||| An advanced computer sends information to its various parts via infrared light pulses traveling through silicon fibers. To acquire data from memory, the central processing unit sends a light-pulse request to the memory unit. The memory unit processes the request, then sends a data pulse back to the central processing unit. The memory unit takes 0.5 ns to process a request. If the information has to be obtained from memory in 2.0 ns, what is the maximum distance the memory unit can be from the central processing unit?

| A red ball is placed at point A in FIGURE P34.44.

1. How many images are seen by an observer at point O?

2. What are the $\left(x\text{,}y\right)$ coordinates of each image?

FIGURE P34.44

|| A microscope is focused on a black dot. When a 1.00-cm-thick piece of plastic is placed over the dot, the microscope objective has to be raised 0.40 cm to bring the dot back into focus. What is the index of refraction of the plastic?

|| The place you get your hair cut has two nearly parallel mirrors 5.0 m apart. As you sit in the chair, your head is 2.0 m from the nearer mirror. Looking toward this mirror, you first see your face and then, farther away, the back of your head. (The mirrors need to be slightly nonparallel for you to be able to see the back of your head, but you can treat them as parallel in this problem.) How far away does the back of your head appear to be? Neglect the thickness of your head.

|| A light ray in air is incident on a transparent material whose index of refraction is $n$.

1. Find an expression for the (non-zero) angle of incidence whose angle of refraction is half the angle of incidence.

2. Evaluate your expression for light incident on glass.

|| The 80-cm-tall, 65-cm-wide tank shown in FIGURE P34.48 is completely filled with water. The tank has marks every 10 cm along one wall, and the 0 cm mark is barely submerged. As you stand beside the opposite wall, your eye is level with the top of the water.

1. Can you see the marks from the top of the tank (the 0 cm mark) going down, or from the bottom of the tank (the 80 cm mark) coming up? Explain.

2. Which is the lowest or highest mark, depending on your answer to part a, that you can see?

FIGURE P34.48

|| The meter stick in FIGURE P34.49 lies on the bottom of a 100-cm-long tank with its zero mark against the left edge. You look into the tank at a $30\text{°}$ angle, with your line of sight just grazing the upper left edge of the tank. What mark do you see on the meter stick if the tank is (a) empty, (b) half full of water, and (c) completely full of water?

FIGURE P34.49

||| A horizontal meter stick is centered at the bottom of a 3.0-m-deep, 3.0-m-wide pool of water. Suppose you place your eye just above the edge of the pool and look along the direction of the meter stick. What angle do you observe between the two ends of the meter stick if the pool is (a) empty and (b) completely filled with water?

|| It’s nighttime, and you’ve dropped your goggles into a 3.0-m-deep swimming pool. If you hold a laser pointer 1.0 m above the edge of the pool, you can illuminate the goggles if the laser beam enters the water 2.0 m from the edge. How far are the goggles from the edge of the pool?

|| A 4.0-m-wide swimming pool is filled to the top. The bottom of the pool becomes completely shaded in the afternoon when the sun is $20\text{°}$ above the horizon. How deep is the pool?

||| An astronaut is exploring an unknown planet when she accidentally drops an oxygen canister into a 1.50-m-deep pool filled with an unknown liquid. Although she dropped the canister 21 cm from the edge, it appears to be 31 cm away when she peers in from the edge. What is the liquid’s index of refraction? Assume that the planet’s atmosphere is similar to earth’s.

|| Shown from above in FIGURE P34.54 is one corner of a rectangular box filled with water. A laser beam starts 10 cm from side A of the container and enters the water at position $x$. You can ignore the thin walls of the container.

1. If $x=15\text{cm,}$ does the laser beam refract back into the air through side B or reflect from side B back into the water? Determine the angle of refraction or reflection.

2. Repeat part a for $x=25\text{cm}$.

3. Find the minimum value of $x$ for which the laser beam passes through side B and emerges into the air.

FIGURE P34.54

|| A horizontal laser beam enters the glass prism shown in FIGURE P34.55. When the laser beam exits the prism, by what angle will it have been deflected from horizontal?

