apho (asian physics olympiad)

1

Problem 1 Eclipses of the Jupiter’s Satellite A long time ago before scientists could measure the speed of light accurately, O Römer, a Danish astronomer studied the time eclipses of the Jupiter’s satellite. He was able to determine the speed of light from observed periods of a satellite around the planet Jupiter. Figure 1 shows the orbit of the earth E around the sun S and one of the satellites M around the planet Jupiter. (He observed the time spent between two successive emergences of the satellite M from behind Jupiter). A long series of observations of the eclipses permitted an accurate evaluation of the period of M. The observed period T depends on the relative position of the earth with respect to the frame of reference SJ as one of the coordinate axes. The average time of revolution is T0 = 42h 28 m 16s and maximum observed period is ( T0 + 15)s.

Figure 1 : The orbits of the earth E around the sun and a satellite M around Jupiter J. The average distance of the earth E to the Sun is RE = 149.6 x 106 . The maximum distance is RE,max = 1.015 RE. The period of revolution of the earth is 365 days and of Jupiter is 11.9 years. The distance of the satellite M to the planet Jupiter RM= 422 x 103 km.

a. Use Newton’s law of gravitation to estimate the distance of Jupiter to the Sun. Determine the relative angular velocity ω of the earth with respect to the frame of reference Sun-Jupiter (SJ). Calculate the speed of the earth with respect to SJ.

b. Take a new frame which Jupiter is at rest with respect to the Sun. Determine the relative

angular velocity ω of the earth with respect to the frame of reference Sun-Jupiter ( SJ). Calculate the speed of the earth with respect to SJ.

c. Suppose an observed saw M begin to emerge from the shadow when his position was at

θ k and the next emergence when he was at θ k+ 1 , k = 1,2,3,… From these observations he got the apparent periods of revolution T ( t k ) as a function of time t k from Figure 1 and then use an approximate expression to explain how the distance influences the observed periods of revolution of M. Estimate the relative error of your approximate distance.

2

d. Derive the relation between d ( t k ) and T ( t k ). Plot period T ( t k ) as a function of time

of observation t k. Find the positions of the earth when he observed maximum period, minimum period and true period of the satellite M.

e. Estimate the speed of light from the above result. Pont out sources of errors of your

estimation and calculate the order of magnitude of the error. f. We know that the mass of the earth = 5.98 x 1024 kg and 1 month = 27d 7h 3m. Find the

mass of the planet Jupiter.

3

Problem 2 Detection of Alpha Particles We are constantly being exposed to radiation, either natural or artificial. With tha advance of nuclear power reactors and utilization of radioisotopes in agriculture, industry, biology and medicine, the number of man made prepared (artificial) radioactive sources is also increasing every year. One type of the radiation emitted by radioactive materials is alpha (α) particles (doubly ionized helium atom having two units of positive charge and four units of nuclear mass). The detection of α particles by electrical means is based on their ability to produce ionization when passing through gas and other substance. For α particle in air at normal (atmospheric) pressure, there is an empirical relation between the mean range Rα and its energy E

Rα = 0.318 Ε 3/2 ( 1 )

Where Rα is measured in cm and E in MeV. For monitoring α radiation, one can use an ionization chamber, which is a gas-filled detector that operates on the principle of separation of positive and negative charges created during the ionization of gas atoms by the α particle. The collection of charges yields a pulse that can be detected, amplified and then recorded. The voltage difference between anode and cathode is kept sufficiently high so that there is a negligible amount of recombination of charges during their passage to the anodes.

Figure 1 : Schematic diagram of ionization chamber circuit.

4

a. An ionization chamber electrometer system with a capacitance of 45 picofarad is used to detect α particles having a range Rα of 5.50 cm. Assume the energy required to produce an ion-pair (consisting of a light negative electron and a heavier positive ion, each carrying one electronic charges of magnitude e = 1.60 x 10-19 Coulomb) in air is 35 eV. What will be the magnitude of the voltage produced by each α particle?

b. The voltage pulses due to the α particle of the above problem occur across a

resistance R. The smallest detectable saturation current (a condition where the current is more or less constant, indicating that the charge is collected at the same rate at which it is being produced by the incident α particle) with this instrument is 10-12 ampere. Calculate the lowest activity A (disintegration rate of the emitter radioisotope) of the α source that could be detected by this instrument if the range Rα is 5.50 cm assuming a 10 % efficiency for the detector geometry.

