Chapter 25
Optical Instruments
Problem Solutions
25.1
The f-num ber (or focal ratio) of a lens is d efined to be the ratio of focal length of the lens
to its d iam eter. Therefore, th e f-num ber of the given lens is
f -number
25.2
f
D
28 cm
4.0 cm
7.0
If a cam era has a lens w ith focal length of 55 m m and can operate at f -numbers that
range from f 1.2 to f 22 , the aperture d iam eters for the cam era m ust range from
f
f -number
Dmin
max
55 mm
22
2.5 mm
55 mm
1.2
46 mm
to
f
f -number
Dmax
25.3
The thin lens equation,
q
1
q
1
, gives the im age d istance as
f
100 m 52.0 mm
pf
p
1
p
min
f
100 m 52.0 10
3
m
52.0 mm
From the m agnitud e of the lateral m agnification, M
the im age is h
h
h
h h
q p , w here the height of
0.092 0 m 92.0 mm , the height of the object (the build ing) m ust be
p
q
92.0 mm
100 m
52.0 mm
177 m
466
Optical Instruments
25.4
467
Consid er rays com ing from
opposite ed ges of the object and
passing und eviated through the
center of the lens as show n at the
right. For a very d istant object,
the im age d istance equals the
focal length of the lens. If the
angular w id th of the object is ,
the full im age w id th on the film is
h
2 f tan
2
2 55.0 mm tan
20
2
19 mm
so the im age easily fits w ithin a 23.5 m m by 35.0 m m area.
25.5
The exposure tim e is being red uced by a factor of
t2
t1
1 256 s
1 32 s
1
8
Thus, to m aintain correct exposure, the intensity of the light reaching the film should be
increased by a factor of 8. This is d one by increasing the area of the aperture by a factor
of 8, so in term s of the d iam eter, D22 4 8 D12 4 or D2
8 D1 .
The new f-num ber w ill be
f -number
2
f
D2
f
8 D1
f -number
8
1
4.0
1.4 or
8
f 1.4
468
25.6
CH APTER 25
(a) The intensity is a m easure of the
rate at w hich energy is received by
the film per unit area of the image, or
I 1 Aimage . Consid er an object w ith
horizontal and vertical d im ensions
hx and hy as show n at the right. If
the vertical d im ension intercepts
angle , the vertical d im ension of
the im age is hy q , or hy q .
Sim ilarly for the horizontal d im ension, hx
Aimage
q , and the area of the im age is
q . Assum ing a very d istant object, q f , so Aimage f 2 and w e
2
hx hy
conclud e that I
1 f2.
The intensity of the light reaching the film is also proportional to the cross -sectional
area of the lens and hence, to the square of the d iam eter of that lens, or I D2 .
Com bining this w ith our earlier conclusion gives
D2
f2
I
1
f D
1
or I
2
f -number
2
(b) The total light energy hitting the film is proportional to the prod uct of intensity and
exposure tim e, It. Thus, to m aintain correct exposure, this prod uct m ust be kept
constant, or I 2t2 I1t1 giving
t2
25.7
f 2 -number
I1
t1
I2
f1 -number
2
2
4.0
1.8
t1
2
1
s
500
1 100 s
Since the exposure tim e is unchanged , the intensity of the light reaching the film m ust be
d oubled if the energy d elivered is to be d oubled . Using the result of Problem 6 (part a),
w e obtain
f 2 -number
2
I1
I2
Thus, you should use the
f1 -number
2
1
11
2
2
61 , or f2 -number
f 8.0 setting on the cam era.
61 7.8
Optical Instruments
25.8
469
The im age m ust alw ays be focused on the film , so the im age d istance is the d istance
betw een the lens and the film . From the thin lens equation, 1 p 1 q 1 f , the object
d istance is p qf (q f ) , and the range of object d istances this cam era can w ork w ith is
from
pmin
qmax f
qmax f
pmax
qmin f
qmin f
210 mm 175 mm
210 mm 175 mm
1.05 103 mm
1.05 m
6.30 103 mm
6.30 m
to
25.9
180 mm 175 mm
180 mm 175 mm
The corrective lens m ust form an up right, virtual im age at the near point of the eye (i.e.,
q
60.0 cm in this case) for objects located 25.0 cm in front of the eye ( p 25.0 cm ).
From the thin lens equation, 1 p 1 q 1 f , the required focal length of the corrective
lens is
f
pq
p q
25.0 cm
60.0 cm
25.0 cm 60.0 cm
42.9 cm
and the pow er (in d iopters) of this lens w ill be
P
25.10
1
f in meters
1
0.429 m
2.33 diopters
(a) The person is farsighted , able to see d istant objects but unable to focus on objects at
the norm al near point for a hum an eye.
