown or flint glass? Explain.
80.Figure 27.34shows two 7.50-cm-long glass slides illuminated by
pure 589-nm wavelength light incident perpendicularly. The top slide
touches the bottom slide at one end and rests on some debris at the
other end, forming a wedge of air. How thick is the debris, if the dark
bands are 1.00 mm apart?
81.RepeatExercise 27.70, but take the light to be incident at a
82.RepeatExercise 27.71, but take the light to be incident at a
To save money on making military aircraft invisible to radar, an inventor
decides to coat them with a non-reflective material having an index of
refraction of 1.20, which is between that of air and the surface of the
plane. This, he reasons, should be much cheaper than designing
Stealth bombers. (a) What thickness should the coating be to inhibit the
reflection of 4.00-cm wavelength radar? (b) What is unreasonable about
this result? (c) Which assumptions are unreasonable or inconsistent?
84.What angle is needed between the direction of polarized light and
the axis of a polarizing filter to cut its intensity in half?
85.The angle between the axes of two polarizing filters is
how much does the second filter reduce the intensity of the light coming
through the first?
86.If you have completely polarized light of intensity
what will its intensity be after passing through a polarizing filter with its
axis at an
angle to the light’s polarization direction?
87.What angle would the axis of a polarizing filter need to make with
the direction of polarized light of intensity
to reduce the
88.At the end ofExample 27.8, it was stated that the intensity of
polarized light is reduced to
of its original value by passing
through a polarizing filter with its axis at an angle of
direction of polarization. Verify this statement.
89.Show that if you have three polarizing filters, with the second at an
to the first and the third at an angle of
to the first,
the intensity of light passed by the first will be reduced to
value. (This is in contrast to having only the first and third, which
reduces the intensity to zero, so that placing the second between them
increases the intensity of the transmitted light.)
90.Prove that, if
is the intensity of light transmitted by two polarizing
filters with axes at an angle
is the intensity when the axes
are at an angle
the original intensity.
(Hint: Use the trigonometric identities
91.At what angle will light reflected from diamond be completely
92.What is Brewster’s angle for light traveling in water that is reflected
from crown glass?
93.A scuba diver sees light reflected from the water’s surface. At what
angle will this light be completely polarized?
94.At what angle is light inside crown glass completely polarized when
reflected from water, as in a fish tank?
95.Light reflected at
from a window is completely polarized.
What is the window’s index of refraction and the likely substance of
which it is made?
96.(a) Light reflected at
from a gemstone in a ring is completely
polarized. Can the gem be a diamond? (b) At what angle would the light
be completely polarized if the gem was in water?
is Brewster’s angle for light reflected from the top of an
interface between two substances, and
is Brewster’s angle for
light reflected from below, prove that
If a polarizing filter reduces the intensity of polarized light to
its original value, by how much are the electric and magnetic fields
Suppose you put on two pairs of Polaroid sunglasses with their axes at
an angle of
. How much longer will it take the light to deposit a
given amount of energy in your eye compared with a single pair of
sunglasses? Assume the lenses are clear except for their polarizing
(a) On a day when the intensity of sunlight is
, a circular
lens 0.200 m in diameter focuses light onto water in a black beaker.
Two polarizing sheets of plastic are placed in front of the lens with their
axes at an angle of
Assuming the sunlight is unpolarized and
the polarizers are
efficient, what is the initial rate of heating of
the water in
, assuming it is
absorbed? The aluminum
beaker has a mass of 30.0 grams and contains 250 grams of water. (b)
Do the polarizing filters get hot? Explain.
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Figure 28.1Special relativity explains why traveling to other star systems, such as these in the Orion Nebula, is unreasonable using our current level of technology. (credit:
• State and explain both of Einstein’s postulates.
• Explain what an inertial frame of reference is.
• Describe one way the speed of light can be changed.
28.2.Simultaneity And Time Dilation
• Describe simultaneity.
• Describe time dilation.
• Calculate γ.
• Compare proper time and the observer’s measured time.
• Explain why the twin paradox is a false paradox.
• Describe proper length.
• Calculate length contraction.
• Explain why we don’t notice these effects at everyday scales.
