Chapter 26

A plate glass window \((n=1.5)\) has a thickness of \(4.0 \times 10^{-3} \mathrm{~m}\). How long does it take light to pass perpendicularly through the plate?

The refractive indices of materials \(A\) and \(B\) have a ratio of \(n_{\mathrm{A}} / n_{\mathrm{B}}=1.33 .\) The speed of light in material \(A\) is \(1.25 \times 10^{8} \mathrm{~m} / \mathrm{s}\). What is the speed of light in material \(B ?\)

The frequency of a light wave is the same when the light travels in ethyl alcohol as it is when it travels in carbon disulfide. Find the ratio of the wavelength of the light in ethyl alcohol to that in carbon disulfide.

Interactive Solution \(26.7\) at offers one model for problems like this one. In a certain time, light travels \(6.20 \mathrm{~km}\) in a vacuum. During the same time, light travels only \(3.40 \mathrm{~km}\) in a liquid. What is the refractive index of the liquid?

Interactive Solution \(\underline{26.7}\) at offers one model for problems like this one. In a certain time, light travels \(6.20 \mathrm{~km}\) in a vacuum. During the same time, light travels only \(3.40 \mathrm{~km}\) in a liquid. What is the refractive index of the liquid?

A flat sheet of ice has a thickness of \(2.0 \mathrm{~cm} .\) It is on top of a flat sheet of crystalline quartz that has a thickness of \(1.1 \mathrm{~cm} .\) Light strikes the ice perpendicularly and travels through it and then through the quartz. In the time it takes the light to travel through the two sheets, how far (in centimeters) would it have traveled in a vacuum?

A light ray in air is incident on a water surface at a \(43^{\circ}\) angle of incidence. Find (a) the angle of reflection and (b) the angle of refraction.

A ray of light impinges from air onto a block of ice \((n=1.309)\) at a \(60.0^{\circ}\) angle of incidence. Assuming that this angle remains the same, find the difference \(\theta_{2, \text { ice }}-\theta_{2, \text { water }}\) water in the angles of refraction when the ice turns to water \((n=1.333)\).

A spotlight on a boat is \(2.5 \mathrm{~m}\) above the water, and the light strikes the water at a point that is \(8.0 \mathrm{~m}\) horizontally displaced from the spotlight (see the drawing). The depth of the water is \(4.0 \mathrm{~m}\). Determine the distance \(d\), which locates the point where the light strikes the bottom.

Amber \((n=1.546)\) is a transparent brown-yellow fossil resin. An insect, trapped and preserved within the amber, appears to be \(2.5 \mathrm{~cm}\) beneath the surface when viewed directly from above. How far below the surface is the insect actually located?

A beam of light is traveling in air and strikes a material. The angles of incidence and refraction are \(63.0^{\circ}\) and \(47.0^{\circ}\), respectively. Obtain the speed of light in the material.

A scuba diver, submerged under water, looks up and sees sunlight at an angle of \(28.0^{\circ}\) from the vertical. At what angle, measured from the vertical, does this sunlight strike the surface of the water?

Light in a vacuum is incident on a transparent glass slab. The angle of incidence is \(35.0^{\circ} .\) The slab is then immersed in a pool of liquid. When the angle of incidence for the light striking the slab is \(20.3^{\circ},\) the angle of refraction for the light entering the slab is the same as when the slab was in a vacuum. What is the index of refraction of the liquid?

A silver medallion is sealed within a transparent block of plastic. An observer in air, viewing the medallion from directly above, sees the medallion at an apparent depth of \(1.6 \mathrm{~cm}\) beneath the top surface of the block. How far below the top surface would the medallion appear if the observer (not wearing goggles) and the block were under water?

The drawing shows a rectangular block of glass \((n=1.52)\) surrounded by liquid carbon disulfide \((n=1.63)\). A ray of light is incident on the glass at point \(\mathrm{A}\) with a \(30.0^{\circ}\) angle of incidence. At what angle of refraction does the ray leave the glass at point B?

Review Conceptual Example 4 as background for this problem. A man in a boat is looking straight down at a fish in the water directly beneath him. The fish is looking straight up at the man. They are equidistant from the air-water interface. To the man, the fish ap pears to be \(2.0 \mathrm{~m}\) beneath his eyes. To the fish, how far above its eyes does the man appear to be?

A small logo is embedded in a thick block of crown glass \((n=1.52), 3.20 \mathrm{~cm}\) beneath the top surface of the glass. The block is put under water, so there is \(1.50 \mathrm{~cm}\) of water above the top surface of the block. The logo is viewed from directly above by an observer in air. How far beneath the top surface of the water does the logo appear to be?

A beaker has a height of \(30.0 \mathrm{~cm}\). The lower half of the beaker is filled with water, and the upper half is filled with oil \((n=1.48)\). To a person looking down into the beaker from above, what is the apparent depth of the bottom?

One method of determining the refractive index of a trans parent solid is to measure the critical angle when the solid is in air. If \(\theta_{\mathrm{c}}\) is found to be \(40.5^{\circ},\) what is the index of refraction of the solid?

