Answer: The critical angle for total internal reflection

Two plane mirrors intersect at right angles. A laser beam strikes the first of them at a point 11.5 cm from their point of intersection, as shown in Fig. E33.1 For what angle of incidence at the first mirror will this ray strike the midpoint of the second mirror (which is 28.0 cm long) after reflecting from the first mirror?

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 380 nm (violet) to 750 nm (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?

A beam of light has a wavelength of 650 nm in vacuum. (a) What is the speed of this light in a liquid whose index of refraction at this wavelength is 1.47? (b) What is the wavelength of these waves in the liquid?

Light with a frequency of 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?

A light beam travels at in quartz. The wavelength of the light in quartz is 355 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?

Light of a certain frequency has a wavelength of 438 nm in water. What is the wavelength of this light in benzene?

A parallel beam of light in air makes an angle of with the surface 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 of the glass?

A laser beam shines along the surface of a block of transparent material (see Fig. E33.8.). 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?

Light traveling in air is incident on the surface of a block of plastic at an angle of to the normal and is bent so that it makes a angle with the normal in the plastic. Find the speed of light in the plastic.

(a) A tank containing methanol has walls 2.50 cm thick made of glass of refractive index 1.550. Light from the outside air strikes the glass at a \(41.3^{\circ}\) angle with the normal to the glass. Find the angle the light makes with the normal in the methanol. (b) The tank is emptied and refilled with an unknown liquid. If light incident at the same angle as in part (a) enters the liquid in the tank at an angle of \(20.2^{\circ}\) from the normal, what is the refractive index of the unknown liquid?

As shown in Fig. E33.11, a layer of water covers a slab of material X in a beaker. A ray of light traveling upward 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

A horizontal, parallel-sided plate of glass having a refractive index of 1.52 is in contact with the surface of water in a tank. A ray coming from above in air makes an angle of incidence of \(35.0^{\circ}\) with the normal to the top surface of the glass. (a) What angle does the ray refracted into the water make with the normal to the surface? (b) What is the dependence of this angle on the refractive index of the glass? Text Transcription: 35.0^circ

In a material having an index of refraction a light ray has frequency wavelength and speed What are the frequency, wavelength, and speed of this light (a) in vacuum and (b) in a material having refractive index In each case, express your answers in terms of only

A ray of light traveling in water is incident on an interface with a flat piece of glass. The wavelength of the light in the water is 726 nm and its wavelength in the glass is 544 nm. If the ray in water makes an angle of \(42.0^{\circ}\) with respect to the normal to the interface, what angle does the refracted ray in the glass make with respect to the normal?

A ray of light is incident on a plane surface separating two sheets of glass with refractive indexes 1.70 and 1.58. The angle of incidence is and the ray originates in the glass with Compute the angle of refraction.

A flat piece of glass covers the top of a vertical cylinder that is completely filled with water. If a ray of light traveling in the glass is incident on the interface with the water at an angle of , the ray refracted into the water makes an angle of with the normal to the interface. What is the smallest value of the incident angle for which none of the ray refracts into the water?

Light Pipe. Light enters a solid pipe made of plastic having an index of refraction of 1.60. The light travels parallel to the upper part of the pipe (Fig. E33.17). You want to cut the face AB so that all the light will reflect back into the pipe after it first strikes that face. (a) What is the largest that \(\theta\) can be if the pipe is in air? (b) If the pipe is immersed in water of refractive index 1.33, what is the largest that \(\theta\) can be?

A beam of light is traveling inside a solid glass cube having index of refraction 1.53. It strikes the surface of the cube from the inside. (a) If the cube is in air, at what minimum angle with the normal inside the glass will this light not enter the air at this surface? (b) What would be the minimum angle in part (a) if the cube were immersed in water?

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 at the interface of \(35.0^{\circ}\), 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 at the interface of \(35.0^{\circ}\) what angle does the refracted ray in the liquid make with the normal?

At the very end of Wagners series of operas Ring of the Nibelung, Brnnhilde 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 compared to the depth of the river to be treated as a point and that the Rhine is 10.0 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?

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 larger than \(48.7^{\circ}\) no light is refracted into the water. What is the refractive index of the glass?

Light is incident along the normal on face of a glass prism of refractive index 1.52, as shown in Fig. E33.22. Find the largest value the angle can have without any light refracted out of the prism at face if (a) the prism is immersed in air and (b) the prism is immersed in water

A piece of glass with a flat surface is at the bottom of a tank of water. If a ray of light traveling in the glass is incident on the interface with the water at an angle with respect to the normal that is greater than , no light is refracted into the water. For smaller angles of incidence, part of the ray is refracted into the water. If the light has wavelength 408 nm in the glass, what is the wavelength of the light in the water?

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. Snells law then applies to the refraction of sound waves. The speed of a sound wave is 344 m s in air and 1320 m s in water. (a) Which medium has the higher index of refraction for sound? (b) What is the critical angle for a sound wave incident on the surface between air and water? (c) For total internal reflection to occur, must the sound wave be traveling in the air or in the water? (d) Use your results to explain why it is possible to hear people on the opposite shore of a river or small lake extremely clearly.

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 Fig. E33.25. For the transmitted light inside the glass, through what angle \(\Delta \theta\) is the portion of the visible spectrum between 400 nm and 700 nm dispersed? (Consult the graph in Fig. 33.18.)

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?

Unpolarized light with intensity is incident on two polarizing filters. The axis of the first filter makes an angle of with the vertical, and the axis of the second filter is horizontal. What is the intensity of the light after it has passed through the second filter?

(a) At what angle above the horizontal is the sun if sunlight reflected from the surface of a calm lake is completely polarized? (b) What is the plane of the electric-field vector in the reflected light?

A beam of unpolarized light of intensity passes through a series of ideal polarizing filters with their polarizing directions turned to various angles as shown in Fig. E33.29. (a) What is the light intensity (in terms of ) at points and (b) If we remove the middle filter, what will be the light intensity at point C?

Light traveling in water strikes a glass plate at an angle of incidence of \(53.0^{\circ}\) part of the beam is reflected and part is refracted. If the reflected and refracted portions make an angle of \(90.0^{\circ}\) with each other, what is the index of refraction of the glass?

