The desired overall magnication of a compound microscope

A camera used by a professional photographer to shoot portraits has a focal length of 25.0 cm. The photographer takes a portrait of a person 1.50 m in front of the camera. Where is the image formed, and what is the lateral magnication?

The lens of a certain 35 mm camera (35 mm is the width of the lm strip) has a focal length of 55 mm and a speed (an f-number) of f/1.8. Determine the diameter of the lens.

A photographic image of a building is 0.092 0 m high. The image was made with a lens with a focal length of 52.0 mm. If the lens was 100 m from the building when the photograph was made, determine the height of the building.

The full Moon is photographed using a camera with a 120-mm-focal-length lens. Determine the diameter of the Moons image on the lm. [Note: The radius of the Moon is 1.74  106 m, and the distance from the Earth to the Moon is 3.84  108 m.]

A camera is being used with the correct exposure at f/4 and a shutter speed of 1/32 s. In order to stop a fastmoving subject, the shutter speed is changed to 1/256 s. Find the new f-stop that should be used to maintain satisfactory exposure, assuming no change in lighting conditions.

(a) Use conceptual arguments to show that the intensity of light (energy per unit area per unit time) reaching the lm in a camera is proportional to the square of the reciprocal of the f-number, as (b) The correct exposure time for a camera set to f/1.8 is (1/500) s. Calculate the correct exposure time if the f-number is changed to f/4 under the same lighting conditions.

A certain type of lm requires an exposure time of 0.010 s with an f/11 lens setting. Another type of lm requires twice the light energy to produce the same level of exposure. What f-stop does the second type of lm need with the 0.010-s exposure time?

Assume that the camera in Figure 25.1 has a xed focal length of 65.0 mm and is adjusted to properly focus the image of a distant object. How far and in what direction must the lens be moved to focus the image of an object that is 2.00 m away?

A retired bank president can easily read the ne print of the nancial page when the newspaper is held no closer than arms length, 60.0 cm from the eye. What should be the focal length of an eyeglass lens that will allow her to read at the more comfortable distance of 24.0 cm?

A person has far points 84.4 cm from the right eye and 122 cm from the left eye. Write a prescription for the powers of the corrective lenses.

The accommodation limits for Nearsighted Nicks eyes are 18.0 cm and 80.0 cm. When he wears his glasses, he is able to see faraway objects clearly. At what minimum distance is he able to see objects clearly?

The near point of an eye is 100 cm. A corrective lens is to be used to allow this eye to clearly focus on objects 25.0 in front of it. (a) What should be the focal length of the lens? (b) What is the power of the needed corrective lens?

An individual is nearsighted; his near point is 13.0 cm and his far point is 50.0 cm. (a) What lens power is needed to correct his nearsightedness? (b) When the lenses are in use, what is this persons near point?

A certain childs near point is 10.0 cm; her far point (with eyes relaxed) is 125 cm. Each eye lens is 2.00 cm from the retina. (a) Between what limits, measured in diopters, does the power of this lenscornea combination vary? (b) Calculate the power of the eyeglass lens the child should use for relaxed distance vision. Is the lens converging or diverging?

An articial lens is implanted in a persons eye to replace a diseased lens. The distance between the articial lens and the retina is 2.80 cm. In the absence of the lens, an image of a distant object (formed by refraction at the cornea) falls 2.53 cm behind the retina. The lens is designed to put the image of the distant object on the retina. What is the power of the implanted lens? [Hint: Consider the image formed by the cornea to be a virtual object.]

A person is to be tted with bifocals. She can see clearly when the object is between 30 cm and 1.5 m from the eye. (a) The upper portions of the bifocals (Fig. P25.16) should be designed to enable her to see distant objects clearly. What power should they have? (b) The lower portions of the bifocals should enable her to see objects comfortably at 25 cm. What power should they have?

A stamp collector uses a lens with 7.5-cm focal length as a simple magnier. The virtual image is produced at the normal near point (25 cm). (a) How far from the lens should the stamp be placed? (b) What is the expected angular magnication?

A lens having a focal length of 25 cm is used as a simple magnier. (a) What is the angular magnication obtained when the image is formed at the normal near point (q  25 cm)? (b) What is the angular magnication produced by this lens when the eye is relaxed?

A biology student uses a simple magnier to examine the structural features of the wing of an insect. The wing is held 3.50 cm in front of the lens, and the image is formed 25.0 cm from the eye. (a) What is the focal length of the lens? (b) What angular magnication is achieved?

