Measuring the Speed of Light: Michelson In 1926, Albert

Problem 115IP Suppose the incident beam of light is linearly polarized in the vertical direction. In addition, the transmission axis of the analyzer is at an angle of 80.0° to the vertical. What angle should the transmission axis of the polarizer make with the vertical if the transmitted intensity is to be a maximum?

Problem 1P If the electric field in an electromagnetic wave is increasing in magnitude at a particular time, is the magnitude of the magnetic field at the same time increasing or decreasing? Explain.

Problem 1CQ Explain why the “invisible man” would be unable to see.

Problem 2CQ The magnitude of the Doppler effect tells how rapidly a weather system is moving. What determines whether the system is approaching or receding?

Problem 3CQ Explain why radiation pressure is more significant on a grain of dust in interplanetary space when the grain is very small.

Problem 2P The electric field of an electromagnetic wave points in the positive y direction. At the same time, the magnetic field of this wave points in the positive z direction. In what direction is the wave traveling?

Problem 3P An electric charge on the x axis oscillates sinusoidally about the origin. A distant observer is located at a point on the +y axis. (a) In what direction will the electric field oscillate at the observer’s location? (b) In what direction will the magnetic field oscillate at the observer’s location? (c) In what direction will the electromagnetic wave propagate at the observer’s location?

Problem 4CQ While wearing your Polaroid sunglasses at the beach, you notice that they reduce the glare from the water better when you are sitting upright than when you are lying on your side. Explain.

Problem 5CQ You want to check the time while wearing your Polaroid sunglasses. If you hold your forearm horizontally, you can read the time easily. If you hold your forearm vertically, however, so that you are looking at your watch sideways, you notice that the display is black. Explain.

Problem 5P Give the direction (N, S, E, W, up, or down) of the missing quantity for each of the four electromagnetic waves listed in Table 25-1. TABLE 25-1 Direction of propagation Direction of electric field Direction of magnetic field N W (a) N (b) W up S (c) (d) down S

Problem 6CQ Polarization and the Ground Spider The ground spider Drassodes cupreus, like many spiders, has several pairs of eyes. It has been discovered that one of these pairs of eyes acts as a set of polarization filters, with one eye’s polarization direction oriented at 90° to the other eye’s polarization direction. In addition, experiments show that the spider uses these eyes to aid in navigating to and from its burrow. Explain how such eyes might aid navigation.

Problem 4P An electric charge on the z axis oscillates sinusoidally about the origin. A distant observer is located at a point on the +y axis. (a) In what direction will the electric field oscillate at the observer’s location? (b) In what direction will the magnetic field oscillate at the observer’s location? (c) In what direction will the electromagnetic wave propagate at the observer’s location?

Problem 6P Give the direction (±x, ±y, ±z) of the missing quantity for each of the four electromagnetic waves listed in Table 25-2. TABLE 25-2 Direction of propagation Direction of electric field Direction of magnetic field +x +y (a) +x (b) +y ?y +z (c) (d) +z +y

Problem 7CQ The electromagnetic waves we pick up on our radios are typically polarized. In contrast, the indoor light we see every day is typically unpolarized. Explain.

At a particular instant of time, a light beam traveling in the positive z direction has an electric field given by \(\bar{E}=(6.22 \mathrm{~N} / \mathrm{C}) \hat{x}+(2.87 \mathrm{~N} / \mathrm{C}) \hat{y}\). The magnetic field in the beam has a magnitude of \(2.28 \mathrm{c} \times 10^{-8} \mathrm{~T}\) at the same time. (a) Does the magnetic field at this time have a z component that is positive,negative, or zero? Explain. (b) Write \(\bar{B}\) in terms of unit vectors. Equation Transcription: Text Transcription: \bar E=(6.22 N / C) \hat x+(2.87 N / C) \hat y 2.28 c x 10-8 T \bar B

Problem 8CQ You are given a sheet of Polaroid material. Describe how to determine the direction of its transmission axis if none is indicated on the sheet.

A light beam traveling in the negative z direction has a magnetic field \(\bar{B}=\left(3.02 \times 10^{-9} \mathrm{~T}\right) \hat{x}+\left(-5.28 \times 10^{-9} \mathrm{~T}\right) \hat{y}\) at a given instant of time. The electric field in the beam has a magnitude of 1.82 N/C at the same time. (a) Does the electric field at this time have a component that is positive, negative, or zero? Explain. (b) Write in terms of unit vectors. Equation Transcription: Text Transcription: \bar B=(3.02 \times 10^-9 T) \hat x+(-5.28 \times 10^-9 T) \hat y

Problem 9CQ Can sound waves be polarized? Explain.