A parallel 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?

Light of original intensity \(I_{0}\) passes through two ideal polarizing filters having their polarizing axes oriented as shown in Fig. E33.32. 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? Text Transcription: I_0 phi I_0/10

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 the intensity of the emerging beam is If you now want the intensity to be what should be the angle (in terms of ) between the polarizing angle of the filter and the original direction of polarization of the light?

The refractive index of a certain glass is 1.66. For what incident angle is light reflected from the surface of this glass completely polarized if the glass is immersed in (a) air and (b) water?

Unpolarized light of intensity \(20.0 \mathrm{\ W}/\mathrm{cm}^2\) is incident on two polarizing filters. The axis of the first filter is at an angle of \(25.0^{\circ}\) counterclockwise from the vertical (viewed in the direction the light is traveling), and the axis of the second filter is at \(62.0^{\circ}\) counterclockwise from the vertical. What is the intensity of the light after it has passed through the second polarizer?

Three 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?

Three Polarizing Filters. Three polarizing filters are stacked with the polarizing axes of the second and third at and respectively, with that of the first. (a) If unpolarized light of intensity is incident on the stack, find the intensity and tate of polarization of light emerging from each filter. (b) If the second filter is removed, what is the intensity of the light emerging from each remaining filter?

A beam of white light passes through a uniform thickness of air. If the intensity of the scattered light in the middle of the green part of the visible spectrum is I, find the intensity (in terms of I) of scattered light in the middle of (a) the red part of the spectrum and (b) the violet part of the spectrum. Consult Table 32.1.

The Corner Reflector. An inside corner of a cube is lined with mirrors to make a corner reflector (see Example 33.3 in Section 33.2). A ray of light is reflected successively from each of three mutually perpendicular mirrors; show that its final direction is always exactly opposite to its initial direction.

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?

BIO Heart Sonogram. Physicians use high-frequency (f=1-5MHz) sound waves, called ultrasound, to image internal organs. The speed of these ultrasound waves is 1480 m s in muscle and 344 m 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. Snells 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 m s.) (b) What is the critical angle for sound waves in air incident on muscle?

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?

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 Fig. P33.43)?

A light ray in air strikes the rightangle prism shown in Fig. P33.44. The prism angle at B is . This ray consists of two different wavelengths. When it emerges at face it has been split into two different rays that diverge from each other by Find the index of refraction of the prism for each of the two wavelengths.

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?

A glass plate 2.50 mm thick, with an index of refraction of 1.40, is placed between a point source of light with wavelength 540 nm (in vacuum) and a screen. The distance from source to screen is 1.80 cm. How many wavelengths are there between the source and the screen?

Old photographic plates were made of glass with a lightsensitive emulsion on the front surface. This emulsion was somewhat transparent. When a bright point source is focused on the front of the plate, the developed photograph will show a halo around the image of the spot. If the glass plate is 3.10 mm thick and the halos have an inner radius of 5.34 mm, what is the index of refraction of the glass? (Hint: Light from the spot on the front surface is scattered in all directions by the emulsion. Some of it is then totally reflected at the back surface of the plate and returns to the front surface.)

After a long day of driving you take a late-night swim in a motel swimming pool. When you go to your room, you realize that you have lost your room key in the pool. You borrow a powerful flashlight and walk around the pool, shining the light into it. The light shines on the key, which is lying on the bottom of the pool, when the flashlight is held 1.2 m above the water surface and is directed at the surface a horizontal distance of 1.5 m from the edge (Fig. P33.48). If the water here is 4.0 m deep, how far is the key from the edge of the pool?

You sight along the rim of a glass with vertical sides so that the top rim is lined up with the opposite edge of the bottom (Fig. P33.49a). The glass is a thin-walled, hollow cylinder 16.0 cm high. The diameter of the top and bottom of the glass is 8.0 cm. While you keep your eye in the same position, a friend fills the glass with a transparent liquid, and you then see a dime that is lying at the center of the bottom of the glass (Fig. P33.49b). What is the index of refraction of the liquid?

A prism is immersed in water. A ray of light is incident normally on one of its 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?

A thin layer of ice floats on the surface of water 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 icewater interface and still pass out into the air above the ice? (b) What is this angle after the ice melts?

A thin layer of ice floats on the surface of water 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 icewater interface and still pass out into the air above the ice? (b) What is this angle after the ice melts?

The prism shown in Fig. P33.53 has a refractive index of 1.66, and the angles are Two light rays and are parallel as they enter the prism. What is the angle between them after they emerge?

A horizontal cylindrical tank 2.20 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 (Fig. P33.54). You observe that when the laser has moved a distance S = 1.09 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) What minimum time 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 m?

When the sun is either rising or setting and appears to be just on the horizon, it is in fact below the horizon. The explanation for this seeming paradox is that light from the sun bends slightly when entering the earth’s atmosphere, as shown in Fig. P33.55. Since our perception is based on the idea that light travels in straight lines, we perceive the light to be coming from an apparent position that is an angle \(\delta\) above the sun’s true position. (a) Make the simplifying assumptions that the atmosphere has uniform density, and hence uniform index of refraction n, and extends to a height h above the earth’s surface, at which point it abruptly stops. Show that the angle \(\delta\) is given by \(\delta=\arcsin \left(\frac{n R}{R+h}\right)-\arcsin \left(\frac{R}{R+h}\right)\) where R = 6378 km is the radius of the earth. (b) Calculate \(\delta\) using n = 1.0003 and h = 20 km. How does this compare to the angular radius of the sun, which is about one quarter of a degree?

CALC Fermats Principle of Least Time. A ray of light traveling with speed leaves point 1 shown in Fig. P33.56 and is reflected to point 2. The ray strikes the reflecting surface a horizontal distance from point 1. (a) Show that the time required for the light to travel from 1 to 2 is (b) Take the derivative of with respect to Set the derivative equal to zero to show that this time reaches its minimum value when which is the law of reflection and corresponds to the actual path taken by the light. This is an example of Fermats principle of least time, which states that among all possible paths between two points, the one actually taken by a ray of light is that for which the time of travel is a minimum. (In fact, there are some cases in which the time is a maximum rather than a minimum.)

