NATIONAL UNIVERSITY OF SINGAPORE

PC1142 Physics ll(Semester l: AY 2009-10, 28 November 2009)

Time Allowed:2 Hours

INSTRUCTIONS TO CANDIDATES

1. This examination paper comprises EIGHT (8) printed pages with FIVE (5) short questions in

Part 'l and THREE (3) long questions in Part 2.

2. Answer ALL questions.

3. The total marks for Part 1 is 40 and for Part 2 is 60.

4. Answers to the questions are to be written in the answer booklet provided.

5, This is a CLOSED BOOK examination. Calculators are allowed.

6. A list of physical constants and formulae is given on pages 2 and 3,

Thermal physics and kinetic theory:(i) Kinetic mean free path is given by

4I = --: - , where nv is the number density

J Z x . n , ' d 2

and d is the molecular diameter.

(ii) Maxwell-Boltzmann distribution in 3D:

/ 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).One form of adiabat: p .Vv = Constanf .

(vi) Stefan-Boltzmann equation: P,"a = 6 . A.e .Ta

(vii) Planck radiation equation:

2 n . h . c 2foa l r , )=m

r ' . [ *o[ ̂ n, ' ' = l-r l[ ( . r ' k , ' T ) )

(viii) Wien displacement law:

" 2.898 x10-3 m .Klu peak

=

(ix) Convection equation: Pror, = h . A. LT .

(x) Conduction equation: Pcond = -k A +L

(xi) Linear thermal expansion: AL = cr .L. LT .

(b) Average speed: v,, -

(c) Most-probable speed: ,," = ̂ l2krT' l l m

( i i i ) ldealgas equation: p .V = n .R.T .

(iv) Van der Waals equation:^ ^ 2

( p + o ' l ' ) . ( v - n . b ) = n ' R . T ." v z

Thermodynamics:

(i) First law of thermodynamics: LU = qin + Win. Gas

expansion work done by the gas: W,,, = [p

.dV .

(ii) Carnot heat-engine efficiency e. =1-L." T ,

Geometric optics:

(i) p and q are object and image distances measured

respectively on opposite sides of the lens or of the

refracting surface:

(a) Object-image relation (thin

where l=(n-1) ff -f)t ' \ . R ,

R r )

(b) Refracting-surface equation:

bns): 1* 1 = 1 ,

p q t

f r t , D z _ f l 2 - D tp q R

(iii) Enhopy dS = dg',n .

T( iv)Enthalpy H= E+ PV.

(v) Helmholtz free energy F = E- IS.

(vi) Gibbs free energy G = H- IS,

(ii) Gullstrand equation:

(a) Effective power: P = P, + p,

(b) Fronfvertex refracting power:

P. -P. + P2. t . ,

n p r . d

n

(c) Back-vertex refracting power:

P . - P * q

. o . , n q . d

n

-P ' P r ' !' - n

SkBT

nn

(iii) Spherical-minor equation: , where

, F f( lv) f-numDer: -=-.# D

(v) Numerical aperture: NA = n. sin 0 .

Wave optics:

(i) Circular aperture (Airy's disc): first diffraction

1.221mlnlmum lS aI Sln U - - .

a

(ii) Slit: first diffraction minimum is at sin B = 4 .a

(iii) N-slit intensity pattern: / - /^ . sin'z ( l'0 /?)

," sin'(g / 2)

)*where g = ' " . d . s ing .' 1 .

General:

(i) For a circle, the arc length subtended by conical

angle a is given by s = ru.

v-- A

(ii) For a sphere, the surface area subtended by

conicalangle 20 is given by A= 2 rrz ( 1 -cos0 ) .

Universal constants:

Gas constant R = 8.314 J K-1 mol-1

Boltzmann constant ke = 1.381x10-23 J K-1

Stefan-Boltzmann constant o = 5.670x1 0a W m-2 Ka

Speed of light in vacuum 6 = l.$$$xl Qe rn 5-t

(v i) Snel l 's law: n, .Sin 0, = n, .s in 0, .

(a). Critical angle: sin 0c = rz / nr.

(b) Brewster angle: tan 0p = nz /nr.

(vii) Wave relation: v = f .), .

(viii) Abbe number: v = flo - 1

f l r - f rc

(iv) Single-slit diffraction pattern:

r = r^ . t i l ' (s, /^2), where o =4. a. sin0.- l6t2)' l.

(iii) /og" (b c)= log'b + logac.

(iv) /og' b= loy b / loy a.

. , n + 1

(v) lxndx =a _ +c except for n = -1 forr n + 1

wnich Jx- 'dx=lnx+c.

Avogadro's number Nr = 6.022x1023 mol-1

Permittivity of free space s' = $,$${x'l$-12 [rn-l

Planck constant h : 6.626x10-34 J s

1 1 1p q t

1 2- = - ,f R

PC1142 Physics ll

Part 1 Answer all FIVE questions. All questions carry 8 marks each.

