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Chapter 36: Q97P (page 1115)

A spy satellite orbiting at 160 km above Earth鈥檚 surface has a lens with a focal length of 3.6 m and can resolve objects on the ground as small as 30 cm. For example, it can easily measure the size of an aircraft鈥檚 air intake port. What is the effective diameter of the lens as determined by diffraction consideration alone? Assume 位=550nm.

Short Answer

Expert verified

The effective diameter of the lens is 0.36 m.

Step by step solution

01

Describe the diffraction by a circular aperture or a lens

The expression to calculate the effective diameter of the lens is given by,

sin胃=1.22位d

Here, 位 is the wavelength, d is diameter, and 胃 is angle.

Let D be the distance between two objects, and L be the distance between Earth and satellite. Then,

L胃=D胃=DL

From the above equation,

胃=1.22位dDL=1.22位dd=1.22位LD 鈥�.(1)

02

Determine the effective diameter of the lens

Substitute 550脳10-9m for 位, 160脳103m for L, and 0.30m for D in equation (1).

d=1.22550脳10-9m160脳103m0.30m=0.36m

Therefore, the effective diameter of the lens is 0.36 m.

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Most popular questions from this chapter

The full width at half-maximum (FWHM) of a central diffraction maximum is defined as the angle between the two points in the pattern where the intensity is one-half that at the center of the pattern. (See Fig.36-8b.) (a) Show that the intensity drops to one-half the maximum value when sin2伪=伪22. (b) Verify that 伪=1.39rad. (about 80掳) is a solution to the transcendental equation of (a). (c) Show that the FWHM is 螖胃=2sin-10.443位/awhere a is the slit width. Calculate the FWHM of the central maximum for slit width (d) 1.00位 ,(e) 5.00位,and (f) 10.00位.

In Fig. 36-48, let a beam of x-rays of wavelength 0.125 nm be incident on an NaCl crystal at angle 胃 = 45.0掳 to the top face of the crystal and a family of reflecting planes. Let the reflecting planes have separation d = 0.252 nm. The crystal is turned through angle 蠒 around an axis perpendicular to the plane of the page until these reflecting planes give diffraction maxima. What are the (a) smaller and (b) larger value of 蠒 if the crystal is turned clockwise and the (c) smaller and (d) larger value of 蠒 if it is turned counter-clockwise

A circular obstacle produces the same diffraction pattern as a circular hole of the same diameter (except very near u 0).Airborne water drops are examples of such obstacles. When you see the Moon through suspended water drops, such as in a fog, you intercept the diffraction pattern from many drops. The composite of the central diffraction maxima of those drops forms a white region that surrounds the Moon and may obscure it. Figure 36-43 is a photograph in which the Moon is obscured. There are two faint, colored rings around the Moon (the larger one may be too faint to be seen in your copy of the photograph). The smaller ring is on the outer edge of the central maxima from the drops; the somewhat larger ring is on the outer edge of the smallest of the secondary maxima from the drops (see Fig. 36-10).The color is visible because the rings are adjacent to the diffraction minima (dark rings) in the patterns. (Colors in other parts of the pattern overlap too much to be visible.) (a) What is the color of these rings on the outer edges of the diffraction maxima? (b) The colored ring around the central maxima in Fig. 36-43 has an angular diameter that is 1.35 times the angular diameter of the Moon, which is 0.50掳. Assume that the drops all have about the same diameter. Approximately what is that diameter?

In three arrangements, you view two closely spaced small objects that are the same large distance from you. The angles that the objects occupy in your field of view and their distances from you are the following: (1) 2蠒and ; (2) 2蠒and 2R; (3) 蠒/2and R/2. (a) Rank the arrangements according to the separation between the objects, with the greatest separation first. If you can just barely resolve the two objects in arrangement 2, can you resolve them in (b) arrangement 1 and (c) arrangement 3?

If Superman really had x-ray vision at 0.10nm wavelength and a 4.0mm pupil diameter, at what maximum altitude could he distinguish villains from heroes, assuming that he needs to resolve points separated by 5.0cm to do this?
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