CALC A ray of light goes from point in a medium in which the speed of light is to point in a medium in which the speed is (Fig. P33.57). The ray strikes the interface a horizontal distance to the right of point (a) Show that the time required for the light to go from to is A B x A. (b) Take the derivative of with respect to Set this derivative equal to zero to show that this time reaches its minimum value when This is Snells law and corresponds to the actual path taken by the light. This is another example of Fermats principle of least time (see Problem 33.56).

Light is incident in air at an angle (Fig. P33.58) on the upper surface of a transparent plate, the surfaces of the plate being plane and parallel to each other. (a) Prove that (b) Show that this is true for any number of different parallel plates. (c) Prove that the lateral displacement of the emergent beam is given by the relationship where is the thickness of the plate. (d) A ray of light is incident at an angle of on one surface of a glass plate 2.40 cm thick with an index of refraction of 1.80. The medium on either side of the plate is air. Find the lateral displacement between the incident and emergent rays.

Angle of Deviation. The incident angle \(\theta_{a}\) shown in Fig. P33.59 is chosen so that the light passes symmetrically through the prism, which has refractive index n and apex angle A. (a) Show that the angle of deviation (the angle between the initial and final directions of the ray) is given by \(\sin \frac{A+\delta}{2}=n \ \sin \frac{A}{2}\) (When the light passes through symmetrically, as shown, the angle of deviation is a minimum.) (b) Use the result of part (a) to find the angle of deviation for a ray of light passing symmetrically through a prism having three equal angles \(\left(A=60.0^{\circ}\right)\) and n = 1.52. (c) A certain glass has a refractive index of 1.61 for red light (700 nm) and 1.66 for violet light (400 nm). If both colors pass through symmetrically, as described in part (a), and if \(A=60.0^{\circ}\) find the difference between the angles of deviation for the two colors.

A thin beam of white light is directed at a flat sheet of silicate flint glass at an angle of to the surface of the sheet. Due to dispersion in the glass, the beam is spread out in a spectrum as shown in Fig. P33.60. The refractive index of silicate flint glass versus wavelength is graphed in Fig. 33.18. (a) The rays and shown in Fig. P33.60 correspond to the extreme wavelengths shown in Fig. 33.18. Which corresponds to red and which to violet? Explain your reasoning. (b) For what thickness of the glass sheet will the spectrum be 1.0 mm wide, as shown (see Problem 33.58)?

A beam of light traveling horizontally is made of an unpolarized component with intensity \(\boldsymbol{I}_{0}\) and a polarized component with intensity \(\boldsymbol{I}_{p}\). The plane of polarization of the polarized component is oriented at an angle of \(\theta\) with respect to the vertical. The data in the table give the intensity measured through a polarizer with an orientation of \(\phi\) with respect to the vertical. (a) What is the orientation of the polarized component? (That is, what is the angle \(\theta\)?) (b) What are the values of \(\boldsymbol{I}_{0}\) and \(\boldsymbol{I}_{p}\)? Text Transcription: I_0 I_p theta phi

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 cm: From these data, find the relationship between the concentration C (in grams per 100 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 each amino acid.)

A beam of unpolarized sunlight strikes the vertical plastic wall of a water tank at an unknown angle. Some of the light reflects from the wall and enters the water (Fig. P33.63). The refractive index of the plastic wall is 1.61. If the light that has been reflected from the wall into the water is observed to be completely polarized, what angle does this beam make with the normal inside the water?

A certain birefringent material has indexes of refraction \(n_1\) and \(n_2\) for the two perpendicular components of linearly polarized light passing through it. The corresponding wavelengths are \(\lambda_{1}=\lambda_{0} / n_{1}\) and \(\lambda_{0} / n_{2}\) where \(\lambda_{0}\) is the wavelength in vacuum. (a) If the crystal is to function as a quarter-wave plate, the number of wavelengths of each component within the material must differ by \(\frac{1}{4}\). Show that the minimum thickness for a quarter-wave plate is \(d=\frac{\lambda_{0}}{4\left(n_{1}-n_{2}\right)}\) (b) Find the minimum thickness of a quarter-wave plate made of siderite \(\left(\mathrm{FeO} \cdot \mathrm{CO}_{2}\right)\) if the indexes of refraction are \(n_{1}=1.875\) and \(n_{2}=1.635\) and the wavelength in vacuum \(\lambda_0=589\mathrm{\ nm}\).

Consider two vibrations of equal amplitude and frequency but differing in phase, one along the and the other along the These can be written as follows: (1) (2) (a) Multiply Eq. (1) by and Eq. (2) by and then subtract the resulting equations. (b) Multiply Eq. (1) by and Eq. (2) by and then subtract the resulting equations. (c) Square and add the results of parts (a) and (b). (d) Derive the equation where (e) Use the above result to justify each of the diagrams in Fig. P33.65. In the figure, the angle given is the phase difference between two simple harmonic motions of the same frequency and amplitude, one horizontal (along the ) and the other vertical (along the ). The figure thus shows the resultant motion from the superposition of the two perpendicular harmonic motions.

CALC A rainbow is produced by the reflection of sunlight by spherical drops of water in the air. Figure P33.66 shows a ray that refracts into a drop at point is reflected from the back surface of the drop at point and refracts back into the air at point The angles of incidence and refraction, and are shown at points and and the angles of incidence and reflection, and are shown at point (a) Show that and (b) Show that the angle in radians between the ray before it enters the drop at and after it exits at (the total angular deflection of the ray) is (Hint: Find the angular deflections that occur at and and add them to get ) (c) Use Snells law to write in terms of and the refractive index of the water in the drop. (d) A rainbow will form when the angular deflection is stationary in the incident angle that is, when If this condition is satisfied, all the rays with incident angles close to will be sent back in the same direction, producing a bright zone in the sky. Let be the value of for which this occurs. Show that (Hint: You may find the derivative formula helpful.) (e) The index of refraction in water is 1.342 for violet light and 1.330 for red light. Use the results of parts (c) and (d) to find and for violet and red light. Do your results agree with the angles shown in Fig. 33.20d? When you view the rainbow, which color, red or violet, is higher above the horizon?