1. The dark nebula in interstellar space is thought to contain 50 hydrogen (H) atoms per cubic

centimetre. These have a root-mean-square Speed of 1400 m s-1. [Nofe: H has an

atomic weight of 1,0 g mol-t and a diameter of 1.0 A.l

(i) Compute the mean free path of these H atoms,

(ii) Compute their kinetic temperature.

2. Consider a black body in vacuum comprising a thermally-conductive solid core of radius rc

and having a uniform temperature I', surrounded by a thermally-insulating solid shell of

thickness d with thermal conductivity k, as shown below. Heat is produced internally in the

core at a rate of P. Derive an expression for the value of I, at steady state, in terms of P,

k, 16, d, and other necessary constants.

vacuum

1,0 kg of helium gas is heated at constant pressure from 300 K to 600 K. Compute the

change in entropy of the gas. [Nofe: He has an atomic weight of 4.0 g moFt,]

3.

4. A camera with an effective focal length of 300 mm and F-number of 5.6 is used to recordthe optical image of the ground from an aeroplane flying 10 km above ground. Can thiscamera resolve a car on the ground, if the lens system is free from aberrations, based onRayleigh's criteria? [Nofe: The optical spectrum spans 450-650 nm.]

5. Wave plates are optical elements designed to differentially retard the phase of thecomponents of a polarised beam. One such wave plate is made of birefringent crystallinequartz with an ordinary refractive index (no) of 1.5426 and an extraordinary refractive index(ne) of 1.5517 at a wavelength of 632.8 nm, and oriented as shown below. The fast axishas the index no, while the slow axis has index nr, The thickness of the plate is 1.00g3mm.

A linearly polarised beam of 632.8-nm light is incident normally with the plane ofpolarisation at an angle 0 measured clockwise from the fast axis. Compute the phase shiftbetween lhe component polarised parallel to the fast axis and that polarised parallel to theslow axis, and give the state of polarisation after passing this wave plate.

Part2 Answer all THREE questions. All questions carry 20 marks each.

6. A Stirling engine can be modeled as a thermodynamic gas cycle comprising two isotherrnalprocesses at temperatures L and Tz, and two isochoric (i.e., constant volume) processes

Vr and Vz, as shown below.

Assume that the working gas comprises n moles of an ideal diatomic gas with molar heatcapacity at constant volume cv = 512 R, molar heat capacity at constant pressure cp, ofid

heat capacity ratio y, The rate of cycles is Ycycles per unit time.

(i) Derive an expression for the net power input into the engine by heat,

[8 marks](ii) Derive an expression for the thermal efficiency of this engine.

[B marks](iii) Briefly outline how to maximise the efficiency of this engine cycle (you are allowedto adjust parameters but not the nature of the gas processes themselves).

[4 marks]

7. Achromatic doublet lenses have significantly better optical performance than singlet lensesfor imaging applications. An achromat (general structure shown below) consists of apositive low-index crown Elass lens element with radii of curvature h ?nd h, and centrethickness fcr, ceffiertod to a negative high-index flint glass lens element with radil ofcurvature h and rl, ?fld centre thickness fcz. These lens elements are chosen so as tocancel chromatic abenation at the red (630-nm wavelength) and blue (450 nm)wavelengths. One such achromat has a diameter of 6.35 mm and is constructed with rr =

8.609 mm, rz =-5,134 mm, ta = 2.7 mm, and tcz = 0] mm. lt has an effective focal length(EFL) of 12.7 mm at 589-nm wavelength. The refractive index for the crown glass used is1,52 and for flint glass is 1.66 at this wavelength. BFL = back focal length, FFL = frontfocal length.

(i) Compute the value of rr in the thin-lens paraxial ray approximation.

[8 marks]

(ii) Compute the lateral magnification of the image of an object located 12,7 m in frontof the lens,

(iii) Briefly explain the meaning of "chromatic aberration"

[8 marks]

[4 marks]

8. When two or more waves are present in the same region of space, the resultant wave is a

superposition of the individual waves.

(i) Consider the superposition of two monochromatic coherent plane waves with

amplitudes 0r ?fld az rospectively, and phases 0r and $z respectively, Show that

the intensity of the resultant wave is given by:

I = 11+ t2 +2r[tfucosA, where 4 = Q2 - gr. [B marks]

Consider now that these two waves have equal intensities /0, but they travel at an

angle with each other. One wave travels along the +z-direction, while the other

wave travels at an angle 0 with the z-axis in the xz plane, as shown below. At the

overlap on the xy plane, a sinusoidal pattern of bright and dark fringes is formed.

(a) Show that the phase difference between the two waves measured along x

varies according to A = x4sing t8 marksl

(b) From the result of (i) above or othenruise, derive the equation that

describes the intensity variation in the x-direction of this interference pattern.

[4 marks]

- END OF PAPER -

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