CE Three electromagnetic waves have electric and magnetic fields pointing in the directions shown in Figure 25–23. For each of the three cases, state whether the wave propagates in the \(+x,-x,+y,-y,+z \text { or }-z \text { direction }\). Equation Transcription: Text Transcription: +x, -x, +y, -y, +z or -z direction

Problem 10CQ At a garage sale you find a pair of “Polaroid” sunglasses priced to sell. You are not sure, however, if the glasses are truly Polaroid, or if they simply have tinted fences. How can you tell which is the case? Explain.

Problem 10P The light-year (ly) is a unit of distance commonly used in astronomy. It is defined as the distance traveled by light in a vacuum in one year. (a) Express 1 ly in km. (b) Express the speed of light, c, in units of ly per year. (c) Express the speed of light in feet per nanosecond.

Problem 11CQ 3-D Movies Modern-day 3-D movies are produced by projecting two different images onto the screen, with polarization directions that are at 90° relative to one another. Viewers must wear headsets with polarizing filters to experience the 3-D effect. Explain how this works.

Problem 11P Alpha Centauri, the closest star to the sun, is 4.3 ly away. How far is this in meters?

Problem 12P Mars Rover When the Mars rover was deployed on the surface of Mars in July 1997, radio signals took about 12 min to travel from Earth to the rover. How far was Mars from Earth at that time?

Problem 13P A distant star is traveling directly away from Earth with a speed of 37,500 km/s. By what factor are the wavelengths in this star’s spectrum changed?

Problem 14P A distant star is traveling directly toward Earth with a speed of 37,500 km/s. (a) When the wavelengths in this star’s spectrum are measured on Earth, are they greater or less than the wavelengths we would find if the star were at rest relative to us? Explain. (b) By what fraction are the wavelengths in this star’s spectrum shifted?

Problem 15P The frequency of light reaching Earth from a particular galaxy is 12% lower than the frequency the light had when it was emitted. (a) Is this galaxy moving toward or away from Earth? Explain. (b) What is the speed of this galaxy relative to the Earth? Give your answer as a fraction of the speed of light.

Measuring the Speed of Light Galileo attempted to measure the speed of light by measuring the time elapsed between his opening a lantern and his seeing the light return from his assistant’s lantern. The experiment is illustrated in Figure 25–24.What distance, d, must separate Galileo and his assistant inorder for the human reaction time, \(\Delta t=0.2 \mathrm{~s}\), to introduce no more than a \(15 \%\) error in the speed of light? Equation Transcription: Text Transcription: \Delta t=0.2 s 15 \%

Measuring the Speed of Light: Michelson In 1926, Albert Michelson measured the speed of light with a technique similar to that used by Fizeau. Michelson used an eight-sided mirror rotating at \(528 \mathrm{rev} / \mathrm{s}\) in place of the toothed wheel, as illustrated in Figure 25–25. The distance from the rotating mirror to a distant reflector was \(35.5 \mathrm{~km}\). If the light completed the \(71.0-\mathrm{km}\) round trip in the time it took the mirror to complete one-eighth of a revolution, what is the speed of light? Equation Transcription: Text Transcription: 528 rev/s 71.0-km 35.5 km

Problem 18P Communicating with the Voyager Spacecraft When the Voyager I and Voyager II spacecraft were exploring the outer planets, NASA flight controllers had to plan the crafts’ moves well in advance. How many seconds elapse between the time a command is sent from Earth and the time the command is received by Voyager at Neptune? Assume the distance from. Earth to Neptune is 4.5 × 1012 m.

Problem 19P A father and his daughter are interested in the same baseball game. The father sits next to his radio at home and listens to the game; his daughter attends the game and sits in the outfield bleachers. In the bottom of the ninth inning a home run is hit. If the father’s radio is 132 km from the radio station, and the daughter is 115 m from home plate, who hears the home run first? (Assume that there is no time delay between the baseball being hit and its sound being broadcast by the radio station. In addition, let the speed of sound in the stadium be 343 m/s.)

Problem 20P (a) How fast would a motorist have to be traveling for a yellow (? = 590 nm) traffic light to appear green (? = 550 nm) because of the Doppler shift? (b) Should the motorist be traveling toward or away from the traffic light to see this effect? Explain.

Problem 21P Most of the galaxies in the universe are observed to be moving away from Earth. Suppose a particular galaxy emits orange light with a frequency of 5.000 × 1014 Hz. If the galaxy is receding from Earth with a speed of 3325 km/s, what is the frequency of the light when it reaches Earth?

Problem 22P Two starships, the Enterprise and the Constitution, are approaching each other head-on from a great distance. The separation between them is decreasing at a rate of 782.5 km/s. The Enterprise sends a laser signal toward the Constitution. If the Constitution observes a wavelength ? = 670.3 nm, what wavelength was emitted by the Enterprise?