CALC A secondary rainbow is formed when the incident light undergoes two internal reflections in a spherical drop of water as shown in Fig. 33.20e. (See Challenge Problem 33.66.) (a) In terms of the incident angle and the refractive index of the drop, what is the angular deflection of the ray? That is, what is the angle between the ray before it enters the drop and after it exits? (b) What is the incident angle for which the derivative of with respect to the incident angle is zero? (c) The indexes of refraction for red and violet light in water are given in part (e) of Challenge Problem 33.66. Use the results of parts (a) and (b) to find and for violet and red light. Do your results agree with the angles shown in Fig. 33.20e? When you view a secondary rainbow, is red or violet higher above the horizon? Explain.

A beam of unpolarized sunlight strikes the vertical plastic wall of a water tank at an unknown angle. Some of the reflects from the wall and enters the water (Fig. P33.63). The refractive index of the plastic wall is 1.61. If the light that has been reflected from the wall into the water is observed to be completely polarized, what angle does this beam make with the normal inside the water?

A certain birefringent material has indexes of refraction \(n_{1}\) and \(n_{2}\) for the two perpendicular components of linearly polarized light passing through it. The corresponding wavelengths are \(\lambda_{1}=\lambda_{0} / n_{1}\) and \(\lambda_{0} / n_{2}\), where \(\lambda_{0}\) is the wavelength in vacuum. (a) If the crystal is to function as a quarter-wave plate, the number of wavelengths of each component within the material must differ by \(\frac{1}{4}\). Show that the minimum thickness for a quarter-wave plate is \(d=\frac{\lambda_{0}}{4\left(n_{1}-n_{2}\right)}\) (b) Find the minimum thickness of a quarter-wave plate made of siderite \(\left(\mathrm{FeO} \cdot \mathrm{CO}_{2}\right)\) if the indexes of refraction are \(n_{1}=1.875\) and \(n_{2}=1.635\) and the wavelength in vacuum is \(\lambda_{0}=589 \mathrm{~nm}\). Equation Transcription: Text Transcription: n_1 n_2 lambda_1=lambda_0/n_1 lambda_0/n_2 lambda_0 1/4 d=lambda_0 over 4(n_1-n_2) (FeO cdot CO2) n_1=1.875 n_2=1.635 lambda0=589 nm

Problem 1DQ A spherical mirror is cut in half horizontally. Will an image be formed by the bottom half of the mirror? If so, where will the image be formed?

Problem 1E A candle 4.85 cm tall is 39.2 cm to the left of a plane mirror. Where is the image formed by the mirror, and what is the height of this image?

Sunlight or starlight passing through the earth’s atmosphere is always bent toward the vertical. Why? Does this mean that a star is not really where it appears to be? Explain.

Problem 2E The image of a tree just covers the length of a plane mirror 4.00 cm tall when the mirror is held 35.0 cm from the eye. The tree is 28.0 m from the mirror. What is its height?

Problem 3DQ A beam of light goes from one material into another. On physical grounds, explain why the wavelength changes but the frequency and period do not.

Problem 3E A beam of light has a wavelength of 650 nm in vacuum. (a) What is the speed of this light in a liquid whose index of refraction at this wavelength is 1.47? (b) What is the wavelength of these waves in the liquid?

Problem 4DQ A student claimed that, because of atmospheric refraction (see Discussion Question Q33.2), the sun can be seen after it has set and that the day is therefore longer than it would be if the earth had no atmosphere. First, what does she mean by saying that the sun can be seen after it has set? Second, comment on the validity of her conclusion. Q33.2 Sunlight or starlight passing through the earth’s atmosphere is always bent toward the vertical. Why? Does this mean that a star is not really where it appears to be? Explain.

Problem 4E Light with a frequency of 5.80 × 1014 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?

When hot air rises from a radiator or heating duct, objects behind it appear to shimmer or waver. What causes this?

A light beam travels at 1.94 \(\times\) 108 m/s in quartz. The wavelength of the light in quartz is 355 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?

Problem 6DQ Devise straightforward experiments to measure the speed of light in a given glass using (a) Snell’s law; (b) total internal reflection; (c) Brewster’s law.

Problem 6E Light of a certain frequency has a wavelength of 438 nm in water. What is the wavelength of this light in benzene?

Sometimes when looking at a window, you see two reflected images slightly displaced from each other. What causes this?

Problem 7E A parallel beam of light in air makes an angle of 47.5o with the surface 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 of the glass?

Problem 8DQ If you look up from underneath toward the surface of the water in your aquarium, you may see an upside-down reflection of your pet fish in the surface of the water. Explain how this can happen.

A laser beam shines along the surface of a block of transparent material (see Fig. E33.8.). 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? \(n=?\) \(\leftarrow 2.50 m \rightarrow\) Equation transcription: Text transcription: n=? \leftarrow 2.50 m \rightarrow

A ray of light in air strikes a glass surface. Is there a range of angles for which total internal reflection occurs? Explain.

Problem 9E Light traveling in air is incident on the surface of a block of plastic at an angle of 62.7o to the normal and is bent so that it makes a 48.1o angle with the normal in the plastic. Find the speed of light in the plastic.

Problem 10DQ When light is incident on an interface between two materials, the angle of the refracted ray depends on the wavelength, but the angle of the reflected ray does not. Why should this be?

Problem 10E (a) A tank containing methanol has walls 2.50 cm thick made of glass of refractive index 1.550. Light from the outside air strikes the glass at a 41.3o angle with the normal to the glass. Find the angle the light makes with the normal in the methanol. (b) The tank is emptied and refilled with an unknown liquid. If light incident at the same angle as in part (a) enters the liquid in the tank at an angle of 20.2o from the normal, what is the refractive index of the unknown liquid?

Problem 11DQ A salesperson at a bargain counter claims that a certain pair of sunglasses has Polaroid filters; you suspect that the glasses are just tinted plastic. How could you find out for sure?

As shown in Fig. E33.11, a layer of water covers a slab of material X in a beaker. A ray of light traveling upward 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.

