Your digital camera lens, with an effective focal length

What is an optical instrument?

How does a camera work?

How does vision work?

What optical systems are used to magnify things?

Is color important in optics?

What is the resolution of an optical system?

The lens on a camera is usually a combination of two or more single lenses. Consider a camera in which light passes first through a diverging lens, with f1 = -120 mm, then a converging lens, with f2 = 42 mm, spaced 60 mm apart. A reasonable definition of the effective focal length of this lens combination is the focal length of a single lens that could produce an image in the same location if placed at the midpoint of the lens combination. A 10-cm-tall object is 500 mm from the first lens. a. What are the location, size, and orientation of the image? b. What is the effective focal length of the double-lens system used in this camera?

The second lens in this optical instrument a. Causes the light rays to focus closer than they would with the first lens acting alone. b. Causes the light rays to focus farther away than they would with the first lens acting alone. c. Inverts the image but does not change where the light rays focus. d. Prevents the light rays from reaching a focus.

Your digital camera lens, with an effective focal length of 10.0 mm, is focused on a flower 20.0 cm away. You then turn to take a picture of a distant landscape. How far, and in which direction, must the lens move to bring the landscape into focus?

A photographer has adjusted his camera for a correct exposure with a short-focal-length lens. He then decides to zoom in by increasing the focal length. To maintain a correct exposure without changing the shutter speed, the diameter of the lens aperture should a. Be increased. b. Be decreased. c. Stay the same.

Before a race, a photographer finds that she can make a perfectly exposed photo of the track while using a shutter speed of 1/250 s and a lens setting of f/8.0. To freeze the sprinters as they go past, she plans to use a shutter speed of 1/1000 s. To what f@number must she set her lens?

You need to improvise a magnifying glass to read some very tiny print. Should you borrow the eyeglasses from your hyperopic friend or from your myopic friend? a. The hyperopic friend b. The myopic friend c. Either will do. d. Neither will work.

Sanjay has hyperopia. The near point of his left eye is 150 cm. What prescription lens will restore normal vision?

A biologist rotates the turret of a microscope to replace a 20* objective with a 10* objective. To keep the same overall magnification, the focal length of the eyepiece must be a. Doubled. b. Halved. c. Kept the same. d. The magnification cannot be kept the same if the objective is changed

Martina has myopia. The far point of her left eye is 200 cm. What prescription lens will restore normal vision?

Four diffraction-limited lenses focus plane waves of light with the same wavelength l. Rank in order, from largest to smallest, the spot sizes wa to wd. f = 10 mm 2 mm f = 5 mm 2 mm f = 10 mm 4 mm f = 24 mm a b c d 8 mm

A pathologist inspects a sample of 7@mm@diameter human blood cells under a microscope. She selects a 40* objective and a 10* eyepiece. What size object, viewed from 25 cm, has the same apparent size as a blood cell seen through the microscope?

Example 34.4 found that a ray incident on a 30 prism is deflected by 22.6 if the prisms index of refraction is 1.59. Suppose this is the index of refraction of deep violet light and deep red light has an index of refraction of 1.54. a. What is the deflection angle for deep red light? b. If a beam of white light is dispersed by this prism, how wide is the rainbow spectrum on a screen 2.0 m away?

A 12-cm-diameter telescope lens has a focal length of 1.0 m. What is the diameter of the image of a star in the focal plane if the lens is diffraction limited and if the earths atmosphere is not a limitation?

The normal human eye has maximum visual acuity with a pupil diameter of about 3 mm. For larger pupils, acuity decreases due to increasing aberrations; for smaller pupils, acuity decreases due to increasing diffraction. If your pupil diameter is 2.0 mm, as it would be in bright light, what is the smallest-diameter circle that you should be able to see as a circle, rather than just an unresolved blob, on an eye chart at the standard distance of 20 ft? The index of refraction inside the eye is 1.33

Suppose a cameras exposure is correct when the lens has a focal length of 8.0 mm. Will the picture be overexposed, underexposed, or still correct if the focal length is zoomed to 16.0 mm without changing the diameter of the lens aperture? Explain.