A glass block \((n=1.56)\) is immersed in a liquid. A ray of light within the glass hits a glassliquid surface at a \(75.0^{\circ}\) angle of incidence. Some of the light enters the liquid. What is the smallest possible refractive index for the liquid?

Interactive Solution \(\underline{26.25}\) at provides one model for solving problems such as this. A glass block \((n=1.56)\) is immersed in a liquid. A ray of light within the glass hits a glassliquid surface at a \(75.0^{\circ}\) angle of incidence. Some of the light enters the liquid. What is the smallest possible refractive index for the liquid?

Concept Simulation 26.1 at illustrates the concepts that are pertinent to this problem. A ray of light is traveling in glass and strikes a glass-liquid interface. The angle of incidence is \(58.0^{\circ},\) and the index of refraction of glass is \(n=1.50 .\) (a) What must be the index of refraction of the liquid such that the direction of the light entering the liquid is not changed? (b) What is the largest index of refraction that the liquid can have, such that none of the light is transmitted into the liquid and all of it is reflected back into the glass?

The drawing shows a ray of light traveling from point \(A\) to point \(B,\) a distance of \(4.60 \mathrm{~m}\) in a material that has an index of refraction \(n_{1}\). At point \(B\), the light encounters a different substance whose index of refraction is \(n_{2}=1.63 .\) The light strikes the interface at the critical angle of \(\theta_{c}=48.1^{\circ} .\) How much time does it take for the light to travel from \(A\) to \(B\) ?

A layer of liquid \(B\) floats on liquid \(A\). A ray of light begins in liquid \(A\) and undergoes total internal reflection at the interface between the liquids when the angle of incidence exceeds \(36.5^{\circ} .\) When liquid \(B\) is replaced with liquid \(C,\) total internal reflection occurs for angles of incidence greater than \(47.0^{\circ} .\) Find the ratio \(n_{B} / n_{C}\) of the refractive indices of liquids \(B\) and \(C\).

Light is reflected from a glass coffee table. When the angle of incidence is \(56.7^{\circ},\) the reflected light is completely polarized parallel to the surface of the glass. What is the index of refraction of the glass?

For light that originates within a liquid and strikes the liquid-air interface, the critical angle is \(39^{\circ} .\) What is Brewster's angle for this light?

Sunlight strikes a diamond surface. At what angle of incidence is the reflected light completely polarized?

Light is incident from air onto the surface of a liquid. The angle of incidence is \(53.0^{\circ}\), and the angle of refraction is \(34.0^{\circ} .\) At what angle of incidence would the reflected light be \(100 \%\) polarized?

When light strikes the surface between two materials from above, the Brewster angle is \(65.0^{\circ} .\) What is the Brewster angle when the light encounters the same surface from below?

When red light in a vacuum is incident at the Brewster angle on a certain type of glass, the angle of refraction is \(29.9^{\circ} .\) What are (a) the Brewster angle and (b) the index of refraction of the glass?

In Section 26.4 it is mentioned that the reflected and refracted rays are perpendicular to each other when light strikes the surface at the Brewster angle. This is equivalent to saying that the angle of reflection plus the angle of refraction is \(90^{\circ} .\) Using Snell's law and Brewster's law, prove that the angle of reflection plus the angle of refraction is \(90^{\circ} .\)

A ray of sunlight is passing from diamond into crown glass; the angle of incidence is \(35.00^{\circ} .\) The indices of refraction for the blue and red components of the ray are: blue \(\left(n_{\text {diamond }}=2.444, n_{\text {crown glass }}=1.531\right),\) and red \(\left(n_{\text {diamond }}=2.410, n_{\text {crown glass }}=1.520\right)\) Determine the angle between the refracted blue and red rays in the crown glass.

Red light \((n=1.520)\) and violet light \((n=1.538)\) traveling in air are incident on a slab of crown glass. Both colors enter the glass at the same angle of refraction. The red light has an angle of incidence of \(30.00^{\circ} .\) What is the angle of incidence of the violet light?

Violet light and red light travel through air and strike a block of plastic at the same angle of incidence. The angle of refraction is \(30.400^{\circ}\) for the violet light and \(31.200^{\circ}\) for the red light. The index of refraction for violet light in plastic is greater than that for red light by \(0.0400 .\) Delaying any rounding off of calculations until the very end, find the index of refraction for violet light in plastic.

A diverging lens has a focal length of \(-32 \mathrm{~cm}\). An object is placed \(19 \mathrm{~cm}\) in front of this lens. Calculate (a) the image distance and (b) the magnification. Is the image (c) real or virtual, (d) upright or inverted, and (e) enlarged or reduced in size?

A macroscopic (or macro) lens for a camera is usually a converging lens of normal focal length built into a lens barrel that can be adjusted to provide the additional lens-tofilm distance needed when focusing at very close range. Suppose that a macro lens \((f=\) \(50.0 \mathrm{~mm}\) ) has a maximum lens-to- film distance of \(275 \mathrm{~mm}\). How close can the object be located in front of the lens?