Problem 12DQ Does it make sense to talk about the polarization of a longitudinal wave, such as a sound wave? Why or why not?

A horizontal, parallel- sided plate of glass having a refractive index of 1.52 is in contact with the surface of water in a tank. A ray coming from above in air makes an angle of incidence of 35.0° with the normal to the top surface of the glass. (a) What angle does the ray refracted into the water make with the normal to the surface? (b) What is the dependence of this angle on the refractive index of the glass?

How can you determine the direction of the polarizing axis of a single polarizer?

Problem 13E In a material having an index of refraction n, a light ray has frequency f, wavelength ?, and speed ?. What are the frequency, wavelength, and speed of this light (a) in vacuum and (b) in a material having refractive index n’? In each case, express your answers in terms of only f, ?, v, n, and n’.

Problem 14DQ It has been proposed that automobile windshields and headlights should have polarizing filters to reduce the glare of oncoming lights during night driving. Would this work? How should the polarizing axes be arranged? What advantages would this scheme have? What disadvantages?

Problem 14E A ray of light traveling in water is incident on an interface with a flat piece of glass. The wavelength of the light in the water is 726 nm and its wavelength in the glass is 544 nm. If the ray in water makes an angle of 42.0° with respect to the normal to the interface, what angle does the refracted ray in the glass make with respect to the normal?

Problem 15DQ When a sheet of plastic food wrap is placed between two crossed polarizers, no light is transmitted. When the sheet is stretched in one direction, some light passes through the crossed polarizers. What is happening?

Problem 15E A ray of light is incident on a plane surface separating two sheets of glass with refractive indexes 1.70 and 1.58. The angle of incidence is 62.0o, and the ray originates in the glass with n = 1.70. Compute the angle of refraction.

Problem 16DQ If you sit on the beach and look at the ocean through Polaroid sunglasses, the glasses help to reduce the glare from sunlight reflecting off the water. But if you lie on your side on the beach, there is little reduction in the glare. Explain why there is a difference.

Problem 16E A flat piece of glass covers the top of a vertical cylinder that is completely filled with water. If a ray of light traveling in the glass is incident on the interface with the water at an angle of ?a = 36.2o, the ray refracted into the water makes an angle of 49.8o with the normal to the interface. What is the smallest value of the incident angle ?a for which none of the ray refracts into the water?

Problem 17DQ When unpolarized light is incident on two crossed polarizers, no light is transmitted. A student asserted that if a third polarizer is inserted between the other two, some transmission will occur. Does this make sense? How can adding a third filter increase transmission?

Light Pipe. Light enters a solid pipe made of plastic having an index of refraction of 1.60. The light travels parallel to the upper part of the pipe (Fig. E33.17). You want to cut the face so that all the light will reflect back into the pipe after it first strikes that face. (a) What is the largest thatcan be if the pipe is in air? (b) If the pipe is immersed in water of refractive index 1.33, what is the largest that can be?

Problem 18DQ For the old “rabbit-ear” style TV antennas, it’s possible to alter the quality of reception considerably simply by changing the orientation of the antenna. Why?

A beam of light is traveling inside a solid glass cube that has index of refraction 1.62. It strikes the surface of the cube from the inside. (a) If the cube is in air, at what minimum angle with the normal inside the glass will this light not enter the air at this surface? (b) What would be the minimum angle in part (a) if the cube were immersed in water?

In Fig. 33.32, since the light that is scattered out of the incident beam is polarized, why is the transmitted beam not also partially polarized?

Problem 19E The critical angle for total internal reflection at a liquid– air interface is 42.5o. (a) If a ray of light traveling in the liquid has an angle of incidence at the interface of 35.0o, 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 at the interface of 35.0o, what angle does the refracted ray in the liquid make with the normal?

Problem 20DQ You are sunbathing in the late afternoon when the sun is relatively low in the western sky. You are lying flat on your back, looking straight up through Polaroid sunglasses. To minimize the amount of sky light reaching your eyes, how should you lie: with your feet pointing north, east, south, west, or in some other direction? Explain your reasoning.

At the very end of Wagner’s series of operas Ring of the Nibelung, Brünnhilde 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 compared to the depth of the river to be treated as a point and that the Rhine is 10.0 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?

Problem 21DQ Light scattered from blue sky is strongly polarized because of the nature of the scattering process described in Section 33.6. But light scattered from white clouds is usually not polarized. Why not?

Problem 21E 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 larger than 48.7o, no light is refracted into the water. What is the refractive index of the glass?

Problem 22DQ Atmospheric haze is due to water droplets or smoke particles (“smog”). Such haze reduces visibility by scattering light, so that the light from distant objects becomes randomized and images become indistinct. Explain why visibility through haze can be improved by wearing red-tinted sunglasses, which filter out blue light.

Light is incident along the normal on face AB of a glass prism of refractive index 1.52, as shown in Fig. E33.22. Find the largest value the angle can have without any light refracted out of the prism at face AC if (a) the prism is immersed in air and (b) the prism is immersed in water.

Problem 23DQ The explanation given in Section 33.6 for the color of the setting sun should apply equally well to the rising sun, since sunlight travels the same distance through the atmosphere to reach your eyes at either sunrise or sunset. Typically, however, sunsets are redder than sunrises. Why? (Hint: Particles of all kinds in the atmosphere contribute to scattering.)

Problem 23E A piece of glass with a flat surface is at the bottom of a tank of water. If a ray of light traveling in the glass is incident on the interface with the water at an angle with respect to the normal that is greater than 62.0°, no light is refracted into the water. For smaller angles of incidence, part of the ray is refracted into the water. If the light has wavelength 408 nm in the glass. what is the wavelength of the light in the water?

Huygens’s principle also applies to sound waves. During the day, the temperature of the atmosphere decreases with increasing altitude above the ground. But at night, when the ground cools, there is a layer of air just above the surface in which the temperature increases with altitude. Use this to explain why sound waves from distant sources can be heard more clearly at night than in the daytime. (Hint: The speed of sound increases with increasing temperature. Use the ideas displayed in Fig. 33.37 for light.)

