Problem Solving Techniques
You should be able to calculatethe image position for an object in ffont ofa piane or spherical
mirror. Use the mirror equations:i: - p for a plane mirror and (1ip) + (lii) : (1/f, where f = % R
for a spherioal mirror. Be careful about the signs of the various quantities. Be sure you know the
location of the image formed by a plane mirror and how to tell if the image is visible to an
observer at a given location. Recall that the lateral magnification is given by m - - i/p and that
the lateral sizeh' of the image is relatedto the lateral sizeh ofthe object by h'= mh. Krow how
to interpret signs.
You should know how to find the image formed by a spherical refracting surface and by a lens.
If the incoming rays make small angleswith t1lecentralaxis, use (n1/p)+ (nzli) = (nz - n1/r for a
single surfaceand (1/p) + (1/D = (1/0 for a thin lens in air. You should also-knowhow to
computethe focal length, given the radii of the two surfacesofa lens. SeeEq. 34-10.
You should understandcompoundlens systems,such as microscopesand telescopes.Be su.re
you car calculate the angular magnification for each ofthese instruments,
Questionsand Example Problemsfrom Chapter 34
QuestionI
Lake monsters,mermen,andmermaidshavelong been"sighted" by observerslocatedeitheron
a shoreor on a low deck of a ship.From sucha low point, an observercan interceptraysof light
that leavea floating object (say,a 1ogor a porpoise)andbendslightly back downwmdtoward
the observer(onesuchrefractedray, exaggerated,is shownin figue a below). The observerthen
perceivestlle objectas being elongatedupwardfrom the water (andprobablyoscillatingbecause
of air turbulence)in a miragethat might easilyresembleoneofthe fabled creatures.Figureb
beiow gives severalplots of heightfrom the water surfaceversusair temperature.Which oneof
the plots bestillustratestlre air-temperatureconditionsthat canbendthe light rays so asto create
this miraee?
Temperature
(t,
Question2
In the figure betowyou look into a systemoffwo
verticarparallel mirorc A and B separatedby
-Each
distanced' A griruringgargoyleis pe^rched
at p"irn o-,]ail*"e
0.2d
from
mirror
A.
minor
producesa fust (leastdeeo)imageotthe gargoyf".
ffr""
_i,'orp.od;;;;;;";;;;;"g"
with the object beingthe iist iniage in th;dd;;;;Then "u"f,
eachmirrorproducesa third
imagewith the objectbeine tl
in theoppositemirror, *a ,o oo_vo"-Jgirt
r""
hrmdre.sof orinninr,-"._^_,,,1::::ld,image
rmagesm mrror A? C.
images.
H#;";;;;dl;;;.;;"1il?,il;"1#ffi
*9:t :f,ry-*rgargoyre
l.et
\'/
4.0.d
J
0-.rar
-Tl
($l
'
,il:.
d
Problem I
You look throrigha carneratowardan imageof
a hummingbirdin a planemirror. The camera
is
4'30 m in front of the minor. The bird is aicum"ra
t"uet, 5.00m to your right and 3.30m aom
isthedistance
between
thecamera
andtheapparent
prriu-* ,rtrr" tird,simage
*"r#frT#"t
4'3Dh n
S.oD'\i
?'lont
ll
"/1'
: 5.oo."r
J:'l,,;b.".,..-2,d1le,^nc,q,{rre
i
ll i,z'-i
ArYr!4.tu
(zl.76rq-r Z.3o'..)
5.oe^
d* = ("t,go.,'3.ior.,)a r ( 5-oa^r1!
d"
\./
d = g.lory)
I
Problem2
The imagebehinda convexmirror (radiusof cwvature= 68 cm) is located22 cm ftom the
miuor. (a) Whereis the objectlocatedand (b) what is the magnificationof the minor?
Detenninewhetherthe imageis (c) upright or invertedand (d) largeror smallerthan the object.
O=
-&8c"\
i = - l). "..r
Pt+Pr;
- -J-
+/P= -J'lcrq
P=?
---+J =t-!
