A swimmer at point S in Figure 31-53 develops a leg cramp

A ray of light reflects from a plane mirror. The angle between the incoming ray and the reflected ray is What is the angle of reflection? (a) (b) (c) (d) Not enough information is given to determine the reflection angle. SSM

A ray of light in air is incident on the surface of a piece of glass. The angle between the normal to the surface and the incident ray is and the angle between the normal and the refracted ray is . What is the angle between the incident ray and the refracted ray? (a) 12, (b) 28, (c) 40, (d) 68

ENGINEERING APPLICATION During a physics experiment, you are measuring refractive indices of different transparent materials using a red heliumneon laser beam. For a given angle of incidence, the beam has an angle of refraction equal to in material A, and an angle of refraction equal to in material B. Which material has the larger index of refraction? (a) A, (b) B, (c) The indices of refraction are the same. (d) You cannot determine the relative magnitudes of the indices of refraction from the data given.

A ray of light passes from air into water, striking the surface of the water at an angle of incidence of Which, if any, of the following four quantities changes as the light enters the water: (a) wavelength, (b) frequency, (c) speed of propagation, (d) direction of propagation, (e) none of the above?

Earths atmosphere decreases in density as the altitude increases. As a consequence, the index of refraction of the atmosphere also decreases as altitude increases. Explain how one can see the Sun when it is below the horizon. (The horizon is the extension of a plane that is tangent to the Earths surface.) Why does the setting Sun appear flattened?

A physics student playing pocket billiards wants to strike her cue ball so that it hits a cushion and then hits the eight ball squarely. She chooses several points on the cushion and then measures the distances from each point to the cue ball and to the eight ball. She aims at the point for which the sum of these distances is least. (a)Will her cue ball hit the eight ball? (b) How is her method related to Fermats principle? Neglect any effects due to ball rotation.

A swimmer at point S in Figure 31-53 develops a leg cramp while swimming near the shore of a calm lake and calls for help. A lifeguard at point hears the call. The lifeguard can run and swim She knows physics and chooses a path that will take the least time to reach the swimmer. Which of the paths shown in the figure does the lifeguard take? SSM

Material Ahas a higher index of refraction than material B. Which material has the larger critical angle for total internal reflection when the material is in air? (a) A, (b) B, (c) The angles are the same. (d) You cannot compare the angles based on the data given.

BIOLOGICAL APPLICATION A human eye perceives color using a structure which is called a cone that is located on the retina. Three types of molecules compose these cones and each type of molecule absorbs either red, green, or blue light by resonance absorption. Use this fact to explain why the color of an object that appears blue in air appears blue underwater, in spite of the fact that the wavelength of the light is shortened in accordance with Equation 31-6. SSM

Let be the angle between the transmission axes of two polarizing sheets. Unpolarized light of intensity is incident on the first sheet. What is the intensity of the light transmitted through both sheets? (a) (b) (c) (d) (e) (f) none of the above

Draw a diagram to explain how Polaroid sunglasses reduce glare from sunlight reflected from a smooth horizontal surface, such as the surface found on a pool of water. Your diagram should clearly indicate the direction of polarization of the light as it propagates from the Sun to the reflecting surface and then through the sunglasses into the eye.

BIOLOGICAL APPLICATION Why is it far less dangerous to stand in front of an intense beam of red light than in front of a very low-intensity beam of gamma rays?

Three energy states of an atom are A, B, and C. State B is above state A and state C is above state B. Which atomic transition results in the emission of the shortest wavelength of light? (a) (b) (c) (d)

In Problem 13, if the atom is initially in state A, which transition results in the emission of the longest wavelength of light? (a) (b) (c) (d)

What role does the helium play in a heliumneon laser?

When a beam of visible white light that passes through a gas of atomic hydrogen at room temperature is viewed with a spectroscope, dark lines are observed at the wavelengths of the hydrogen atom emission series. The atoms that participate in the resonance absorption then emit light of same wavelength as they return to the ground state. Explain why the observed spectrum nevertheless exhibits pronounced dark lines.

Which of the following types of light would have the highest energy photons? (a) red, (b) infrared, (c) blue, (d) ultraviolet

Estimate the time required for light to make the round trip during Galileos experiment to measure the speed of light. Compare the time of the round trip to typical human response times. How accurate do you think this experiment is?

Estimate the time delay in receiving a light on your retina when you are wearing eyeglasses compared to when you are not wearing your eyeglasses.