(b) With the corrective lens 2.00 cm in front of the eye, the object d istance for an object
20.0 cm in front of the eye is p 20.0 cm 2.00 cm 18.0 cm .
(c) The upright, virtual im age form ed by the corrective lens w ill serve as the object for
the eye, and this object m ust be 40.0 cm in front of the eye. With the lens 2.00 cm in
front of the eye, the m agnitud e of the im age d istance for the lens w ill be
q 40.0 cm 2.00 cm 38.0 cm .
(d ) The im age m ust be located in front of the corrective lens, so it is a virtual image and
the im age d istance is negative . Thus, q
38.0 cm .
470
CH APTER 25
(e) From the thin lens equation, 1 p 1 q 1 f , the required focal length of the
corrective lens is
f
(f)
pq
p q
18.0 cm
38.0 cm
18.0 cm 38.0 cm
34.2 cm
The pow er of the corrective lens is then
P
1
f in meters
1
0.342 m
2.92 diopters
(g) With a contact lens, the lens to eye d istance w ould be zero, so w e w ould have
p 20.0 cm , q
40.0 cm , giving a required focal length of
f
pq
p q
20.0 cm
40.0 cm
20.0 cm 40.0 cm
40.0 cm
and a pow er in d iopters of
P
25.11
1
f in meters
1
0.400 m
2.50 diopters
H is lens m ust form an upright, virtual im age of a very d istant object ( p
) at his far
point, 80.0 cm in front of the eye. Therefore, the focal length is f q 80.0 cm .
If this lens is to form a virtual im age at his near point ( q
m ust be
p
18.0 cm
qf
q
f
18.0 cm
80.0 cm
80.0 cm
23.2 cm
18.0 cm ), the object d istance
Optical Instruments
25.12
471
(a) When the child clearly sees objects at her far point pmax 125 cm the lens-cornea
com bination has assum ed a focal length suitable of form ing the im age on the retina
q 2.00 cm . The thin lens equation gives the optical pow er und er these co nd itions
as
Pfar
1
f in meters
1
p
1
q
1
1.25 m
1
0.020 0 m
50.8 diopters
When the eye is focused q 2.00 cm on objects at her near point pmin 10.0 cm
the optical pow er of the lens-cornea com bination is
Pnear
1
1
p
f in meters
1
q
1
0.100 m
1
0.020 0 m
(b) If the child is to see very d istant objects p
clearly, her eyeglass lens m ust
form an erect virtual im age at the far point of her eye q
pow er of the required lens is
P
1
f in meters
1
p
1
q
0
60.0 diopters
1
1.25 m
125 cm . The optical
0.800 diopters
Since the p ow er, and hence the focal length, of this lens is negative, it is diverging
25.13
(a) The lens should form an upright, virtual im age at the far point q
very d istant objects p
P
1
f
1
0.500 m
. Therefore, f
q
50.0 cm for
50.0 cm , and the required pow er is
2.00 diopters
(b) If this lens is to form an upright, virtual im age at the near point of the unaid ed eye
q 13.0 cm , the object d istance should be
p
25.14
(a)
13.0 cm
qf
q
f
13.0 cm
50.0 cm
50.0 cm
17.6 cm
Yes, a single lens can correct the patient's vision. The patient need s corrective action in
both the near vision (to allow clear view ing of objects betw een 45.0 cm and the
norm al near point of 25 cm ) and the d istant vision (to allow clear view ing of object s
m ore than 85.0 cm aw ay). A single lens solution is for the patient to w ear a bifocal
or progressive lens. Alternately, the patient m ust purchase tw o pairs of glasses, one
for read ing, and one for d istant vision.
472
CH APTER 25
(b) To correct the near vision, the lens m ust form an upright, virtual im age at the
patient’ s near point ( q
45.0 cm ) w hen a real object is at the norm al near point
(p
).
