Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments
Chapter 36. Lenses and Optical Instruments Focal Length and the Lensmaker’s Equation (Assume n = 1.! unless told other"ise.# 36$1. A plano$con%e& plano$con%e& lens lens is to 'e construc constructed ted out o )lass )lass so that that it has a ocal ocal len)th len)th o *! *! 4 - = ∞
cm. +hat is the radius radius o cur%ature o the cur%ed surace surace 1 f
1 1 1 1 = (n − 1# + = (1. − 1# + R R *! cm ∞ 1 -
1 = (!.# *! cm R1 1
R1
= !.(*! cm#
4 1 = n = 1.
R1 = -!.! cm
36$-. I one uses uses a )lass, )lass, dou'le$c dou'le$con%e& on%e& lens lens to o'tain o'tain a ocal ocal len)th len)th o 3! 3! cm, "hat "hat must 'e the cur%ature o each con%e& surace 1 f
/ R1 = R2 = R 0
1 1 = (n − 1# + = (1. − 1# R = R R R
R2
R1
f R = 3! cm
36$3. he cur%ed cur%ed surace surace o a plano$c plano$conca%e onca%e lens lens has has a radius radius o 21- cm. +hat is is the ocal len)th i the lens is made rom a material "ith a reracti%e inde& o 1.*. 1 f 1 f
1 1 1 1 1# + = (n − 1# = (1.* − 1# + ∞ R R $1cm 1 - 1 = (1.* − 1# −1- cm
R1 R2 =
1 f
=
!. !.*!
−1- cm
∞
f = $--.- cm
36$*. A con%er)in) con%er)in) meniscu meniscuss lens (n (n = 1.# has a conca%e conca%e surace surace "hose "hose radius radius is 2-! cm and and a con%e& surace "hose radius is 1- cm. +hat is the ocal len)th 1 1 = (!.# −-! cm 1- cm = (1. − 1# + (1- cm#($-! cm# f 1- cm −-! cm 1
-*1
f = 3! cm
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments
36$. A con%er)in) lens such as sho"n in 5i). 36$a is made o )lass. he irst surace has a radius o 1 cm and the second surace has a radius o 1! cm. +hat is the ocal len)th Both surfaces are convex, and therefore positive: ( R1 = +15 cm, R2 = + 10 cm, n = 1.5 )
1! cm 1 cm 1 1 = (1. − 1# + = (!.# (1! cm#(1 cm# f 1 cm 1! cm 1
f = 1- cm
36$6. A meniscus lens has a con%e& surace "hose radius is -! cm and a conca%e surace "hose radius is 2 3! cm. +hat is the ocal len)th i the reracti%e inde& is 1.*. iven: R1 = -! cm , R2 = $3! cm , n = 1.*: !ind: f = "
−3! cm -! cm 1 1 (!.*# = (1.* − 1# + = (-! cm#($3! cm# f -! cm $3! cm 1
f = 1.11 m
36$7. A plano$con%e& lens is )round rom cro"n )lass (n = 1.-#. +hat should 'e the radius o the cur%ed surace i the desired ocal len)th is to 'e *!! mm 1 f
1 1 = (n − 1# + R1 R-
1 = (!.-# *!! mm R1 1
1 1 = (1.- − 1# + *!! mm R1 ∞ 1
4 - = ∞ = *!! mm n = 1.-
R1
= !.-(*!! mm#
R1 = -! mm
36$. he ma)nitudes o the conca%e and con%e& suraces o a )lass lens are -!! and 6!! mm, respecti%ely. +hat is the ocal len)th Is it di%er)in) or con%er)in) iven: R1 = #-! cm , R2 = 6! cm , n = 1.: !ind: f = "
6! cm $ -! cm 1 1 = (1. − 1# + = (!.# f $-! cm 6! cm ($-! cm#(6! cm# 1
f = $6! cm
$he %ens is a diver&in& %ens, since the foca% %en&th is ne&ative.
-*-
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments
36$8. A plastic lens (n = 1.*# has a con%e& surace o radius - cm and a conca%e surace o 27! cm. +hat is the ocal len)th Is it di%er)in) or con%er)in)
$7! cm - cm 1 1 = (1.* − 1# + = (!.*# f - cm $7! cm (- cm#($7! cm# 1
f = 7-.! cm
$he %ens is a conver&in& %ens, since the foca% %en&th is positive.