c. The above ionization chamber is to be used for pulse counting with a time

constant τ = 10-3 seconds. Calculate the resistance and also the necessary voltage pulse amplification required to produce 0.25 V signal.

d. Ionization chamber has geometry such as cylindrical counter, the central metal

wire (anode) and outer thin metal sheath (cathode) have diameter d and D, respectively. Derive the expression for the electric field E(τ ) and potential V (τ )

at a radial distance 2 2d Dwithτ τ ≤ ≤

from the central axis when the wire carries

a charge per unit length λ. Then deduce the capacitance per unit length of the tube. The breakdown field strength of air E b is 3 MV m-1 (breakdown field strengths greater than E b , maximum electric field in the substance ). If d=1 mm and D = 1 cm , calculate the potential difference between wire and sheat at which breakdown occurs.

Data : 1 MeV = 106 eV ; 1 picoFarad = 10-12 F ; 1 Ci = 3.7 x 1010 disintegration/second = 106 µ Ci ( Curie, the fundamental SI unit of activity A );

lndrr Cτ= +∫

5

Problem 3 Stewart-Tolman Effect In 1917, Stewart and Tolman discovered a flow of current through a coil wound around a cylinder rotated axially with certain angular acceleration. Consider a great number of rings, with the radius τ each, made of a thin metallic wire with resistance R. The rings have been put in a uniform way on very long glass cylinder, which is vacuum inside. Their positions on the cylinder are fixed by gluing the rings to the cylinder. The number of rings per unit of length along the symmetry axis is n. The planes containing the rings are perpendicular to the symmetry axis of the cylinder. At some moment the cylinder starts a rotational movement around its symmetry axis with an acceleration α .Find the value of the magnetic field B at the center of the cylinder (after a sufficiently long time). We assume that the electric charge e of an electron, and the electron mass m are known.

APHO II 2001 Theoretical Question 1 p. 1 / 3

1

Theoretical Question 1

When will the Moon become a Synchronous Satellite? The period of rotation of the Moon about its axis is currently the same as its

period of revolution about the Earth so that the same side of the Moon always faces the Earth. The equality of these two periods presumably came about because of actions of tidal forces over the long history of the Earth-Moon system.

However, the period of rotation of the Earth about its axis is currently shorter than the period of revolution of the Moon. As a result, lunar tidal forces continue to act in a way that tends to slow down the rotational speed of the Earth and drive the Moon itself further away from the Earth.

In this question, we are interested in obtaining an estimate of how much more time it will take for the rotational period of the Earth to become equal to the period of revolution of the Moon. The Moon will then become a synchronous satellite, appearing as a fixed object in the sky and visible only to those observers on the side of the Earth facing the Moon. We also want to find out how long it will take for the Earth to complete one rotation when the said two periods are equal.

Two right-handed rectangular coordinate systems are adopted as reference frames. The third coordinate axes of these two systems are parallel to each other and normal to the orbital plane of the Moon. (I)The first frame, called the CM frame, is an inertial frame with its origin located at

the center of mass C of the Earth-Moon system. (II)The second frame, called the xyz frame, has its origin fixed at the center O of the

Earth. Its z-axis coincides with the axis of rotation of the Earth. Its x-axis is along the line connecting the centers of the Moon and the Earth, and points in the direction of the unit vector as shown in Fig.1a. The Moon remains always on the negative x-axis in this frame.

Note that distances in Fig.1a are not drawn to scale. The curved arrows show the directions of the Earth's rotation and the Moon's revolution. The Earth-Moon distance is denoted by r.

M

C

O

Earth Moon

y

x Fig. 1a


2

The following data are given: (a) At present, the distance between the Moon and the Earth is and

increases at a rate of 0.038 m per year. (b) The period of revolution of the Moon is currently = 27.322 days. (c) The mass of the Moon is . (d) The radius of the Moon is . (e) The period of rotation of the Earth is currently = 23.933 hours. (f) The mass of the Earth is . (g) The radius of the Earth is . (h) The universal gravitational constant is .

The following assumptions may be made when answering questions: (i) The Earth-Moon system is isolated from the rest of the universe. (ii) The orbit of the Moon about the Earth is circular. (iii) The axis of rotation of the Earth is perpendicular to the orbital plane of the Moon. (iv) If the Moon is absent and the Earth does not rotate, then the mass distribution of

the Earth is spherically symmetric and the radius of the Earth is . (v) For the Earth or the Moon, the moment of inertia I about any axis passing through

its center is that of a uniform sphere of mass M and radius R, i.e. . (vi) The water around the Earth is stationary in the xyz frame.