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. The speed of a sound wave is 344 m / s in air and 1320 m / s in water. (a) Which medium has the higher index of refraction for sound? (b) What is the critical angle for a sound wave incident on the surface between air and water? (c) For total internal reflection to occur, must the sound wave be traveling in the air or in the water? (d) Use your results to explain why it is possible to hear people on the opposite shore of a river or small lake extremely clearly.

Can water waves be reflected and refracted? Give examples. Does Huygens’s principle apply to water waves? Explain.

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 Fig. E33.25. For the transmitted light inside the glass, through what angle is the portion of the visible spectrum between 400 nm and 700 nm dispersed? (Consult the graph in Fig. 33.18.) \(\mid \Delta \Theta=?\) \(55.0^{0}\) Equation transcription: Text transcription: mid Delta Theta=? 55.0^{0}

Problem 26E A beam of light strikes a sheet of glass at an angle of 57.0o with the normal in air. You observe that red light makes an angle of 38.1o with the normal in the glass, while violet light makes a 36.7o 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?

Problem 27E Unpolarized light with intensity I0 is incident on two polarizing filters. The axis of the first filter makes an angle of 60.0o with the vertical, and the axis of the second filter is horizontal. What is the intensity of the light after it has passed through the second filter?

(a) At what angle above the horizontal is the sun if sunlight reflected from the surface of a calm lake is completely polarized? (b) What is the plane of the electric-field vector in the reflected light?

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 Fig. E33.29. (a) What is the light intensity (in terms of \(I_{0}\) ) at points , and ? (b) If we remove the middle filter, what will be the light intensity at point ? Equation transcription: Text transcription: I{0}

Problem 30E Light traveling in water strikes a glass plate at an angle of incidence of 53.0°; part of the beam is reflected and part is reflected. If the reflected and refracted portions make an angle of 90.0° with each other, what is the index of refraction of the glass?

Problem 31E A parallel beam of unpolarized light in air is incident at an angle of 54.5o (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?

Light of original intensity \(I_{0}\) passes through two ideal polarizing filters having their polarizing axes oriented as shown in Fig. E33.32. You want to adjust the angle so that the intensity at point is equal to \(I_{0} / 10\) (a) If the original light is unpolarized, what should be? (b) If the original light is linearly polarized in the same direction as the polarizing axis of the first polarizer the Equation transcription: Text transcription: I{0} I{0} / 10

Problem 33E 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 ?, the intensity of the emerging beam is I. If you now want the intensity to be I/2, what should be the angle (in terms of ?) between the polarizing angle of the filter and the original direction of polarization of the light?

The refractive index of a certain glass is 1.66. For what incident angle is light reflected from the surface of this glass completely polarized if the glass is immersed in (a) air and (b) water?

Problem 35E Unpolarized light of intensity 20.0 W/cm2 is incident on two polarizing filters. The axis of the first filter is at an angle of 25.0o counterclockwise from the vertical (viewed in the direction the light is traveling), and the axis of the second filter is at 62.0o counterclockwise from the vertical. What is the intensity of the light after it has passed through the second polarizer?

Problem 36E Three polarizing filters are stacked, with the polarizing axis of the second and third filters at 23.0° and 62.0°, respectively, to that of the first. If unpolarized light is incident on the stack, the light has intensity 75.0 W/cm2 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?

Problem 37E Three Polarizing Filters. Three polarizing filters are stacked with the polarizing axes of the second and third at 45.0o and 90.0o, respectively, with that of the first. (a) If unpolarized light of intensity I0 is incident on the stack, find the intensity and state of polarization of light emerging from each filter. (b) If the second filter is removed, what is the intensity of the light emerging from each remaining filter?

A beam of white light passes through a uniform thickness of air. If the intensity of the scattered light in the middle of the green part of the visible spectrum is I find the intensity (in terms of I) of scattered light in the middle of (a) the red part of the spectrum and (b) the violet part of the spectrum. Consult Table 32

The Corner Reflector. An inside corner of a cube is lined with mirrors to make a corner reflector (see Example 33.3 in Section 33.2). A ray of light is reflected successively from each of three mutually perpendicular mirrors; show that its final direction is always exactly opposite to its initial direction.

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?

Problem 41P BIO Heart Sonogram. Physicians use high-frequency (f = 195 MHz) sound waves, called ultrasound, to image internal organs. The speed of these ultrasound waves is 1480 m/s in muscle and 344 m/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 9.73o from the normal to the heart wall? (Assume that the speed of sound in the lungs is 344 m/s.) (b) What is the critical angle for sound waves in air incident on muscle?

Problem 42P 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?

A ray of light is incident in air on a block of a transparent solid whose index of refraction is If \(n=1.38\), what is the largest angle of incidence \(\Theta_{a}\) for which total internal reflection will occur at the vertical face (noint shown in Fig. P33.43)? Equation transcription: Text transcription: n=1.38 Theta{a}

A light ray in air strikes the right-angle prism shown in Fig. P33.44. The prism angle at B is \(30.0^{0}\). This ray consists of two different wavelengths. When it emerges at AB face 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. Equation transcription: Text transcription: 30.0^{0} 8.50^{\circ}

Problem 45P 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° 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?

Problem 46P A glass plate 2.50 mm thick, with an index of refraction of 1.40, is placed between a point source of light with wavelength 540 nm (in vacuum) and a screen. The distance from source to screen is 1.80 cm. How many wavelengths are there between the source and the screen?

Problem 47P Old photographic plates were made of glass with a light-sensitive emulsion on the front surface. This emulsion was somewhat transparent. When a bright point source is focused on the front of the plate, the developed photograph will show a halo around the image of the spot. If the glass plate is 3.10 mm thick and the halos have an inner radius of 5.34 mm. what is the index of refraction of the glass? (Hint: Light from the spot on the front surface is scattered in all directions by the emulsion. Some of it is then totally reflected at the hack surface of the plate and returns to the front surface.)