Problem 23P Baseball scouts often use a radar gun to measure the speed of a pitch. One particular model of radar gun emits a microwave signal at a frequency of 10.525 GHz. What will be the increase in frequency if these waves are reflected from a 90.0-mi/h fastball headed straight toward the gun? (Note: 1 mi/h = 0.447 m/s)

Problem 24P A state highway patrol car radar unit uses a frequency of 8.00 × 109 Hz. What frequency difference will the unit detect from a car receding at a speed of 44.5 m/s from a stationary patrol car?

Consider a spiral galaxy that is moving directly away from Earth with a speed \(V=3.600 \times 10^{5} \mathrm{~m} / \mathrm{s}\) at its center, as shown in Figure 25–26. The galaxy is also rotating about its center, so that points in its spiral arms are moving with a speed \(v-6.400 \times 10^{5} \mathrm{~m} / \mathrm{s}\) relative to the center. If light with a frequency of \(8.230 \times 10^{14} \mathrm{~Hz}\) is emitted in both arms of the galaxy, what frequency is detected by astronomers observing the arm that is moving (a) toward and (b) away from Earth? (Measurements of this type are used to map out the speed of various regions in distant, rotating galaxies.) Equation Transcription: Text Transcription: V=3.600 x 10^5 m/s v-6.400 x 10^5 m/s 8.230 x 10^14 Hz

Problem 26P A highway patrolman sends a 24.150-GHz radar beam toward a speeding car. The reflected wave is lower in frequency by 4.04 kHz. (a) Is the car moving toward or away from the radar gun? Explain. (b) What is the speed of the car? [Hint:For small values of x, the following approximation may be used: (1 + x)2? 1 + 2x.]

Problem 27P Dental X-rays X-rays produced in the dentist’s office typically have a wavelength of 0.30 nm. What is the frequency of these rays?

Problem 28P Find the frequency of blue light with a wavelength of 460 nm.

Problem 29P Yellow light has a wavelength ? = 590 nm. How many of these waves would span the 1.0-mm thickness of a dime?

Problem 30P How many red wavelengths (? = 705 nm) tall are you?

Problem 31P A cell phone transmite at a frequency of 1.75 × 108 Hz. What is the wavelength of the electromagnetic wave used by this phone?

Problem 32P Human Radiation Under normal conditions, humans radiate electromagnetic waves with a wavelength of about 9.0 microns. (a) What is the frequency of these waves? (b) To what portion of the electromagnetic spectrum do these waves belong?

Problem 33P UV Radiation. Ultraviolet light is typically divided into three categories. UV-A, with wavelengths between 400 nm and 320 nm, has been linked with malignant melanomas. UV-B radiation, which is the primary cause of sunburn and other skin cancers, has wavelengths between 320 nm. and 280 nm. Finally, the region known as UV-C extends to wavelengths of 100 nm. (a) Find the range of frequencies for UV-B radiation. (b) In which of these three categories does radiation with a frequency of 7.9 × 1014 Hz belong?

Problem 34P Communicating with a Submarine Normal radiofrequency waves cannot penetrate more than a few meters below the surface of the ocean. One method of communicating with submerged submarines uses very low frequency (VLF) radio waves. What is the wavelength (in air) of a 10.0-kHz VLF radio wave?

When an electromagnetic wave travels from one medium to another with a different speed of propagation, the frequency of the wave remains the same. Its wavelength, however,changes. (a) If the wave speed decreases, does the wavelength increase or decrease? Explain. (b) Consider a case where the wave speed decreases from c to \(\frac{3}{4} c\). By what factor does the wavelength change? Equation Transcription: Text Transcription: \frac{3}{4} c

Problem 36P (a) Which color of light has the higher frequency, red or violet? (b) Calculate the frequency of blue light with a wavelength of 470 nm, and red light with a wavelength of 680 nm.

Problem 37P ULF (ultra low frequency) electromagnetic waves, produced in the depths of outer space, have been observed with wavelengths in excess of 29 million kilometers. What is the period of such a wave?

Problem 39P An AM radio station’s antenna is constructed to be ?/4 tall, where ? is the wavelength of the radio waves. How tall should the antenna be for a station broadcasting at a frequency of 810 kHz?

Problem 38P A television is tuned to a station broadcasting at a frequency of 6.60 × 107 Hz. For best reception, the rabbit-ear antenna used by the TV should be adjusted to have a tip-to-tip length equal to half a wavelength of the broadcast signal. Find the optimum length of the antenna.

Problem 40P As you drive by an AM radio station, you notice a sign saying that its antenna is 112 m high. If this height represents one quarter-wavelength of its signal, what is the frequency of the station?

Problem 42P Find the difference in frequency (f1 ? f2) for each of the following pairs of radio waves: (a) ?1 = 300.0 m and ?2 = 300.5 m, (b) ?1 = 30.0 m and ?2 = 30.5 m.