A camera has a circular aperture immediately behind the lens. Reducing the aperture diameter to half its initial value will A. Make the image blurry. B. Cut off the outer half of the image and leave the inner half unchanged. C. Make the image less bright. D. All the above. Explain your choice

Suppose you wanted special glasses designed to let you see underwater without a face mask. Should the glasses use a converging or diverging lens? Explain

A red card is illuminated by red light. What color will the card appear? What if its illuminated by blue light?

The center of the galaxy is filled with low-density hydrogen gas that scatters light rays. An astronomer wants to take a picture of the center of the galaxy. Will the view be better using ultraviolet light, visible light, or infrared light? Explain

A friend lends you the eyepiece of his microscope to use on your own microscope. He claims the spatial resolution of your microscope will be halved, since his eyepiece has the same diameter as yours but twice the magnification. Is his claim valid? Explain.

A diffraction-limited lens can focus light to a 10@mm@diameter spot on a screen. Do the following actions make the spot diameter larger, make it smaller, or leave it unchanged? A. Decreasing the wavelength of the light. B. Decreasing the lens diameter. C. Decreasing the lens focal length. D. Decreasing the lens-to-screen distance

To focus parallel light rays to the smallest possible spot, should you use a lens with a small f@number or a large f@number? Explain

An astronomer is trying to observe two distant stars. The stars are marginally resolved when she looks at them through a filter that passes green light with a wavelength near 550 nm. Which of the following actions would improve the resolution? Assume that the resolution is not limited by the atmosphere. A. Changing the filter to a different wavelength. If so, should she use a shorter or a longer wavelength? B. Using a telescope with an objective lens of the same diameter but a different focal length. If so, should she select a shorter or a longer focal length? C. Using a telescope with an objective lens of the same focal length but a different diameter. If so, should she select a larger or a smaller diameter? D. Using an eyepiece with a different magnification. If so, should she select an eyepiece with more or less magnification?

Two converging lenses with focal lengths of 40 cm and 20 cm are 10 cm apart. A 2.0-cm-tall object is 15 cm in front of the 40-cm-focal-length lens. a. Use ray tracing to find the position and height of the image. Do this accurately using a ruler or paper with a grid, then make measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a

A converging lens with a focal length of 40 cm and a diverging lens with a focal length of -40 cm are 160 cm apart. A 2.0-cm-tall object is 60 cm in front of the converging lens. a. Use ray tracing to find the position and height of the image. Do this accurately using a ruler or paper with a grid, then make measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a.

A 2.0-cm-tall object is 20 cm to the left of a lens with a focal length of 10 cm. A second lens with a focal length of 15 cm is 30 cm to the right of the first lens. a. Use ray tracing to find the position and height of the image. Do this accurately using a ruler or paper with a grid, then make measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a.

A 2.0-cm-tall object is 20 cm to the left of a lens with a focal length of 10 cm. A second lens with a focal length of 5 cm is 30 cm to the right of the first lens. a. Use ray tracing to find the position and height of the image. Do this accurately using a ruler or paper with a grid, then make measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a.

A 2.0-cm-tall object is 20 cm to the left of a lens with a focal length of 10 cm. A second lens with a focal length of -5 cm is 30 cm to the right of the first lens. a. Use ray tracing to find the position and height of the image. Do this accurately using a ruler or paper with a grid, then make measurements on your diagram. b. Calculate the image position and height. Compare with your ray-tracing answers in part a.

A 2.0-m-tall man is 10 m in front of a camera with a 15-mmfocal-length lens. How tall is his image on the detector?

What is the f@number of a lens with a 35 mm focal length and a 7.0-mm-diameter aperture?

What is the aperture diameter of a 12-mm-focal-length lens set to f/4.0?

A camera takes a properly exposed photo at f/5.6 and 1/125 s. What shutter speed should be used if the lens is changed to f/4.0?

A camera takes a properly exposed photo with a 3.0-mmdiameter aperture and a shutter speed of 1/125 s. What is the appropriate aperture diameter for a 1/500 s shutter speed?

Ramon has contact lenses with the prescription +2.0 D. a. What eye condition does Ramon have? b. What is his near point without the lenses?

