Solved: Reading glasses of what power are needed for a

Problem 8Q Is the image formed on the retina of the human eye upright or inverted? Discuss the implications of this for our perception of objects.

Problem 9Q The human eye is much like a camera—yet, when a camera shutter is left open and the camera is moved, the image will be blurred. But when you move your head with your eyes open, you still see clearly. Explain.

Problem 10P (I) A human eyeball is about 2.0 cm long and the pupil has a maximum diameter of about 8.0 mm. What is the "speed" of this lens?

.Problem 10Q Reading glasses use converging lenses. A simple magnifier is also a converging lens. Are reading glasses therefore magnifiers? Discuss the similarities and differences between converging lenses as used for these two different purposes.

Problem 11Q Inexpensive microscopes for children's use usually produce images that are colored at the edges. Why?

Problem 13Q Which aberrations present in a simple lens are not present (or are greatly reduced) in the human eye?

(II) A person struggles to read by holding a book at arm's length, a distance of 45 cm away (= near point). What power of reading glasses should be prescribed for him, assuming they will be placed 2.0 cm from the eye and she wants to read at the normal near point of 25 cm?

Spherical aberration in a thin lens is minimized if rays are bent equally by the two surfaces. If a planoconvex lens is used to form a real image of an object at infinity, which surface should face the object? Use ray diagrams to show why.

Problem 14Q Explain why chromatic aberration occurs for thin lenses but not for mirrors.

By what factor can you improve resolution, other things being equal, if you use blue light \((\lambda=450 \mathrm{~nm})\) rather than red (700 nm)?

(II) About how much longer is the nearsighted eye in Example 25-6 than the \(2.0 \mathrm{~cm}\) of a normal eye?

Problem 17Q Which color of visible light would give the best resolution in a microscope? Explain. 10-6 cm. (I = 450 nm)

Problem 18Q Atoms have diameters of about Can visible light be used to "see" an atom? Explain.

(II) The Earth and Moon are separated by about \(400 \times 10^{6} \mathrm{~m}\). When Mars is \(8 \times 10^{10} \mathrm{~m}\) from Earth, could a person standing on Mars resolve the Earth and its Moon as two separate objects without a telescope? Assume a pupil diameter of \(5 \mathrm{~mm} \text { and } \lambda=550 \mathrm{~mm}\). Equation Transcription: Text Transcription: 400 x 10^6m 8 x 10^10m 5 mm and \lambda=550 mm

Problem 58GP Sam purchases +3.50-D eyeglasses which correct his faulty vision to put his near point at 25 cm. (Assume he wears the lenses 2.0 cm from his eyes.) Calculate (a) the focal length of Sam's glasses, (b) Sam's near point without glasses, (c) Pam, who has normal eyes with near point at 25 cm, puts on Sam's glasses. Calculate Pam's near point with Sam's glasses on.

Problem 64GP What is the magnifying power of a lens used as a +4.0-D magnifier? Assume a relaxed normal eye.

Problem 70GP The Hubble Space Telescope, with an objective diameter of 2.4 m, is viewing the Moon. Estimate the minimum distance between two objects on the Moon that the Hubble can distinguish. Consider diffraction of light with wavelength 550 nm. Assume the Hubble is near the Earth.

Problem 73GP You want to design a spy satellite to photograph license plate numbers. Assuming it is necessary to resolve points separated by 5 cm with 550-nm light, and that the satellite orbits at a height of 130 km, what minimum lens aperture (diameter) is required?

Problem 1Q Why is the depth of field greater, and the image sharper, when a camera lens is "stopped down" to a larger f-number? Ignore diffraction.

Problem 2Q Describe how diffraction affects the statement of Question 2. [Hint: See Eq. 24-3 or 25-7.]

Problem 3Q Why must a camera lens be moved farther from the sensor or film to focus on a closer object?

Problem 5Q Will a nearsighted person who wears corrective lenses in her glasses be able to see clearly underwater when wearing those glasses? Use a diagram to show why or why not.