Images Formed by hin Lenses 36$1!. A 7$cm tall pencil is placed 3 cm rom a thin con%er)in) lens o ocal len)th - cm. +hat are the nature, si9e, and location o the ima)e ormed '=
pf p − f
=
−' p
=
=
(3 cm#(- cm# 3 cm $ - cm
:
: =
' = 7. cm
!
2!
− ' : −(7. cm#(7 m# = p
3 cm
' = +7. cm, y; = $17. cm , rea%, inverted, and en%ar&ed.
36$11. An o'
pf p − f
=
−' p
=
=
(3! cm#(1- cm#
:
3! cm $ 1- cm : =
' = -! cm
!
2!
− ' : −(-! cm#( m# = p
3! cm
' = +-!.! cm, y; = $.33 cm , rea%, inverted, and diminished.
36$1-. A %irtual, erect ima)e appears to 'e located *! cm in ront o a lens o ocal len)th 1 cm. +hat is the o'
$he foca% %en&th must *e positive (conver&in&), since
rea% o*ects cannot produce erect virtua% ima&es farther aa from %ens than the o*ect.
-*3
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments 36$1-. (Cont.#. 5ind o'
'f ' − f
=
(−*! cm#(1 cm# $*! cm $ 1 cm
!
p = 1!.8 cm
2!
-ote ou shou%d sho that the ma&nification is +-.67.
36$13. A ! mm tall o'ee dra"in) or Pro'. 36$1- a'o%e.# ' =
(1- cm#(-! cm# 1- cm $ -! cm
' = #3! cm
=
−' p
=
:
: =
− ' : −(−3! cm#(! mm# = p
1- cm
' = #3!.! cm, y; = 1- mm , virtua%, erect, and en%ar&ed.
36$1*. An o'
p' p + '
=
(3! cm#(6! cm# 3! cm 6! cm
f = -! cm
!
2!
$he foca% %en&th is positive, so the %ens is conver&in&. f = -! cm , conver&in&
36$1. A li)ht source is 6!! mm rom a con%er)in) lens o ocal len)th 1! mm. Construct the ima)e usin) ray dia)rams. +hat is the ima)e distance Is the ima)e real or %irtual '=
pf p − f
=
(6! cm#(1 cm# 6! cm $ 1 cm
' = -.7 cm
' = +-.7cm, rea%, inverted, and diminished. ?ote /ho that the ma&nification of the ima&e is = $ !.*-.
-**
!
2!
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments
36$16. A plano$con%e& lens is held *! mm rom a 6$cm o'
(*! mm#(6! mm# *! cm $ 6! cm
' = #1-! mm : =
−' : −(−1-! mm#(6 cm# = p
*! mm
' = #1-! mm, y; = 1.! cm , virtua%, erect, and en%ar&ed.
36$17. An o'
pf p − f
=
−' p
=
=
(* cm#($-* cm# * cm $ ($-* cm#
:
: =
' = $3.*3 cm
− ' : −(−3.*3 cm#(6 m# = p
* cm
' = $3.*3 cm, y; = .1* cm , virtua%, erect, and diminished.
36$1. he ocal len)th o a con%er)in) lens is -!! mm. An o'
pf p − f
(a# ' =
(1 cm#(-! cm#
('# ' =
(-! cm#(-! cm#
(c# ' =
1 cm $ -! cm
-! cm $ -! cm (3! cm#(-! cm# 3! cm $ -! cm
)
=
( : (
=
−' p
= $6! cm, virtua% : =
= ∞, no ima&e
: =
: =
= 6! cm, rea%
-*
f = -! cm, y = 6 cm (:=
− '( p
−(−6! cm#(6 cm# (1 cm#
−(∞#(6 cm# (-! cm#
= ∞, no ima)e
−(6! cm#(6 cm# (3! cm#
= -* cm, erect
= $1- cm, inverted
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments 36$1. (Cont.# ma&es formed * conver&in& mirror, f = 20 cm, = cm. (d# ' =
(*! cm#(-! cm#
(e# ' =
(6! cm#(-! cm#
*! cm $ -! cm
6! cm $ -! cm
= *! cm, rea%
: =
= 3! cm, rea%
: =
−(*! cm#(6 cm#
= $6 cm, inverted
(*! cm#
−(3! cm#(6 cm# (6! cm#
= $3 cm, inverted
36$18. An o'
p' p + '
=
(* cm#(8! cm# * cm 8! cm
f = 3! cm
!agni"ication 36$-!. An o'
=
−' p
= −-
' = - p = -(-! cm#
' = *! cm, rea%, inverted, and en%ar&ed .
