Chapter 33

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 \(\mathrm{nm}\) (violet) to 750 \(\mathrm{nm}(\) 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 \(\mathrm{nm}\) 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 \(5.80 \times 10^{14} \mathrm{Hz}\) 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?

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

A parallel beam of light in air makes an angle of \(47.5^{\circ}\) 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?

Light traveling in air is incident on the surface of a block of plastic at an angle of \(62.7^{\circ}\) to the normal and is bent so that it makes a \(48.1^{\circ}\) angle with the normal in the plastic. Find the speed of light in the plastic.

(a) A tank containing methanol has walls 2.50 \(\mathrm{cm}\) 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?

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?

In a material having an index of refraction \(n,\) a light ray has frequency \(f,\) wavelength \(\lambda,\) and speed \(v .\) What are the frequency, wavelength, and speed of this light (a) in vacuum and (b) in a material having refractive index \(n^{\prime} ?\) In each case, express your answers in terms of only \(f, \lambda, v, n,\) and \(n^{\prime} .\)

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 \(\mathrm{nm}\) and its wavelength in the glass is 544 \(\mathrm{nm} .\) 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 withrespect 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 \(62.0^{\circ},\) and the ray originates in the glass with \(n=1.70 .\) 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 \(\theta_{a}=36.2^{\circ},\) the ray refracted into the water makes an angle of \(49.8^{\circ}\) with the normal to the interface. What is the smallest value of the incident angle \(\theta_{a}\) for which none of the ray refracts into the water?

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

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?

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 \(\mathrm{m} / \mathrm{s}\) in air and 1320 \(\mathrm{m} / \mathrm{s}\) 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 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 \(I_{0}\) is incident on two polarizing filters. The axis of the first filter makes an angle of \(60.0^{\circ}\) 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?

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?

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

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 \(45.0^{\circ}\) and \(90.0^{\circ},\) respectively, with that of the first. (a) If unpolarized light of intensity \(I_{0}\) is incident on the stack, find the intensity and state of polarization of light emerging from each fitter. (b) If the second filter is removed, what is the intensity of the light emerging from each remaining filter?

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?

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

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 \(\mathrm{ns}\) to travel from the laser to the photocell. What is the wavelength of the light in the glass?

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

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 \(\mathrm{mm}\) thick and the halos have an inner radius of \(5.34 \mathrm{mm},\) 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.)

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 (Fhin-walled, hollow cylinder 16.0 \(\mathrm{cm}\) high. The diameter of the top and bottom of the glass is 8.0 \(\mathrm{cm} .\) 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?

\(\mathrm{A} 45^{\circ}-45^{\circ}-90^{\circ}\) 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 \((n=1.309)\) 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 \(30^{\circ}-\) \(60^{\circ}-90^{\circ}\) prism (Fig. \(\mathrm{P} 33.52 ) . \mathrm{A}\) 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.

A certain birefringent material has indexes of refraction \(n_{1}\) and \(n_{2}\) for the two per- pendicular 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 \(\lambda_{0}=589 \mathrm{nm} .\)