.
Questions and Example Problems from Chapter 35
Question1
Is therean interferencemaximum, a minimum, an intermediatestatecloserto a maximum,or an
intermediatestatecloserto a minimum at point P in the figure below if the path length diflerence
of the two raysis (a) 2.D,,,(b) 3.51,,(c) 1.8X,and(d) 1.01,?
For eachsituation,givethevalueoi
m associatedwith the maximum or minimum involved.
D-----4
T
)
I
Question2
The figure showstwo sources51and Szthat emit radio wavesofwaveiength l, in all directions.
The sourcesareexactlyin phaseand areseparatedby a distanceequalto 1.51,.The vertical
brokenline is the perpendicularbisectorof the distancebetweenthe sources.(a) If we startat the
indicatedstart point andtravel alongpath 1, doesthe interferenceproducea maximum all along
the path, a minimum all along the path,or altematingmaximaandminima? Repeatfor (b) path 2
and(c) path 3. Explainyour reasoning.
- - - - - { - - - j
Smrr
(q)
(b)
(lte<-'r", AL = /.5>,iq !,fu
F*t)
Problem1
Two waves.oflight in air, of wavelength600.0nm, areinitially in phase.They then travel
:
throughplasticlayersasshownin the figurebelow,with Lr : 4'00 ir.m,Lz= 3'50 pm, n1 1'40,
andnz : t.OO.(a) In wavelengths,what is their phasedifferenceafter they both haveemerged
from the layers?(b) If the waveslater arrive at somecommonpoint, what type of interferencedo
they undergo?
2----------^;.#
Jyt\ Wr bdA',
+ $unuun4*
.!-a
N= t/x
\=6oonq
i_l
LO*
N*= 44- * e::l,X-'J= G.EryCl
I
(b.boo/,.')
4. oo i,rrq)( l" tc'
)
o,5o6rr'1)( l.o)
(a,uoo'rn)
4ae
N*-N,:
==t:
(0.teoo,.^'"\)
= 6.booqrr
O"Sz)- klo.4k ^b4&Y' ,\9.4'^re?
->
g.?13
)^
tt,o:r< cz6ttY<-f,U*'---t
Problem2
.
In the figure below,two pulsesof light aresentt}rough layersof plastic with the indexesof P*(:)l:@
refractionindicatedandwith thicknessesof eitherL or 2L asshown.(a) Which pulsetrav.it I
throughthe plasticin lesstime? (b) In termsof L/c, what is the differencein the traversaltimes
of thepulses?
rr*rr-rr-l.-r*l
f)=
t'
+or p"/4" A => t ^=
**
P^'l* f >
X,-t^=
d/v = /('/n)
= nVt
-r l,'la4//c-*l'@7" + l"\S Y.
fr = l.5r(ery
633LL-
v= t/o
Vu -
+Lt"glc+t.SoL/. = 6.zst1
to':'tt/. -
6 -o3
L/q.
Problem3
In a double-slitexperimentinterferenceexperiment,the slit separationis 2.00 pm, the light
waveiengtliis 500nm, andthe separationbetweenthe slits and the screenis 4.b0 m. (a) "what is
the anglebetweent}recentera:rdthe third bright fringe? (b) If we decreasethe frequencyof light
by 10.0%of its initial value,how far along the screenand in what direction doesthe bri-ghtfringe
shift?
'fiu^Jop----- d-o-^g = n
ft:o,, r1{,..,
\
O =/v",-t ( ^ Va )
,c,a.*r(..,=otl. dt g =o.
-lr"rw
al= f,"ooum
= l.oor lf orq
=) 3d
\ = 5oo..^
: 5'ox 16?.
(cn*=
s)^-a
*| g . ,.a.ln-'
fun4rgr\r,,,L
| 1ffffi
P = 4-oot
I " O,t.rs^l
(^) o*ga jrAt*"*,
Ler,fu.,r*;
S"Al^.A}.f e'r^At
(b) 4 f,",,-= o.9Df, Jfir" \;
O.trtl^A_ .= qr.G"
-SoD
= Yu"ro
=
n.r
-
O^,,*=-a.I/n-'f:I_::_tr"*;) ] = O'1Xb^.",L->
Y - D'r"q0 ay = D(uno^.-- tr^o)
p r o b l e4
m
I Dbnrn
"==:-
o"9o
5{o.5"
Ay --/.sl,."r
= (+.oo-) (,U, sn.t'-]*n ttt,u) -)
aUna$,r.arn cur*a
S-uppose
that Young'sexperimentis performedwith blue-greenlight of wavelengthsoo ,*E
slils are 1'20 nrm apat, andthe viewing screenis 5.40m fiom the slits. How far"apart
areae
bright fringes?