Problem 4Q are bifocals needed mainly by older persons and not generally by younger people?

You can tell whether a person is nearsighted or farsighted by looking at the width of the face through their glasses. If the person's face appears narrower through the glasses (Fig. ), is the person farsighted or nearsighted?

In attempting to discern distant details, people will sometimes squint. Why does this help?

Problem 13P (II) A person has a far point of 14 cm. What power glasses would correct this vision if the glasses were placed 2.0 cm from the eye? What power contact lenses, placed on the eye, would the person need?

Problem 1P A 55-mm-focal-length lens has f-stops ranging from f/l.4 to f/22. What is the corresponding range of lens diaphragm diameters?

A television camera lens has a 14-cm focal length and a lens diameter of 6.0 cm. What is its f-number?

(I) A light meter reports that a camera setting of \(\frac{1}{250}\mathrm s\) at \(f/5.6\) will give a correct exposure. But the photographer wishes to use \(f/11\) to increase the depth of field. What should the shutter speed be? Equation Transcription: Text Transcription: {1 over 250}s f/5.6 f/11

(I) A properly exposed photograph is taken at \(f/16\) and \(\frac{1}{60}\mathrm s\). What lens opening would be required if the shutter speed were \(\frac{1}{1000}\mathrm s\)? ________________ Equation Transcription: Text Transcription: f/16 {1 over 60}s {1 over 1000}s

Problem 5P If an f = 135-mm telephoto lens is designed to cover object distances from 1.2 m to ?, over what distance must the lens move relative to the plane of the film?

Problem 6P A 200-mm-focal-length lens can be adjusted so that it is 200.0 mm to 206.0 mm from the film. For what range of object distances can it be adjusted?

(II) A nature photographer wishes to photograph a 28-m-tall tree from a distance of 58 m. What focal-length lens should be used if the image is to fill the 24-mm height of the film?

(II) A “pinhole” camera uses a tiny pinhole instead of a lens. Show, using ray diagrams, how reasonably sharp images can be formed using such a pinhole camera. In particular, consider two point objects 2.0 cm apart that are 1.0 m from a 1.0-mm-diameter pinhole. Show that on a piece of film 7.0 cm behind the pinhole the two objects produce two separate circles that do not overlap.

(III) Suppose that a correct exposure is \(\frac{1}{250}\mathrm s\) at \(f/11\). Under the same conditions, what exposure time would be needed for a pinhole camera (Problem 8) if the pinhole diameter is 1.0 mm and the film is 7.0 cm from the hole? ________________ Equation Transcription: Text Transcription: 1 over 250 f/11

(I) If the nearsighted person in Example 25–6 wore contact lenses corrected for the far point \((=~\infty)\), show that the near point would be 41 cm. (Would glasses be better in this case?) ________________ Equation Transcription: Text Transcription: (= infinity)

Problem 12P Reading glasses of what power are needed for a person whose near point is 115 cm, so that he can read a computer screen at 55 cm? Assume a lens–eye distance of 1.8 cm.

A person’s left eye is corrected by a ?3.50-diopter lens, 2.0 cm from the eye. (a) Is this person’s left eye near- or farsighted? (b) What is this eye’s far point without glasses?

Problem 16P A person’s right eye can see objects clearly only if they are between 25 cm and 75 cm away. (a) What power of contact lens is required so that objects far away are sharp? (b) What will be the near point with the lens in place?

Give at least two advantages for the use of large reflecting mirrors in astronomical telescopes.

(II) One lens of a nearsighted person’s eyeglasses has a focal length of ?22.0 cm, and the lens is 1.8 cm from the eye. If the person switches to contact lenses that are placed directly on the eye, what should be the focal length of the corresponding contact lens?

Problem 19P What is the focal length of the eye-lens system when viewing an object (a) at infinity, and (b) 33 cm from the eye? Assume that the lens-retina distance is 2.0 cm.

(III) A nearsighted person has near and far points of 10.0 and \(20.0 \mathrm{~cm}\), respectively. If she puts on contact lenses with power \(P=-4.00 \mathrm{D}\), what are her new near and far points?

