College Physics

Hugh D. Young

Chapter 23

Electromagnetic Waves and Propagationof Light - all with Video Answers

Educators

Chapter Questions

Problem 1

$\bullet$ When a solar flare erupts on the surface of the sun, how many minutes after it occurs does its light show up in an astronomer's telescope on earth? (Consult Appendix E.)

Ze-Han Lee

Ze-Han Lee

Numerade Educator

Problem 2

$\bullet$ TV ghosting. In a TV picture, faint, slightly offset ghost images are formed when the signal from the transmitter travels to the receiver both directly and indirectly after reflection from a building or some other large metallic mass. In a 25 inch set, the ghost is about 1.0 $\mathrm{cm}$ to the right of the principal image if the reflected signal arrives 0.60$\mu$ s after the principal signal. In this case, what is the difference in the distance traveled by the two signals?

Ryan Hood

Numerade Educator

Problem 3

$\bullet$ (a) How much time does it take light to travel from the moon to the earth, a distance of $384,000 \mathrm{km} ?$ (b) Light from the star Sirius takes 8.61 years to reach the earth. What is the distance to Sirius in kilometers?

Ze-Han Lee

Numerade Educator

Problem 4

$\bullet$$\bullet$ A geostationary communications satellite orbits the earth directly above the equator at an altitude of $35,800 \mathrm{km} .$ Calculate the time it takes for a signal to travel from a point on the equator to the satellite and back to the ground at another point on the equator exactly halfway around the earth. (See Appendix E.)

Ryan Hood

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Problem 5

$\bullet$ Consider electromagnetic waves propagating in air. (a) Determine the frequency of a wave with a wavelength of (i) $5.0 \mathrm{km},$ (ii) $5.0 \mu \mathrm{m},$ (iii) 5.0 $\mathrm{nm}$ . (b) What is the wavelength (in meters and nanometers) of (i) gamma rays of frequency $6.50 \times 10^{21} \mathrm{Hz}$ , (ii) an AM station radio wave of frequency 590 $\mathrm{kHz} ?$

Ze-Han Lee

Numerade Educator

Problem 6

$\bullet$ Most people perceive light having a wavelength between 630 $\mathrm{nm}$ and 700 $\mathrm{nm}$ as red and light with a wavelength between 400 $\mathrm{nm}$ and 440 $\mathrm{nm}$ as violet. Calculate the approximate frequency ranges for (a) violet light and (b) red light.

Ryan Hood

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Problem 7

$\bullet$ The electric field of a sinusoidal electromagnetic wave obeys the equation $E=-(375 \mathrm{V} / \mathrm{m}) \sin [(5.97 \times$ $10^{15} \operatorname{rad} / \mathrm{s} ) t+\left(1.99 \times 10^{7} \mathrm{rad} / \mathrm{m}\right) x ] .$ (a) What are the amplitudes of the electric and magnetic fields of this wave? (b) What are the frequency, wavelength, and period of the wave? Is this light visible to humans? (c) What is the speed of the wave?

Ze-Han Lee

Numerade Educator

Problem 8

$\bullet$ A sinusoidal electromagnetic wave having a magnetic field of amplitude 1.25 $\mu$ and a wavelength of 432 nm is traveling in the $+x$ direction through empty space. (a) What is the frequency of this wave? (b) What is the amplitude of the associated electric field? (c) Write the equations for the electric and magnetic fields as functions of $x$ and $t$ in the form of Equations $(23.3) .$

Ryan Hood

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Problem 9

$\bullet$ Visible light. The wavelength of visible light ranges from 400 nm to 700 nm. Find the corresponding ranges of this light's (a) frequency, (b) angular frequency, (c) wave number.

Ze-Han Lee

Numerade Educator

Problem 10

$\bullet$ Ultraviolet radiation. There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 $\mathrm{nm}$ to 400 nm. It is not so harmful to the skin and is necessary for the production of vitamin D. UVB, with a wavelength between 280 $\mathrm{nm}$ and $320 \mathrm{nm},$ is much more dangerous, because it causes skin cancer. (a) Find the frequency ranges of UVA and UVB. (b) What are the ranges of the wave numbers for UVA and UVB?

Ryan Hood

Numerade Educator

Problem 11

$\bullet$ Medical rays. Medical xays are taken with electromagnetic waves having a wavelength around 0.10 nm. What are the frequency, period, and wave number of such waves?

Ze-Han Lee

Numerade Educator

Problem 12

$\bullet$ Radio station $\mathrm{WCCO}$ in Minneapolis broadcasts at a frequency of 830 $\mathrm{kHz}$ . At a point some distance from the transmitter, the magnetic-field amplitude of the electromagnetic wave from $\mathrm{WCCO}$ is $4.82 \times 10^{-11} \mathrm{T}$ . Calculate (a) the wavelength, (b) the wave number, (c) the angular frequency, and (d) the electric-field amplitude.