Problem 41P Find the difference in wavelength (?1 ? ?2) for each of the following pairs of radio waves: (a) f1 = 50 kHz and f2 = 52 kHz, (b) f1 = 500 kHz and f2 = 502 kHz.

Problem 43P If the rms value of the electric field in an electromagnetic wave is doubled, (a) by what factor does the rms value of the magnetic field change? (b) By what factor does the average intensity of the wave change?

Problem 45P The maximum magnitude of the electric field in an electromagnetic wave is 0.0400 V/m. What is the maximum magnitude of the magnetic field in this wave?

Problem 46P What is the rms value of the electric field in a sinusoidal electromagnetic wave that has a maximum electric field of 88 V/m?

Problem 44P The radiation pressure exerted by beam of light 1 is half the radiation pressure of beam of light 2. If the rms electric field of beam 1 has the value E0, what is the rms electric field in beam 2?

Problem 47P The magnetic field in an electromagnetic wave has a peak value given by B = 3.7 ? T. For this wave, find (a) the peak electric field strength, (b) the peak intensity, and (c) the average intensity.

Problem 48P What is the maximum value of the electric field in an electromagnetic wave whose maximum intensity is 5.00 W/m2?

Problem 49P What is the maximum value of the electric field in an eletro magnetic wave whose average intensity is 5.00 W/m2?

Problem 51P A 65-kW radio station broadcasts its signal uniformly in all directions. (a) What is the average intensity of its signal at a distance of 250 m from the antenna? (b) What is the average intensity of its signal at a distance of 2500 m from the antenna?

Problem 50P Electromagnetic wave 1 has a maximum electric field of E0 = 52 V/m, and electromagnetic wave 2 has a maximum magnetic field of B0 = 1.5 ? T.(a) Which wave has the greater intensity? (b) Calculate the intensity of each wave.

Problem 52P At what distance will a 45-W lightbulb have the same apparent brightness as a 120-W bulb viewed from a distance of 25 m? (Assume that both bulbs convert electrical power to light with the same efficiency, and radiate light uniformly in all directions.)

Problem 53P What is the ratio of the sunlight intensity reaching Pluto compared with the sunlight intensity reaching Earth? (On average, Pluto is 39 times as far from the Sun as is Earth.)

Problem 54P In the following, assume that light bulbs radiate uniformly in all directions and that 5.0% of their power is converted to light. (a) Find the average intensity of light at a point 2.0 m from a 120-W red lightbulb (? = 710 nm). (b) Is the average intensity 2.0 m from a 120-W blue light bulb (? = 480 nm) greater than, less than, or the same as the intensity found in part (a)? Explain. (c) Calculate the average intensity for part (b).

Problem 55P A 5.0-mW laser produces a narrow beam of light. How much energy is contained in a 1.0-m length of its beam?

Problem 56P What length of a 5.0-mW laser’s beam will contain 9.5 mJ of energy?

Problem 57P Sunlight Intensity After filtering through the atmosphere, the Sun’s radiation illuminates Earth’s surface with an average intensity of 1.0 kW/m2. Assuming this radiation strikes the 15-m × 45-m black, flat roof of a building at normal incidence, calculate the average force the radiation exerts on the roof.

Problem 58P (a) Find the electric and magnetic field amplitudes in an electromagnetic wave that has an average energy density of 1.0 J/m3. (b) By what factor must the field amplitudes be increased if the average energy density is to be doubled to 2.0 J/m3?

Problem 59P Lasers for Fusion Two of the most powerful lasers in the world are used in nuclear fusion experiments. The NOVA laser produces 40.0 kJ of energy in a pulse that lasts 2.50 ns, and the N1F laser (under construction) will produce a 10.0-ns pulse with 3.00 MJ of energy. (a) Which laser produces more energy in each pulse? (b) Which laser produces the greater average power during each pulse? (c) If the beam diameters are the same, which laser produces the greater average intensity?

Problem 60P You are sanding 2.5 m from a 150-W light bulb. (a) If the pupil of your eye is a circle 5.0 mm in diameter, how much energy enters your eye per second? (Assume that 5.0% of the light bulb’s power is converted to light.) (b) Repeat part (a) for the case of a 1.0-mm-diameter laser beam with a power of 0.50 mW.

Laser Safety A \(0.75-m W\) laser emits a narrow beam of light that enters the eye, as shown in Figure 25–27. (a) How much energy is absorbed by the eye in 0.2 s? (b) The eye focuses this beam to a tiny spot on the retina, perhaps \(5.0 \mu m\) in diameter. What is the average intensity of light \(\left(i n W / c m^{2}\right)\) at this spot? (c) Damage to the retina can occur if the average intensity of light exceeds \(1.0 \times 10^{-2} W / c m^{2}\). By what factor has the intensity of this laser beam exceeded the safe value? Equation Transcription: Text Transcription: 0.75-mW 5.0 \mu m (in W/ cm^2) 1.0 x 10^-2 W/cm^2

Problem 62P Find the rms electric and magnetic fields at a point 2.50 m from a lightbulb that radiates 75.0 W of light uniformly in all directions.

