termsof tlre angularseparationof lines of diflerent wavelenglh(D = A'0|AZ)and in termsof the
You shouldknow themeaningofthe resolvingpowerandhow
gratingpropsrties (D: mldcos?1.
to calculateit in termsof the differencein wavelength(R = il LX) and in termsof grating
properties(R: Nm).
You shouldknow that the diftraction anglefor the first minimum of a circular apertureis given
by dn: sin-r(l.22)/d),whered is tle diameterof the aperhre.You shouldalsoknow that this
alsogivesthe Rayleighcriterionfor the resolutionof two far-awayobjects. If d> 1
expression
thendnis givenin radiansby 0n- 1.ZT"/d.
2dsinq: tnA.
You shouldbe ableto usethe Braggconditionfor diffractionfrom a crysta1:
fiom the reflectingplanes,nottheirnormal.Someptoblemsask
tlat dis measured
Remember
you to usethe Braggconditionto find the atomic separationin a crystal.
Questions and Example Problems from Chapter 36
QuestionI
You areconductinga single-slit diffraction experimentwith light of wavelength1..what
appears,on a distantviewing screen,at a point at which the top andbottom raystlrough the slit
havea path lengthdifferenceequalto (a) 5X ard (b) 4.51"?
rq\
a-A,'-;1..rg=
ol
---+
.\
AL=rclA
O n = 1 1" { 1 . . .
f
(h)i.r,;t^*
3^-';il(ffi
,Y
Le.D",
.,-' Question2
(a) Figurea below showsthe lines producedby diffraction gratingsA andB using light of the
t
samewavelength;the lines me of the sameorderand appearat the sameangies0. Which grating
hasthe greaternumberofrulings? (b) Figureb below showslines of two ordersproducedby a
singlediffraction gratingusing light oftwo wavelengths,both in the red region of the spectrum.
Which 1ines,the left pair or right pair, arein the order with gteaterm? Is the centerof the
diffraction patternto the left or to the right in (c) Fig. a and (d) Fig. b?
,b,a
^,^
Aor*"tr-'
AI--T-J'3 [-]-[-=_-l
\a)
(")E
(b)l4l
(.)lw(l)@
(r)
lt lu'onN) ^r*.0.h a e
nD=-r.dwtO
44
nt
,__-,,,
"oJ&
bqt^ J&+&a.t/
Problem I
Monochromaticlight of wavelength441 nm is incident on a narrow siit. On a screen2.00 m
away,the ilistancebetweenthe seconddiffraction minimum andthe centralmaximumis 1.50cm.
(a) Calculatethe argle of diffraction 0 of the secondminimum. (b) Find the width of the slit.
= 4{t*1d1m
\
p e i,.0orr
1".,
=
).., /- Fanlo
(b)
rbD"rS=YnlA -,
$tn.a*'r.-ra.€
(a) I
--- lO - O, Do?5 /LaA 't 1
_ O"OlB6r'r
A'oot"i
r-n=?
Aig'riS
q=?
=.9L
=
I
o'7Yp
c.tao"
(
lY\ ,\
'}.xr"r O
Problem2
A slit 1.00mm wide is illunrinatedby light of wavelength589 nm. We seea diffraction pattem
on a screen3.00 m away.Wlat is the distancebetweenthe first two dif&action minima on the
samesideof the centraldiffraction maximum?
*lc"g=
n Vq
#ar.ret = Y^/O
Y^=e>D
A) = Y.*Y,-- \Dy'
Y,= r\D/q
A) = t,l?xld?cq
= l.??o'n
( l.oo. to tm)
Problem3
A single-slit diffraction experimentis setup with light of wavelenglh420 nm, incident
perpendicularlyon a slit of width 5.10 pm. The viewing screenis 3.20m distant.On the screen,
what is the distancebetweenthe centerof the diffraction pattemandthe seconddiffraction
minimum?
\ = 41o"1o
I,
a j
o, = 5)D*
br.,
Fo.'" d = f -/p
D
lD
= 3'rot"''
a)=?
.,g=
)6/>a l$.rrY".tJU *49
^\^=l-/v
o
n
*yq
----+-oiqg=
m)
>
.bmg=L/.;Oa€
)-
=-
\rr
lQ z\ L"'
-."..--*
q
5,d
/ l\\ fi\
Problem4
The figure below gives cr versusthe sine ofthe angle0 in a single-slit dif&action experiment
using light Of wavelength6i 0 nm. The vertical axis scaleis setby ascr,: I 2rad. What are (a) the
slit width, ft)the totd numberof diffiaction minima in thepattern(countthem on both sidesof
the cenjg{qf.tlE diffr-actionpattem), (c) t@ leaq!q4glefora minimum, and(d) the greatestatgle
for a minimum.