Ryan Hood

Numerade Educator

Problem 13

$\bullet$$\bullet \mathrm{A}$ sinusoidal electromagnetic wave of frequency $6.10 \times 10^{14} \mathrm{Hz}$ travels in vacuum in the $+x$ -direction. The magnetic field is parallel to the $y$ -axis and has amplitude $5.80 \times 10^{-4} \mathrm{T} .$ (a) Find the magnitude and direction of the electric field. (b) Write the wave functions for the electric and magnetic fields in the form of Equations $(23.3).$

Shoukat Ali

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Problem 14

$\bullet$ Consider each of the electric- and magnetic-field orientations given next. In each case, what is the direction of propagation of the wave? (a) $\vec{E}$ in the $+x$ direction, $\vec{B}$ in the $+y$ direction. (b) $\vec{E}$ in the $-y$ direction, $\vec{B}$ in the $+x$ direction. (c) $\vec{\boldsymbol{E}}$ in the $+z$ direction, $\vec{\boldsymbol{B}}$ in the $-x$ direction. (d) $\vec{\boldsymbol{E}}$ in the $+y$ direction, $\vec{\boldsymbol{B}}$ in the $-z$ direction.

Ryan Hood

Numerade Educator

Problem 15

$\bullet$$\bullet$ An electromagnetic wave has a magnetic field given by $B=\left(8.25 \times 10^{-9} \mathrm{T}\right) \sin \left[\left(\omega t+1.38 \times 10^{4} \mathrm{rad} / \mathrm{m}\right) x\right],$ with the magnetic field in the $+y$ direction. (a) In which direction is the wave traveling? (b) What is the frequency $f$ of the wave? (c) Write the wave function for the electric field.

Ze-Han Lee

Numerade Educator

Problem 16

$\bullet$ Laboratory lasers. He-Ne lasers are often used in physics demonstrations. They produce light of wavelength 633 $\mathrm{nm}$ and a power of 0.500 $\mathrm{mW}$ spread over a cylindrical beam 1.00 $\mathrm{mm}$ in diameter (although these quantities can vary). (a) What is the intensity of this laser beam? (b) What are the maximum values of the electric and magnetic fields? (c) What is the average energy density in the laser beam?

Ryan Hood

Numerade Educator

Problem 17

$\bullet$ Fields from a lightbulb. We can reasonably model a 75 $\mathrm{W}$ incandescent lightbulb as a sphere 6.0 $\mathrm{cm}$ in diameter. Typically, only about 5$\%$ of the energy goes to visible light; the rest goes largely to nonvisible infrared radiation. (a) What is the visible light intensity (in $\mathrm{W} / \mathrm{m}^{2} )$ at the surface of the bulb? (b) What are the amplitudes of the electric and magnetic fields at this surface, for a sinusoidal wave with this intensity?

Ze-Han Lee

Numerade Educator

Problem 18

$\bullet$ Threshold of vision. Under controlled darkened conditions in the laboratory, a light receptor cell on the retina of a person's eye can detect a single photon (more on photons in Chapter 28 ) of light of wavelength 505 $\mathrm{nm}$ and having an energy of $3.94 \times 10^{-19} \mathrm{J} .$ We shall assume that this energy is absorbed by a single cell during one period of the wave. Cells of this kind are called rods and have a diameter of approximately 0.0020 $\mathrm{mm} .$ What is the intensity (in $\mathrm{W} / \mathrm{m}^{2} )$ delivered to a rod?

Ryan Hood

Numerade Educator

Problem 19

$\bullet$ $\cdot$ High-energy cancer treatment. Scientists are working on a new technique to kill cancer cells by zapping them with ultrahigh-energy (in the range of $10^{12}$ W) pulses of light that last for an extremely short time (a few nanoseconds). These short pulses scramble the interior of a cell without causing it to explode, as long pulses would do. We can model a typical such cell as a disk 5.0$\mu \mathrm{m}$ in diameter, with the pulse lasting for 4.0 $\mathrm{ns}$ with an average power of $2.0 \times 10^{12} \mathrm{W}$ . We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. (a) How much energy is given to the cell during this pulse? (b) What is the intensity (in $\mathrm{W} / \mathrm{m}^{2} )$ delivered to the cell? (c) What are the maximum values of the electric and magnetic fields in the pulse?

Ze-Han Lee

Numerade Educator

Problem 20

$\bullet$ At the floor of a room, the intensity of light from bright
overhead lights is 8.00 $\mathrm{W} / \mathrm{m}^{2} .$ Find the radiation pressure on a
totally absorbing section of the floor.

Ryan Hood

Numerade Educator

Problem 21

$\bullet$ The intensity at a certain distance from a bright light source is 6.00 $\mathrm{W} / \mathrm{m}^{2} .$ Find the radiation pressure (in pascals and in atmospheres) on (a) a totally absorbing surface and (b) a totally reflecting surface.

Ze-Han Lee

Numerade Educator

Problem 22

$\bullet$$\bullet$ A sinusoidal electromagnetic wave from a radio station passes perpendicularly through an open window that has area 0.500 $\mathrm{m}^{2} .$ At the window, the electric field of the wave has rea value 0.0200 $\mathrm{V} / \mathrm{m} .$ How much energy does this wave carry through the window during a 30.0 s commercial?

Ryan Hood

Numerade Educator

Problem 23

$\bullet$$\bullet$ Two sources of sinusoidal electromagnetic waves have average powers of 75 $\mathrm{W}$ and 150 $\mathrm{W}$ and emit uniformly in all directions. At the same distance from each source, what is the ratio of the maximum electric field for the 150 $\mathrm{W}$ source to
that of the 75 $\mathrm{W}$ source?