28.4.Relativistic Addition of Velocities
• Calculate relativistic velocity addition.
• Explain when relativistic velocity addition should be used instead of classical addition of velocities.
• Calculate relativistic Doppler shift.
• Calculate relativistic momentum.
• Explain why the only mass it makes sense to talk about is rest mass.
• Compute total energy of a relativistic object.
• Compute the kinetic energy of a relativistic object.
• Describe rest energy, and explain how it can be converted to other forms.
• Explain why massive particles cannot reach C.
Introduction to Special Relativity
Have you ever looked up at the night sky and dreamed of traveling to other planets in faraway star systems? Would there be other life forms? What
would other worlds look like? You might imagine that such an amazing trip would be possible if we could just travel fast enough, but you will read in
this chapter why this is not true. In 1905 Albert Einstein developed the theory of special relativity. This theory explains the limit on an object’s speed
and describes the consequences.
Relativity. The wordrelativitymight conjure an image of Einstein, but the idea did not begin with him. People have been exploring relativity for many
centuries. Relativity is the study of how different observers measure the same event. Galileo and Newton developed the first correct version of
classical relativity. Einstein developed the modern theory of relativity. Modern relativity is divided into two parts.Special relativitydeals with observers
who are moving at constant velocity.General relativitydeals with observers who are undergoing acceleration. Einstein is famous because his
theories of relativity made revolutionary predictions. Most importantly, his theories have been verified to great precision in a vast range of
experiments, altering forever our concept of space and time.
CHAPTER 28 | SPECIAL RELATIVITY Y 997
Figure 28.2Many people think that Albert Einstein (1879–1955) was the greatest physicist of the 20th century. Not only did he develop modern relativity, thus revolutionizing
our concept of the universe, he also made fundamental contributions to the foundations of quantum mechanics. (credit: The Library of Congress)
It is important to note that although classical mechanic, in general, and classical relativity, in particular, are limited, they are extremely good
approximations for large, slow-moving objects. Otherwise, we could not use classical physics to launch satellites or build bridges. In the classical limit
(objects larger than submicroscopic and moving slower than about 1% of the speed of light), relativistic mechanics becomes the same as classical
mechanics. This fact will be noted at appropriate places throughout this chapter.
Figure 28.3Special relativity resembles trigonometry in that both are reliable because they are based on postulates that flow one from another in a logical way. (credit: Jon
Have you ever used the Pythagorean Theorem and gotten a wrong answer? Probably not, unless you made a mistake in either your algebra or your
arithmetic. Each time you perform the same calculation, you know that the answer will be the same. Trigonometry is reliable because of the certainty
that one part always flows from another in a logical way. Each part is based on a set of postulates, and you can always connect the parts by applying
those postulates. Physics is the same way with the exception thatallparts must describe nature. If we are careful to choose the correct postulates,
then our theory will follow and will be verified by experiment.
Einstein essentially did the theoretical aspect of this method forrelativity. With two deceptively simple postulates and a careful consideration of how
measurements are made, he produced the theory ofspecial relativity.
Einstein’s First Postulate
The first postulate upon which Einstein based the theory of special relativity relates to reference frames. All velocities are measured relative to some
frame of reference. For example, a car’s motion is measured relative to its starting point or the road it is moving over, a projectile’s motion is
measured relative to the surface it was launched from, and a planet’s orbit is measured relative to the star it is orbiting around. The simplest frames of
reference are those that are not accelerated and are not rotating. Newton’s first law, the law of inertia, holds exactly in such a frame.
Inertial Reference Frame
Aninertial frame of referenceis a reference frame in which a body at rest remains at rest and a body in motion moves at a constant speed in a
straight line unless acted on by an outside force.
The laws of physics seem to be simplest in inertial frames. For example, when you are in a plane flying at a constant altitude and speed, physics
seems to work exactly the same as if you were standing on the surface of the Earth. However, in a plane that is taking off, matters are somewhat
more complicated. In these cases, the net force on an object,
, is not equal to the product of mass and acceleration,
is equal to
plus a fictitious force. This situation is not as simple as in an inertial frame. Not only are laws of physics simplest in inertial frames, but they
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