The
thin
lens
equation gives the need ed focal length as
25.0 cm
f
25.0 cm
pq
p q
45.0 cm
25.0 cm 45.0 cm
56.3 cm
so the required pow er in d iopters is
P
1
f in meters
1
0.563 m
1.78 diopters
(c) To correct the d istant vision, the lens m ust form an upright, virtual im age at the
patient’ s far point ( q
). The thin
85.0 cm ) for the m ost d istant objects ( p
lens equation gives the need ed focal length as f q
85.0 cm , so the need ed
pow er is
P
25.15
1
f in meters
1
0.850 m
Consid ering the im age form ed by the cornea as a virtual object for the im planted lens,
w e have p
2.80 cm 2.53 cm
5.33 cm and q
2.80 cm . The thin lens equation
then gives the focal length of the im planted lens as
f
pq
p q
5.33 cm 2.80 cm
5.33 cm 2.80 cm
P
so the pow er is
25.16
1.18 diopters
1
f
1
0.059 0 m
5.90 cm
17.0 diopters
(a) The upper portion of the lens should form an upright, virtual im age of very d istant
objects p
at the far point of the eye q
1.5 m . The thin lens equation then
gives f q
1.5 m , so the need ed pow er is
P
1
f
1
1.5 m
0.67 diopters
Optical Instruments
473
(b) The low er part of the lens should form an upright, virtual im age at the near point of
the eye q
30 cm w hen the object d istance is p 25 cm . From the thin lens
equation,
f
pq
p q
25 cm
30 cm
25 cm 30 cm
Therefore, the pow er is P
25.17
1
f
1.5 102 cm
1
1.5 m
1.5 m
0.67 diopters
The corrective lens should form an upright, virtual im age at the w om an’ s far point
(q
). The thin lens equation gives the
40.0 cm ) for a very d istant object ( p
required focal length as f
q
40.0 cm
0.400 m . Since f
0 , it is a diverging lens ,
and the required pow er is
P
25.18
(a)
f
1
fin meters
1
P
1
0.400 m
1
4.00 diopters
2.50 diopters
0.250 m
25.0 cm
(b) The corrective lens form virtual im ages of very d istant objects ( p
q
) at
25.0 cm . Thus, the person m ust be very nearsighted , unable to see objects
f
clearly w hen they are over (25.0 2.00) cm 27.0 cm in front of the eye.
(c) If contact lenses are to be w orn, the far point of the eye w ill be 27.0 cm in front of
the lens, so the need ed focal length w ill be f q 27.0 cm , and the pow er is
P
25.19
1
f in meters
1
0.270 m
3.70 diopters
(a) The sim ple m agnifier (a converging lens) is to form an upright, virtual im age
located 25 cm in front of the lens q
25 cm . The thin lens equation then gives
p
25 cm 7.5 cm
qf
q
f
25 cm 7.5 cm
5.8 cm
so the stam p should be placed 5.8 cm in front of the lens
474
CH APTER 25
(b) When the im age is at the near p oint of the eye, the angular m agnification prod uced
by the sim ple m agnifier is
m
25.20
mmax
1
25 cm
25 cm
1
f
7.5 cm
4.3
(a) The m axim um m agnification of a sim ple m agnifier is mmax 1 (25 cm) f . Thus, if
mmax
6.0 , the focal length of the lens is
25 cm
m max 1
f
25 cm
6.0 1
5.0 cm
(b) While using a sim ple m agnifier, the eye is m ost relaxed if the lens form s the virtual
im age at infinity (so parallel rays em erge from the lens) rather than at the near point
of the eye. Und er these cond itions, the m agnification prod uced is
25 cm
f
m
25.21
25 cm
5.0 cm
5.0
(a) From the thin lens equation,
3.50 cm
pq
p q
f
25.0 cm
4.07 cm
3.50 cm 25.0 cm
(b) With the im age at the norm al near point, the angular m agnification is
m
25.22
mmax
1
25.0 cm
25.0 cm
1
f
4.07 cm
7.14
(a) When the object is at the focal point of the m agnifying lens, a virtual im age is
form ed at infinity and parallel rays em erge from the lens. Und er these cond itions,
the eye is m ost relaxed and the m agnification prod uced is
m
25 cm
f
25 cm
5.0 cm
5.0
(b) When the object is positioned so the m agnifier form s a virtual im age at the near
point of the eye ( q 25 cm ), m axim um m agnification is prod uced and this is
mmax
1
25 cm
25 cm
1
f
5.0 cm
6.0
Optical Instruments
475
(c) From the thin lens equation, the object d istance need ed to yield the m axim um
m agnification com puted in part (b) above is
p
25.23
25 cm 5.0 cm
qf
q f
4.2 cm
25 cm 5.0 cm
(a) From the thin lens equation, a real inverted im age is form ed at an im age d istance of
71.0 cm 39.0 cm
pf
q
p
f
71.0 cm 39.0 cm
86.5 cm