Problem 63P A 0.50-mW laser produces a beam of light with a diameter of 1.5 mm. (a) What is the average intensity of this beam? (b) At what distance does a 150-W light bulb have the same average intensity as that found for the laser beam in part (a)? (Assume that 5.0% of the bulb’s power is converted to light.)

Problem 64P A laser emits a cylindrical beam of light 2.4 mm in diameter. If the average power of the laser is 2.8 mW, what is the rms value of the electric field in the laser beam?

Problem 65P (a) If the laser in Problem 64 shines its light on a perfectly absorbing surface, how much energy does the surface receive in 12 s? (b) What is the radiation pressure exerted by the beam?

Problem 66P Laser Surgery Each pulse produced by an argon-fluoride excimer laser used in PRK and LASIK ophthalmic surgery lasts only 10.0 ns but delivers an energy of 2.50 mJ. (a) What is the power produced during each pulse? (b) If the beam has a diameter of 0.850 mm, what is the average intensity of the beam during each pulse? (c) If the laser emits 55 pulses per second, what is the average power it generates?

Problem 67P A pulsed laser produces brief bursts of light. One such laser emits pulses that carry 0.350 J of energy but last only 225 fs. (a) What is the average power during one of these pulses? (b) Assuming the energy is emitted in a cylindrical beam of light 2.00 mm in diameter, calculate the average intensity of this laser beam. (c) What is the rms electric field in this wave?

CE Predict/Explain Consider the two polarization experiments shown in Figure 25–28. (a) If the incident light is unpolarized, is the transmitted intensity in case A greater than, less than, or the same as the transmitted intensity in case B? (b) Choose the best explanation from among the following: 1. The transmitted intensity is the same in either case; the first polarizer lets through one-half the incident intensity, and the second polarizer is at an angle \(\theta\) relative to the first. 2. Case A has a smaller transmitted intensity than case B because the first polarizer is at an angle \(\theta\) relative to the incident beam. 3. Case B has a smaller transmitted intensity than case A because the direction of polarization is rotated by an angle \(\theta\) in the clockwise direction in case B. Equation Transcription: Text Transcription: \theta \theta \theta

Problem 69P Consider the two polarization experiments shown in Figure 25–28. (a) If the incident light is polarized in the horizontal direction, is the transmitted intensity in case A greater than, less than, or the same as the transmitted intensity in case B? (b) Choose the best explanation from among the following: I. The two cases have the same transmitted intensity because the angle between the polarizers is ? in each case. II. The transmitted intensity is greater in case B because all of the initial beam gets through the first polarizer. III. The transmitted intensity in case B is smaller than in case A; in fact, the transmitted intensity in case B is zero because the first polarizer is oriented vertically.

Problem 70P Suppose linearly polarized light is incident on the polarization experiments shown in Figure 25–28. In what direction, relative to the vertical, must the incident light be polarized if the transmitted intensity is to be the same in both experiments? Explain.

CE An incident beam of light with an intensity passes through a polarizing filter whose transmission axis is at an angle \(\theta\) to the vertical. As the angle is changed from to \(\theta=0 \text { to } \theta=90^{\circ}\), the intensity as a function of angle is given by one of the curves in Figure 25–29. Give the color of the curve corre- sponding to an incident beam that is (a) unpolarized, (b) verti-cally polarized, and (c) horizontally polarized. Equation Transcription: Text Transcription: \theta \theta=0 to \theta=90^\circ

Problem 72P Vertically polarized light with an intensity of 0.55 W/m2 passes through a polarizer whose transmission axis is at an angle of 65.0° with the vertical. What is the intensity of the transmitted light?

Problem 73P A person riding in a boat observes that the sunlight reflected by the water is polarized parallel to the surface of the water. The person is wearing polarized sunglasses with the polarization axis vertical. If the wearer leans at an angle of 21.5° to the vertical, what fraction of the reflected light intensity will pass through the sunglasses?

Problem 74P Unpolarized light passes through two polarizers whose transmission axes are at an angle of 30.0° with respect to each other. What fraction of the incident intensity is transmitted through the polarizers?

Problem 75P In Problem, what should be the angle between the transmission axes of the polarizers if it is desired that one-tenth of the incident intensity be transmitted?

Problem 76P Unpolarized light with intensity I0 falls on a polarizing filter whose transmission axis is vertical. The axis of a second polarizing filter makes an angle of ? with the vertical. Plot a graph that shows the intensity of light transmitted by the second filter (expressed as a fraction of I0) as a function of ?. Your graph should cover the range ? = 0° to ? = 360°.