BIOLOGICAL APPLICATION Estimate the number of photons that enter your eye if you look for a tenth of a second at the Sun. What energy is absorbed by your eye during that time, assuming that all the photons are absorbed? The total power output of the Sun is

Rmer was observing the eclipses of Jupiters moon Io with the hope that they would serve as a highly accurate clock that would be independent of longitude. (Prior to GPS, such a clock was needed for accurate navigation.) Io eclipses (enters the umbra of Jupiters shadow) every Assuming an eclipse of Io is observed on Earth on June 1 at midnight when Earth is at location A (as shown in Figure 31-54), predict the expected time of observation of an eclipse one-quarter of a year later when Earth is at location assuming (a) the speed of light is infinite and (b) the speed of light is .

If the angle of incidence is small enough, the small angle approximation may be used to simplify Snells law of refraction. Determine the maximum value of the angle that would make the value for the angle differ by no more than one percent from the value for the sine of the angle. (This approximation will be used in connection with image formation by spherical surfaces in Chapter 32.)

Mission Control sends a brief wake-up call to astronauts in a spaceship that is far from Earth. At a time after the call is sent, Mission Control can hear the groans of the astronauts. How far from Earth is the spaceship? (a) (b) (c) (d) (e) The spaceship is on the moon.

ENGINEERING APPLICATION The distance from a point on the surface of Earth to a point on the surface of the moon is measured by aiming a laser light beam at a reflector on the surface of the moon and measuring the time required for the light to make a round trip. The uncertainty in the measured distance is related to the uncertainty in the measured time by If the time intervals can be measured to (a) find the uncertainty of the distance. (b) Estimate the percentage uncertainty in the distance.

Ole Rmer discovered the finiteness of the speed of light by observing Jupiters moons. Approximately how sensitive would the timing apparatus need to be in order to detect a shift in the predicted time of the moons eclipses that occur when the moon happens to be at perigee and those that occur when the moon is at apogee Assume that an instrument should be able to measure to at least one-tenth the magnitude of the effect it is to measure.

Calculate the fraction of light energy reflected from an airwater interface at normal incidence.

A ray of light is incident on one of two mirrors that are set at right angles to each other. The plane of incidence is perpendicular to both mirrors. Show that after reflecting from each mirror, the ray will emerge traveling in the direction opposite to the incident direction, regardless of the angle of incidence.

SPREADSHEET (a) A ray of light in air is incident on an airwater interface. Using a spreadsheet or graphing program, plot the angle of refraction as a function of the angle of incidence from to (b) Repeat Part (a), but for a ray of light in water that is incident on a waterair interface. [For Part (b), there is no reflected ray for angles of incidence that are greater than the critical angle.]

The red light from a heliumneon laser has a wavelength of in air. Find the (a) speed, (b) wavelength, and (c) frequency of heliumneon laser light in air, water, and glass. (The glass has an index of refraction equal to 1.50.)

The index of refraction for silicate flint glass is 1.66 for violet light that has a wavelength in air equal to and for red light that has a wavelength in air equal to A ray of red light and a ray of 400-nm-wavelength violet light both have angles of refraction equal to upon entering the glass from air. (a) Which is greater, the angle of incidence of the ray of red light or the angle of incidence of the ray of violet light? Explain your answer. (b) What is the difference between the angles of incidence of the two rays?

Aslab of glass that has an index of refraction of 1.50 is submerged in water that has an index of refraction of 1.33. Light in the water is incident on the glass. Find the angle of refraction if the angle of incidence is (a) (b) and (c)

Repeat Problem 31 for a beam of light initially in the glass that is incident on the glasswater interface at the same angles.

A beam of light in air strikes a glass slab at normal incidence. The glass slab has an index of refraction of 1.50. (a) Approximately what percentage of the incident light intensity is transmitted through the slab (in one side and out the other)? (b) Repeat Part (a) if the glass slab is immersed in water.

This problem is a refraction analogy. A band is marching down a football field with a constant speed About midfield, the band comes to a section of muddy ground that has a sharp boundary making an angle of with the line, as shown in Figure 31-55. In the mud, each marcher moves at a speed equal to in a direction perpendicular to the row of marchers they are in. (a) Diagram how each line of marchers is bent as it encounters the muddy section of the field so that the band is eventually marching in a different direction. Indicate the original direction by a ray and the final direction by a second ray. (b) Find the angles between these rays and the line normal to the boundary. Is their direction of motion bent toward the normal or away from it? Explain your answer in terms of refraction.