Ze-Han Lee

Numerade Educator

Problem 24

$\bullet$$\bullet$ Radiation falling on a perfectly reflecting surface produces an average pressure $p .$ If radiation of the same intensity falls on a perfectly absorbing surface and is spread over twice the area, what is the pressure at that surface in terms of $p ?$

Ryan Hood

Numerade Educator

Problem 25

$\bullet$$\bullet$ A sinusoidal electromagnetic wave emitted by a cellular phone has a wavelength of 35.4 $\mathrm{cm}$ and an electric field amplitude of $5.40 \times 10^{-2} \mathrm{V} / \mathrm{m}$ at a distance of 250 $\mathrm{m}$ from the antenna. Calculate: (a) the frequency of the wave; (b) the
magnetic-field amplitude; (c) the intensity of the wave.

Shoukat Ali

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Problem 26

$\bullet$ Two plane mirrors intersect at right angles. A laser beam strikes the first of them at a point 11.5 $\mathrm{cm}$ from their point of intersection, as shown in Figure $23.49 .$ For what angle of incidence at the first mirror will this ray strike the midpoint of the second mirror (which is 28.0 $\mathrm{cm}$ long) after reflecting from the first mirror?

Ryan Hood

Numerade Educator

Problem 27

$\bullet$ Three plane mirrors intersect at right angles. A beam of laser light strikes the first of them at an angle $\theta$ with respect to the normal. (See Figure $23.50 . )$ (a) Show that when this ray is reflected off of the other two mirrors and crosses the original ray, the angle $\alpha$ between these two rays will be $\alpha=180^{\circ}-2 \theta$ . (b) For what angle $\theta$ will the two rays be perpendicular when they cross?

Shoukat Ali

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Problem 28

$\bullet$ Two plane mirrors $A$ and $B$ intersect at a $45^{\circ}$ angle. Three rays of light leave point $P$ (see Figure 23.51 ) and strike one of the mirrors. What is the subsequent path of each of the following rays until they no longer strike either of the mirrors? (a) Ray $1,$ which strikes rors? (a) Ray $1,$ which strikes $A$ at $45^{\circ}$ with respect to the normal. (b) Ray $2,$ which strikes $B$ traveling perpendicular to mirror $A .$ (c) Ray 3 , which strikes $B$ perpendicular to its surface.

Ryan Hood

Numerade Educator

Problem 29

$\bullet$ . Prove that when a ray of light travels at any angle into the corner formed by two mirrors placed at right angles to each other, the reflected ray emerges parallel to the original ray (see Figure 23.52).

Shoukat Ali

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Problem 30

$\bullet$ A light beam travels at $1.94 \times 10^{8} \mathrm{m} / \mathrm{s}$ in quartz. The wavelength of the light in quartz is 355 $\mathrm{nm}$ . (a) What is the index of refraction of quartz
at this wavelength? (b) If this same light travels through air, what is its wavelength there?

Ryan Hood

Numerade Educator

Problem 31

$\bullet$$\bullet$ Using a fast-pulsed laser and electronic timing circuitry, you find that light travels 2.50 $\mathrm{m}$ within a plastic rod in 11.5 $\mathrm{ns} .$ What is the refractive index of the plastic?

Shoukat Ali

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Problem 32

$\bullet$ Light with a frequency of $5.80 \times 10^{14}$ Hz travels in a block of glass that has an index of refraction of $1.52 .$ What is the wavelength of the light (a) in vacuum and (b) in the glass?

Ryan Hood

Numerade Educator

Problem 33

$\bullet$ The speed of light with a wavelength of 656 $\mathrm{nm}$ in heavy
flint glass is $1.82 \times 10^{8} \mathrm{m} / \mathrm{s} .$ What is the index of refraction of
the glass at this wavelength?

Shoukat Ali

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Problem 34

$\bullet$ Light inside the eye. The vitreous humor, a transparent, gelatinous fluid that fills most of the eyeball, has an index of refraction of 1.34 . Visible light ranges in wavelength from
400 $\mathrm{nm}$ (violet) to $700 \mathrm{nm}(\mathrm{red}),$ as measured in air. This light travels through the vitreous humor and strikes the rods and cones at the surface of the retina. What are the ranges of (a) the wavelength, (b) the frequency, and (c) the speed of the light just as it approaches the retina within the vitreous humor?

Ryan Hood

Numerade Educator

Problem 35

$\bullet$$\bullet$ Light of a certain frequency has a wavelength of 438 $\mathrm{nm}$ inwater. What is the wavelength of this light (a) in benzene, (b) in air? (See Table $23.1 . )$

Shoukat Ali

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Problem 36

$\bullet$$\bullet$ A 1.55 -m-tall fisherman stands at the edge of a lake, being watched by a suspicious trout who is 3.50 $\mathrm{m}$ from the fisherman in the horizontal direction and 45.0 $\mathrm{cm}$ below the surface of the water. At what angle from the vertical does the fish see the top of the fisherman's head?

Ryan Hood

Numerade Educator

Problem 37

$\bullet$ Show that when a light ray travels from air through a sheet of glass with parallel surfaces and back into the air, it emerges traveling parallel to its original direction, although slightly displaced. (Note: The result of this problem is important, because it shows us that a sheet with parallel faces does not change the direction of a ray. It can also be shown that a very thin sheet does not displace the beam significantly. This is the case with a thin lens, since its opposite faces near its center are essentially parallel to each other. Rays that strike the center are essentially go essentially straight through.)