A lens that has a focal length of 5.00 cm is used as a magnifying glass. (a) To obtain maximum magnication, where should the object be placed? (b) What is the magnication?

A leaf of length h is positioned 71.0 cm in front of a converging lens with a focal length of 39.0 cm. An observer views the image of the leaf from a position 1.26 m behind the lens, as shown in Figure P25.21. (a) What is the magnitude of the lateral magnication (the ratio of the image size to the object size) produced by the lens? (b) What angular magnication is achieved by viewing the image of the leaf rather than viewing the leaf directly?

The objective lens in a microscope with a 20.0-cm-long tube has a magnication of 50.0, and the eyepiece has a magnication of 20.0. What are the focal lengths of (a) the objective and (b) the eyepiece? (c) What is the overall magnication of the microscope?

The desired overall magnication of a compound microscope is 140. The objective alone produces a lateral magnication of 12. Determine the required focal length of the eyepiece.

A microscope has an objective lens with a focal length of 16.22 mm and an eyepiece with a focal length of 9.50 mm. With the length of the barrel set at 29.0 cm, the diameter of a red blood cells image subtends an angle of 1.43 mrad with the eye. If the nal image distance is 29.0 cm from the eyepiece, what is the actual diameter of the red blood cell?

The length of a microscope tube is 15.0 cm. The focal length of the objective is 1.00 cm, and the focal length of the eyepiece is 2.50 cm. What is the magnication of the microscope, assuming it is adjusted so that the eye is relaxed? [Hint: To solve this question go back to basics and use the thin lens equation.]

A certain telescope has an objective of focal length 1 500 cm. If the Moon is used as an object, a 1.0-cm-long image formed by the objective corresponds to what distance, in miles, on the Moon? Assume 3.8  108 m for the EarthMoon distance.

The lenses of an astronomical telescope are 92 cm apart when adjusted for viewing a distant object with minimum eyestrain. The angular magnication produced by the telescope is 45. Compute the focal length of each lens.

An elderly sailor is shipwrecked on a desert island, but manages to save his eyeglasses. The lens for one eye has a power of 1.20 diopters, and the other lens has a power of 9.00 diopters. (a) What is the magnifying power of the telescope he can construct with these lenses? (b) How far apart are the lenses when the telescope is adjusted for minimum eyestrain?

Astronomers often take photographs with the objective lens or mirror of a telescope alone, without an eyepiece. (a) Show that the image size h for a telescope used in this manner is given by h fh/( f  p), where h is the object size, f is the objective focal length, and p is the object distance. (b) Simplify the expression in part (a) if the object distance is much greater than the objective focal length. (c) The wingspan of the International Space Station is 108.6 m, the overall width of its solar panel conguration. When it is orbiting at an altitude of 407 km, nd the width of the image formed by a telescope objective of focal length 4.00 m.

Galileo devised a simple terrestrial telescope that produces an upright image. It consists of a converging objective lens and a diverging eyepiece at opposite ends of the telescope tube. For distant objects, the tube length is the objective focal length less the absolute value of the eyepiece focal length. (a) Does the user of the telescope see a real or virtual image? (b) Where is the nal image? (c) If a telescope is to be constructed with a tube of length 10.0 cm and a magnication of 3.00, what are the focal lengths of the objective and eyepiece?

A person decides to use an old pair of eyeglasses to make some optical instruments. He knows that the near point in his left eye is 50.0 cm and the near point in his right eye is 100 cm. (a) What is the maximum angular magnication he can produce in a telescope? (b) If he places the lenses 10.0 cm apart, what is the maximum overall magnication he can produce in a microscope? (Go back to basics and use the thin-lens equation to solve part (b).)

If the distance from the Earth to the Moon is 3.8  108 m, what diameter would be required for a telescope objective to resolve a Moon crater 300 m in diameter? Assume a wavelength of 500 nm.

A converging lens with a diameter of 30.0 cm forms an image of a satellite passing overhead. The satellite has two green lights (wavelength 500 nm) spaced 1.00 m apart. If the lights can just be resolved according to the Rayleigh criterion, what is the altitude of the satellite?

The pupil of a cats eye narrows to a vertical slit of width 0.500 mm in daylight. What is the angular resolution for a pair of horizontally separated mice? (Use 500-nm light in your calculation.)