A beam of vertically polarized light encounters two polarizing filters, as shown in Figure 25–30. (a) Rank the three cases, A, B, and C, in order of increasing transmitted intensity. Indicate ties where appropriate. (b) Calculate the transmitted intensity for each of the cases in Figure 25–30, assuming that the incident intensity is \(37.0 \mathrm{~W} / \mathrm{m}^{2}\). Verify that your numerical results agree with the rankings in part (a). Equation Transcription: Text Transcription: 37.0 W/m^2

Problem 78P This time assuming that the polarizers to the left in Figure 25–30 are at an angle of 22.5° to the vertical rather than 45°. The incident intensity is again 37.0 W/m2.

Problem 79P BIO Optical Activity Optically active molecules have the property of rotating the direction of polarization of linearly polarized light. Many biologically important molecules have this property, some causing a counterclockwise rotation (negative rotation angle), others causing a clockwise rotation (positive rotation angle). For example, a 5.00 gram per 100 mL solution of l-leucine causes a rotation of ?0.550°; the same concentration of d-glutamic acid causes a rotation of 0.620”. (a) If placed between crossed polarizers, which of these solutions transmits the greater intensity? Explain. (b) Find the transmitted intensity for each of these solutions when placed between crossed polarizers. The incident beam is unpolarized and has an intensity of 12.5 W/m2.

A helium–neon laser emits a beam of unpolarized light that passes through three Polaroid filters, as shown in Figure 25–31. The intensity of the laser beam is \(I_{0}\). (a) What is the intensity of the beam at point A? (b) What is the intensity of the beam at point B? (c) What is the intensity of the beam at point C? (d) If filter 2 is removed, what is the intensity of the beam at point C? Equation Transcription: Text Transcription I_0 0^\circ \theta_2=30^\circ \theta_3=90^\circ

Referring to Figure 25–31, suppose that filter 3 is at a general angle \(\theta\) with the vertical, rather than the angle \(90^{\circ}\). (a) Find an expression for the transmitted intensity as a function of \(\theta\) . (b) Plot your result from part (a), and determine the maximum transmitted intensity. (c) At what angle \(\theta\) does maximum transmission occur? Equation Transcription: Text Transcription: \theta 90° \theta \theta

Problem 82GP Suppose the magnitude of the electric field in an electro-magnetic wave is doubled. (a) By what factor does the magnitude of the magnetic field change? (b) By what factor does the maximum intensity of the wave change?

Problem 83GP If “sailors” of the future use radiation pressure to propel their ships, should the surfaces of their sails be absorbing or re fleeting? Explain.

Problem 84GP Sunlight at the surface of Earth has an average intensity of about 1.00 × 103 W/m2. Find the rms values of the electric and magnetic fields in the sunlight.

Problem 85GP A typical medical X-ray has a frequency of 1.50 × 1019 Hz. What is the wavelength of such an X-ray?

Problem 86GP How many hydrogen atoms, 0.10 nm in diameter, must be placed end to end to fit into one wavelength of 410-nm violet light?

Problem 87GP Radiofrequency Ablation In radiofrequency (RF) ablation, a small needle is inserted into a cancerous tumor. When radio frequency oscillating currents are sent into the needle, ions in the neighboring tissue respond by vibrating rapidly, causing local heating to temperatures as high as 100°C. This kills the cancerous cells and, because of the small size of the needle, relatively few of the surrounding healthy cells. A typical RF ablation treatment uses a frequency of 750 kHz. What is the wavelength that such radio waves would have in a vacuum?

CE Figure 25–32 shows four polarization experiments in which unpolarized incident light passes through two polarizing filters with different orientations. Rank the four cases in order of increasing amount of transmitted light. Indicate ties where appropriate.

Problem 89GP (a) What minimum intensity must a laser beam have if it is to levitate a tiny black (100% absorbing) sphere of radius r = 0.5 ? m and mass = 1.6 × 10?15 kg? Comment on the feasibility of such levitation. (b) If the radius of the sphere is doubled but its mass remains the same, will the minimum intensity be greater than, less than, or equal to the value found in part (a)? Explain. (c) Find the minimum intensity for the situation described in part (b).

Problem 90GP The Apollo 11 Reflector One of the experiments placed on the Moon’s surface by Apollo 11 astronauts was a reflector that is used to measure the Earth–Moon distance with high accuracy. A laser beam on Earth is bounced off the reflector, and its round-trip travel time is recorded. If the travel time can be measured to within an accuracy of 0.030 ns, what is the uncertainty in the Earth–Moon distance?

Problem 91GP The H? line of the hydrogen atom’s spectrum has a normal wavelength ?? = 486 nm. This same line is observed in the spectrum of a distant quasar, but lengthened by 20.0 nm. What is the speed of the quasar relative to Earth, assuming it is moving along our line of sight?