Shoukat Ali

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Problem 38

$\bullet$ A glass plate having parallel faces and a refractive index of 1.58 lies at the bottom of a liquid of refractive index $1.70 . \mathrm{A}$ ray of light in the liquid strikes the top of the glass at an angle of incidence of $62.0^{\circ} .$ Compute the angle of refraction of this light in the glass.

Ryan Hood

Numerade Educator

Problem 39

$\bullet$ A beam of light in air makes an angle of $47.5^{\circ}$ with the surface (not the normal) of a glass plate having a refractive index of 1.66 (a) What is the angle between the reflected part of the beam and the surface of the glass? (b) What is the angle between the refracted beam and the surface (not the normal) of the glass?

Shoukat Ali

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Problem 40

$\bullet$$\bullet$ A laser beam shines along the surface of a block of transparent material. (See Figure $23.53 .)$ Half of the beam goes straight to a detector, while the other half travels through the block and then hits the detector. The time delay between the arrival of the two light beams at the detector is 6.25 ns. What is the index of refraction of this material?

Ryan Hood

Numerade Educator

Problem 41

$\bullet$$\bullet$ Reversibility of rays. Ray 1 of light in medium $A$ (see Figure 23.54 ) strikes the surface at $51.0^{\circ}$ with respect to the normal. (a) Show that the angle of refraction of ray 1 with respect to the normal in medium $B$ is $35.8^{\circ} .$ (b) Now suppose that ray 2 is the reverse of ray $1,$ so that it strikes the surface at $35.8^{\circ}$ with the normal in $B$ . Show that ray 2 will come out in $A$ at $51.0^{\circ}$ with the normal. In other words, show that rays 1 and 2 follow the same path, except reversed from each other.

Shoukat Ali

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Problem 42

$\bullet$$\bullet$ You (height of your eyes above the water, 1.75 $\mathrm{m}$ ) are standing 2.00 $\mathrm{m}$ from the edge of a 2.50 -m-deep swimming pool. You notice that you can barely see your cell phone, which went missing a few minutes before, on the bottom of the pool. How far from the side of the pool is your cell phone?

Ryan Hood

Numerade Educator

Problem 43

$\bullet$$\bullet$ A parallel-sided plate of glass having a refractive index of 1.60 is in contact
with the surface of water in a tank. A ray coming from above makes an angle of incidence of $32.0^{\circ}$ with the top surface of the glass. What angle does this ray make with the normal in the water?

Shoukat Ali

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Problem 44

$\bullet$$\bullet$ As shown in Figure 23.56 , a layer of water covers a slab of material $X$ in a beaker. A ray of light traveling upwards follows the path indicated. Using the information on the figure, find (a) the index of refraction of material $X$ and (b) the angle the light makes with the normal in the air.

Ryan Hood

Numerade Educator

Problem 45

$\bullet$ A ray of light in diamond (index of refraction 2.42$)$ is incident on an interface with air. What is the largest angle the ray can make with the normal and not be totally reflected back into
the diamond?

Vishal Gupta

Numerade Educator

Problem 46

$\bullet$ The critical angle for total internal reflection at a liquid-air interface is $42.5^{\circ} .$ (a) If a ray of light traveling in the liquid has an angle of incidence of $35.0^{\circ}$ at the interface, what angle does the refracted ray in the air make with the normal? (b) If a ray of light traveling in air has an angle of incidence of $35.0^{\circ}$ at the interface, what angle does the refracted ray in the liquid make with the normal?

Ryan Hood

Numerade Educator

Problem 47

$\bullet$ At the very end of Wagner's series of operas The Ring of the Nibelung, Brunnhilde takes the golden ring from the finger of the dead Siegfried and throws it into the Rhine, where it sinks to the bottom of the river. Assuming that the ring is small enough to be treated as a point compared with the depth of the river and that the Rhine is 10.0 $\mathrm{m}$ deep where the ring goes in, what is the area of the largest circle at the surface of the water over which light from the ring could escape from the water?

Shoukat Ali

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Problem 48

$\bullet$$\bullet$ A ray of light is traveling in a glass cube that is totally immersed in water. You find that if the ray is incident on the glass-water interface at an angle to the normal greater than $48.7^{\circ},$ no light is refracted into the water. What is the refractive index of the glass?

Ryan Hood

Numerade Educator

Problem 49

$\bullet$$\bullet$ Light is incident along the normal to face AB of a glass prism of refractive index 1.52 as shown in Figure $23.57 .$ Find the largest value the angle $\alpha$ can have without any light
refracted out of the prism at face $A C$ if (a) the prism is immersed in air and (b) the prism is immersed in water.

Shoukat Ali

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Problem 50

$\bullet$$\bullet$ Light pipe. Light enters a solid tube made of plastic having an index of refraction
of $1.60 .$ The light travels parallel to the upper part of the tube. (See Figure $23.58 . )$ You want to cut the face $A B$ so that all the light will reflect back into the tube after it first strikes that face.
(a) What is the largest that $\theta$ can be if the tube is in air? (b) If the tube is immersed in water of refractive index $1.33,$ what is the largest that $\theta$ can be?