)-*L=t
-- -3.fo",
9= /"c
?
=
- o.olL crq*t
- I l.rq
---)
,/o.orr.;t
n\
6 A c,"t
rr
(b)
(.)
(r)
Problem3
A small cup of greentea is positionedon the centralaxis ofa sphericalmirror. The lateral
magnifrcationofthe cup is +0.250,andthe distancebetweenthe mirror andits focal point is
2.00 cm. (a) What is the distancebetweenthe mirror and the imageit produces?(b) Is the focal
lengthpositive or negative?(c) Is the imagereal or virtual?
C = -o.l.sop
ln=*tl=+O-\5o
'
lY
= -q'oo t
?
9 -- t
2..-bDctn
\
,a /wLe
-O.lSOp
l=
t.-
f= -1.oo".
P=
- 9 - ' - ^- - A " ' . '
./'
I.t/
L
lrn\aA, lA
axrfi.,r! ( i'
)y+-*
a/u-t{
Snagl
-L *t
Vt
3/..
/aL
tDE:ffi
:
(.)
"4-*,
I
-A.oocrn
--)
o)
( 1,.1.
r)
t't
-++=4
II
()
3 (-r'oo".
Jt)i"""
rr.J/v\J1
Probfem 4
A concavemirror hasa radiusof curvat're of 24 cm.How
far is an objectfrom the mirror if an
rmageisformedrtr is (a) virtual ano:.0 timesae
si;;L
object,(b) rearand 3.0 times rhe
sizeofthe object?
f- = ?,rl cr*, i
(o)
q a..r^,c,orr'r rvurvrar
4pr
rurdf^..1 )J;*4e^
[1 :-c/v=3.o
rr
,/
.tt
/t p *I ' / ; =/ r /
-
^,i€ t+7';+$.l*,14
---
t/
/ Ct - )
tl
'ri o
:
ht', > O
i = -a"of
>= )tacr.,
_ | -; tl^ ----+
z-o? ( y
3f
..
(U)
2."-n *.yU^
= l),e rq
f='/xr
o"" ,r^*,*)
= 8.ocrq
2o rq < o
tr)> - i/,= -3"o -,-.+ i = 3.ol
t/
|t
probrem5/P* /i=
tl
tt
I
/S ---" /f '#r=
tl/
t/-
/+ n
_ -)-
= fG..,
-ip
)/27=-n.n
Fill in thetableberow,eachrowofwhich refersto a difrerent
combination
ofan objectand
eitheraplane
a spherical
----y,'.ro.,
_mirror,
concave
mirror.Distances
9o"u9*
arein
centimeters.
If a numberlacksa sign,
find the,t*"r ".rpl#*
Type
fri
p
m
RealImage? InvertedImaee?
.o aO - b +10 +1.0
Plo"t
O) e-t..-- +20 +tto 60 +30 - e
(c) Convex -Lo-40 -4.0 t5a 0.&
(d) Gn.... +g +lb ta.. +24-O.SO
(a)
(")
No.
No
Yus
Y"'
N"
No
Ve3
Yes
n: .!1P n a=-sP - - ( + l . o ) ( + r o ) ) ( ' = - r o
/e =b"{y =+.-+ -+ 7l= o ---J t="D
V+-'/r- *.*, *
/t
n=-%=g
*
fi=k"==:
rl
*;+
(b)
(o)
(d)
-o,it-
,1""+
," a-;"bJ,-,r.lp
rrrr-u+i-0.ora^.fq <o )4
v\---% . c"=-mp=-(-o.s")().r)
-JL
k=**i
=
[- = "D'
=-d
=lo,ro'
r.=......-
'--r-*l
lI*a"
,
--; \-: s
Problem 6
sphereof indexof
A beamofparaliellight raysfrom a laseris incidenton a solidtransparent
refractionn (seethe figurebelow).(a)Ifa point imageis producedat thebackofthe sphere,
a
what is the index ofrefraction of the sphere?(b) what index of refraction,if any,will produce
point image at the center ofthe sphere?