so the lateral m agnification prod uced by the lens is
M
h
h
q
p
86.5 cm
71.0 cm
1.22
and the m agnitud e is
M
1.22
(b) If h is the actual length of the leaf, the sm all angle approxim ation gives the angular
w id th of the leaf w hen view ed by the unaid ed eye from a d istance of
d 126 cm 71.0 cm 197 cm as
0
h
h
d
197 cm
The length of the im age form ed by the lens is h
w id th w hen view ed from a d istance of d
h
1.22 h
d
39.5 cm
Mh
1.22 h , and its angular
126 cm q 39.5 cm is
The angular m agnification achieved by view ing the im age instead of view ing the
leaf d irectly is
1.22 h 39.5 cm
1.22 197 cm
h 197 cm
39.5 cm
0
25.24
6.08
(a) With the im age at the norm al near point q
m 1
25 cm
25 cm
1
f
25 cm
2.0
25 cm , the angular m agnification is
476
CH APTER 25
(b) When the eye is relaxed , parallel rays enter the eye and
m
25.25
25 cm
f
25 cm
25 cm
1.0
The overall m agnification is m
M 1me
M1
25 cm
fe
w here M1 is the lateral m agnification p rod uced by the objective lens. Therefore, the
required focal length for the eye piece is
fe
25.26
M 1 25 cm
12 25 cm
m
140
The approxim ate overall m agnification of a com pound m icroscope is given by
m (L fo )(25.0 cm fe ) , w here L is the d istance betw een the objective and eyepiece
lenses, w hile fo and fe are the focal lengths of the objective and eyepiece lenses
respectively. Thus, the d escribed m icroscope should have an approxim ate overall
m agnification of
L 25.0 cm
fo
fe
m
25.27
20.0 cm
0.500 cm
25.0 cm
1.70 cm
588
The m agnitud e of the m agnification of a telescope is m fo fe , w here fo and fe are the
focal lengths of the objective elem ent and the eyepiece respectively. Thus, if m 45 and
fe 4.0 cm , the focal length of the objective m ust be fo mfe (45)(4.0 cm) 180 cm . The
overall length of the telescope w ill therefore be
L
25.28
2.1 cm
fo
fe 180 cm 4.0 cm 184 cm
1.84 m
It is specified that the final im age the m icroscope form s of the blood cell is 29.0 cmin
front of the eye and that the d iam eter of this im age intercepts an angle of
1.43 mrad .
The d iam eter of this final im age m ust then be
he
r
29.0 10
2
m 1.43 10
3
rad
4.15 10
4
m
Optical Instruments
477
At this point, it is tem pting to use Equation 25.7 from the textbook for the overall
m agnification of a com pound m icroscope, and com pute h he m as the size of the blood
cell serving as the object for the m icroscope. H ow ever, the d erivation of that equation is
based on several assum ptions, one of w hich is that the eye is relaxed and view ing a final
im age located an infinite d istance in front of the eyepiece. This is clearly not true in this
case, and the use of Equation 25.7 w ould introd uce consid erable error. Instead , w e shall
return to basics and use the thin lens equation to find the size of the original object.
The im age form ed by the objective lens is the object for the eyepiece, and w e label the
size of this im age as h . The lateral m agnification of the objective lens is
M1 h h
q1 p1 and that of the eyepiece is Me he h
qe pe .The overall
m agnification prod uced by the m icroscope is
M
he
h
h
h
he
h
w hich gives the size of the original object as h he M .
From the thin lens equation, the required object d istance for the eyepiece is
pe
qe f e
qe f e
29.0 cm 0.950 cm
29.0 cm 0.950 cm
0.920 cm
and the m agnification prod uced by the eyepiece is
Me
qe
pe
29.0 cm
0.920 cm
31.5
The im age d istance for the objective lens is then
q1
L pe
29.0 cm 0.920 cm 28.1 cm
and the object d istance for this lens is
p1
q1 f o
q1 f o
28.1 cm 1.622 cm
28.1 cm 1.622 cm
1.72 cm
The m agnification by the objective lens is
M1
q1
p1
28.1 cm
1.72 cm
16.3
and the overall lateral m agnification is M
M1M e
16.3
31.5
513
478
CH APTER 25
The size of the red blood cell serving as the original object is
he
M
h
25.29
4.15 10
513
4
m
8.09 10
7
m
0.809 m
Som e of the approxim ations m ad e in the textbook w hile d eriving the overall
m agnification of a com pound m icroscope are not valid in this case. Therefore, w e start
w ith the eyepiece and w ork backw ard s to d eterm ine the overall m agnification.