To increase the resolving power of a microscope, the object and the objective are immersed in oil (n  1.5). If the limiting angle of resolution without the oil is 0.60  rad, what is the limiting angle of resolution with the oil? [Hint: The oil changes the wavelength of the light.]

(a) Calculate the limiting angle of resolution for the eye, assuming a pupil diameter of 2.00 mm, a wavelength of 500 nm in air, and an index of refraction for the eye of 1.33. (b) What is the maximum distance from the eye at which two points separated by 1.00 cm could be resolved?

Two stars in a binary system are 8.0 lightyears away from the observer and can just be resolved by a 20-in. telescope equipped with a lter that allows only light of wavelength 500 nm to pass. What is the distance between the two stars?

A spy satellite circles the Earth at an altitude of 200 km and carries out surveillance with a special high-resolution telescopic camera having a lens diameter of 35 cm. If the angular resolution of this camera is limited by diffraction, estimate the separation of two small objects on the Earths surface that are just resolved in yellow-green light (   550 nm).

Suppose a 5.00-m-diameter telescope were constructed on the Moon, where the absence of atmospheric distortion would permit excellent viewing. If observations were made using 500-nm light, what minimum separation between two objects could just be resolved on Mars at closest approach (when Mars is 8.0  107 km from the Moon)?

The H  line in hydrogen has a wavelength of 656.20 nm. This line differs in wavelength from the corresponding spectral line in deuterium (the heavy stable isotope of hydrogen) by 0.18 nm. (a) Determine the minimum number of lines a grating must have to resolve these two wavelengths in the rst order. (b) Repeat part (a) for the second order.

A 15.0-cm-long grating has 6 000 slits per centimeter. Can two lines of wavelengths 600.000 nm and 600.003 nm be separated with this grating? Explain.

Light of wavelength 550 nm is used to calibrate a Michelson interferometer. With the use of a micrometer screw, the platform on which one mirror is mounted is moved 0.180 mm. How many fringe shifts are counted?

An interferometer is used to measure the length of a bacterium. The wavelength of the light used is 650 nm. As one arm of the interferometer is moved from one end of the cell to the other, 310 fringe shifts are counted. How long is the bacterium?

Mirror M1 in Figure 25.15 is displaced a distance L. During this displacement, 250 fringe shifts are counted. The light being used has a wavelength of 632.8 nm. Calculate the displacement L.

A thin sheet of transparent material has an index of refraction of 1.40 and is 15.0  m thick. When it is inserted in the light path along one arm of an interferometer, how many fringe shifts occur in the pattern? Assume that the wavelength (in a vacuum) of the light used is 600 nm. [Hint: The wavelength will change within the material.]

The Michelson interferometer can be used to measure the index of refraction of a gas by placing an evacuated transparent tube in the light path along one arm of the device. Fringe shifts occur as the gas is slowly added to the tube. Assume that 600-nm light is used, that the tube is 5.00 cm long, and that 160 fringe shifts occur as the pressure of the gas in the tube increases to atmospheric pressure. What is the index of refraction of the gas? [Hint: The fringe shifts occur because the wavelength of the light changes inside the gas-lled tube.]

The light path in one arm of a Michelson interferometer includes a transparent cell that is 5.00 cm long. How many fringe shifts would be observed if all the air were evacuated from the cell? The wavelength of the light source is 590 nm and the refractive index of air is 1.000 29. (See the hint in Problem 46.)

A person with a nearsighted eye has near and far points of 16 cm and 25 cm, respectively. (a) Assuming a lens is placed 2.0 cm from the eye, what power must the lens have to correct this condition? (b) Suppose that contact lenses placed directly on the cornea are used to correct the persons eye. What is the power of the lens required in this case, and what is the new near point? [Hint: The contact lens and the eyeglass lens require slightly different powers because they are at different distances from the eye.]

The near point of an eye is 75.0 cm. (a) What should be the power of a corrective lens prescribed to enable the eye to see an object clearly at 25.0 cm? (b) If, using the corrective lens, the person can see an object clearly at 26.0 cm, but not at 25.0 cm, by how many diopters did the lens grinder miss the prescription?

If a typical eyeball is 2.00 cm long and has a pupil opening that can range from about 2.00 mm to 6.00 mm, what are (a) the focal length of the eye when it is focused on objects 1.00 m away, (b) the smallest f-number of the eye when it is focused on objects 1.00 m away, and (c) the largest f-number of the eye when it is focused on objects 1.00 m away?