After a long day of driving you take a late-night swim in a motel swimming pool. When you go to your room, you realize that you have lost your room key in the pool. You borrow a powerful flashlight and walk around the pool, shining the light into it. The light shines on the key, which is lying on the bottom of the pool, when the flashlight is held 1.2 m above the water surface and is directed at the surface a horizontal distance of 1.5 m from the edge (Fig. P33.48). If the water here is 4.0 m deep, how far is the key from the edge of the pool?

You sight along the rim of a glass with vertical sides so that the top rim is lined up with the opposite edge of the bottom (Fig. P33.49a). The glass is a thin-walled, hollow cylinder 16.0 cm high. The diameter of the top and bottom of the glass is 8.0 cm. While you keep your eye in the same position, a friend fills the glass with a transparent liquid, and you then see a dime that is lying at the center of the bottom of the glass (Fig. P33.49b). What is the index of refraction of the liquid?

Problem 50P A 45o-45o-90o prism is immersed in water. A ray of light is incident normally on one of its 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?

Problem 51P 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?

Light is incident normally on the short face of a \(30^{0}-60^{0}-90^{0}\) prism (Fig. P33.52). A drop of liquid is placed on the hypotenuse of the prism. If the index of refraction of the prism is 1.62, find the maximum index that the liquid may have if the light is to be totally reflected. Equation transcription: Text transcription: 30^{0}-60^{0}-90^{0}

The prism shown in Fig. P33.53 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?

A horizontal cylindrical tank 2.20 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 (Fig. P33.54). You observe that when the laser has moved a distance 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) What minimum time does it take the light beam to travel from the laser to the rim of the tank when \(\text { (i) } S>1.09 m\) and \(\text { (ii) } S<1.09 \mathrm{~m}\) ? Equation transcription: Text transcription: { (i) } S>1.09 m { (ii) } S<1.09{~m}

When the sun is either rising or setting and appears to be just on the horizon, it is in fact below the horizon. The explanation for this seeming paradox is that light from the sun bends slightly when entering the earth’s atmosphere, as shown in Fig. P33.55. Since our perception is based on the idea that light travels in straight lines, we perceive the light to be coming from an apparent position that is an angle \(\delta\) above the sun’s true position. (a) Make the simplifying assumptions that the atmosphere has uniform density, and hence uniform index of refraction n and extends to a height h above the earth’s surface, at which point it abruptly stops. Show that the angle \(\delta\) is given by \(\delta=\arcsin \left(\frac{n R}{R+h}\right)-\arcsin \left(\frac{R}{R+h}\right)\) where \(R=6378\) km is the radius of the earth. (b) Calculate \(\delta\) using \(n=1.0003\) and \(h=20\) km. How does this compare to the angular radius of the sun, which is about one quarter of a degree? (In actuality a light ray from the sun bends gradually, not abruptly, since the density and refractive index of the atmosphere change gradually with altitude.)

Fermat's Principle of Least Time. A ray of light traveling with speed leaves point 1 shown in Fig. P33.56 and is reflected to point 2 . The ray strikes the reflecting surface horizontal distance from point 1 . (a) Show that the time required for the light to travel from 1 to 2 is \(t=\frac{\sqrt{y_{1}^{2}+x^{2}} \sqrt{y_{2}^{2}+(1-x)^{2}}}{C}\) (b) Take the derivative of with respect to . Set the derivative equal to zero to show that this time reaches its minimum value when \(\Theta_{1}=\Theta_{2}\), which is the law of reflection and corresponds to the actual path taken by the light. This is an example of Fermat's principle of least time, which states that among all possible paths between two points, the one actually taken by a ray of light is that for which the time of travel is a minimum. (In fact, there are some cases in which the time is a maximum rather than a minimum.) Equation transcription: Text transcription: t=\frac{sqrt{y{1}^{2}+x^{2}} sqrt{y{2}^{2}+(1-x)^{2}}}{C} Theta{1}=Theta{2}

A ray of light goes from point A in a medium which the speed of light is \(v_{1}\) to point B in a medium in which speed is \(v_{2}\) (Fig. P33.57). The ray strikes the interface a horizontal distance x to the right of point A. (a) Show that the time required for the light to go from A to B is $$t=\frac{\sqrt{h_{1}^{2}+x^{2}}}{v_{1}}+\frac{\sqrt{h_{2}^{2}+(l-x)^{2}}}{v_{2}}$$ (h) Take the derivative of t with respect to x. Set this derivative equal to zero to show that this time reaches its minimum value when \(n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}\). This is Snell's law and corresponds to the actual path taken by the light. This is another example of Fermat's principle of least time (see Problem 33.56). Equation Transcription: Text Transcription: v_1 v_2 t = h sub 1 squared + x^2/v_1 + h sub 2 squared + (l-x)^2/v_2 n_1 sin theta_1 = n_2 sin theta_2

Light is incident in air at an angle \(\theta_{a}\) (Fig. P33.58) on the upper surface of a transparent plate, the surfaces of the plate being plane and parallel to each other. (a) Prove that \(\theta_{a}=\theta_{a}^{\prime}\). (b) Show that this is true for any number of different parallel plates. (c) Prove that the lateral displacement d of the emergent beam is given by the relationship $$d=t \frac{\sin \left(\theta_{s}-\theta_{b}^{\prime}\right)}{\cos \theta_{b}^{\prime}}$$ where t is the thickness of the plate. (d) A ray of light is incident at an angle of \(66.0^{\circ}\) on one surface of a glass plate 2.40 cm thick with an index of refraction of The medium on either side of the plate is air. Find the lateral displacement between the incident and emergent rays. Equation Transcription: ° Text Transcription: theta _a Theta_a = theta_a prime d = t sin(theta_a - theta_b)/cos theta_b^prime 66.0 degrees

Angle of Deviation. The incident angle \(\theta_{a}\) shown in Fig. P33.59 is chosen so that the light passes symmetrically through the prism, which has refractive index n and apex angle A.(a) Show that the angle of deviation \(\delta\) (the angle between the initial and final directions of the ray) is given by(When the light passes through symmetrically, as shown, the angle of deviation is a minimum.) (b) Use the result of part (a) to find the angle of deviation for a ray of light passing symmetrically through a prism having three equal angles \(\left(A=60.0^{2}\right)\) and n=1.52. (c) A certain glass has a refractive index of 1.61 for red light (700 nm) and 1.66 for violet light (400 nm) If both colors pass through symmetrically, as described in part (a), and if A=60.0\(^{\circ}\), find the difference between the angles of deviation for the two colors.