If the eye is relaxed , the eyepiece im age is at infinity qe
pe
, so the object d istance is
2.50 cm , and the angular m agnification by the eyepiece is
fe
me
25.0 cm
fe
25.0 cm
10.0
2.50 cm
The im age d istance for the objective lens is then,
q1
L pe 15.0 cm 2.50 cm 12.5 cm
and the object d istance is p1
q1 f o
q1 f o
12.5 cm 1.00 cm
12.5 cm 1.00 cm
The m agnification by the objective lens is M 1
q1
p1
1.09 cm
12.5 cm
1.09 cm
11.5 , and the overall
m agnification of the m icroscope is
m M1me
25.30
11.5 10.0
115
(a) For a refracting telescope, the overall length is L fo fe and the m agnification
prod uced is m fo fe , w here fo and fe are the focal lengths of the objective elem ent
and the eyepiece respectively. Thus, w e m ay w rite fe fo m to obtain
L
fo
fo
m
f0 1
1
m
fo
m 1
m
Optical Instruments
479
(b) Using the result of part (a), the required change in the length of the telescope w ill be
L
fo
m 1
m
m 1
m
2.00 m
101
100
51.0
50.0
2.00 10
2
cm
2.00 cm
or the telescope m ust be shortened by m oving the eyepiece 2.00 cm forward tow ard
the objective lens.
25.31
The length of the telescope is
L
fo
fe
and the angular m agnification is
m
fo
fe
45
Therefore, fo
fe
25.32
45 fe and L
2.0 cm
fo
and
fe
fo
45 fe
92 cm
fe
fe
92 cm
46 fe
92 cm , giving
or
The m oon m ay be consid ered an
infinitely d istant object p
w hen
view ed w ith this lens, so the im age
d istance w ill be q fo 1500 cm .
Consid ering the rays that pass
und eviated through the center of this
lens as show n in the sketch, observe
that the angular w id ths of the im age
and the object are equal. Thus, if w is the
linear w id th of an object form ing a 1.00 cm
w id e im age, then
w
3.8 108 m
or
w
3.8 108 m
1.0 cm
fo
1.0 cm
1 500 cm
1.0 cm
1 500 cm
1 mi
1 609 m
1.6 10 2 mi
fo
90 cm
480
25.33
CH APTER 25
pf
(a) From the thin lens equation, q
lens is M
h
(b) If p
h h
q p
fh
p f
Mh
f , then f
f
p
p
f
, so the lateral m agnification by the objective
f . Therefore, the im age size w ill be
fh
f p
fh
p
p and h
p
(c) Suppose the telescope observes the space station at the zenith.
25.34
fh
p
h
Then,
4.00 m 108.6 m
407 103 m
1.07 10-3 m
1.07 mm
Use the larger focal length (low est pow er) lens as the objective elem ent and the shorter
focal length (largest pow er) lens for the eye piece. The focal len gths are
1
1.20 diopters
fo
0.833 m , and f e
1
9.00 diopters
0.111 m
(a) The angular m agnification (or m agnifying pow er) of the telescope is then
m
(b)
0.833 m
0.111 m
7.50
The length of the telescope is
L
25.35
fo
fe
fo
fe
0.833 m 0.111 m
0.944 m
The lens for the left eye form s an upright, virtual im age at qL
50.0 cm w hen the object
d istance is pL 25.0 cm , so the thin lens equation gives its focal length as
fL
pL qL
pL qL
25.0 cm
50.0 cm
25.0 cm 50.0 cm
Sim ilarly for the other lens, qR
50.0 cm
100 cm w hen pR
(a) Using the lens for the left eye as the objective,
m
fo
fe
fL
fR
50.0 cm
33.3 cm
1.50
25.0 cm , and f R
33.3 cm .
Optical Instruments
(b) Using the lens for the right eye as the eyepiece and , for m axim um m agnification,
requiring that the final im age be form ed at the norm al near point qe
25.0 cm
gives
pe
qe f e
qe f e
25.0 cm 33.3 cm
25.0 cm 33.3 cm
14.3 cm
The m axim um m agnification by the eyepiece is then
me
1
25.0 cm
25.0 cm
1
fe
33.3 cm
1.75
and the im age d istance for the objective is
q1
L pe 10.0 cm 14.3 cm
4.3 cm
The thin lens equation then gives the object d istance for the objective as
p1
q1 f1
q1 f1
4.3 cm 50.0 cm
4.3 cm 50.0 cm
4.0 cm
The m agnification by the objective is then
4.3 cm
q1
p1
M1
1.1
4.0 cm
and the overall m agnification is m M1me
25.36
1.1
1.75
1.9
N ote: We solve part (b) before answ ering part (a) in this problem .
(b) The objective form s a real,
d im inished , inverted im age of a
very d istant object at q1 fo .