Problem 21P What is the magnification of a lens used with a relaxed eye if its focal length is 12 cm?

Problem 22P What is the focal length of a magnifying glass of 3.5× magnification for a relaxed normal eye?

Problem 23P A magnifier is rated at 2.5× for a normal eye focusing on an image at the near point. (a) What is its focal length? (b) What is its focal length if the 2.5× refers to a relaxed eye?

Problem 24P Sherlock Holmes is using a 9.00-cm-focal-length lens as his magnifying glass. To obtain maximum magnification, where must the object be placed (assume a normal eye), and what will be the magnification?

Problem 25P A 3.30-mm-wide beetle is viewed with a 9.50-cm-focal-length lens. A normal eye views the image at its near point. Calculate (a) the angular magnification, (b) the width of the image, and (c) the object distance from the lens.

(II) A small insect is placed \(5.55 \mathrm{~cm}\) from a \(+6.00-\mathrm{cm}-\) focal-length lens. Calculate (a) the position of the image, and (b) the angular magnification.

Problem 27P Problem A magnifying glass with a focal length of 8.5 cm is used to read print placed at a distance of 7.5 cm. Calculate (a) the position of the image; (b) the angular magnification.

Problem 28P Problem A magnifying glass is rated at 3.0× for a normal eye that is relaxed. What would be the magnification of this lens for a relaxed eye whose near point is (a) 55 cm, and (b) 16 cm? Explain the differences.

Problem 29P What is the magnification of an astronomical telescope whose objective lens has a focal length of 76 cm and whose eyepiece has a focal length of 2.8 cm? What is the overall length of the telescope when adjusted for a relaxed eye?

Problem 30P The overall magnification of an astronomical telescope is desired to be 25×. If an objective of 78-cm focal length is used, what must be the focal length of the eyepiece? What is the overall length of the telescope when adjusted for use by the relaxed eye?

Problem 31P An 8.0× binocular has 2.8-cm-focal-length eyepieces. What is the focal length of the objective lenses?

Problem 32P An astronomical telescope has an objective with focal length 85 cm and a +35-D eyepiece. What is the total magnification?

Problem 33P An astronomical telescope has its two lenses spaced 75.2 cm apart. If the objective lens has a focal length of 74.5 cm, what is the magnification of this telescope? Assume a relaxed eye.

Problem 34P A Galilean telescope adjusted for a relaxed eye is 32.8 cm long. If the objective lens has a focal length of 36.0 cm, what is the magnification?

Problem 35P What is the magnifying power of an astronomical telescope using a reflecting mirror whose radius of curvature is 6.0 m and an eyepiece whose focal length is 3.2 cm?

Problem 36P The Moon’s image appears to be magnified 120× by a reflecting astronomical telescope with an eyepiece having a focal length of 3.2 cm. What are the focal length and radius of curvature of the main (objective) mirror?

Problem 37P A 170× astronomical telescope is adjusted for a relaxed eye when the two lenses are 1.25 m apart. What is the focal length of each lens?

(III) A reflecting telescope (Fig. 25–21 b) has a radius of curvature of 3.0 m for its objective mirror and a radius of curvature of ?1.50 m for its eyepiece mirror. If the distance between the two mirrors is 0.90 m, how far in front of the eyepiece should you place the photographic film to record the image of a star?

Problem 39P A microscope uses an eyepiece with a focal length of 1.40 cm. Using a normal eye with a final image at infinity, the tube length is 17.5 cm and the focal length of the objective lens is 0.65 cm. What is the magnification of the microscope?

Problem 41P A 17-cm-long microscope has an eyepiece with a focal length of 2.5 cm and an objective with a focal length of 0.28 cm. What is the approximate magnification?

Problem 40P A 620× microscope uses a 0.40-cm-focal-length objective lens. If the tube length is 17.5 cm, what is the focal length of the eyepiece? Assume a normal eye and that the final image is at infinity.