FIGURE P34.55

|| Optical engineers need to know the cone of acceptance of an optical fiber. This is the maximum angle that an entering light ray can make with the axis of the fiber if it is to be guided down the fiber. What is the cone of acceptance of an optical fiber for which the index of refraction of the core is 1.55 while that of the cladding is 1.45? You can model the fiber as a cylinder with a flat entrance face.

|| One of the contests at the school carnival is to throw a spear at an underwater target lying flat on the bottom of a pool. The water is 1.0 m deep. You’re standing on a small stool that places your eyes 3.0 m above the bottom of the pool. As you look at the target, your gaze is $30\text{°}$ below horizontal. At what angle below horizontal should you throw the spear in order to hit the target? Your raised arm brings the spear point to the level of your eyes as you throw it, and over this short distance you can assume that the spear travels in a straight line rather than a parabolic trajectory.

|| There’s one angle of incidence $\beta$ onto a prism for which the light inside an isosceles prism travels parallel to the base and emerges at angle $\beta$.

1. Find an expression for $\beta$ in terms of the prism’s apex angle $\alpha$ and index of refraction $n$.

2. A laboratory measurement finds that $\beta =52.2\text{°}$ for a prism shaped like an equilateral triangle. What is the prism’s index of refraction?

FIGURE P34.58

|| You’re visiting the shark tank at the aquarium when you see a 2.5-m-long shark that appears to be swimming straight toward you at 2.0 m/s. What is the shark’s actual speed through the water? You can ignore the glass wall of the tank.

|| Paraxial light rays approach a transparent sphere parallel to an optical axis passing through the center of the sphere. The rays come to a focus on the far surface of the sphere. What is the sphere’s index of refraction?

|| To determine the focal length of a lens, you place the lens in front of a small lightbulb and then adjust a viewing screen to get a sharply focused image. Varying the lens position produces the following data:

Use the best-fit line of an appropriate graph to determine the focal length of the lens.

| BIO A dentist uses a curved mirror to view the back side of teeth in the upper jaw. Suppose she wants an upright image with a magnification of 1.5 when the mirror is 1.2 cm from a tooth. Should she use a convex or a concave mirror? What focal length should it have?

|| BIO The illumination lights in an operating room use a concave mirror to focus an image of a bright lamp onto the surgical site. One such light uses a mirror with a 30 cm radius of curvature. If the mirror is 1.2 m from the patient, how far should the lamp be from the mirror?

||| BIO A keratometer is an optical device used to measure the radius of curvature of the eye’s cornea—its entrance surface. This measurement is especially important when fitting contact lenses, which must match the cornea’s curvature. Most light incident on the eye is transmitted into the eye, but some light reflects from the cornea, which, due to its curvature, acts like a convex mirror. The keratometer places a small, illuminated ring of known diameter 7.5 cm in front of the eye. The optometrist, using an eyepiece, looks through the center of this ring and sees a small virtual image of the ring that appears to be behind the cornea. The optometrist uses a scale inside the eyepiece to measure the diameter of the image and calculate its magnification. Suppose the optometrist finds that the magnification for one patient is 0.049. What is the absolute value of the radius of curvature of her cornea?

|| A 2.0-cm-tall candle flame is 2.0 m from a wall. You happen to have a lens with a focal length of 32 cm. How many places can you put the lens to form a well-focused image of the candle flame on the wall? For each location, what are the height and orientation of the image?

||| A 25-cm-long rod lies along the optical axis of a converging lens, perpendicular to the lens plane. The lens has a 30 cm focal length. The rod’s real image, along the optical axis on the other side of the lens, is also 25 cm long. What is the distance from the lens to the nearest end of the rod?

|| An old-fashioned slide projector needs to create a 98-cm-high image of a 2.0-cm-tall slide. The screen is 300 cm from the slide.