36$-1. A pencil is held -! cm rom a di%er)in) lens o ocal len)th 21! cm. +hat is the ma)niication =
−' −(−1! cm# = p
-! cm
= + 5
36$--. A ma)niyin) )lass has a ocal len)th o -7 cm. Bo" close must this )lass 'e held to an o'
=
−' p
$3( p 3 f) = f4
= +3
' = −3 p ' =
#p + f = f4
p=
-
-*6
3
pf p− f
− 3p =
f = - 3 (-7 cm#
pf p − f p = 1.! cm
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments
36$-3. A ma)niyin) )lass held *! mm rom a specimen produces an erect ima)e that is t"ice the o'
=
−' p
$-( p 3 f) = f4
= +-
pf
' = −- p ' =
#2p + 2f = f4
p− f
− -p=
f = 2p = 2(*! mm#
pf
p − f
f = !.! mm
36$-*. +hat is the ma)niication o a lens i the ocal len)th is *! cm and the o'
(6 cm#(*! cm# 6 cm $ *! cm
= 1!* cm
=
−' −(1!* cm# = p
(6 cm#
= #1.6!
Challenge #roblems 36$-. he radius o the cur%ed surace in a plano$conca%e lens is -! cm. +hat is the ocal len)th i n = 1.* / R1 = # -! cm, R2 = ∞ 0 1 f 1 f
R1
1 1 1 1 = (n − 1# + = (1.* − 1# + $-! cm ∞ R1 R- 1 = (1.* − 1# −-! cm
1 f
=
!.*!
−-! cm
R2 =
∞
f = $37.* cm
36$-6. A thin meniscus lens i ormed "ith a conca%e surace o radius 2*! cm and a con%e& surace o radius 3! cm. I the resultin) ocal len)th is 78.! cm, "hat "as the inde& o reraction o the transparent material 1 f
1 1 = (n − 1# + R1 R-
1 1 = ( n − 1# + 78 cm −*! cm 3! cm 1
−1! cm = (n − 1# 78 cm $1-!! cm- 1
-*7
n − 1 = 1.-
n = -.-
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments
36$-7. A con%er)in) lens has a ocal len)th o -! cm. An o'
pf p − f
=
(1 cm#(-! cm# 1 cm $ -! cm
6*ect ' = $6! cm !
$he ima&e is erect, virtua%, and en%ar&ed.
2!
ma&e
36$-. Bo" ar rom a source o li)ht must a lens 'e placed i it is to orm an ima)e !! mm rom the lens he ocal len)th is -!! mm. ! p =
'f ' − f
=
(! cm#(-! cm# ! cm $ -! cm
2!
p = -6.7 cm
36$-8. A source o li)ht 36 cm rom a lens pro
p' p + '
=
(36 cm#(1 cm# 36 cm 1 cm
f = 1- cm, con%er)in).
36$3!. +hat is the minimum ilm si9e needed to pro
pf p − f
=
(-!!! mm#( mm# -!!! mm $ mm
' = 6.6 mm
-*
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments
36$31. +hen parallel li)ht stri@es a lens, the li)ht d i%er)es, apparently comin) rom a point ! mm 'ehind the lens. Bo" ar rom an o'
=
−' p
=+
1 *
p = − *' = − *(− ! mm#
p = -*! mm
36$3-. Bo" ar rom a di%er)in) lens should an o'
=
−' p
=
1 *
'=
$( p 3 f) = 7f4
−p *
'=
#p + f = 7f4
pf p− f
−p
=
*
pf
p − f
p = #f=#(#3 cm#
5 p = 1! cm
36$33. he irst surace o a thin lens has a con%e& radius o -! cm. +hat should 'e the radius o the second surace to produce a con%er)in) lens o ocal len)th .!! cm 1 f
1 1 = (n − 1# + R1 R-
R + -! cm = !. cm (-! cm# R - 1
1 1 = (1. − 1# + cm -! cm R- 1
-. R-
= !.R- + 1! cm
R2 = .!! cm
36$3* "o thin con%er)in) lenses are placed 6! cm apart and ha%e the same a&is. he irst lens has a ocal len)th 1! cm, and the second has a ocal len)th o 1.! cm. I an o'
=
(-! cm#(1! cm#
'-
=
-! cm $ 1! cm
= -! cm
(*! cm#(1 cm# *! cm $ 1 cm
p-
= -* cm -*8
= 6! cm $ -! cm = *! cm ' = -*.! cm, real ima)e
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments 36$3*. (Cont.#
6! cm
'1 = -! cm4 '2 = -* cm = 1 & -
−'1 − '- p = p 1 -
=
'1'p1 p-
f 2 = 1 cm
f 1 = 1! cm cm
)
=
(-! cm#(-* cm# -! cm#(*! cm#
= +!.6
( : = )( = (!.6#(6 cm#
( : = 3.6 cm
' = -* cm *eond %ens 2, 8 = 3.6 cm, rea%, erect, and diminished.