Ryan Hood

Numerade Educator

Problem 51

$\bullet$$\bullet$ An optical fiber consists of an outer "cladding" layer and an inner core with a slightly higher index of refraction. Light rays entering the core are trapped inside by total internal reflection and forced to travel along the fiber (see Figure $23.59 ) .$ Suppose the cladding has an index of refraction of 1.46 and the core has an index of refraction of $1.48 .$ Calculate the largest angle $\theta$ between a light ray and the longitudinal axis of the fiber ( see the figure) for which the ray will be totally internally reflected at the core/cladding boundary.

Shoukat Ali

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Problem 52

$\bullet$ A beam of light strikes a sheet of glass at an angle of $57.0^{\circ}$ with the normal in air. You observe that red light makes an angle of $38.1^{\circ}$ with the normal in the glass, while violet light makes a $36.7^{\circ}$ angle. (a) What are the indexes of refraction of this glass for these colors of light? (b) What are the speeds of red and violet light in the glass?

Ryan Hood

Numerade Educator

Problem 53

$\bullet$ Use the information from the graph in Figure 23.29 to construct a graph of the index of refraction of silicate flint glass as a function of the frequency of light.

Shoukat Ali

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Problem 54

$\bullet$ A narrow beam of white light strikes one face of a slab of silicate flint glass. The light is traveling parallel to the two adjoining faces, as shown in Figure $23.60 .$ For the transmitted light inside the glass, through what angle $\Delta \theta$ is the complete visible spectrum of light dispersed? (Consult the graph in Figure $23.29 .$)

Ryan Hood

Numerade Educator

Problem 55

$\bullet$$\bullet$ Use the graph in Figure 23.29 for silicate flint glass. (a) What are the indexes of refraction of this glass for extreme violet light of wavelength 400 $\mathrm{nm}$ and for extreme red light of wavelength 700 $\mathrm{nm}$ ? (b) What are the wavelengths of 400 nm violet light and 700 $\mathrm{nm}$ red light in this glass? (c) Calculate the ratio of the speed of extreme red light to that of extreme violet light in the glass. Which of these travels faster in the glass? (d) If a beam of white light in air strikes a sheet of this glass at $65.0^{\circ}$ with the normal in air, what will be the angle of dispersion between the extremes of visible light in the glass? In other words, what will be the angle between extreme red and extreme violet light in the glass?

Shoukat Ali

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Problem 56

$\bullet$ The indices of refraction for violet light $(\lambda=400 \mathrm{nm})$ and red light $(\lambda=700 \mathrm{nm})$ in diamond are 2.46 and $2.41,$ respectively. A ray of light traveling through air strikes the diamond surface at an angle of $53.5^{\circ}$ to the normal. Calculate the angular separation between these two colors of light in the refracted ray.

Ryan Hood

Numerade Educator

Problem 57

$\bullet$ Unpolarized light with intensity $I_{0}$ is incident on an ideal polarizing filter. The emerging light strikes a second ideal polarizing filter whose axis is at $41.0^{\circ}$ to that of the first.
Determine (a) the intensity of the beam after it has passed through the second polarizer and (b) its state of polarization.

Shoukat Ali

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Problem 58

$\bullet$ Two ideal polarizing filters are oriented so that they transmit the maximum amount of light when unpolarized light is shone on them. To what fraction of its maximum value $I_{0}$ is the intensity of the transmitted light reduced when the second filter is rotated through (a) $22.5^{\circ},$ (b) $45.0^{\circ},$ and (c) $67.5^{\circ} ?$

Ryan Hood

Numerade Educator

Problem 59

$\bullet$ A beam of unpolarized light of intensity $I_{0}$ passes through a series of ideal polarizing filters with their polarizing directions turned to various angles as shown in Figure 23.61 (a) What is the light intensity (in terms of $I_{0}$ ) at points $A, B$ , and $C ?$ (b) If we remove the middle filter, what will be the light intensity at point $C$ ?

Shoukat Ali

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Problem 60

$\bullet$ Three ideal polarizing filters are stacked, with the polarizing axis of the second and third filters at $23.0^{\circ}$ and $62.0^{\circ}$ , respectively, to that of the first. If unpolarized light is incident on the stack, the light has intensity 75.0 $\mathrm{W} / \mathrm{cm}^{2}$ after it passes through the stack. If the incident intensity is kept constant, what is the intensity of the light after it has passed through the stack if the second polarizer is removed?

Ryan Hood

Numerade Educator

Problem 61

$\bullet$ Light of original intensity $I_{0}$ passes through two ideal polarizing filters having their polarizing axes oriented as shown in Figure $23.62 .$ You want to adjust the angle $\phi$ so that the intensity at point $P$ is equal to $I_{0} / 10 .$ (a) If the original light is unpolarized, what should $\phi$ be? (b) If the original light is linearly polarized in the same direction as the polarizing axis
of the first polarizer the light reaches, what should $\phi$ be?

Shoukat Ali

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Problem 62

$\bullet$ The polarizing angle for light in air incident on a glass plate is $57.6^{\circ} .$ What is the index of refraction of the glass?

