9. THINK A concave mirror has a positive value of focal length.

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1 9. THINK A concave mirror has a positive value o ocal length. EXPRESS For spherical mirrors, the ocal length is related to the radius o curvature r by r/2. The object distance p, the image distance i, and the ocal length are related by Eq. 34-4:. p i The value o i is positive or a real images, and negative or virtual images. The corresponding lateral magniication is m i / p. The value o m is positive or upright (not inverted) images, and negative or inverted images. Real images are ormed on the same side as the object, while virtual images are ormed on the opposite side o the mirror. ANALYZE (a) With = +2 cm and p = +8 cm, the radius o curvature is r = 2 = 2(2 cm) = + 24 cm. (b) The image distance is p (8 cm)(2 cm) i 36 cm. p 8 cm 2 cm (c) The lateral magniication is m = i/p = (36 cm)/(8 cm) = 2.0. (d) Since the image distance i is positive, the image is real (R). (e) Since the magniication m is negative, the image is inverted (I). () A real image is ormed on the same side as the object. LEARN The situation in this problem is similar to that illustrated in Fig. 34-0(c). The object is outside the ocal point, and its image is real and inverted.

2 . THINK A convex mirror has a negative value o ocal length. EXPRESS For spherical mirrors, the ocal length is related to the radius o curvature r by r/2. The object distance p, the image distance i, and the ocal length are related by Eq. 34-4:. p i The value o i is positive or a real images, and negative or virtual images. The corresponding lateral magniication is i m. p The value o m is positive or upright (not inverted) images, and negative or inverted images. Real images are ormed on the same side as the object, while virtual images are ormed on the opposite side o the mirror. ANALYZE (a) With = 0 cm and p = +8 cm, the radius o curvature is r = 2 = 20 cm. (b) The image distance is p (8 cm)( 0 cm) i 4.44 cm. p 8 cm ( 0) cm (c) The lateral magniication is m = i/p = (4.44 cm)/(8.0 cm) = (d) Since the image distance is negative, the image is virtual (V). (e) The magniication m is positive, so the image is upright [not inverted] (NI). () A virtual image is ormed on the opposite side o the mirror rom the object.

3 LEARN The situation in this problem is similar to that illustrated in Fig. 34-(c). The mirror is convex, and its image is virtual and upright.

4 45. Let the diameter o the Sun be d s and that o the image be d i. Then, Eq leads to

5 i d m d d d p p m m 3 i s s s.860 m.86 mm..500 m

6 47. THINK Our lens is o double-convex type. We apply lens maker s equation to analyze the problem. EXPRESS The lens maker s equation is given by Eq. 34-0: b g F HG n r r 2 where is the ocal length, n is the index o reraction, r is the radius o curvature o the irst surace encountered by the light and r 2 is the radius o curvature o the second surace. Since one surace has twice the radius o the other and since one surace is convex to the incoming light while the other is concave, set r 2 = 2r to obtain F HG ( n ) r 2r I KJ I KJ 3( n ). 2r ANALYZE (a) We solve or the smaller radius r : r 3( n ) 3( 5. )( 60 mm) 45 mm. 2 2 (b) The magnitude o the larger radius is r2 2r 90 mm. LEARN An image o an object can be ormed with a lens because it can bend the light rays, but the bending is possible only i the index o reraction o the lens is dierent rom that o its surrounding medium.

7 9. THINK This problem is about human eyes. We model the cornea and eye lens as a single eective thin lens, with image ormed at the retina. EXPRESS When the eye is relaxed, its lens ocuses ar-away objects on the retina, a distance i behind the lens. We set p = in the thin lens equation to obtain /i = /, where is the ocal length o the relaxed eective lens. Thus, i = = 2.50 cm. When the eye ocuses on closer objects, the image distance i remains the same but the object distance and ocal length change. ANALYZE (a) I p is the new object distance and ' is the new ocal length, then We substitute i = and solve or ':. p i p b p cmgb2. 50 cmg cm cm cm. (b) Consider the lens maker s equation b g F HG n r r 2 where r and r 2 are the radii o curvature o the two suraces o the lens and n is the index o reraction o the lens material. For the lens pictured in Fig , r and r 2 have about the same magnitude, r is positive, and r 2 is negative. Since the ocal length decreases, the combination (/r ) (/r 2 ) must increase. This can be accomplished by decreasing the magnitudes o both radii. LEARN When ocusing on an object near the eye, the lens bulges a bit (smaller radius o curvature), and its ocal length decreases. I KJ