Ellen wears eyeglasses with the prescription -1.0 D. a. What eye condition does Ellen have? b. What is her far point without the glasses?

What is the f-number of a relaxed eye with the pupil fully dilated to 8.0 mm? Model the eye as a single lens 2.4 cm in front of the retina.

A magnifier has a magnification of 5*. How far from the lens should an object be held so that its image is seen at the near-point distance of 25 cm? Assume that your eye is immediately behind the lens

A microscope has a 20 cm tube length. What focal-length objective will give total magnification 500* when used with an eyepiece having a focal length of 5.0 cm?

A standardized biological microscope has an 8.0-mm-focallength objective. What focal-length eyepiece should be used to achieve a total magnification of 100*?

A 6.0-mm-diameter microscope objective has a focal length of 9.0 mm. What object distance gives a lateral magnification of -40?

A 20* telescope has a 12-cm-diameter objective lens. What minimum diameter must the eyepiece lens have to collect all the light rays from an on-axis distant source?

A reflecting telescope is built with a 20-cm-diameter mirror having a 1.00 m focal length. It is used with a 10* eyepiece. What are (a) the magnification and (b) the f@number of the telescope?

A sheet of glass has nred = 1.52 and nviolet = 1.55. A narrow beam of white light is incident on the glass at 30. What is the angular spread of the light inside the glass?

A narrow beam of white light is incident on a sheet of quartz. The beam disperses in the quartz, with red light (l 700 nm) traveling at an angle of 26.3 with respect to the normal and violet light (l 400 nm) traveling at 25.7. The index of refraction of quartz for red light is 1.45. What is the index of refraction of quartz for violet light?

A hydrogen discharge lamp emits light with two prominent wavelengths: 656 nm (red) and 486 nm (blue). The light enters a flint-glass prism perpendicular to one face and then refracts through the hypotenuse back into the air. The angle between these two faces is 35. a. Use Figure 35.18 to estimate to {0.002 the index of refraction of flint glass at these two wavelengths. b. What is the angle (in degrees) between the red and blue light as it leaves the prism?

Infrared telescopes, which use special infrared detectors, are able to peer farther into star-forming regions of the galaxy because infrared light is not scattered as strongly as is visible light by the tenuous clouds of hydrogen gas from which new stars are created. For what wavelength of light is the scattering only 1% that of light with a visible wavelength of 500 nm?

A scientist needs to focus a helium-neon laser beam (l = 633 nm) to a 10@mm@diameter spot 8.0 cm behind a lens. a. What focal-length lens should she use? b. What minimum diameter must the lens have?

Two lightbulbs are 1.0 m apart. From what distance can these lightbulbs be marginally resolved by a small telescope with a 4.0-cm-diameter objective lens? Assume that the lens is diffraction limited and l = 600 nm.

In FIGURE P35.26, parallel rays from the left are focused to a point at two locations on the optical axis. Find the position of each location, giving your answer as a distance left or right of the lens.

The rays leaving the two-component optical system of FIGURE P35.27 produce two distinct images of the 1.0-cm-tall object. What are the position (relative to the lens), orientation, and height of each image?

A common optical instrument in a laser laboratory is a beam expander. One type of beam expander is shown in FIGURE P35.28. The parallel rays of a laser beam of width w1 enter from the left. a. For what lens spacing d does a parallel laser beam exit from the right? b. What is the width w2 of the exiting laser beam?

A common optical instrument in a laser laboratory is a beam expander. One type of beam expander is shown in FIGURE P35.29. The parallel rays of a laser beam of width w1 enter from the left. a. For what lens spacing d does a parallel laser beam exit from the right? b. What is the width w2 of the exiting laser beam?

In FIGURE P35.30, what are the position, height, and orientation of the final image? Give the position as a distance to the right or left of the lens.

A 1.0-cm-tall object is 110 cm from a screen. A diverging lens with focal length -20 cm is 20 cm in front of the object. What are the focal length and distance from the screen of a second lens that will produce a well-focused, 2.0-cm-tall image on the screen?

A 15-cm-focal-length converging lens is 20 cm to the right of a 7.0-cm-focal-length converging lens. A 1.0-cm-tall object is distance L to the left of the first lens. a. For what value of L is the final image of this two-lens system halfway between the two lenses? b. What are the height and orientation of the final image?