Ryan Hood

Numerade Educator

Problem 63

$\bullet$$\bullet$ A beam of polarized light passes through a polarizing filter. When the angle between the polarizing axis of the filter and the direction of polarization of the light is $\theta$ , the intensity of the emerging beam is $I$ . If you instead want the intensity to be $I / 2,$ what should be the angle (in terms of $\theta )$ between the polarizing angle of the filter and the original direction of polarization of the light?

Shoukat Ali

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Problem 64

$\bullet$$\bullet$ A beam of unpolarized light in air is incident at an angle of $54.5^{\circ}$ (with respect to the normal) on a plane glass surface. The reflected beam is completely linearly polarized. (a) What is the refractive index of the glass? (b) What is the angle of refraction of the transmitted beam?

Ryan Hood

Numerade Educator

Problem 65

$\bullet$$\bullet$ Plane-polarized light passes through two polarizers whose axes are oriented at $35.0^{\circ}$ to each other. If the intensity of the original beam is reduced to $15.0 \%,$ what was the polarization direction of the original beam, relative to the first polarizer?

Shoukat Ali

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Problem 66

$\bullet$$\bullet$ The energy flow to the earth from sunlight is about 1.4 $\mathrm{kW} / \mathrm{m}^{2}$ . (a) Find the maximum values of the electric and magnetic fields for a sinusoidal wave of this intensity. (b) The distance from the earth to the sun is about $1.5 \times 10^{11} \mathrm{m} .$ Find the total power radiated by the sun.

Ryan Hood

Numerade Educator

Problem 67

$\bullet$$\bullet$ A plane sinusoidal electromagnetic wave in air has a wave-length of 3.84 $\mathrm{cm}$ and an $\vec{\boldsymbol{E}}$ field amplitude of 1.35 $\mathrm{V} / \mathrm{m}$ .
(a) What is the frequency of the wave? (b) What is the $\vec{\boldsymbol{B}}$ field amplitude? (c) What is the intensity? (d) What average force does this radiation exert perpendicular to its direction of propagation on a totally absorbing surface with area 0.240 $\mathrm{m}^{2}$ ?

Shoukat Ali

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Problem 68

$\bullet$$\bullet$ A powerful searchlight shines on a man. The man's cross- sectional area is 0.500 $\mathrm{m}^{2}$ perpendicular to the light beam, and the intensity of the light at his location is 36.0 $\mathrm{kW} / \mathrm{m}^{2}$ . He is wearing black clothing, so that the light incident on him is
totally absorbed. What is the magnitude of the force the light beam exerts on the man? Do you think he could sense this force?

Ryan Hood

Numerade Educator

Problem 69

$\bullet$$\bullet$ Laser surgery. Very short pulses of high-intensity laser beams are used to repair detached portions of the retina of the eye. The brief pulses of energy absorbed by the retina welds the detached portion back into place. In one such procedure, a laser beam has a wavelength of 810 $\mathrm{nm}$ and delivers 250 $\mathrm{mW}$ of power spread over a circular spot 510$\mu \mathrm{m}$ in diameter. The vitreous humor (the transparent fluid that fills most of the eye) has an index of refraction of 1.34 . (a) If the laser pulses are each 1.50 $\mathrm{ms}$ long, how much energy is delivered to the retina with each pulse? (b) What average pressure does the pulse of the laser beam exert on the retina as it is fully absorbed by the circular spot? (c) What are the wavelength and frequency of the laser light inside the vitreous humor of the eye? (d) What are the maximum values of the electric and magnetic fields in the laser beam?

Shoukat Ali

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Problem 70

$\bullet$$\bullet$ A small helium-neon laser emits red visible light with a power of 3.20 $\mathrm{mW}$ in a beam that has a diameter of 2.50 $\mathrm{mm}$ . (a) What are the amplitudes of the electric and magnetic fields of the light? (b) What are the average energy densities associated with the electric field and with the magnetic field? (c) What is the total energy contained in a 1.00 $\mathrm{m}$ length of the beam?

Ryan Hood

Numerade Educator

Problem 71

$\bullet$$\bullet$ Radio receivers can comfortably pick up a broadcasting station's signal when the electric field strength of the signal is about 10.0 $\mathrm{mV} / \mathrm{m} .$ If a radio station broadcasts in all directions with an average power of $50.0 \mathrm{kW},$ what would be the maximum distance at which you could easily pick up its transmissions? (Atmospheric conditions can have major effects on this distance.)

Shoukat Ali

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Problem 72

$\bullet$$\bullet$ The 19 th-century inventor Nikola Tesla proposed to transmit electric power via sinusoidal electromagnetic waves. Sup- pose power is to be transmitted in a beam of cross-sectional
area 100 $\mathrm{m}^{2} .$ What electric- and magnetic-field amplitudes are required to transmit an amount of power comparable to that handled by modern transmission lines (which carry voltages and currents of the order of 500 $\mathrm{kV}$ and 1000 $\mathrm{A} )$ ?

Ryan Hood

Numerade Educator

Problem 73

$\bullet$$\bullet$ Solar sail. NASA is doing research on the concept of solar sailing. A solar sailing craft uses a large, low-mass sail and the energy and momentum of sunlight for propulsion. (a) Should the sail be absorptive or reflective? Why? (b) The total power output of the sun is $3.9 \times 10^{26} \mathrm{W}$ . How large a sail is necessary to propel a $10,000$ kg spacecraft against the gravitational force of the sun? Express your result in square kilometers.(c) Explain why your answer to part (b) is independent of the distance from the sun.