Problem 42P A microscope has a 12.0× eyepiece and a 59.0× objective lens 20.0 cm apart. Calculate (a) the total magnification, (b) the focal length of each lens, and (c) where the object must be for a normal relaxed eye to see it in focus.

Problem 43P A microscope has a 1.8-cm-focal-length eyepiece and a 0.80-cm objective lens. Assuming a relaxed normal eye, calculate (a) the position of the object if the distance between the lenses is 16.0 cm, and (b) the total magnification.

Problem 44P Repeat Problem 43 assuming that the final image is located 25 cm from the eyepiece (near point of a normal eye). Problem 43 A microscope has a 1.8-cm-focal-length eyepiece and a 0.80-cm objective lens. Assuming a relaxed normal eye, calculate (a) the position of the object if the distance between the lenses is 16.0 cm, and (b) the total magnification.

(III) The eyepiece of a compound microscope has a focal length of 2.70 cm, and the objective lens has f = 0.740 cm. If an object is placed 0.790 cm from the objective lens, calculate (a) the distance between the lenses when the microscope is adjusted for a relaxed eye, and (b) the total magnification.

Problem 46P An achromatic lens is made of two very thin lenses, placed in contact, that have focal lengths of f1 = ?28 cm and f2 = +23 cm. (a) Is the combination converging or diverging? (b) What is the net focal length?

(III) Let’s examine spherical aberration in a particular situation. A planoconvex lens of index of refraction 1.50 and radius of curvature \(R=12.0~\mathrm {cm}\) is shown in Fig. 25–48. Consider an incoming ray parallel to the principal axis and a height h above it as shown. Determine the distance d, from the flat face of the lens, to where this ray crosses the principal axis if \((a)\ h=1.0~\mathrm {cm}\), and \((b)\ h=6.0~\mathrm {cm}\). (c) How far apart are these “focal points”? (d) What is the radius of the “circle of least confusion” produced by the \(h=6.0 \text{-cm}\) ray at the “focal point” for \(h=1.0 \mathrm {\ cm}\)? ________________ Equation Transcription: Text Transcription: R=12.0 cm (a) h=1.0 cm (b) h=6.0 cm h=6.0-cm h=1.0 cm

Problem 48P What is the angular resolution limit (degrees) set by diffraction for the 100-in. (254-cm mirror diameter) Mt. Wilson telescope (? = 550 nm)?

Problem 49P Suppose that you wish to construct a telescope that can resolve features 7.0 km across on the Moon, 384,000 km away. You have a 2.0-m-focal-length objective lens whose diameter is 11.0 cm. What focal-length eyepiece is needed if your eye can resolve objects 0.10 mm apart at a distance of 25 cm? What is the resolution limit (radians) set by the size of the objective lens (that is, by diffraction)? Use ? = 550 nm.

Problem 50P The normal lens on a 35-mm camera has a focal length of 50.0 mm. Its aperture diameter varies from a maximum of 25 mm (f/2) to a minimum of 3.0 mm (f/16). Determine the resolution limit set by diffraction for f/2 and f/16. Specify as the number of lines per millimeter resolved on the film. Take ? = 550 nm.

Problem 51P Two stars 15 light-years away are barely resolved by a 55-cm (mirror diameter) telescope. How far apart are the stars? Assume ? = 550 nm and that the resolution is limited by diffraction.

Problem 52P (a) How far away can a human eye distinguish two car headlights 2.0 m apart? Consider only diffraction effects and assume an eye pupil diameter of 5.0 mm and a wavelength of 550 nm. (b) What is the minimum angular separation an eye could resolve when viewing two stars, considering only diffraction effects? In reality, it is about 1? of arc. Why is it not equal to your answer in (b)?

(II) X-rays of wavelength 0.133 nm fall on a crystal whose atoms, lying in planes, are spaced 0.280 nm apart. At what angle \(\phi\) (relative to the surface, Fig. 25–37) must the X-rays be directed if the first diffraction maximum is to be observed? ________________ Equation Transcription: ???? Text Transcription: phi

Problem 55P X-rays of wavelength 0.0973 nm are directed at an unknown crystal. The second diffraction maximum is recorded when the X-rays are directed at an angle of 23.4° relative to the crystal surface. What is the spacing between crystal planes?