Yang can focus on objects 150 cm away with a relaxed eye. With full accommodation, she can focus on objects 20 cm away. After her eyesight is corrected for distance vision, what will her near point be while wearing her glasses?

The cornea, a boundary between the air and the aqueous humor, has a 3.0 cm focal length when acting alone. What is its radius of curvature?

The objective lens of a telescope is a symmetric glass lens with 100 cm radii of curvature. The eyepiece lens is also a symmetric glass lens. What are the radii of curvature of the eyepiece lens if the telescopes magnification is 20*?

Mars (6800 km diameter) is viewed through a telescope on a night when it is 1.1 * 108 km from the earth. Its angular size as seen through the eyepiece is 0.50, the same size as the full moon seen by the naked eye. If the eyepiece focal length is 25 mm, how long is the telescope?

Youve been asked to build a telescope from a 2.0* magnifying lens and a 5.0* magnifying lens. a. What is the maximum magnification you can achieve? b. Which lens should be used as the objective? Explain. c. What will be the length of your telescope

Marooned on a desert island and with a lot of time on your hands, you decide to disassemble your glasses to make a crude telescope with which you can scan the horizon for rescuers. Luckily youre farsighted, and, like most people, your two eyes have different lens prescriptions. Your left eye uses a lens of power +4.5 D, and your right eyes lens is +3.0 D. a. Which lens should you use for the objective and which for the eyepiece? Explain. b. What will be the magnification of your telescope? c. How far apart should the two lenses be when you focus on distant objects?

A microscope with a tube length of 180 mm achieves a total magnification of 800* with a 40* objective and a 20* eyepiece. The microscope is focused for viewing with a relaxed eye. How far is the sample from the objective lens?

Modern microscopes are more likely to use a camera than human viewing. This is accomplished by replacing the eyepiece in Figure 35.14 with a photo-ocular that focuses the image of the objective to a real image on the sensor of a digital camera. A typical sensor is 22.5 mm wide and consists of 5625 4.0@mm@ wide pixels. Suppose a microscopist pairs a 40* objective with a 2.5* photo-ocular. a. What is the field of view? That is, what width on the microscope stage, in mm, fills the sensor? b. The photo of a cell is 120 pixels in diameter. What is the cells actual diameter, in mm?

White light is incident onto a 30 prism at the 40 angle shown in FIGURE P35.41. Violet light emerges perpendicular to the rear face of the prism. The index of refraction of violet light in this glass is 2.0% larger than the index of refraction of red light. At what angle f does red light emerge from the rear face?

A beam of white light enters a transparent material. Wavelengths for which the index of refraction is n are refracted at angle u2. Wavelengths for which the index of refraction is n + dn, where dn V n, are refracted at angle u2 + du. a. Show that the angular separation of the two wavelengths, in radians, is du = -1dn/n2 tan u2. b. A beam of white light is incident on a piece of glass at 30.0. Deep violet light is refracted 0.28 more than deep red light. The index of refraction for deep red light is known to be 1.552. What is the index of refraction for deep violet light?

High-power lasers are used to cut and weld materials by focusing the laser beam to a very small spot. This is like using a magnifying lens to focus the suns light to a small spot that can burn things. As an engineer, you have designed a laser cutting device in which the material to be cut is placed 5.0 cm behind the lens. You have selected a high-power laser with a wavelength of 1.06 mm. Your calculations indicate that the laser must be focused to a 5.0@mm@diameter spot in order to have sufficient power to make the cut. What is the minimum diameter of the lens you must install?