35-g:
,ft"o"r -r-5,-,';Jf"ar
/^\
.i
\-./
lJt-
-
!.-\
rlo,.-
_,
n
1''b
-.2-+-.
,\
,.r
-
n
D-l"P
--:---.{-
-
&ac
I la
Da
t)r
D^-D'
", =
{*
Yvt-
=
::-
r'
i i
n/
'/\
z = d' ('
( b) D,=l. o
',.-= f)
{-
: n-l
-------) = D::
!9 * ll f-'- - r1-)g
+
t
-d=
-,
l1a:-
(
" /v-r^'J4
I D'Jg
p+r:'t'k *' "-}-^* i
\-y -* aJ
* ,
i d-t
,tJ'",ih -veA *,'-+-0'r
I
o --> dc o't r -> o1
I
Problem7
machinethat canbe.setto grinda
you havea supplyofflat glass1n= 1.5)anda lens-grinding
like
radius of curvatureof either40 cm or 60 cm. You areaskedto prepmea set of six lenses
a choice
thoseshownbelow. what will be the focal len$h of eachlens?Q'Jote:Whereyou have
of radii of curvature.selectthe smallerone.)
) "tr*^r
$.o,cna< r,o^Ne4
"t**
<
/*.,&,"j
) r)O 5 c,oo,c4^r{ -Ou,'$a,.eet- A
(r)
Bi-conYex
Planar
convex
'/+=
("-')(+,-d) C, = ,lil
(.)
Meniscus
convex
t/9, 6s-')(*.*-*..)
(o)
n ea"-..,g*1;
: o"oesct'-r
$ . /.,ru,;' ----o[=l^l
Bi-concale
(c)
(r)
(l\
Plana|
conca\/e
Nleniscrrs
concaYe
(")
/+-('4(G;
i9" -qo'"'j
r/(:(t"r)(-L.,,--J_t
.l0eF,
tJ
r\
t{occ.'
1
= D"otrsc.nI
f: Zocm
- (r.s-r)(+o.--*.*)
Y= )-9o.-
r d\
-,)
I r.s (*
rV.=
Y
( b)
J'
- -#.)
(4)
t/, = ( r.r-t)
)
r-)
a- - l 10cr.)
T-
Problem8
A bright object and a viewing screenareseparatedby a distanceof 76.0 cm. At what location(s)
betweenthe object andthe screenshoulda lensof focal length 16.0cm be placedin orderto
producea crisp imageon the screen?
J)Fr\"g'/\
ll*
-k,\tfuL)4j,\M\a
,J,.ibr**
u,-rti^* ^"rJ #*
AA/"*j/, 4"""r,,4*
-!4.la"tx"Q 1". P
tr
t
t/
+
i
/Y
7
(?b-?)
|
= ?6-o" 'o
= ?{e.. O c.o -
I
*:"'
/*
L+
).--
+-)'=
J6'P n
('71,'P)P
ll"
^!
e
'/t"
p-p = I )li"
'lG
----)
)
|
- =' -E
tb
]b- ?
pt- tra.o".) p + )2]bcroa=
J
=
aa-qcrqo"
D
I
w/a+e,&-a
egt^*fia
o
Problem 9
b 5o I ctrr
A movie camerawith a (single)lens of focal length 75 mm takesa picture of a personstanding
27 m avray.lf the personis 180cm tal1,what is the height of the imageon the film?
f=
?5t,'q
i"r=4
= t5"to?r.,
p = A?..)
t/
I
[
= -7-*
'/t
u
T
lt
--)--
- -)-
/ t
?5*lo',.'
lr') = -c/
/P
/l
---)
=
a"t n
I
I
y
13.3 ---+ i = O.O15 r.r
.-r=--
= l$Dcr.r
| ' -(>
n-L_
r ll
rr= Yh *
tQ=
- ( o"o?5 ,.'
'
a'. / fr)
h,'- mh
tt
h=
( a-?'.ro-')( rro"^)
h'= o'bo.t
tD= A.?*lO3
.-:-
Problem 10
In the figure'below,a peasits at a focal point of the first (nearer)thin diverginglens,4'00 cm
from that lens.The lensesme identical andseparatedby 10.0cm, with a commoncentra.laxis.