36$3. A con%er)in) lens o ocal len)th - cm is placed ! cm in ront o a di%er)in) lens "hose ocal len)th is 2- cm. I an o'
=
p'-
=
(7 cm#(- cm# 7 cm $ - cm
= 37. cm
= 6! cm $ -! cm = *! cm (1-. cm#($- cm# 1-. cm $($- cm# =
'1' p1 p-
=
= −.33 cm
)
= )1
(37. cm#($.33 cm# (7 cm#(1-. cm#
& )-
=
(−'1 #(− '- # p1 p-
= $!.333
' = $.33 cm %eft of %ens 2 virtua%, inverted, diminished4 = $!.333
-!
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments
Critical hinking #roblems 36$36. A camera consists o a con%er)in) lens o ocal len)th ! mm mounted in ront o a li)ht$sensiti%e ilm as sho"n in 5i). 36$18. +hen photo)raphin) ininite o'
('# ' =
1
4
∞
pf
=
∞
p − f
(c# =
1
1
'
f
+ =
(!! mm#(! mm# !! mm $ ! mm
−' −(.6 mm# = p
' = f = ! mm
!! mm
' = .6 mm for o*ect at !! mm.
= #!.111, in%erted, real, and diminished.
36$37. An o'
3! cm $ x =
x( cm# x − cm
pf p − f
(3! − x #( x − # = x
3! x 2 1! $ x2 x = 5x4
3! cm
x 3! cm $ x
x2 30x + 150 = 04 ( /o%ve the 'uadratic e'uation for x)
$o positions are: x = 6.3* cm and -3.7 cm from o*ect .
-1
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments
36$3. A simple pro
=
−' p
=
:
=
( −6! cm# !. cm
= −1-!
p=
' 3 f = 120f4
= −1-!
' 1-!
p=
' 1-!
=
'− f
f =
121f = '4
p =
'f
'
1-!
' 1-1
6!! cm 1-!
=
=
'f '− f
6!! cm 1-1
1 1-!
=
f
' − f
f = *.86 cm
p = .!! cm
36$38. A telescope has an o'
p1 = 000 cm, f 1 = 8! cm, f 2 = 5 cm (a#
' =
(6!!! cm#(8! cm# 6!!! cm $ 8! cm
p2 = x 3 ;1.< cm
x − 81.3 cm =
0 m
= 81.37 cm
p-
=
;0 cm
'- f '- − f -
( −- cm#( cm# $- cm $ cm
x − 81.37 cm = *.167
5 cm 25 cm
x = 8.* cm
$he to %ens must *e separated * a distance of 8. cm.
--
Physics, 6th Edition
Chapter 36. Lenses and Optical Instruments 36$38. (Cont.#
('# p2 = 8.* cm 2 81.37 cm =
'1' p1 p-
=
(81.37 cm#(−- cm# (6!!! cm#(*.17 cm#
p2 = *.17 cm = #0.0;174
= 1 x 2 8 =
8 = ($!.!81* )(3! cm# 8= $-.7* cm, %irtual, in%erted, and diminished
36$*!. he Dalilean telescope consists o a di%er)in) lens as the eyepiece and a con%er)in) lens as the o'
p-
=
=
(*!!! cm#(3! cm# *!!! cm $ 3! cm
'- f '- − f -
=
36$6.
'1' p1 p-
=
x
= 3!.-3 cm
( −- cm#($-. cm# $- cm $ ($-. cm#
p2 = x 3 3!.-3 cm =
*!!! cm
$- cm
p2 = $-.7 cm
x = 3!.-3 cm p2 = 3!.-3 cm ($-.7 cm# x = -7. cm
(3!.-3 cm#($- cm# (*!!! cm#(-.7 cm#
= #0.0
he ocal len)th o the eyepiece o a particular microscope is 3.! cm, and the ocal len)th o the o'
f 2 = 3! mm
p2 = f 2 = 3! mm
'1 = -6 mm 2 3! mm = -3 mm p1
=
'1 f 1 '1 − f 1
=
(-3 mm#(18 mm# -3 mm $ 18 mm
-3
' = -!.7 mm
