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NATIONALUNIVERSITYOFSINGAPORE
PC1142Physics
ll
(Semester l: AY2009-10, 28November2009)
TimeAllowed:2Hours
INSTRUCTIONS
TOCANDIDATES
- Thisexaminationpaper comprisesEIGHT
(8)printedpages withFIVE(5)shortquestions in
Part
'l
andTHREE(3)longquestionsinPart2.
- AnswerALL
questions.
- ThetotalmarksforPart 1 is40andforPart 2 is60.
- Answerstothe
questions are tobewrittenintheanswerbooklet
provided.
5, Thisisa CLOSEDBOOKexamination. Calculatorsareallowed.
A list ofphysicalconstantsandformulaeisgivenonpages 2 and3,
Thermal
physics andkinetictheory:
(i) Kinetic mean free path is
given by
4
I = --: - , where
nvis thenumber density
J Z x. n , ' d 2
andd isthemolecular diameter.
(ii) Maxwell-Boltzmanndistributionin3D:
/ l 3 l 2 r a
P(v)
= 4n.n'-(
l"''.
,rJ-
t '
u'
\ 2 n. k ,. r)
' \ _ r u , - ,
(a)Root-mean-squarespeed:
y-^ = ^ltrt' .
V m
(v).Oneform of adiabat:p .Vv = Constanf .
(vi)Stefan-Boltzmann equation: P,"a
= 6. A.e .Ta
(vii)Planck radiationequation:
2 n. h. c 2
f o a l r , ) = m
r'.[*o[
^
n,''
=l-rl
[
(. r' k , ' T
(viii)Wiendisplacement law:
" 2.898x10-3m
.K
lu
peak
=
(ix) Convectionequation: Pror,
= h. A. LT.
(x)Conductionequation:
Pcond
= -k A +
L
(xi)Linear thermalexpansion: AL= cr.L.LT.
(b)Averagespeed:
v,,
(c)Most-probable speed: ,," =
^l2krT '
l l m
(iii) ldealgasequation:p .V = n .R.T .
(iv) VanderWaalsequation:
^ ^ 2
( p +
o ' l '
). ( v- n. b )
= n ' R. T.
" v z
Thermodynamics:
(i)First lawof thermodynamics:LU
= qin
expansionworkdonebythegas: W,,,
=
[p
.dV.
(ii)Carnotheat-engine efficiency e. =1-L.
" T ,
Geometricoptics:
(i)p and
q are objectandimage distancesmeasured
respectivelyon oppositesidesof thelensor of the
refracting surface:
(a) Object-imagerelation(thin
wherel=(n-1)
ff
-f)
t
' . R ,
R r )
(b)Refracting-surfaceequation:
bns):
1*
1
=
1
,
p q t
f r t
,
D z
_ f l 2 - D t
p q R
(iii)Enhopy
dS
=
dg',n
.
T
(iv)EnthalpyH= E+ PV.
(v)HelmholtzfreeenergyF = E- IS.
(vi)Gibbs freeenergyG
= H- IS,
(ii) Gullstrand equation:
(a)Effectivepower:
P = P, +
p,
(b)Fronfvertexrefractingpower:
P. -P. +
P
. t. ,
n
p r. d
n
(c)Back-vertex refractingpower:
P. - P *
q
. o. ,
n
q. d
n
' - n
SkBT
nn
PC1142Physicsll
Part1 Answer allFIVE
questions. Allquestionscarry 8 markseach.
- Thedarknebulaininterstellarspaceisthought tocontain50hydrogen(H)atomspercubic
centimetre.Thesehavea root-mean-squareSpeedof 1400 m s-1.
[Nofe:
H hasan
atomicweightof1,
g mol-t anda diameterof1.0A.l
(i) Compute themeanfree
path oftheseHatoms,
(ii) Compute theirkinetictemperature.