(a) Whereis the imageof the peaproducedby the secondlens?(b) Is that imageinvertedor does
it havethe sameorientationasthe pea?(c) Is it real or virhsal?
+'
a -a
Ta-'ir
z--------------
$^=$,
'
a-------------+A-----------)
{.oocrr;
)M^D.a"y'*r@lou
lo jotpl
'/r"* Vp=/+ - t1c'=/+ Yp
t '= f'P'
'-ilf
P'-f'
i
g- !-oOc-s
)1lJLt='
=-g.oocn
o-
4 . o o c$
?
{,'..
(-ar,DD.n)( q, oo".n)
L, =
4.Do.,q- (-4.*-t
/,= -A.oO.n
\ "m^u;-r{
rd a.oocntb }pe!-#*lrfu
p"*
!e\4 }rAi.2 ^-alAcm
-A )te ^"i'* ,t Pu ^r&").!Y^'
l/^
*
ll cnq
I
- a.l.O0clt
, C\=
.(.'
Lo=
l^f2-
(-q.e-,xB.d
---)
le"r, -(-1'ooca)
P',i
?
--)
"d?"'a1
3.ooont )no 1"4*otr nt
attzA-!e")
(c) A^/,,"*
aq(O,$n^*r*,"€, -\ffil
(,b)M=n',".=
(+)(ii) =(ffi)(tr) =/, -,
,{v\42
))
Problem 11
In the figurg below, an object is piacedin front ofa converginglensat a distanceequalto twice
the focal length/ of the lens.On the other sideof the lensis a concavemiror of focal lengthB
separatedfrom tJrelensby a distance2(fl +/.2). Light from the objectpassesrightwardthrough
the lens,reflectsfrom the mirror, passesleftward throughthe lens,andforrnsa final imageof the
object.What are(a) the distancebetweenthe lens andthat fina1imageand (b) the overall lateral
magniftcationM of the object?Is the image(c) real or virtual (if it is virnral, it requiressomeone
looking throughthe lenstowardthe mirror), (d) to the left or right ofthe lens,and (e) inverted.or
non-inverted
relativeto the obiect?
oo
4r,ut 94ot\ D" -QN
-IT--
,l
C
)l
D,. --
o\)l
L1-
cl
--
(
.\,
?
'o =
c-^
j-YL
R-f'
^,.}
Y-
- Lt=
n r - dt l
)t
d.tt
4\c
dlr
o ,Jtf.o^r" Af, b )et ^q\t
fl\, ^It(\'o.42
+\,a,\\fu^\t )l
-al.^a;fri"
4 Ue
Ar rrA"*pdMt )}rt.rnrr^4
lxr-na
^ //'
/'\
^a
/\ /'
'J-ra
cI (T,+TxJ- cLT, =
I
4
la.-
a
I ). f r
(,?\-
?
LLf')
:
= Afa.
;I'tr- Ta.
f^-T r
Lf= 2.far
I
W'o .vu^lin ',bta
?z
tt::
)3
^ *aia,na Af{ }t mo lt'l'* a6
fr,, Nn;4"#/\s,\r%r ;^ ;'t
--
fle Arg*$+ffia
-A9^A
(\
^r-lC
a\ Tr{T*,1 - }Tr = }l-r
^f(
=f,
" C .=
Lg
io= ?
(q)
l'a
-J'-n
A
^
=
J:}--fi-
a
-5E
._.
Ay$ NW\AA
)/rt\D\,/W u
U-
_J3t)---
f{,-f ,,
La..<-
c\ lt
(b) M = rn,$x rD3
'(-#)i-#)(f)
M,=(-*)(.tX-*)=
(t)
r --+ 11=-l
.-A,we,la2o, }*,
$/^-l "-^gr:"@
(,d)+^^,/
,ttvtot4t )a
b, -bg*a fu Ps,,
Ce-) -rt^*
!
.
II
iI
'!
I
{
l
i
II
I
\-.I