In Figure 31-56, light is initially in a medium that has an index of refraction It is incident at angle on the surface of a liquid that has an index of refraction The light passes through the layer of liquid and enters glass that has an index of refraction If is the angle of refraction in the glass, show that That is, show that the second medium can be neglected when finding the angle of refraction in the third medium. SSM

On a safari, you are spearfishing while wading in a river. You observe a fish gliding by you. If your line of sight to the fish is below the horizontal in air, and assuming the spear follows a straight-line path through the air and water after it is released, determine the angle below the horizontal that you should aim your spear gun in order to catch dinner. Assume the spear gun barrel is above the water surface, the fish is below the surface, and the spear travels in a straight line all the way to the fish.

You are standing on the edge of a swimming pool and looking directly across at the opposite side. You notice that the bottom edge of the opposite side of the pool appears to be at an angle of below the horizontal. However, when you sit on the pool edge, the bottom edge of the opposite side of the pool appears to be at an angle of only below the horizontal. Use these observations to determine the width and depth of the pool. Hint: You will need to estimate the height of your eyes above the surface of the water when standing and sitting.

Figure 31-57 shows a beam of light incident on a glass plate of thickness and index of refraction (a) Find the angle of incidence so that the separation b between the ray reflected from the top surface and the ray reflected from the bottom surface and exiting the top surface is a maximum. (b) What is this angle of incidence if the index of refraction of the glass is 1.60? (c) What is the separation of the two beams if the thickness of the glass plate is 4.0 cm?

What is the critical angle for light traveling in water that is incident on a waterair interface? SSM

A glass surface has a layer of water on it. Light in the glass is incident on the glasswater interface. Find the critical angle for total internal reflection.

A point source of light is located below the surface of a large pool of water. Find the area of the largest circle on the pools surface through which light coming directly from the source can emerge.

Light traveling in air strikes the largest face of an isosceles-right-triangle prism at normal incidence. What is the speed of light in this prism if the prism is just barely able to produce total internal reflection?

A point source of light is located at the bottom of a steel tank, and an opaque circular card of radius is placed horizontally over it. Atransparent fluid is gently added to the tank so that the card floats on the fluid surface with its center directly above the light source. No light is seen by an observer above the surface until the fluid is deep. What is the index of refraction of the fluid?

ENGINEERING APPLICATION An optical fiber allows rays of light to propagate long distances by using total internal reflection. Optical fibers are used extensively in medicine and in digital communications. As shown in Figure 31-58 the fiber consists of a core material that has an index of refraction and radius b surrounded by a cladding material that has an index of refraction The numerical aperture of the fiber is defined as where is the angle of incidence of a ray of light that impinges on the center of the end of the fiber and then reflects off the corecladding interface just at the critical angle. Using the figure as a guide, show that the numerical aperture is given by assuming the ray is initially in air. Hint: Use of the Pythagorean theorem may be required.

ENGINEERING APPLICATION Find the maximum angle of incidence of a ray that would propagate through an optical fiber that has a core index of refraction of 1.492, a core radius of and a cladding index of 1.489. See Problem 44.

ENGINEERING APPLICATION Calculate the difference in time needed for two pulses of light to travel down of the fiber that is described in Problem 44. Assume that one pulse enters the fiber at normal incidence and the second pulse enters the fiber at the maximum angle of incidence calculated in Problem 45. In fiber optics, this effect is known as modal dispersion.

Investigate how a thin film of water on a glass surface affects the critical angle for total reflection. Use for glass and for water. (a) What is the critical angle for total internal reflection at the glasswater interface? (b) Does a range of incident angles exist such that the angles are greater than for glassto- air refraction and for which the light rays will leave the glass, travel through the water, and then pass into the air?

A laser beam is incident on a plate of glass that is thick (Figure 31-57). The glass has an index of refraction of 1.5 and the angle of incidence is The top and bottom surfaces of the glass are parallel. What is the distance b between the beam formed by reflection off the top surface of the glass and the beam reflected off the bottom surface of the glass.

A beam of light strikes the plane surface of silicate flint glass at an angle of incidence of The index of refraction of the glass varies with wavelength (see Figure 31-59). How much smaller is the angle of refraction for violet light of wavelength than the angle of refraction for red light of wavelength 700 nm?