This im age is a virtual object for
the eyepiece at pe
fe ,
giving
1
qe
and
1
pe
1
fe
qe
1
fe
1
fe
0
481
482
CH APTER 25
(a) Parallel rays em erge from the eyepiece,
so the eye observes a virtual image
fo
fe
(c) The angular m agnification is m
3.00 , giving
fo 3.00 fe . Also, the length of the telescope is L
giving
fe
25.37
10.0 cm
2.00
fe
1.22
D
h
Thus, the altitud e is
500 10 9 m
0.300 m
2.03 10
1.00 m
2.03 10 6 rad
4.92 105 m
1.22
d
6
3.00 fe
15.0 cm
rad
For a narrow slit, Rayleigh’ s criterion gives
min
25.39
3.00 fe
fe
If just resolved , the angular separation is
min
25.38
5.00 cm and fo
fo
a
500 10 9 m
1.00 10
0.500 10 3 m
The lim it of resolution in air is
min air
3
1.22
1.00 mrad
D
0.60 rad
In oil, the lim iting angle of resolution w ill be
min oil
or
1.22
oil
D
min air
min oil
noil
1.22
noil
D
0.60 rad
1.5
1.22
1
D noil
0.40 rad
492 km
fe
10.0 cm ,
Optical Instruments
25.40
483
(a) The w avelength of the light w ithin the eye is n
n . Thus, the lim iting angle of
resolution for light passing through the pupil (a circular aperture w it h d iam eter
D 2.00 mm ), is
1.22
min
n
D
1.22
1.22
nD
500 10
9
m
1.33 2.00 10
3
2.29 10
m
4
rad
(b) From s r , the d istance from the eye that tw o points separated by a d istance
s 1.00 cm w ill intercept this m inim um angle of resolution is
s
r
1.00 cm
2.29 10-4 rad
min
25.41
4.36 103 cm
43.6 m
The angular separation of the head lights w hen view ed from a d istance of r 10.0 km is
s
r
2.00 m
10.0 103 m
2.00 10
4
rad
If the head lights are to be just resolved , this separation m ust equal the lim iting a ngle of
resolution for the circular aperture, min 1.22 D , so the d iam eter of the aperture is
D
1.22
min
25.42
1.22
1.22 885 10
2.00 10
4
9
m
rad
5.40 10
3
m
Diffraction occurs w hen w aves pass through an aperture,
causing the intensity to go through m axim a and m inim a as
one goes from the center of the beam outw ard as illustrated
in the figure at the right. The angular separation of the first
m inim um from the central m axim um is a constant
d eterm ined by the d im ension of the aperture, the
w avelength of the w ave, and the shape of the aperture. For a
circular aperture, this angular separation is given by
1.22 D , w here D is the d iam eter of the aperture. The
min
full angular w id th of the central m axim um is then
2 min 2.44 D .
5.40 mm
484
CH APTER 25
The lateral w id th of the central m axim um , d, increases as the d istance r from the
aperture increases. When a beam of laser light having w avelength
632.8 nm d iffracts
through a circular opening of d iam eter D 0.200 cm , w e estim ate the d iam eter of the
beam at d istance r 3.00 km past the opening as equal to the d iam eter of the central
m axim um in the d iffraction pattern at this location. This gives
d
25.43
r
3.00 10 m
2.44 632.8 10
0.200 10
2
9
m
m
500 10 9 m
5.00 m
8.0 107 km 1.22
200 103 m 1.22
550 10 9 m
0.35 m
1 cm
1.67 10
6 000
600 nm light that can be observed is
The grating spacing is d
mmax
d sin 90
1.67 10
6
600 10
4
m 1
9
m
0.38 m
cm 1.67 10
2.78
Nm
9.00 104 2
min
1.22
min
1.22
600.000 nm
0.003 nm
6
m , and the highest ord er of
9.00 104 , and the resolving
1.80 105
2.0 105
These lines cannot be separated w ith this grating.
D
38 cm
The resolving pow er required to separate the given spectral lines is
Rneeded
D
2 orders
The total num ber of slits is N 15.0 cm 6 000 slits cm
pow er of the grating in the second ord er is
Ravailable
2.32 m
9.8 km
If just resolved , the angular separation of the objects is
and s r
25.45
3
If just resolved , the angular separation of the objects is
and s r
25.44
2.44
r
D
Optical Instruments
25.46
The resolving pow er of a d iffraction grating is
R
485
Nm
(a) The num ber of lines the grating m ust have to resolve the H line in the first ord er is
N
R
m
656.2 nm
0.18 nm
1
3.6 103 lines
R
2
(b) In the second ord er m 2 , N
25.47
1.8 103 lines
A fringe shift occurs w hen the m irror m oves d istance 4 . Thus, if the m irror m oves
d istance L 0.180 mm , the num ber of fringe shifts observed is
4
L
4
Nshifts
25.48
656.2 nm
2 0.18 nm
4 0.180 10 3 m
L
550 10
9
1.31 103 fringe shifts
m
(a) When the central spot in the interferom eter pattern goes through a full cycle from
bright to d ark and back to bright, tw o fringe shifts have occurred and the m ovable
m irror has m oved a d istance of 2( 4)
2 . Thus, if Ncycles 1700 such cycles are
observed as the m irror m oves d istance d
d
N cycles
2d
N cycles
or
2
0.382 mm , it m ust be true that
and the w avelength of the light illum inating the interferom eter is
2 0.382 10 3 m
1700
4.49 10
7
m
449 nm
w hich is in the blue region of the visible spectrum .