Problem 56P First-order Bragg diffraction is observed at 25.2° related to the crystal surface, with spacing between atoms of 0.24 nm. (a) At what angle will second order be observed? (b) What is the wavelength of the X-rays?

(II) (a) Suppose for a conventional X-ray image that the X-ray beam consists of parallel rays. What would be the magnification of the image? (b) Suppose, instead, the X-rays come from a point source (as in Fig. 25–41) that is 15 cm in front of a human body 25 cm thick, and the film is pressed against the person’s back. Determine and discuss the range of magnifications that results.

As early morning passed toward midday, and the sunlight got more intense, a photographer noted that, if she kept her shutter speed constant, she had to change the f-number from f/5.6 to f/22. By what factor had the sunlight intensity increased during that time?

Problem 60GP Show that for objects very far away (assume infinity), the magnification of a camera lens is proportional to its focal length.

Problem 61GP For a camera equipped with a 50-mm-focal-length lens, what is the object distance if the image height equals the object height? How far is the object from the film?

Problem 62GP A woman can see clearly with her right eye only when objects are between 45 cm and 155 cm away. Prescription bifocals should have what powers so that she can see distant objects clearly (upper part) and be able to read a book 25 cm away (lower part) with her right eye? Assume that the glasses will be 2.0 cm from the eye.

Problem 63GP A child has a near point of 15 cm. What is the maximum magnification the child can obtain using an 8.0-cm-focal-length magnifier? What magnification can a normal eye obtain with the same lens? Which person sees more detail?

A physicist lost in the mountains tries to make a telescope using the lenses from his reading glasses. They have powers of \(+2.0 \mathrm{D}\) and \(+4.5 \mathrm{D}\), respectively. (a) What maximum magnification telescope is possible? (b) Which lens should be used as the eyepiece?

A 50-year-old man uses +2.5-diopter lenses to read a newspaper 25 cm away. Ten years later, he must hold the paper 35 cm away to see clearly with the same lenses. What power lenses does he need now in order to hold the paper 25 cm away? (Distances are measured from the lens.)

Problem 67GP Spy planes fly at extremely high altitudes (25 km) to avoid interception. Their cameras are reportedly able to discern features as small as 5 cm. What must be the minimum aperture of the camera lens to afford this resolution? (Use ? = 550 nm.)

Problem 68GP When shooting pictures at very short distances, exposure times must be increased because of the increased distance of the lens from the film for a focused image. (a) Show that when the object is so close to the camera that the image height equals the object height, the exposure time must be four times longer (or 2 f-stops) than when the object is a long distance away (say, ?), given the same illumination and f-stop. (b) Show that if do is at least four or five times the focal length f of the lens, the exposure time is increased by less than half an f-stop relative to the same object being a great distance away.

Problem 69GP The objective lens and the eyepiece of a telescope are spaced 85 cm apart. If the eyepiece is +23 diopters, what is the total magnification of the telescope?

Two converging lenses, one with f = 4.0 cm and the other with f = 44 cm, are made into a telescope. (a) What are the length and magnification? Which lens should be the eyepiece? (b) Assume these lenses are now combined to make a microscope; if the magnification needs to be \(25\times\), how long would the microscope be?

Problem 72GP An astronomical telescope has a magnification of 8.0. If the two lenses are 28 cm apart, determine the focal length of each lens.

A Lucite planoconvex lens (Fig. 23–29a) has one flat surface and the other has \(R=18.4\ \mathrm {cm}\). This lens is used to view an object, located 66.0 cm away from the lens, which is a mixture of red and yellow. The index of refraction of the glass is 1.5106 for red light and 1.5226 for yellow light. What are the locations of the red and yellow images formed by the lens? ________________ Equation Transcription: Text Transcription: R=18.4 cm