ENGINEERING APPLICATION In many transparent materials, dispersion causes different colors (wavelengths) of light to travel at different speeds. This can cause problems in fiber-optic communications systems where pulses of light must travel very long distances in glass. Assuming a fiber is made of silicate crown glass (see Figure 31-59), calculate the difference in travel times that two short pulses of light take to travel in the fiber if the first pulse has a wavelength of and the second pulse has a wavelength of

What is the polarizing angle for light in air that is incident on (a)water and (b) glass

Light that is horizontally polarized is incident on a polarizing sheet. It is observed that only 15 percent of the intensity of the incident light is transmitted through the sheet. What angle does the transmission axis of the sheet make with the horizontal?

Two polarizing sheets have their transmission axes crossed so that no light gets through. A third sheet is inserted between the first two so that its transmission axis makes an angle with the transmission axis of the first sheet. Unpolarized light of intensity is incident on the first sheet. Find the intensity of the light transmitted through all three sheets if (a) and (b)

A horizontal laser beam that is vertically polarized is incident on a polarizing sheet that is oriented with its transmission axis vertical. Behind the first sheet is a second sheet that is oriented so that its transmission axis makes an angle of with respect to vertical. What is the power of the beam transmitted through the second sheet?

The polarizing angle for light in air that is incident on a certain substance is (a) What is the angle of refraction of light incident at this angle? (b) What is the index of refraction of this substance?

Two polarizing sheets have their transmission axes crossed so that no light is transmitted. A third sheet is inserted so that its transmission axis makes an angle with the transmission axis of the first sheet. (a) Derive an expression for the intensity of the transmitted light as a function of (b) Show that the intensity transmitted through all three sheets is maximum when

If the middle polarizing sheet in Problem 56 is rotating at an angular speed about an axis parallel with the light beam, find an expression for the intensity transmitted through all three sheets as a function of time. Assume that at time

SPREADSHEET Astack of ideal polarizing sheets is arranged so that each sheet is rotated by an angle of with respect to the preceding sheet. A linearly polarized light wave of intensity is incident normally on the stack. The incident light is polarized along the transmission axis of the first sheet and is therefore perpendicular to the transmission axis of the last sheet in the stack. (a) Show that the intensity of the light transmitted through the entire stack is given by (b) Using a spreadsheet or graphing program, plot the transmitted intensity as a function of for values of from 2 to 100. (c) What is the direction of polarization of the transmitted beam in each case?

SPREADSHEET, ENGINEERING APPLICATION The device described in Problem 58 could serve as a polarization rotator, which changes the linear plane of polarization from one direction to another. The efficiency of such a device is measured by taking the ratio of the output intensity at the desired polarization to the input intensity. The result of Problem 58 suggests that the highest efficiency is achieved by using a large value for the number A small amount of intensity is lost regardless of the input polarization when using a real polarizer. For each polarizer, assume the transmitted intensity is 98 percent of the amount predicted by the law of Malus and use a spreadsheet or graphing program to determine the optimum number of sheets you should use to rotate the polarization

Show mathematically that a linearly polarized wave can be thought of as a superposition of a right and a left circularly polarized wave.

Suppose that the middle sheet in Problem 53 is replaced by two polarizing sheets. If the angles between the transmission axes of the second, third and fourth sheets in the stack make angles of and respectively, with the transmission axis of the first sheet, (a) what is the intensity of the transmitted light? (b) How does this intensity compare with the intensity obtained in part (a) of Problem 53?

Show that the electric field of a circularly polarized wave propagating parallel to the axis can be expressed by .

A circularly polarized wave is said to be right circularly polarized if the electric and magnetic fields rotate clockwise when viewed along the direction of propagation and left circularly polarized if the fields rotate counterclockwise. (a) What is the sense of the circular polarization for the wave described by the expression in Problem 62? (b) What would be the expression for the electric field of a circularly polarized wave traveling in the same direction as the wave in Problem 60, but with the fields rotating in the opposite sense?

A heliumneon laser emits light that has a wavelength equal to and has a power output of How many photons are emitted per second by this laser?

The first excited state of an atom of a gas is above the ground state. (a) What is the maximum wavelength of radiation for resonance absorption by atoms of the gas that are in the ground state? (b) If the gas is irradiated with monochromatic light that has a wavelength of what is the wavelength of the Raman scattered light?

A gas is irradiated with monochromatic ultraviolet light that has a wavelength of Scattered light that has a wavelength equal to is observed, and scattered light that has a wavelength of is also observed. Assuming that the gas atoms were in their ground state prior to irradiation, find the energy difference between the ground state and the excited state obtained by the irradiation.