(b) Red light has a longer w avelength than blue light, so few er w avelengths w ould
cover the given d isplacem ent, hence Ncycles would be smaller .
25.49
A fringe shift occurs w hen the m irror m oves d istance
(length of the bacterium ) as 310 shifts occur is
L
N shifts
4
310
650 10
4
9
m
5.04 10
5
4 . Thus, the d istance m oved
m
50.4 m
486
25.50
CH APTER 25
A fringe shift occurs w hen the m irror m oves d istance
m oves as 250 fringe shifts are counted is
L
25.51
N shifts
250
4
632.8 10
4
9
m
3.96 10
4 . Thus, the d istance the m irror
5
m
39.6 m
When the optical path length that light m ust travel as it goes d ow n one arm of a
Michelson’ s interferom eter changes by one w avelength, four fringe shifts w ill occur
(one shift for every quarter-w avelength change in path length).
The num ber of w avelengths (in a vacuum ) that fit in a d istance equal to a thickness t is
Nvac t . The num ber of w avelengths that fit in this thickness w hile traveling through
the transparent m aterial is Nn t n t
n nt . Thus, the change num ber of
w avelengths that fit in the path d ow n this arm of the interferom eter is
N
Nn
N vac
n 1
t
and the num ber of fringe shifts that w ill occur as the sheet is inserted w ill be
# fringe shifts
25.52
4
N
4 1.40 1
15.0 10 6 m
600 10 9 m
40
A fringe shift w ill occur each tim e the effective length of the tube changes by a quarter
of a w avelength (that is, for each ad d itional w avelength fitted into the length of the tube,
4 fringe shifts occur). If L is the length of the tube, the num ber of fringe shifts observed
as the tube is filled w ith gas is
Nshifts
4
L
L
4
n
H ence, ngas 1
25.53
t
4 n 1
4L
Nshifts 1
L
ngas
L
600 10
4L
9
4 5.00 10
ngas 1
m
2
m
160
1.000 5
(a) For a refracting telescope, the m agnification is m fo fe , w here fo and fe are the
focal lengths of the objective lens and the eyepiece, respectively. Thus, w hen the
Yerkes telescope uses an eyepiece w ith fe 2.50 cm , the m agnification is
m
fo
fe
20.0 m
2.50 10 2 m
8.00 102
800
Optical Instruments
487
(b) Stand ard astronom ical telescopes form inverted im ages. Thus, the observer Martian
polar caps are upside down .