1. What focal length does the lens need? Assume that it is a thin lens.

2. How far should you place the lens from the slide?

|| A lightbulb is 3.0 m from a wall. What are the focal length and the position (measured from the bulb) of a lens that will form an image on the wall that is twice the size of the lightbulb?

|| CALC A converging lens with focal length $f$ creates a real image of an object. What is the minimum possible distance between the object and its image? Your answer will be a multiple of $f$.

|| A plano-concave glass lens (flat on one side, concave on the other) creates an image with magnification $+0.40$ of an object 75 cm from the lens. What is the radius of curvature of the lens’s curved surface?

|| An object is 60 cm from a screen. What are the radii of a symmetric converging plastic lens (i.e., two equally curved surfaces) that will form an image on the screen twice the height of the object?

|| A lens placed 10 cm in front of an object creates an upright image twice the height of the object. The lens is then moved along the optical axis until it creates an inverted image twice the height of the object. How far did the lens move?

|| CALC A wildlife photographer with a 200-mm-focal-length telephoto lens on his camera is taking a picture of a rhinoceros that is 100 m away. Suddenly, the rhino starts charging straight toward the photographer at a speed of 5.0 m/s. What is the speed, in $\mu \text{m}/s$, of the image of the rhinoceros? Is the image moving toward or away from the lens?

|| Some electro-optic materials can change their index of refraction in response to an applied voltage. Suppose a plano-convex lens (flat on one side, a 15.0 cm radius of curvature on the other), made from a material whose normal index of refraction is 1.500, is creating an image of an object that is 50.0 cm from the lens. By how much would the index of refraction need to be increased to move the image 5.0 cm closer to the lens?

|| A concave mirror has a 40 cm radius of curvature. How far from the mirror must an object be placed to create an upright image three times the height of the object?

|| A 2.0-cm-tall object is placed in front of a mirror. A 1.0-cm-tall upright image is formed behind the mirror, 150 cm from the object. What is the focal length of the mirror?

|| A cosmetic mirror has a magnification of 2.0 when you look into it from 30 cm away. What is the mirror’s radius of curvature?

|| A spherical mirror of radius $R$ has its center at C, as shown in FIGURE P34.78. A ray parallel to the axis reflects through F, the focal point. Prove that $f=R/2$ if $\varphi \ll 1\text{rad}$.

FIGURE P34.78

||| The mirror in FIGURE CP34.79 is covered with a piece of glass whose thickness at the center equals the mirror’s radius of curvature. A point source of light is outside the glass. How far from the mirror is the image of this source?

FIGURE CP34.79

||| Consider a lens having index of refraction $n_2$ and surfaces with radii $R_1$ and $R_2$. The lens is immersed in a fluid that has index of refraction $n_1$.

1. Derive a generalized lens maker’s equation to replace Equation 34.26 when the lens is surrounded by a medium other than air. That is, when $n_1\ne 1$.

2. A symmetric converging glass lens (i.e., two equally curved surfaces) has two surfaces with radii of 40 cm. Find the focal length of this lens in air and the focal length of this lens in water.

||| CALC FIGURE CP34.81 shows a light ray that travels from point A to point B. The ray crosses the boundary at position $x$, making angles ${\theta }_1$ and ${\theta }_2$ in the two media. Suppose that you did not know Snell’s law.

1. Write an expression for the time t it takes the light ray to travel from A to B. Your expression should be in terms of the distances $a$, $b$, and $w$; the variable $x$; and the indices of refraction $n_1$ and $n_2$.

2. The time depends on $x$. There’s one value of $x$ for which the light travels from A to B in the shortest possible time. We’ll call it $x_{\text{min}}$. Write an expression (but don’t try to solve it!) from which $x_{\text{min}}$ could be found.

3. Now, by using the geometry of the figure, derive Snell’s law from your answer to part b.

FIGURE CP34.81

You’ve proven that Snell’s law is equivalent to the statement that “light traveling between two points follows the path that requires the shortest time.” This interesting way of thinking about refraction is called Fermat’s principle.

||| A fortune teller’s “crystal ball” (actually just glass) is 10 cm in diameter. Her secret ring is placed 6.0 cm from the edge of the ball.

1. An image of the ring appears on the opposite side of the crystal ball. How far is the image from the center of the ball?

2. Draw a ray diagram showing the formation of the image.

3. The crystal ball is removed and a thin lens is placed where the center of the ball had been. If the image is still in the same position, what is the focal length of the lens?

||| CALC Consider an object of thickness ds (parallel to the axis) in front of a lens or mirror. The image of the object has thickness $ds\prime$. Define the longitudinal magnification as $M=ds\prime /ds$. Prove that $M=-m^2$, where $m$ is the lateral magnification.