Shoukat Ali

Other Schools

Problem 74

$\bullet$$\bullet$ A thick layer of oil is floating on the surface of water in a tank. A beam of light traveling in the oil is incident on the water interface at an angle of $30.0^{\circ}$ from the normal. The refracted beam travels in the water at an angle of $45.0^{\circ}$ from the normal. What is the refractive index of the oil?

Ryan Hood

Numerade Educator

Problem 75

$\bullet$$\bullet$ A thin beam of light in air is incident on the surface of a lanthanum flint glass plate having a refractive index of 1.80 . What is the angle of incidence, $\theta_{a}$ of the beam with this plate, for which the angle of refraction is $\theta_{a} / 2 ?$ Both angles are measured relative to the normal.

Shoukat Ali

Other Schools

Problem 76

$\bullet$$\bullet$ You want to support a sheet of fireproof paper horizontally, using only a vertical upward beam of light spread uniformly over the sheet. There is no other light on this paper. The sheet measures 22.0 $\mathrm{cm}$ by 28.0 $\mathrm{cm}$ and has a mass of 1.50 $\mathrm{g}$ . (a) If the paper is black and hence absorbs all the light that hits it, what must be the intensity of the light beam? (b) For the light in part (a), what are the maximum values of its electric and magnetic fields? (c) If the paper is white and hence reflects all the light that hits it, what intensity of light beam is needed to support it? (d) To see if it is physically reasonable to expect to support a sheet of paper this way, calculate the intensity in a typical 0.500 $\mathrm{mW}$ laser beam that is 1.00 $\mathrm{mm}$ in diameter and compare this value with your answer in part (a).

Ryan Hood

Numerade Educator

Problem 77

$\bullet$$\bullet$ A light ray in air strikes the right-angle prism shown in Figure 23.63 . This ray consists of two different wavelengths. When it emerges at face $A B,$ it has been split into two different rays that diverge from each other by $8.50^{\circ} .$ Find the index of refraction of the prism for each of the two wavelengths.

Shoukat Ali

Other Schools

Problem 78

$\bullet$$\bullet$ A ray of light is incident in air on a block of a transparent solid whose index of
refraction is $n .$ If $n=1.38,$ what is the largest angle of incidence, $\theta_{a},$ for which total internal reflection will occur at the vertical face (point $A$ shown in Figure 23.64 )?

Ryan Hood

Numerade Educator

Problem 79

$\bullet$$\bullet$ A light beam is directed parallel to the axis of a hollow cylindrical tube. When the
tube contains only air, it takes the light 8.72 ns to travel the length of the tube, but when the tube is filled with a transparent jelly, it takes the light 2.04 ns longer to travel its length. What is the refractive index of this jelly?

Shoukat Ali

Other Schools

Problem 80

$\bullet$$\bullet$ Heart sonogram. Physicians use high-frequency $(f=$ 1 MHz to 5 MHz) sound waves, called ultrasound, to image internal organs. The speed of these ultrasound waves is 1480 $\mathrm{m} / \mathrm{s}$ in muscle and 344 $\mathrm{m} / \mathrm{s}$ in air. We define the index of refraction of a material for sound waves to be the ratio of the speed of sound in air to the speed of sound in the material. Snell's law then applies to the refraction of sound waves. (a) At what angle from the normal does an ultrasound beam enter the heart if it leaves the lungs at an angle of $9.73^{\circ}$ from the normal to the heart wall? (Assume that the speed of sound in the lungs is 344 $\mathrm{m} / \mathrm{s} .$ ) (b) What is the critical angle for sound waves in air incident on muscle?

Ryan Hood

Numerade Educator

Problem 81

$\bullet$$\bullet$ The prism shown in Figure 23.65 has a refractive index of $1.66,$ and the angles $A$ are $25.0^{\circ} .$ Two light rays $m$ and $n$ are parallel as they enter the prism. What is the angle between them after they emerge?

Shoukat Ali

Other Schools

Problem 82

$\bullet$$\bullet \mathrm{A} 45^{\circ}-45^{\circ}-90^{\circ}$ prism is immersed in water. A ray of light is incident normally on one of the prism's shorter faces. What is the minimum index of refraction that the prism must have if this ray is to be totally reflected within the glass at the long face of the prism?

Ryan Hood

Numerade Educator

Problem 83

$\bullet$$\bullet$ A beaker with a mirrored bottom is filled with a liquid whose index of refraction is $1.63 .$ A light beam strikes the top surface of the liquid at an angle of $42.5^{\circ}$ from the normal. At what angle from the normal will the beam exit from the liquid after traveling down through it, reflecting from the mirrored bottom, and returning to the surface?

Shoukat Ali

Other Schools

Problem 84

$\bullet$$\bullet$ A ray of light traveling in a block of glass $(n=1.52)$ is incident on the top surface at an angle of $57.2^{\circ}$ with respect to the normal in the glass. If a layer of oil is placed on the top
surface of the glass, the ray is totally reflected. What is the maximum possible index of refraction of the oil?