A thin beam of white light is directed at a flat sheet of silicate flint glass at an angle of to the surface of the sheet. Due to dispersion in the glass, the beam is spread out in a spectrum as shown in Fig. P3i3.60. The refractive index of silicate flint glass versus wavelength is graphed in Fig. 33.18. (a) The rays and shown in Fig. P33.60 correspond to the extreme wavelengths shown in Fig. 33.18. Which corresponds to red and which to violet? Explain your reasoning. (b) For what thickness of the glass sheet will the spectrum be wide, as shown (see Problem )?

Problem 61P A beam of light traveling horizontally is made of an unpolarized component with intensity I0 and a polarized component with intensity Ip. The plane of polarization of the polarized component is oriented at an angle of ? with respect to the vertical. The data in the table give the intensity measured through a polarizer with an orientation of ? with respect to the vertical. (a) What is the orientation of the polarized component? (That is, what is the angle ??) (b) What are the values of I0 and Ip? ?(°) Itotal(W/m2) ?(°) Itotal(W/m2) 0 18.4 100 8.6 10 21.4 110 6.3 20 23.7 120 5.2 30 24.8 130 5.2 40 24.8 140 6.3 50 23.7 150 8.6 60 21.4 160 11.6 70 18.4 170 15.0 80 15.0 180 18.4 90 11.6

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 : Rotation l-leucine -glutamic acid Concentration From these data, find the relationship between the concentration (in grams per ) and the rotation of the polarization (in degrees) of each amino acid. (Hint: Graph the concentration as a function of the rotation angle for each amino acid.)

Consider two vibrations of equal amplitude and frequency but differing is phase, one along the x-axis, \(x=a \sin (\omega t-\alpha)\) and the other along the y-axis, \(y=a \sin (\omega t-\beta)\) These can be written as follows: \(\frac{x}{a}=\sin \omega t \cos \alpha-\cos \omega t \sin \alpha\) [1] \(\frac{y}{a}=\sin \omega t \cos \beta-\cos \omega t \sin \beta\) [2] (a) Multiply Eq. (1) by sin \(\beta\) and . (2) by sin \(\alpha\), and then subtract the resulting equations. (b) Multiply Eq. (1) by cos \(\beta\) and Eq. (2) by cos \(\alpha\), and then subtract the resulting equations. (c) Square and add the results of parts (a) and (b). (d) Derive the equation \(x^{2}+y^{2}-2 x y \cos \delta=a^{2} \sin ^{2} \delta\), where \(\delta=\alpha-\beta\). (e) Use the above result to justify each of the diagrams in Fig. P33.65. In the figure, the angle given is the phase difference between two simple harmonic motions of the same frequency and amplitude, one horizontal (along the x-axis) and the other vertical (along the y-axis). The figure thus shows the resultant motion from the superposition of the two perpendicular harmonic motions.

A rainbow is produced by the reflection of sunlight by spherical drops of water in the air. Figure P33.66 shows a ray that refracts into a drop at point A, is reflected from the back surface of the drop at point B and refracts back into the air at point C. The angles of incidence and refraction, \(\theta_{a}\) and \(\theta_{b}\), are shown at points A and C, and the angles of incidence and reflection, \(\theta_{a}\) and \(\theta_{r}\), are shown at point B. (a) Show that \(\theta_{a}^{B}=\theta_{b}^{A}, \ \theta_{a}^{C}=\theta_{b}^{A}\), and \(\theta_{b}{ }^{C}=\theta_{a}{ }^{A}\). (b) Show that the angle in radians between the ray before it enters the drop at A and after it exits at C (the total angular deflection of the ray) is \(\Delta=2 \theta_{a}{ }^{A}-4 \theta_{b}{ }^{A}+\pi\). (Hint: Find the angular deflections that occur at A, B and C, and add them to get \(\Delta\).) (c) Use Snell’s law to write \(\Delta\) in terms of \(\theta_{a}{ }^{A}\) and n, the refractive index of the water in the drop. (d) A rainbow will form when the angular deflection \(\Delta\) is stationary in the incident angle \(\theta_{a}{ }^{A}\) —that is, when \(d \Delta / d \theta_{a}{ }^{A}=0\). If this condition is satisfied, all the rays with incident angles close to \(\theta_{a}{ }^{A}\) will be sent back in the same direction, producing a bright zone in the sky. Let \(\theta_{1}\) be the value of \(\theta_{a}{ }^{A}\) for which this occurs. Show that \(\cos ^{2} \theta_{1}=\frac{1}{3}\left(n^{2}-1\right)\). (Hint: You may find the derivative formula \(d(\arcsin u(x)) / d x=\left(1-u^{2}\right)^{-1 / 2}(d u / d x)\) helpful.) (e) The index of refraction in water is 1.342 for violet light and 1.330 for red light. Use the results of parts (c) and (d) to find \(\theta_{1}\) and \(\Delta\) for violet and red light. Do your results agree with the angles shown in Fig. 33.20d? When you view the rainbow, which color, red or violet, is higher above the horizon?

A secondary rainbow is formed when the incident light undergoes two internal reflections in a spherical drop of water as shown in Fig. e. (See Challenge Problem 33.66.) (a) In terms of the incident angle \(\Theta_{a}^{A}\) and the refractive index of the drop, what is the angular deflection of the ray? That is, what is the angle between the ray before it enters the drop and after it exits? (b) What is the incident angle \(\Theta_{2}\) for which the derivative of with respect to the incident angle \(\Theta_{a}^{A}\) is zero? (c) The indexes of refraction for red and violet light in water are given in part (e) of Challenge Problem Use the results of parts (a) and (b) to find \(\Theta_{2}\) and \(\Delta\) for violet and red light. Do your results agree with the angles shown in Fig. ? When you view a secondary rainbow, is red or violet higher above the horizon? Explain. Equation transcription: Text transcription: Theta{a}^{A} Theta{2} Delta