X -- 6 toott''
a (rad)
> <>(: le ^''d'
Jr,e\r\ a*n g =-1
sin I
t
o(=
(rr"-.t)--ny\
-4
/r- I
q = X,:3o n"r
-Tr
A -
(b)
rr
t,:3x10-or,
orvnO=rq\
--+-a.vnI= N\Y
., /x
6 ^;*;
(")
!rD'*
04/tt
(1)
mYo< |
1",< A?3on"o
frr < 3"S1'
(lr),rm
---->
r\ill
a*Ab -'--+
9= rn\
^
n
/>O<ate
frrt--lrAr +3 orr erfur4A^t
'-ll
)bOL
O il-rL
h
fn =L
-= 0, 1,{,e.
-.+ -A'\O:rnA
oi
?3ionq
fe=iil;l
t
8" O)r;gt
(J) q"fu4 o",4k')' A-t^'}"
l=
(f)lto,onflzzanm =
Jr^.,g=
A GGI
rt'D
-|
0.?sS --''IC'-
I
Problem5
A double-slitsystemwith individualslit widthsof 0.030mm anda slit separation
of 0.i 8 mm is
illuminated with.5001rmlight directedperpendicularto the plane of the slits. What is the total
Q= OoO3orqm
n= l^ + X^*ry,t-r.Y*:W#
d=O.lr$-
@*l)u /*,dgfu +t
W^,"h^
*VU/tUntvt
m=
O.l9orl
:!
O'O3Oortr
m -- 6 #,@
?
*a asrww{
)rQ -,MLJ9u Fl =o z'^J, rn = l, 1,3,q ,+ g
^t ;fu"l /v)s
&,ntn2r.
ll -t.r€l*&r,"€Problem 6
In a double-slitexperiment,
the slit separation
d is 2.00timesthe slit width w. How manybright
interferencefringesarein the centraldiffraction envelope?
-d = e"oor,r- A.obq
* l'"",*r*
)
;*^"-U*
u{ 4-*t
d
r'y,.wr,.vy..a
t'.= l) ;
r'*'r.4,/-i'*--< (-lr-*;*
dd,+,"*-,b,-qp",b-t
_2r".g = Y^
4",*;-?jr ) ;*,
l-
d /J-/\,t'.r ty'
= 51 7a
''/**-h
sr.
$tn ". -+';"*
J(Y") : -)
hI = JA:
= )\
: a4;O
e.ooq
;v$4'--'fi--',
+ 2)'D4* 4"r,4h = 3
:,^fit\ ti)s
)s*n,a,L4
u4
r*,tt,,v'"<,
gr'-a rg*,
',o',rt)
#,t'+u
Problem7
Light of wavelength600nm is incident normally on a diffraction grating.Two adjacentmaxima
occurat anglesgivenby sin 0 : 0.2 andsin 0 = 0.3.Thefourth-ordermaximaaremissing.(a)
What is the separationbetweenadjacentslits? (b) What is the smallestslit width this grating can
have?(c) which ordersof intensitymaximaareproducedby the grating,assumingthe values
derivedin (a) and (b)?
Nr\,a{\N\4 ct/u
d*^t"rf
(')
d (o'a,)- 'q\
J to.g)=(n-r)X (),)
(")
(b)
n
-4
It
%q
q--
(.)
1
(') {-ra'q (a)
A.olA^srf
(o,r)tr=\
J=
^r1\,O4^;ftY,\ n\r.a,t'%
Jta a r.r- 1
)"p'z*.J
-t
l o \ = ( o o o o r q - 6.o^lo-bm
o ^^+^Jot ,tf ;-
=Ll
;atay^ba
p.re"L
J/q = / . 5 . [ d P r n
(
,n X ( I --. m S I/^' ---r p
-oi" g = nrf,
6.oorto-6 rq
looOrt O-rrn
1q --O, | ) ?) 3 151to2211
h< to
*
Probfem
ttt =J
A diffraction graring20.0mm wide has6000rulings. (a) Cilcuiate the distanced between
T
/ror''^€
if
the
viewing
screen
on
a
maxima
occul
intensity
(b)
0
will
At
what
angles
adjacentrulings.
'b"arrar &l!4
radiationincident on the gratingias a waveiengthof 589nm?
ys = !D.or",--- to.o* lo-)..,
N=6ooD
-oy^*n
Jl *"^l^.
j = yN
^l""dldh"&
(a) J=VN =
6 ooo
"''9:li3
rr --6.r\r A,- -'
( U) -4/'\ e = *
%.
-4,k uLa,tj^E
r., 5
jw;'<
S- ), l,r,'" ^,*,l,ol
3-33,1o
-*
) |
t t €L
,/
r'-l :
d'/
'/ ^
----+ m < 5'?
5?.1 *ldqm
O = -4.r"{'
\ /
r'iy'd
m C 5 s 3* 1 d i ^ )
3.33 x lD:@rn
-=====
1'
.m=o o=o"i m=lg=lo.e" ; r",-- l g:1o.")" I rn=3 O' al..a "1
tn-- q o-!5'j
f'r=5 o= 6 4 . 1 "
c,a/ru,^,0\4
Problem9
A gratinghas600 rulings/rnmand is 5.0 mm wide. (a) What is the smallestwavelengthinterval
it car resolvein the third orderat l, = 500 nm? (b) How manyhigher ordersof maximacanbe
seen?