25.54
When view ed from a d istance of 50 m eters, the angular length of a m ouse (assum ed to
have an actual length of 10 cm ) is
s
r
0.10 m
50 m
2.0 10
3
radians
Thus, the lim iting angle of resolution of the eye of the haw k m ust be
2.0 10
min
25.55
3
rad
The resolving pow er of the grating is R
Nm . Thus, the total num ber of lines
need ed on the grating to resolve the w avelengths in ord er m is
N
R
m
m
(a) For the sod ium d oublet in the first ord er,
N
589.30 nm
1 0.59 nm
1.0 103
(b) In the third ord er, w e need N
25.56
589.30 nm
3 0.59 nm
3.3 102
(a) Since this eye can alread y focus on objects located at the near point of a norm al eye
(25 cm ), no correction is need ed for near objects. To correct the d istant vision, a
corrective lens (located 2.0 cm from the eye) should form virtual im ages of very
d istant objects at 23 cm in front of th e lens (or at the far point of the eye). Thus, w e
m ust require that q 23 cm w hen p
. This gives
P
1
f
1
p
1
q
0
1
0.23 m
4.3 diopters
488
CH APTER 25
(b) A corrective lens in contact w ith the cornea should form virtual im ages of very
d istant objects at the far point of the eye. Therefore, w e require that q 25 cm
w hen p
, giving
1
f
P
1
p
1
q
0
1
0.25 m
4.0 diopters
1
25 cm is in place, the object d istance w hich yield s
P
a virtual im age at the near point of the eye (that is, q 16 cm ) is given by
When the contact lens
p
25.57
16 cm
qf
q
f
f
25 cm
16 cm
44 cm
25 cm
(a) The lens should form an upright, virtual im age at the near point of the eye
q 75.0 cm w hen the object d istance is p 25.0 cm . The thin lens equation then
gives
pq
p q
f
25.0 cm
75.0 cm
25.0 cm 75.0 cm
so the need ed pow er is P
1
f
37.5 cm
1
0.375 m
0.375 m
2.67 diopters
(b) If the object d istance m ust be p 26.0 cm to position the im age at q
actual focal length is
pq
p q
f
and P
1
f
26.0 cm
75.0 cm
26.0 cm 75.0 cm
1
0.398 m
0.398 m
2.51 diopters
The error in the pow er is
P
2.67 2.51 diopters
0.16 diopters too low
75.0 cm , the
Optical Instruments
25.58
489
(a) If q 2.00 cm w hen p 1.00 m 100 cm , the thin lens equation gives the focal
length as
100 cm 2.00 cm
pq
p q
f
1.96 cm
100 cm 2.00 cm
(b) The f-num ber of a lens aperture is the focal length of the lens d ivid ed by the
d iam eter of the aperture. Thus, the sm allest f-num ber occurs w ith the largest
d iam eter of the aperture. For the typical eyeball focused on objects 1.00 m aw ay,
this is
f -number
f
Dmax
min
1.96 cm
0.600 cm
3.27
(c) The largest f-num ber of the typical eyeball focused on a 1.00-m -d istance object is
f -number
25.59
f
Dmin
max
1.96 cm
0.200 cm
9.80
(a) The im planted lens should give an im age d istance of q 22.4 mm for d istant
objects. The thin lens equation then gives the focal length as
p
q 22.4 mm , so the pow er of the im planted lens should be
f
Pimplant
1
f
1
22.4 10
3
44.6 diopters
m
(b) When the object d istance is p 33.0 cm , the corrective lens should prod uce parallel
rays q
. Then the im planted lens w ill focus the final im age on the retina.
From the thin lens equation, the required focal length is f p 33.0 cm , and the
pow er of this lens should be
Pcorrective
25.60
We use
R
q
n1
p
n2
n2
q
n1
n2
1
f
1
0.330 m
n2
n1
R
3.03 diopters
, w ith p
2.00 cm
1.34 1.00
1.34
and q equal to the cornea to retina d istance. Then,
0.507 cm
5.07 mm
490
25.61
CH APTER 25
When a converging lens form s a real im age of a very d istant object, the im age d istance
equals the focal length of the lens. Thus, if the scout started a fire by focusing sunlight
on kind ling 5.00 cm from the lens, f q 5.00 cm .
(a) When the lens is used as a sim ple m agnifier, m axim um m agnification is prod uced
w hen the upright, virtual im age is form ed at the near point of the eye ( q 15 cm
in this case). The object d istance required to form an im age at this location is
p
15 cm 5.0 cm
qf
q
f
15 cm 5.0 cm
15 cm
4.0
and the lateral m agnification prod uced is
M
q
p
15 cm
15 cm 4.0
4.0
(b) When the object is view ed d irectly w hile positioned at the near point of the eye, its
angular size is 0 h 15 cm . When the object is view ed by the relaxed eye w hile
using the lens as a sim ple m agnifier (w ith the object at the focal point so parallel
rays enter the eye), the angular size of the upright, virtual im age is
h f . Thus,
the angular m agnification gained by using the lens is
m
0
25.62
h f
h 15 cm
15 cm
f
15 cm
5.0 cm
3.0
The angular m agnification is m
is the angle subtend ed by the final
o , w here
im age, and o is the angle subtend ed by the object as show n in the figure. When the
telescope is ad justed for m inim um eyestrain, the rays entering the eye are parallel. Thus,
the objective lens m ust form its im age at the focal point of the eyepiece.
From triangle ABC,
h q1 and from triangle DEF,
h f e q1
angular m agnification is then m
h q1 f e
o
o
tan
o
tan
h fe . The
Optical Instruments
From the thin lens equation, the im age d istance of the objective lens in this case is
q1
p1 f1
p1 f1
300 cm 20.0 cm
300 cm 20.0 cm
With an eyepiece of focal length fe
telescope is
m
q1
fe
21.4 cm
2.00 cm
10.7
21.4 cm
2.00 cm , the angular m agnification for this
491