Problem 92GP Suppose the distance to the fixed mirror in Figure 25–25 is decreased to 20.5 km. (a) Should the angular speed of the rotating mirror be increased or decreased to ensure that the experiment works as described in Problem 17? (b) Find the required angular speed, assuming the speed of light is 3.00 × 108 m/s.

Problem 93GP Suppose the speed of the galaxy in Problem is increased by a factor of 10; that is, V = 3.600 × 106 m/s. The speed of the arms, v, and the frequency of the light remain the same. (a) Does the arm near the top of Figure 25–26 show a red shift (toward lower frequency) or a blue shift (toward higher frequency)? Does the lower arm show a red or a blue shift? Explain. What frequency is detected by astronomers observing (b) the upper arm and (c) the lower arm?

Problem 94GP Consider the physical situation illustrated in Figure 25–27. (a) Is Erms in the incident laser beam greater than, less than, or the same as Erms where the beam hits the retina? Explain. (b) If the intensity of the beam at the retina is equal to the damage threshold, 1.0 × 10?2 W/cm2, what is the value of Erms at that location? (c) If the diameter of the spot on the retina is reduced by a factor of 2, by what factor does the intensity increase? By what factor does Erms increase?

Problem 96GP A state highway patrol car radar unit uses a frequency of 9.00 × 109 Hz. What frequency difference will the unit detect from a car approaching a parked patrol car with a speed of 35.0 m/s?

Problem 95GP Polaroid Vision in a Spider Experiments show that the ground spider Drassodes cupreus uses one of its several pairs of eyes as a polarization detector. In fact, the two eyes in this pair have polarization directions that are at right angles to one another. Suppose linearly polarized light with an intensity of 825 W/m2 shines from the sky onto the spider, and that the intensity transmitted by one of the polarizing eyes is 232 W/m2. (a) For this eye, what is the angle between the polarization direction of the eye and the polarization direction of the incident light? (b) What is the intensity transmitted by the other polarizing eye?

Problem 97GP What is the ratio of the sunlight intensity reaching Mercury compared with the sunlight intensity reaching Earth? (On average, Mercury’s distance from the Sun is 0.39 that of Earth’s.)

Problem 98GP What area is needed for a solar collector to absorb 45.0 kW of power from the Sun’s radiation if the collector is 75.0% efficient? (At the surface of Earth, sunlight has an average intensity of 1.00 × 103 W/m2.)

Near-Infrared Brain Scans Light in the near-infrared (close to visible red) can penetrate surprisingly far through human tissue, a fact that is being used to “illuminate” the interior of the brain in a noninvasive technique known as near-infrared spectroscopy (NIRS). In this procedure, illustrated in Figure 25–33, an optical fiber carrying a beam of infrared laser light with a power of 1.5 mW and a cross-sectional diameter of 1.2 mm is placed against the skull. Some of the light enters the brain, where it scatters from hemoglobin in the blood. The scattered light is picked up by a detector and analyzed by a computer. (a) According to the Beer–Lambert law, the intensity of light, I, decreases with penetration distance, d, as \(I=1_{0 e}^{-\mu d} \text { where } I_{0}\) is the initial intensity of the beam \(\mu=4.7 \mathrm{~cm}\) and for a typical case. Find the intensity of the laser beam after it penetrates through 3.0 cm of tissue. (b) Find the electric field of the initial light beam. Equation Transcription: Text Transcription: I=1_0 e^-\mu d where I_0 \mu=4.7 cm

Problem 100GP Three polarizers are arranged as shown in Figure 25–31. If the incident beam of light is unpolarized and has an intensity of 1.60 W/m2, find the transmitted intensify (a) when ?2 = 25.0° and ?3 = 50.0°, and (b) when ?2 = 50.0° and ?3 = 25.0°.

Problem 102GP A light bulb emits light uniformly in all directions. If the rms electric field of this light is 16.0 N/C at a distance of 1.35 m from the bulb, what is the average total power radiated by the bulb?

Problem 101GP This time assuming an incident beam that is vertically polarized. The intensity of the incident beam is 1.60 W/m2.

Problem 103GP A beam of light is a mixture of unpolarized light with intensity Iu and linearly polarized light with intensity Ip. The polarization direction for the polarized light is vertical. When this mixed beam of light is passed through a polarizer that is vertical, the transmitted intensity is 1.6.8 W/m2; when the polarizer is at an angle of 55.0° with the vertical, the transmitted intensity is 8.68 W/m2. (a) Is Iu greater than, less than, or equal to Ip? Explain. (b) Calculate Iu and Ip.

Problem 104GP As mentioned in Problem 95, one pair of eyes in a particular species of ground spider has polarization directions that are at right angles to one another. Suppose that linearly polarized light is incident on such a spider. (a) Prove that the transmitted intensity of one eye plus the transmitted intensity from the other eye is equal to the incident intensity. (b) If the transmitted intensities for the two eyes are 163 W/m2 and 662 W/m2, through what angle must the spider rotate to make the transmitted intensities equal to one another?