A cataract-impaired lens in an eye may be surgically removed and replaced by a manufactured lens. The focal length required for the new lens is determined by the lens-to-retina distance, which is measured by a sonarlike device, and by the requirement that the implant provide for correct distance vision. (a) If the distance from lens to retina is 22.4 mm, calculate the power of the implanted lens in diopters. (b) Since there is no accommodation and the implant allows for correct distance vision, a corrective lens for close work or reading must be used. Assume a reading distance of 33.0 cm, and calculate the power of the lens in the reading glasses.

Estimate the minimum angle subtended at the eye of a hawk ying at an altitude of 50 m necessary to recognize a mouse on the ground.

The wavelengths of the sodium spectrum are  1  589.00 nm and  2  589.59 nm. Determine the minimum number of lines in a grating that will allow resolution of the sodium spectrum in (a) the rst order and (b) the third order.

The text discusses the astronomical telescope. Another type is the Galilean telescope, in which an objective lens gathers light (Fig. P25.54) and tends to form an image at point A. An eyepiece consisting of a diverging lens intercepts the light before it comes to a focus and forms a virtual image at point B. When adjusted for minimum eyestrain, B is an innite distance in front of the lens and parallel rays emerge from the lens, as in Figure P25.54b. An opera glass, which is a Galilean telescope, is used to view a 30.0-cm-tall singers head that is 40.0 m from the objective lens. The focal length of the objective is 8.00 cm, and that of the eyepiece is 2.00 cm. The telescope is adjusted so parallel rays enter the eye. Compute (a) the size of the real image that would have been formed by the objective, (b) the virtual object distance for the diverging lens, (c) the distance between the lenses, and (d) the overall angular magnication.

A laboratory (astronomical) telescope is used to view a scale that is 300 cm from the objective, which has a focal length of 20.0 cm; the eyepiece has a focal length of 2.00 cm. Calculate the angular magnication when the telescope is adjusted for minimum eyestrain. [Note: The object is not at innity, so the simple expression m  fo/fe is not sufciently accurate for this problem. Also, assume small angles, so that tan

If the aqueous humor of the eye has an index of refraction of 1.34 and the distance from the vertex of the cornea to the retina is 2.00 cm, what is the radius of curvature of the cornea for which distant objects will be focused on the retina? (For simplicity, assume that all refraction occurs in the aqueous humor.)

A boy scout starts a re by using a lens from his eyeglasses to focus sunlight on kindling 5.0 cm from the lens. The boy scout has a near point of 15 cm. When the lens is used as a simple magnier, (a) what is the maximum magnication that can be achieved, and (b) what is the magnication when the eye is relaxed? [Caution: The equations derived in the text for a simple magnier assume a normal eye.]

(a) Move this book toward your face until the letters just begin to blur. The distance from the book to your eye is your near point. (b) On a sheet of paper, make a dot near the center. Then place an x about 3 inches to the left of the dot and another x about 3 inches to the right of the dot. With one eye shut and while looking at the dot, move the paper slowly toward your eye. You will notice that a certain distance from your eye, one of the xs will disappear. This is the location of the blind spot of your eyethe point where the optic nerve enters the eye. (c) Stand before a mirror in a darkened room for a few minutes. Then turn on a light in the room and observe your pupils in the mirror as they change size. Such adaptation to the dark also takes place at the rods and cones as they chemically adjust their sensitivity. This adjustment takes 1530 minutes, as you may have noted whenever you entered a darkened movie theater. Iris aperture control takes less than a second and helps protect the retina from overload.

On a sunny day, hold a magnifying glass above a nonammable surface, such as a sidewalk, so the image of the Sun forms a round spot of light on the surface. Note where the spot formed by the lens is most distinct, or smallest. Use a ruler to measure the distance between the glass and the image. The distance is equal to the focal length of the lens.

Hold a pair of prescription glasses about 12 cm from your eye, and look at different objects through the lenses. Try this with different types of glasses, such as those for farsightedness and nearsightedness, and describe what effects the differences have on the image you see. If you have bifocals, how do the images produced by the top and bottom portions of the bifocal lens compare?

If you have never experimented with a 35-mm camera with adjustable f-numbers and shutter speeds, use up a couple of rolls of lm to see what happens. Take several shots of the same object with different settings for these two variables. (You should record your f-numbers and shutter speeds for each photograph.) Explain any differences you see in the nal images in terms of the settings used.