Ryan Hood

Numerade Educator

Problem 85

$\bullet$$\bullet$ A block of glass has a polarizing angle of $60.0^{\circ}$ for red light and $70.0^{\circ}$ for blue light, for light traveling in air and reflecting from the glass. (a) What are the indexes of refraction for red light and for blue light? (b) For the same angle of incidence, which color is refracted more on entering the glass?

Shoukat Ali

Other Schools

Problem 86

$\bullet$$\bullet$ In a physics lab, light with wavelength 490 nm travels in air from a laser to a photocell in 17.0 ns. When a slab of glass 0.840 m thick is placed in the light beam, with the beam incident along the normal to the parallel faces of the slab, it takes the light 21.2 ns to travel from the laser to the photocell. What is the wavelength of the light in the glass?

Ryan Hood

Numerade Educator

Problem 87

$\bullet$$\bullet$ (a) Light passes through three parallel slabs of different thicknesses and refractive indexes. The light is incident in the first slab and finally refracts into the third slab. Show that the middle slab has no effect on the final direction of the light. That is, show that the direction of the light in the third slab is the same as if the light had passed directly from the first slab into the third slab. (b) Generalize this result to a stack of $N$ slabs. What determines the final direction of the light in the last slab?

Shoukat Ali

Other Schools

Problem 88

$\bullet$ The refractive index of a certain glass is $1.66 .$ For what angle of incidence is light that is reflected from the surface of this glass completely polarized if the glass is immersed in (a) air or (b) water?

Ryan Hood

Numerade Educator

Problem 89

$\bullet$$\bullet$ A thin layer of ice $(n=1.309)$ floats on the surface of water $(n=1.333)$ in a bucket. A ray of light from the bottom of the bucket travels upward through the water. (a) What is the largest angle with respect to the normal that the ray can make at the ice-water interface and still pass out into the air above the ice? (b) What is this angle after the ice melts?

Shoukat Ali

Other Schools

Problem 90

$\bullet$$\bullet$ Optical activity of biological molecules. Many biologically important molecules are optically active. When linearly polarized light traverses a solution of compounds containing these molecules, its plane of polarization is rotated. Some compounds rotate the polarization clockwise: others rotate the polarization counterclockwise. The amount of rotation depends on the amount of material in the path of the light. The following data give the amount of rotation through two amino acids over a path length of $100 \mathrm{cm} :$

From these data, find the relationship between the concentration $C($ in grams per 100 $\mathrm{mL})$ and the rotation of the polarization (in degrees) of each amino acid. (Hint: Graph the concentration as a function of the rotation angle for eachamino acid.)

Ryan Hood

Numerade Educator

Problem 91

A horizontal cylindrical tank 2.20 $\mathrm{m}$ in diameter is half full of water. The space above the water is filled with a pressurized gas of unknown refractive index. A small laser can move along the curved bottom of the water and aims a light beam toward the center of the water surface (Figure 23.66 ). You observe that when the laser has moved a distance $S=1.09 \mathrm{m}$ or more (measured along the curved surface) from the lowest point in the water, no light enters the gas. (a) What is the index of refraction of the gas? (b) How long does it take the light beam to travel from the laser to the rim of the tank when (i) $S>1.09$ m and (ii) $S<1.09 \mathrm{m} ?$

When a light ray is incident on a transparent surface we can easily predict the direction of the reflected and refracted rays by using the laws of reflection and refraction. However, the amount of light reflected from a surface is more difficult to determine since this depends on the direction and polarization of the incident ray, and the refractive indices for both surfaces. For example, when light strikes the boundary between two surfaces (with refractive indices of $n_{1}$ and $n_{2}$ ) at an angle of incidence that is near $90^{\circ}$ , the fractional intensity of light reflected from the boundary is given by
$$
\left(\frac{n_{1}-n_{2}}{n_{1}+n_{2}}\right)^{2}
$$
According to this result, typical optical glass (with an index of refraction of 1.5 should reflect about 4$\%(0.04)$ of the light normally incident from the surrounding air, which is in fact the case.

Shoukat Ali

Other Schools

Problem 92

If a light beam strikes a 10 $\mathrm{cm}$ thick slab of glass (which is immersed in air) at an angle of $30^{\circ}$ from the normal to the surface, what will be its angle to the normal when it leaves the
back side of the slab? Assume that the slab has parallel sides and an index of refraction of $1.5 .$
$$
\begin{array}{lllllll}{\text { A. } 0^{\circ}} & {\text { B. } 30^{\circ}} & {\text { C. } 60^{\circ}} & {\text { D. } 58.3^{\circ}}\end{array}
$$

Ryan Hood

Numerade Educator

Problem 93

If a layer of oil, with an index of refraction of $1.8,$ is placed on the top of the glass, what will be the new angle at which the light beam leaves the glass?
$$
\begin{array}{llll}{\text { A. } 0^{\circ}} & {\text { B. } 16^{\circ}} & {\text { C. } 30^{\circ}} & {\text { D. } 46^{\circ}}\end{array}
$$

Shoukat Ali

Other Schools

Problem 94

If the entire slab (without the oil) is submerged in a fluid with an index of refraction of $1.5,$ what will be the effect?
A. The slab will appear to change color.
B. Light striking the slab could be totally reflected.
C. The slab will be very difficult to see.
D. Light exiting the slab could be totally reflected.

Ryan Hood

Numerade Educator