Problem 105GP A typical home may require a total of 2.00 × 103 kWh of energy per month. Suppose you would like to obtain this energy from sunlight, which has an average daylight intensity of 1.00 × 103 W/m2. Assuming that sunlight is available 8.0 h per day, 25 d per month (accounting for cloudy days), and that you have a way to store energy from your collector when the Sun isn’t shining, determine the smallest collector size that will provide the needed energy, given a conversion efficiency of 25%.

Problem 106GP At the top of Earth’s atmosphere, sunlight has an average intensify of 1360 W/m2. If the average distance from Earth to the Sun is 1.50 × 1011 m, at what rate does the Sun radiate energy?

Problem 107GP A typical laser used in introductory physics laboratories produces a continuous beam of light about 1.0 mm in diameter. The average power of such a laser is 0.75 mW. What are (a) the average intensity, (b) the peak intensity, and (c) the average energy density of this beam? (d) If the beam is reflected from a mirror, what is the maximum force the laser beam can exert on it? (e) Describe the orientation of the laser beam relative to the mirror for the case of maximum force.

Optical Activity of Sugar The sugar concentration in a solution (e.g., in a urine specimen) can be measured conveniently by using the optical activity of sugar and other asymmetric molecules. In general, an optically active molecule, like sugar, will rotate the plane of polarization through an angle that is proportional to the thickness of the sample and to the concentration of the molecule. To measure the concentration of a given solution, a sample of known thickness is placed between two polarizing filters that are at right angles to each other, as shown in Figure 25–34. The intensity of light transmitted through the two filters can be compared with a calibration chart to determine the concentration. (a) What percentage of the incident (unpolarized) light will pass through the first filter? (b) If no sample is present, what percentage of the initial light will pass through the second filter? (c) When a particular sample is placed between the two filters, the intensity of light emerging from the second filter is \(40.0 \%\) of the incident intensity. Through what angle did the sample rotate the plane of polarization? (d) A second sample has half the sugar concentration of the first sample. Find the intensity of light emerging from the second filter in this case. Equation Transcription: Text Transcription: 40.0 \%

Problem 108GP Four polarizers are set up so that the transmission axis of each successive polarizer is rotated clockwise by an angle ? relative to the previous polarizer. Find the angle ? for which unpolarized light is transmitted through these four polarizers with its intensity reduced by a factor of 25.

What is the color of the light that is most effective at activating the photoinitiator CPQ? A. red B. yellow C. green D. blue

What is the frequency of the light that is most effective at activating a CPQ molecule? A. \(140 \mathrm{~Hz}\) B. \(1.00 \times 10^{14} \mathrm{~Hz}\) C. \(6.45 \times 10^{14} \mathrm{~Hz}\) D. \(1.55 \times 10^{15} \mathrm{~Hz}\) Equation Transcription: Text Transcription: 140 Hz 1.00 x 10^14 Hz 6.45 x 10^14 Hz 1.55 x 10^15 Hz

Suppose a VLC unit uses an LED that produces light with an average intensity of \(975 \mathrm{~mW} / \mathrm{cm}^{2}\). What is the rms value of the electric field in this beam of light? A. \(606 \mathrm{~N} / \mathrm{C}\) B. \(1920 \mathrm{~N} / \mathrm{C}\) C. \(3.67 \times 10^{6} \mathrm{~N} / \mathrm{C}\) D. \(3.22 \times 10^{7} \mathrm{~N} / \mathrm{C}\) Equation Transcription: Text Transcription: 975 mW/ cm^2 606 N/C 1920 N/C 3.67 x 10^6 N/C 3.22 x 10^7 N/C

How much radiation pressure does the beam of light in Problem 112 exert on a tooth, assuming the tooth absorbs all the light? A. \(3.24 \times 10^{-5} \mathrm{~N} / \mathrm{m}^{2}\) B. \(5.70 \times 10^{-3} \mathrm{~N} / \mathrm{m}^{2}\) C. \(3.67 \times 10^{6} \mathrm{~N} / \mathrm{m}^{2}\) D. \(1.10 \times 10^{15} \mathrm{~N} / \mathrm{m}^{2}\) Equation Transcription: Text Transcription: 3.24 x 10-5 N/m^2 5.70 x 10-3 N/m^2 3.67 x 106 N/m^2 1.10 x 1015 N/m^2

Problem 114IP Suppose the incident beam of light is linearly polarized in the same direction ? as the transmission axis of the analyzer. The transmission axis of the polarizer remains vertical. (a) What value must ? have if the transmitted intensity is to be 0.200 I0? (b) If ? is increased from the value found in part (a), docs the transmitted intensity increase, decrease, or stay the same? Explain.