Unformatted text preview: do2 (2) Substituting Equations (2) into Equation (1) yields
x = d i1 − d i2 = 1
1
1
1
−
=
−
= 0.011 m
1
1
1
1
1
1
1
1
−
−
−
−
f d o1 f d o2 0.2000 m 3.5 m 0.2000 m 50.0 m 55. REASONING AND SOLUTION
a. According to the thinlens equation, we have
111
1
1
=−
=
−
di f do −25 cm 38 cm or di = −15 cm b. The image is virtual since the image distance is negative.
______________________________________________________________________________ Chapter 26 Problems 1383 56. REASONING A converging lens must be used, because a diverging lens cannot produce a
real image.
Since the image is onehalf the size of the object and inverted relative to it, the image height
hi is related to the object height ho by hi = − 1 ho , where the minus sign indicates that the
2
image is inverted.
According to the magnification equation, Equation 26.7, the image distance di is related to
the object distance do by di / d o = − hi / ho .
But we know that hi / ho = − 1 , so
2 () di / d o = − − 1 = 1 .
2
2
SOLUTION
a. Let d be the distance between the object and image, so that d = do + di. However, we
3
know from the REASONING that d i = 1 d o , so d = d o + 1 d o = 2 d o . The object distance
2
2
is, therefore,
d o = = 2 d = 2 ( 90.0 cm ) = 60.0 cm
3
3 b. The thinlens equation, Equation 26.6, can be used to find the focal length f of the lens:
1
1
1
1
1
3
=
+=
+
=
f d o di d o 1 d o d o
2
do 60.0 cm
=
= 20.0 cm
3
3
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f= 57. REASONING AND SOLUTION
a. The sun is so far from the lens that the incident rays are nearly parallel to the principal
axis, so the image distance di is nearly equal to the focal length of the lens. The
magnification of the lens is
m=− di
10.0 × 10 −2 m
=−
= −6.67 × 10−13
do
1.50 × 1011 m The image height hi is
hi = mho = ( −6.67 × 10−13 )(1.39 × 109 m) = −9.27 × 10−4 m The diameter of the sun’s image on the paper is the magnitude of hi, or 9.27 × 10–4 m. The
area A of the image is
A = 1 π (9.27 × 10 −4 m) 2 = 6.74 × 10−7 m 2
4 1384 THE REFRACTION OF LIGHT: LENSES AND OPTICAL INSTRUMENTS b. The intensity I of the light wave is the power P that strikes the paper perpendicularly
divided by the illuminated area A (see Equation 16.8) 0.530 W
P
=
= 7.86 × 105 W/m 2
A 6.74 × 10−7 m 2
______________________________________________________________________________
I= 58. REASONING A diverging lens always produces a virtual image, so that the image distance
di is negative. Moreover, the object distance do is positive. Therefore, the distance between
the object and the image is do + di = 49.0 cm, rather than do – di = 49.0 cm. The equation
do + di = 49.0 cm and the thinlens equation constitute two equations in two unknowns, and
we will solve them simultaneously to obtain values for di and do.
SOLUTION
a. Solving the equation do + di = 49.0 cm for do, substituting the result into the thinlens
equation, and suppressing the units give 111
+=
do di f 1
1
1
+=
49.0 – di di –233.0 or (1) Grouping the terms on the left of Equation (1) over a common denominator, we have
di + 49.0 – d i di ( 49.0 – di ) = 49.0
1
=
di ( 49.0 – di ) –233.0 (2) Crossmultiplying and rearranging in Equation (2) gives
di ( 49.0 – di ) = –11 417 or di2 – 49.0di –11 417 = 0 (3) Using the quadratic formula to solve Equation (3), we obtain di = – ( –49.0 ) ± ( –49.0 )2 – 4 (1.00)( –11 417 )
=
2 (1.00 ) –85.1 cm We have discarded the positive root, because we know that di must be negative for the
virtual image.
b. Using the fact that do + di = 49.0 cm, we find that the object distance is do = 49.0 cm – di = ( 49.0 cm ) – ( –85.1 cm ) = 134.1 cm
______________________________________________________________________________ Chapter 26 Problems 59. 1385 SSM REASONING AND SOLUTION
a. A real image must be projected on the drum; therefore, the lens in the copier must be a
converging lens .
b. If the document and its copy have the same size, but are inverted with respect to one
another, the magnification equation (Equation 26.7) indicates that m = – d i / d o = –1 .
Therefore, d i / d o = 1 or d i = d o . Then, the thinlens equation (Equation 26.6) gives 1
1
1
2
+
==
di do
f do do = di = 2 f or Therefore the document is located at a distance 2 f from the lens.
c. Furthermore, the image is located at a distance of 2 f from the lens.
______________________________________________________________________________
60. REASONING AND SOLUTION The focal length of the lens can be obtained from the
thinlens equation as follows:
1
1
1
=
+
f 4.00 m 0.210 m or f = 0.200 m The same equation applied to the projector gives 1
1
1
=
−
or
do = 0.333 m
do 0.200 m 0.500 m
______________________________________________________________________________
61. SSM REASONING The magnification equation (Equation 26.7) relates the object and
image distances do and di , respectively, to the relative size of the of the image and object:
m = –( d i / d o ) ....
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 Spring '13
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 Physics, Pythagorean Theorem, Force, The Lottery, Right triangle, triangle

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