Sodium has excited states and above the ground state. Assume that the atoms of the gas are all in the ground state prior to irradiation. (a) What is the maximum wavelength of radiation that will result in resonance fluorescence? What is the wavelength of the fluorescent radiation? (b) What wavelength will result in excitation of the state above the ground state? If that state is excited, what are the possible wavelengths of resonance fluorescence that might be observed?

Singly ionized helium is a hydrogen-like atom that has a nuclear charge of Its energy levels are given by where and If a beam of visible white light is sent through a gas of singly ionized helium, at what wavelengths will dark lines be found in the spectrum of the transmitted radiation? (Assume that the ions of the gas are all in the state with energy prior to irradiation.)

A pulse from a ruby laser has an average power of and lasts (a) What is the total energy of the pulse? (b) How many photons are emitted in the pulse?

A beam of red light that has a wavelength of \(700 \mathrm{~nm}\) in air travels in water. (a) What is the wavelength in water? (b) Does a swimmer underwater observe the same color or a different color for this light?

The critical angle for total internal reflection for a substance is What is the polarizing angle for the substance?

Show that when a flat mirror is rotated through an angle about an axis in the plane of the mirror, a reflected beam of light (from a fixed incident beam) that is perpendicular to the rotation axis is rotated through

Use Figure 31-59 to calculate the critical angles for light initially in silicate flint glass that is incident on a glassair interface if the light is (a) violet light of wavelength and (b) red light of wavelength

Light is incident on a slab of transparent material at an angle as shown in Figure 31-60. The slab has a thickness and an index of refraction Show that where d is the distance shown in the figure.

A ray of light begins at the point strikes a mirror in the plane at some point 0, 0), and reflects through the point (a) Find the value of that makes the total distance traveled by the ray a minimum. (b) What is the angle of incidence on the reflecting plane? (c) What is the angle of reflection?

ENGINEERING APPLICATION To produce a polarized laser beam, a plate of transparent material (Figure 31-61) is placed in the laser cavity and oriented so the light strikes it at the polarizing angle. Such a plate is called a Brewster window. Show that if is the polarizing angle for the to interface, then is the polarizing angle for the to n interface.

From the data provided in Figure 31-59, calculate the polarization angle for an airglass interface, using light of wavelength in each of the four types of glass shown.

A light ray passes through a prism with an apex angle of as shown in Figure 31-62. The ray and the bisector of the apex angle intersect at right angles. Show that the angle of deviation is related to the apex angle and the index of refraction of the prism material by sinC12(a _ d)D _ n sinA12 aB.

(a) For light rays inside a transparent medium that is surrounded by a vacuum, show that the polarizing angle and the critical angle for total internal reflection satisfy tan (b) Which angle is larger, the polarizing angle or the critical angle for total internal reflection?

Light in air is incident on the surface of a transparent substance at an angle of with the normal. The reflected and refracted rays are observed to be mutually perpendicular. (a) What is the index of refraction of the transparent substance? (b) What is the critical angle for total internal reflection in this substance?

Alight ray in dense flint glass that has an index of refraction of 1.655 is incident on the glass surface. An unknown liquid condenses on the surface of the glass. Total internal reflection on the glassliquid interface occurs for a minimum angle of incidence on the glassliquid interface of 53.7. (a)What is the refractive index of the unknown liquid? (b) If the liquid is removed, what is the minimum angle of incidence for total internal reflection? (c) For the angle of incidence found in Part (b), what is the angle of refraction of the ray into the liquid film? Does a ray emerge from the liquid film into the air above? Assume the glass and liquid have parallel planar surfaces.

(a) Show that for normally incident light, the intensity transmitted through a glass slab that has an index of refraction of and is surrounded by air is approximately given by (b) Use the Part (a) result to find the ratio of the transmitted intensity to the incident intensity through parallel slabs of glass for light of normal incidence. (c) How many slabs of a glass that has an index of refraction of 1.5 are required to reduce the intensity to 10 percent of the incident intensity? _

Equation 31-14 gives the relation between the angle of deviation of a light ray incident on a spherical drop of water in terms of the incident angle and the index of refraction of water. (a) Assume that and derive an expression for Hint: If then (b) Use the result to show that the angle of incidence for minimum deviation is given by (c) The index of refraction for a certain red light in water is 1.3318 and the index of refraction for a certain blue light in water is 1.3435. Use the result of Part (a) to find the angular separation of these colors in the primary rainbow.

Show that the angle of deviation is a minimum if the angle of incidence is such that the ray and the bisector of the apex angle a (Figure 31-62) intersect at right angles.