= 6oD**r-w!/
n\
N/{.,l)
-- 5'd
\\
-- aoo6
n rr1
(") 4=xT
A\
a\= )*'
z--N
Nr\
A ) = !5b9:O.
[ooo)(a)
v\
(u) frtx(1-rn<
.r\nI :t
Probfem10 /
d=-/u ,- r^Yuou"#..= l,u't,166n
(rulrtdbn)'/
'76oat61^
->
\)
Fra 3'31
cr'v\-AA"4r
.-<=:
TheD iine in the spectrumof sodiumis a doubletffih wavglen@T S956d- A9'6ffi.
Calculatethe minimum numberof lines neededin a grating that will resolvethis doubletin the
second-orderspectrum.
4= I*
= N'q
AX
\" e=
A\ =
Ftq
R ^*
a
5g1,Gn"r - 5??"o
nr.l
-
\_./ .lOU
n m
rq=A
N\_
rqAr\
N_ (s:r.:.-)
| (a"r"on.)
-----a
N = 49l
C,,ft\ JVv+
/a
Problem l l
An x-ray be4mof a ceriainwavelengthis incident on a Nacl crystal,at 30.0" to a certainfamily
of refleiting planesof spacing39.8 pm. If the reflection from thoseplanesis of the fnst order'
what is the wavelengthof the x raYs?
O = 3o'o-
rn : lr 2rZ, -,-
2..Jnr""O = r"t \
-)
+i',}. 01&
I = 39.3 p."r
I
-- 39.x.,^
1d'o.",
t'n=1
1Ja,,"nO=(l)X
\=?
\ = )' t-a'v" g
:
m=-1.
z\ = A (31.1^ )O't.).A,^^3o.D"
)
= ?.qs' lo--"n -'31-g g^
Problem 12
ln the figure below, anx-ray beamof wavelengthsfrom 95.0pm to 140pm is.incidentat 45oto a
-family oi reflectingplaneswith spacirryd 275 pm. At which wavelengthswill theseplanes
produceintensitymaximain their reflections?
).= ?s'op,n-l'logo
I = tts"
-r-
l1
pm= lo- fYl
J = A?5pr"
ta
f
-T.
ld
t
?..J a^.9
rn
\i
\=
x=
"J3-PJL
0rl
e,dair"ro=n\
,r,ad*:
1!:"5r"1tr"
rq
1 t 1- l 1 1 1 3 ; . . .
\-/
371pn, l1'l p'.,,,\3oy., 9?.1pn,
m--l
rrr=l
14=lvr,3,."
n=3
n -*f
lffi
.i
C.,rn -2lvL.t'
/'
Problen 13
The two headlightsof an approachingautomobilearc 1.4m apart.At what (a) angularseparation
and (b) miximum distancewill the eyeresolvethem?Assumethat the pupil diameteris 5.0 mm,
aadusea wavelengthof 550 nm for the light. Also assumethat difffaction effectsalonelimit the
resolutionso that Rayleigh'scriterion can be applied.
J = E"o* 1o-3$
9n = t'ra'y*
(^)
= 55o^go-1r',
\
q.)
V1 = l , ? . , \( B 3 or 1 p
X = l.'l^
E ' o t 1 6 - 3*
r= 7
O^ = l.:t1 '1d1,,,^.t
(u)
'g3r+
/a
\-.
5=4O
----)
A=Yg=
\ =46 &ara^n,cj.4a.-,A,0a.
Ltf ,q
4 = l'o' 16r.
_->
l-3tlr lti'{
= lo Ktr
Problem 14
In June1985,a laserbeamwassentout from the Air ForceOpfical Stationon Maui, Hawaii, and
reflectedbackfrom the shuttleDiscoveryasit spedby, 354 km overhead.The diameterof the
centralmaximumof the beamat the shuttleposilion was saidto be 9.1 m, andthe beam
wavelengthwas 500 nm. What is the effectivediameterof the laseraperhrreat the Maui ground
station?(Hint: A iaserbeamspreadsonly becauseof diffraction; assumea circular exit aperture.)
-b^.a-"^P
6^'bytt"
-=>
g^r-
-o
e= q!\
'lr^
/"t aU
Pv crYLl"i4-0ar/)A h6 dN'4
a/ qk lf^"
.dr\ .N\t
,.1'/\-^^{Vhr\,\rr.I Aa
Q=xb=1(t'aaV+)*
I
352J,.;6=*
dv'aP
e = l")\yJ
q-
2-5 7* tO 5r"^.1
9"1"'
a,AmO=]::l},!
^r/vc^r..a61a4odwru)
&ci
po4",t
""
6!=
1(r.ae)Cso?^16r
0.5 7" lo-e
lo-' a-4
-H
lvvt
au'it c-a4{- )
J = ). (r.la\)
Q
6=^,