A converging lens \((f=12.0 \mathrm{~cm})\) is held \(8.00 \mathrm{~cm}\) in front of a newspaper, the print size of which has a height of \(2.00 \mathrm{~mm}\). Find (a) the image distance (in \(\mathrm{cm}\) ) and (b) the height (in \(\mathrm{mm}\) ) of the magnified print.

A tourist takes a picture of a mountain \(14 \mathrm{~km}\) away using a camera that has a lens with a focal length of \(50 \mathrm{~mm}\). She then takes a second picture when she is only \(5.0 \mathrm{~km}\) away. What is the ratio of the height of the mountain's image on the film for the second picture to its height on the film for the first picture?

\(26.3\) at reviews the concepts that play a role in this problem. A converging lens has a focal length of \(88.00 \mathrm{~cm}\). An object \(13.0 \mathrm{~cm}\) tall is located \(155.0\) \(\mathrm{cm}\) in front of this lens. (a) What is the image distance? (b) Is the image real or virtual? (c) What is the image height? Be sure to include the proper algebraic sign.

Concept Simulation 26.3 at reviews the concepts that play a role in this problem. A converging lens has a focal length of \(88.00 \mathrm{~cm}\). An object \(13.0 \mathrm{~cm}\) tall is located 155.0 \(\mathrm{cm}\) in front of this lens. (a) What is the image distance? (b) Is the image real or virtual? (c) What is the image height? Be sure to include the proper algebraic sign.

A diverging lens has a focal length of \(-25 \mathrm{~cm} .\) (a) Find the image distance when an object is placed \(38 \mathrm{~cm}\) from the lens. (b) Is the image real or virtual?

To focus a camera on objects at different distances, the converging lens is moved toward or away from the film, so a sharp im age always falls on the film. A camera with a telephoto lens \((f=200.0 \mathrm{~mm})\) is to be focused on an object located first at a distance of \(3.5 \mathrm{~m}\) and then at \(50.0 \mathrm{~m}\). Over what distance must the lens be movable?

Consult Interactive Solution \(\underline{26.51}\) at to review the concepts on which this problem depends. A camera is supplied with two interchangeable lenses, whose focal lengths are \(35.0\) and \(150.0 \mathrm{~mm}\). A woman whose height is \(1.60 \mathrm{~m}\) stands \(9.00 \mathrm{~m}\) in front of the camera. What is the height (including sign) of her image on the film, as produced by (a) the \(35.0\) -mm lens and \((b)\) the \(150.0-\mathrm{mm}\) lens?

Consult Interactive Solution 26.51 at to review the concepts on which this problem depends. A camera is supplied with two interchangeable lenses, whose focal lengths are 35.0 and \(150.0 \mathrm{~mm}\). A woman whose height is \(1.60 \mathrm{~m}\) stands \(9.00 \mathrm{~m}\) in front of the camera. What is the height (including sign) of her image on the film, as produced by (a) the 35.0 -mm lens and (b) the 150.0 -mm lens?

\(26.4\) at provides the option of exploring the ray diagram that applies to this problem. The distance between an object and its image formed by a diverging lens is \(49.0 \mathrm{~cm}\). The focal length of the lens is \(-233.0 \mathrm{~cm}\). Find (a) the image distance and (b) the object distance.

Concept Simulation 26.4 at provides the option of exploring the ray diagram that applies to this problem. The distance between an object and its image formed by a diverging lens is \(49.0 \mathrm{~cm} .\) The focal length of the lens is \(-233.0 \mathrm{~cm}\). Find (a) the image distance and (b) the object distance.

The moon's diameter is \(3.48 \times 10^{6} \mathrm{~m},\) and its mean distance from the earth is \(3.85 \times 10^{8} \mathrm{~m}\). The moon is being photographed by a camera whose lens has a focal length of \(50.0 \mathrm{~mm}\). (a) Find the diameter of the moon's image on the slide film. (b) When the slide is projected onto a screen that is \(15.0 \mathrm{~m}\) from the lens of the projector \((f=\) \(110.0 \mathrm{~mm}\) ), what is the diameter of the moon's image on the screen?

An object is \(18 \mathrm{~cm}\) in front of a diverging lens that has a focal length of \(-12 \mathrm{~cm} .\) How far in front of the lens should the object be placed so that the size of its image is reduced by a factor of \(2.0 ?\)

An object is placed in front of a converging lens in such a position that the lens \((f=12.0\) \(\mathrm{cm}\) ) creates a real image located \(21.0 \mathrm{~cm}\) from the lens. Then, with the object remaining in place, the lens is replaced with another converging lens \((f=16.0 \mathrm{~cm})\). A new, real image is formed. What is the image distance of this new image?

A converging lens \((f=25.0 \mathrm{~cm})\) is used to project an image of an object onto a screen. The object and the screen are \(125 \mathrm{~cm}\) apart, and between them the lens can be placed at either of two locations. Find the two object distances.