- Considera blackbodyinvacuumcomprisinga thermally-conductive solidcoreofradiusrc
andhavinga uniformtemperatureI', surroundedbya thermally-insulating solid shell
of
thicknessd withthermalconductivityk,asshownbelow.Heat isproducedinternallyinthe
coreata rateof P. Derive anexpressionforthevalueof I, atsteady state,intermsofP,
k,16,d,andothernecessaryconstants.
vacuum
1,0kgof helium
gasis heatedatconstant
pressure
from 300 K to600K. Computethe
changeinentropy ofthe
gas.[Nofe: Hehasanatomicweightof4.0g moFt,]
- A camerawith aneffectivefocallength of 300 mmandF-number of 5.6is used torecord
theopticalimageof theground fromanaeroplane flying10kmaboveground. Canthis
cameraresolvea carontheground, if thelens systemis freefrom aberrations,basedon
Rayleigh'scriteria?
[Nofe:
The opticalspectrumspans 450-650nm.]
- Waveplates are opticalelements designedto differentially retardthe phase of the
components of a polarisedbeam.One suchwave
plate ismade ofbirefringent crystalline
quartz withanordinary refractiveindex(no) of 1.5426andanextraordinary refractive index
(ne) of 1.5517at a wavelength of632.8nm,and orientedasshown below.Thefast axis
hastheindex no,whiletheslow axishasindexnr, The thicknessof theplateis 1.00g
mm.
A linearlypolarised beamof 632.8-nm lightis incident normallywiththe plane of
polarisation
at anangle0 measured clockwisefromthefastaxis.Compute thephase shift
between lhecomponent
polarised parallel tothefastaxis andthat
polarised parallel tothe
slowaxis,andgive thestateofpolarisation after
passing thiswaveplate.
- Achromaticdoubletlenseshave significantlybetteropticalperformance thansinglet lenses
for imagingapplications.An achromat(general structureshown below)consistsof a
positivelow-index crown
Elass
lenselementwithradii of curvature h ?ndh, andcentre
thickness fcr,ceffiertodto a negativehigh-indexflintglass lenselementwithradil of
curvature h and
rl, ?fldcentre thicknessfcz.Theselens elementsarechosen so asto
cancel chromaticabenationat the red (630-nmwavelength) and blue(450nm)
wavelengths.Onesuchachromat hasa diameterof 6.35mmandis constructedwithrr
8.609mm,rz=-5,134mm,ta = 2.7mm, andtcz
0] mm. lt hasaneffective focallength
(EFL)
of 12.7mmat 589-nmwavelength.Therefractive indexforthecrownglass usedis
1,52andforflint
glass is 1.66at thiswavelength. BFL
= back
focallength, FFL
= front
focallength.
(i) Compute thevalueofrr inthethin-lens
paraxial ray approximation.
[8marks]
(ii) Compute thelateralmagnificationoftheimage ofanobjectlocated12, minfront
ofthelens,
(iii) Brieflyexplainthemeaningof
"chromatic
aberration"
[8marks]
[4marks]
- Whentwoor morewavesarepresentinthesameregion of space,theresultantwaveisa
superposition oftheindividualwaves.
(i) Considerthesuperposition of twomonochromaticcoherent
plane waves with
amplitudes0r ?fldazrospectively, and
phases
0r
and
$zrespectively,
Showthat
theintensityoftheresultantwaveisgivenby:
I = 11+t2 +2r[tfucosA, where 4 =
Q
[Bmarks]
Considernowthatthesetwowaveshave equalintensities/0,buttheytravelatan
anglewitheachother.Onewavetravelsalong the+z-direction,whiletheother
wavetravels at anangle0 withthez-axisin thexzplane,asshown below.Atthe
overlaponthexyplane,a sinusoidal
pattern ofbright anddarkfringesisformed.
(a) Show thatthephasedifferencebetweenthetwowaves measuredalongx
varies accordingto A = x4sing
t8marksl
(b) Fromthe resultof (i) aboveor othenruise,derive the equationthat
describestheintensityvariation inthex-directionofthisinterferencepattern.
[4marks]
- END
OFPAPER
(ii)
PH