X= 5*.td :
\/
= S.oD"1d11",
= r.'\P
/rc)
-{
./.*t=
/
A) = )*.r
-)^
= (rnrr))9-
^)f
A) = >DA
?
fq -o
= ( E.ooxlo 1') (t.?C.
!
!l'=
l
m=1
=
\
r-\
,/\ -l/
I
ol.
|rl=a
rl
*
l"|Onrq = l.lO* l03tq
D = s"nio-
-t
!}-,)XO_
].Aox 1g-3*
W
Problem 5
Laserlight of wavelength632.8nm passesthrougha double-slitarrangementat the front ofa
lectureroom, reflectsoff a mirror 20.0m awayat the backof the room, andthenproducesan
interferencepattemon a screenat the front ofthe room. The distaacebetweenadjacentbright
fiinges is 10.0cm. (a) What is the slit separation?(b) What happensto the pattemwhen the
lecturerplacesa thin cellophanesheetover one slit, therebyincreasingby 2.50the numberof
wavelengtls alongthe path lhat includestl-recdlophane?
\ = 634"9n",'.,
,t'uu;,c* l5) << D; )/"
D=A(f,o.o,q)=4o"O.c
*-'.J
Ay = \o.oc""
-&t^a g R & a ^ , r r l J : * . & o S
-A.M" D
J_
-4,9$\ I
/4.
= /^/o
L\
/J
\t
)
-
-
-!
/m+l
"/
---r
A = rt'-).,,-"r
-+
(7-
n \ /J d.-
)x V/,
/d
=
.1 ,,
^ -.
d.5t
rioq ^Ap Jf,? -tr"",cl'La o"rg& qa1ajAr""abr1
,rr,
:
/'/o
. ...:::..-:
---.:.
c"r'g& S aa zu*.,g-t,,*,"-0{
&=\%) = ( a : e . g , t o o . ' ) ( q o . o - )
(ro.o"16rr.)
r-*9
x lL} rl
a'\-a rr\/6lJ)'
O-99 nt atP*"ra
(U) ;U, i-eAt -k\,a." -yu ):r-0-^"bot^-r^r
--^--'",} 4 p^-r* -eprobrem6
a{Y\-Am,AAr\.q ot 'rr:d
rua&
/\M.^,54
ln a doubleslit experiment(seethe figurebeiow),?,: 546nm, d = 0.10rnm,andD = 20 cm. On
a viewing screen,what is the distancebetweenthe fifth maximumandthe seventhminimum
from the centralmaximum?
5tte* to-l tq
tf\
y = Dfov\O
o.lo * ld3r.l
)
O.lDc.t
4raArnda *@t
,aro g:..r;\A
Y= Dr. & =Dt"g
M^',A,/r;ill',\
-ali\g = .qYd
4
#at A^/t^0^r'u/,rn
^/hoEi= [v.,rtt/r) X/O-_+ y
5 't^.-x--t Y = (s)\DA
?s^*_)
J
j=(b+yl)\%
9r4tfi,'4'
D-Ig
------. y;=
=
"
a)
(6*7')
\D _ (s)\p
\-_--d
+
of nr=o
e,(a,tD*6'",,r)
tI4)
:
D
J
3/+
,J
3r'l
l.t * lo
Problem 7
jewelry areglasswith indexof refraction1.50.To makethemmore
Therhinestolesin costume
reflective,theyareoftencoatedwith a layerof siliconmonoxideofindex ofrefraction2.00.
What is the minimum coatingthicknessneededto ensurethat light of wavelength560 nm andof
perpendicularincidencewill be reflectedfrom the two surfacesofthe coatingwith fully
constructiveinterference?
J.r.rc{^x
L )^-l!"4^-e4 /*tr al**
D,=l'o
//Jrtlr4
o": )'o
\L
.r-+
h
----+
Nu
)NyAJzeA
+X,o4e1llt
W
-prt, @'rt^^ fr,q N'',tr$-&/^/a
:
Da=l'5
/yn-f/l\J n/rwt" ^yv4tvmtaA
\
*4t
= (n*t/*)
m =o)l)a)...
\/n^
nq = O
3.'L= (o*'/^;\,/n..^-
( 56o n^)
4 ( a.o")
n
f : 4 O"
Problem 8
ln the figure to tlte right, light of wavelength600 nm is incident
perpendicularlyon five sectionsof a tramparentstructuresuspended
in
air. The structurehasindexofrefractioni.50. Thethicknessof each
sectionis givenin termsof L = 4.00 pm. For which sectionswill the light
that is reflectedfrom the top and bottom surfacesof that sectionundergo
f,rlly constructiveinterference
?