Once dark adapted, the pupil of your eye is approximately 7 mm in diameter. The headlights of an oncoming car are 120 cm apart. If the lens of your eye is diffraction limited, at what distance are the two headlights marginally resolved? Assume a wavelength of 600 nm and that the index of refraction inside the eye is 1.33. (Your eye is not really good enough to resolve headlights at this distance, due both to aberrations in the lens and to the size of the receptors in your retina, but it comes reasonably close

The resolution of a digital camera is limited by two factors: diffraction by the lens, a limit of any optical system, and the fact that the sensor is divided into discrete pixels. Consider a typical point-and-shoot camera that has a 20-mm-focal-length lens and a sensor with 2.5@mm@wide pixels. a. First, assume an ideal, diffractionless lens. At a distance of 100 m, what is the smallest distance, in cm, between two point sources of light that the camera can barely resolve? In answering this question, consider what has to happen on the sensor to show two image points rather than one. You can use s = f because s W f. b. You can achieve the pixel-limited resolution of part a only if the diffraction width of each image point is no greater than 1 pixel in diameter. For what lens diameter is the minimum spot size equal to the width of a pixel? Use 600 nm for the wavelength of light. c. What is the f-number of the lens for the diameter you found in part b? Your answer is a quite realistic value of the f-number at which a camera transitions from being pixel limited to being diffraction limited. For f-numbers smaller than this (larger-diameter apertures), the resolution is limited by the pixel size and does not change as you change the aperture. For f-numbers larger than this (smaller-diameter apertures), the resolution is limited by diffraction, and it gets worse as you stop down to smaller apertures.

The Hubble Space Telescope has a mirror diameter of 2.4 m. Suppose the telescope is used to photograph stars near the center of our galaxy, 30,000 light years away, using red light with a wavelength of 650 nm. a. Whats the distance (in km) between two stars that are marginally resolved? The resolution of a reflecting telescope is calculated exactly the same as for a refracting telescope. b. For comparison, what is this distance as a multiple of the distance of Jupiter from the su

Alpha Centauri, the nearest star to our solar system, is 4.3 light years away. Assume that Alpha Centauri has a planet with an advanced civilization. Professor Dhg, at the planets Astronomical Institute, wants to build a telescope with which he can find out whether any planets are orbiting our sun. a. What is the minimum diameter for an objective lens that will just barely resolve Jupiter and the sun? The radius of Jupiters orbit is 780 million km. Assume l = 600 nm. b. Building a telescope of the necessary size does not appear to be a major problem. What practical difficulties might prevent Professor Dhgs experiment from succeeding?

Your task in physics laboratory is to make a microscope from two lenses. One lens has a focal length of 2.0 cm, the other 1.0 cm. You plan to use the more powerful lens as the objective, and you want the eyepiece to be 16 cm from the objective. a. For viewing with a relaxed eye, how far should the sample be from the objective lens? b. What is the magnification of your microscope?

The lens shown in FIGURE CP35.49 is called an achromatic doublet, meaning that it has no chromatic aberration. The left side is flat, and all other surfaces have radii of curvature R. a. For parallel light rays coming from the left, show that the effective focal length of this two-lens system is f = R/12n2 - n1 - 12, where n1 and n2 are, respectively, the indices of refraction of the diverging and the converging lenses. Dont forget to make the thin-lens approximation. b. Because of dispersion, either lens alone would focus red rays and blue rays at different points. Define n1 and n2 as nblue - nred for the two lenses. What value of the ratio n1 /n2 makes fblue = fred for the two-lens system? That is, the two-lens system does not exhibit chromatic aberration. c. Indices of refraction for two types of glass are given in the table. To make an achromatic doublet, which glass should you use for the converging lens and which for the diverging lens? Explain. nblue nred Crown glass 1.525 1.517 Flint glass 1.632 1.616 d. What value of R gives a focal length of 10.0 cm?

FIGURE CP35.50 shows a simple zoom lens in which the magnitudes of both focal lengths are f. If the spacing d 6 f, the image of the converging lens falls on the right side of the diverging lens. Our procedure of letting the image of the first lens act as the object of the second lens will continue to work in this case if we use a negative object distance for the second lens. This is called a virtual object. Consider an object very far to the left 1s 2 of the converging lens. Define the effective focal length as the distance from the midpoint between the lenses to the final image. a. Show that the effective focal length is feff = f 2 - fd + 1 2 d2 d b. What is the zoom for a lens that can be adjusted from d = 1 2 f to d = 1 4 f ?

What are the overarching findings of Part VII?

What laws of physics govern optics?

What are the most important models of Part VII?

What are the most important tools introduced in Part VII?