'1
r"i= l.D
n: l.5b
5=).D
,.r,a^J
l-.-> {"0,--"l"&
1,,,*
"b+
\e)
.^'o'-ga --) Art' P't"'a',-*Z:A'"&.|
--(*-'1.) X,
4".[fu .--*do.-,rbv"rrit4r"'n^",* a!'
tq-b1
l.r?,, --
jo q rc'o'&*xt :
,A*} b :Y'L )^l.L% fr
(&-r"--;
K=), b-- K=\r." .r e-: K=t/o)
--r
1(KL) = (m * !.) \/6,.
?
K=
=
(-*
-r-^* u:wr* 6=\'2., t/>r3, ot '/t" , ^o
"fr-,'-
V")(Gosx
16?-)
! . ( i . s " ) ( a t ' o o ^l o - n * )
m,l^,}-Ll-P
Problem9
The reflection ofperpendieularlyincident white light by a soapfilrn in air hasaa interference
maximumd't600 nm and a minimum at 450 nm, with no minimum in between.If n : 1.33for the
film, what is t]le filrn tlickness, assumeduniform?
^,oq7 il,'WA % \/t i ,,.^e A; ^oJ-4,A4id
n = I'o
= t,53
nnaAlr&ti,,nu :
= l.o
\L
(mrIr
/=
.
= (n*'/r) Vnx r4,=arrr),
.
(.
+ [)
( 6oo.'q)
tq : o, lr t r . ..
a ( r.33)
d\,,X
L = l 1 3 n ' q , 3 3 g n r \ , 569nrn,??lnrrrr..,
t
----t-'
/fn
f\.l,|fnlr/fi\
:
Z:
n=o1l1),
Dr\
'troa
rn (.rSao^)
7-:
- y 1c o r l l r ,
11r.:s)
L
; Dl lblnm,
B,
So8o,n,"'
L= 338"'r"r.
>
.
338arqt..r^^."r
p*sa
fuIDflr
Problem 10
In the figure beloq light is incidentperpendiculmlyon four thin layersof thicknessL, The,,
indexesof refractionofthe thin layersandof the mediaaboveand below theselayersaregiven.
Let l, representthe wavelengthof the light in afu,andn2representthe index of refractior of the
thin layerin eachsituation.Consideronly the transmissionof light that undergoesno reflection
or two refleitions, asin figure a below For which of the situationsdoesthe expression
A-
" u " 2 , f o rm : 1 , 2 , 3 , . . . . ,
m
give the wavelengthsof the transmittedlight that undergoesfirlly constructiveinterference?
j/r<
\ = gl(4)
-o
). ^}"&h
(u) L,r9,#
J,rr.td
O er- A
LL= rrr\4^ -tf" /malt",wrrn
(c)
)a.l,g.tl
(t) 7.*'$
61 *^"I
c
Problem 11
In the figure below, a broadbeamoflight of wavelength683 nm is sentdirectly downward
throughthti top plate ofa pair of glassplates.The platesare 120mm long, touch at the left end,
and are separatedby a wire of diameter0.048mm at the rigbt end.The air betweenthe plates
actsas a thin filrn. How manybright fringeswill be seenby an observerlooking down through
the top plate?
Incident light
lil lr
,qlaaa(or= t.a)
or,lrr.6nr= f-e;
,gbaa (lu= l.s)
120mm
). -r"^*
1 - A)"4il 9,"4.Y*r\
,. ,
/
.. \ \ ti
),L = (.nq*'/r)I/"/ "\
ne = I'o
\' !^l -_ n
v)
\r ) . \") '
,.
i
E,L= (m+r/r) ^/r,.)
{
J ll"lu,<&-
r.r u.,M
fr; = 1(o. orr , lo t")
[=D.O9lnrn
-
'/q
fD = /1o.as
( 6r:,^ lo-qm )
Nr= l4o .4
ffiyt uiat g" lt1t".er,^AA*$^rij_ ,t"a
(rn=o,1,1,..., llo)
Problem 12
In the figure below, an airtight chamber5.0 cm long with glasswindows is placedin one arm of
a Michelsoir interferometer.Light of wavelength)" : 500nm is used.Evacuatingthe air from the
chambercausesa shift of60 fringes.From thesedat4 find the index ofrefraction of air at
atmosphericpressure.
4 f,o*"**r*
#"#
il,"-.Jir
$-**J*
oh$.$-0,'0,^.
rr}.'aftglr-&-o*" {oo X
lt --'4": 'Y^
ln
rt
^Dat
q
)QEg'?MA
-lm
n'ir-ata7.:,-,i,
= A LYX
-Lr-l;#&D)r1 *--) hl*
/ll-^Jfi&{t oJ't ---? l\**
= 2-ttty,
lrlr- it.l*"={oo
*?
-ry
=Go
G'o }.
rr- l=
,---nr
o.L
n :
Jo ( Soo* 1s-t*)
5.o ^lo- e m
....?
fi= 3oY+
{n-ii
t
= Ca