Answer the following questions: (1) With respect to the center of mass C, what is the current value of the total angular

momentum L of the Earth-Moon system? (2) When the period of rotation of the Earth and the period of revolution of the Moon

become equal, what is the duration of one rotation of the Earth? Denote the answer as T and express it in units of the present day. Only an approximate solution is required so that iterative methods may be used.

(3) Consider the Earth to be a rotating solid sphere covered with a surface layer of water and assume that, as the Moon moves around the Earth, the water layer is stationary in the xyz-frame. In one model, frictional forces between the rotating solid sphere and the water layer are taken into account. The faster spinning solid Earth is assumed to drag lunar tides along so that the line connecting the tidal bulges is at an angle with the x-axis, as shown in Fig.1b. Consequently, lunar tidal forces acting on the Earth will exert a torque Γ about O to slow down the rotation of the Earth.

The angle δ is assumed to be constant and independent of the Earth-Moon distance r until it vanishes when the Moon's revolution is synchronous with the Earth's rotation so that frictional forces no longer exist. The torque Γ therefore


3

scales with the Earth-Moon distance and is proportional to .

According to this model, when will the rotation of the Earth and the revolution of the Moon have the same period? Denote the answer as and express it in units of the present year.

The following mathematical formulae may be useful when answering questions: (M1) For and :

(M2) If and , then

Moon

Earth

δ O

Fig.1b


1


Motion of an Electric Dipole in a Magnetic Field

In the presence of a constant and uniform magnetic field , the translational motion of a system of electric charges is coupled to its rotational motion. As a result, the conservation laws for the momentum and the component of the angular momentum along the direction of are modified from the usual form. This is illustrated in this problem by considering the motion of an electric dipole made of two particles of equal mass m and carrying charges q and respectively ( q > 0 ). The two particles are connected by a rigid insulating rod of length , the mass of which can be neglected. Let be the position vector of the particle with charge q, that of the other particle and = － . Denote by the angular velocity of the rotation around the center of mass of the dipole. Denote by and the position and the velocity vectors of the center of mass respectively. Relativistic effects and effects of electromagnetic radiation can be neglected.

Note that the magnetic force acting on a particle of charge q and velocity is × , where the cross product of two vectors × is defined, in terms of the x, y, z, components of the vectors, by

( × )x ＝ ( )y ( )z － ( )z ( )y

( × )y ＝ ( )z ( )x － ( )x ( )z ( × )z ＝ ( )x ( )y － ( )y ( )x.

(1) Conservation Laws

(a) Write down the equations of motion for the center of mass of the dipole and for the rotation around the center of mass by computing the total force and the total torque with respect to the center of mass acting on the dipole.

(b) From the equation of motion for the center of mass, obtain the modified form of the conservation law for the total momentum. Denote the corresponding modified conserved quantity by . Write down an expression in terms of and for the conserved energy E.

(c) The angular momentum consists of two parts. One part is due to the motion of the center of mass and the other is due to rotation around the center of mass. From the modified form of the conservation law for the total momentum and the equation of motion of the rotation around the center of mass, prove that the quantity J as defined by

J = ( × ＋I )．

is conserved.


2

Note that

× ＝ ×

‧( × ) ＝ ( × )‧

for any three vectors , and . Repeated application of the above first two formulas may be useful in deriving the conservation law in question.

In the following, let be in the z-direction.

(2) Motion in a Plane Perpendicular to

Suppose initially the center of mass of the dipole is at rest at the origin, points in the x-direction and the initial angular velocity of the dipole is ( is the unit vector in the z-direction).

(a) If the magnitude of is smaller than a critical value , the dipole will not make a full turn with respect to its center of mass. Find .

(b) For a general > 0, what is the maximum distance in the x-direction that the center of mass can reach?

(c) What is the tension on the rod? Express it as a function of the angular velocity .


1

( a,0,0)

(0,0, a)

(0, a, 0)

A

y

B

F G

D

E

C x

z

Fig. 3a


Thermal Vibrations of Surface Atoms

This question considers the thermal vibrations of surface atoms in an elemental

metallic crystal with a face-centered cubic (fcc) lattice structure. The unit cubic cell

of an fcc lattice consists of one atom at each corner and one atom at the center of

each face of the cubic cell, as shown in Fig. 3a. For the crystal under consideration,

we use (a, 0, 0), (0, a, 0) and (0, 0, a) to represent the locations of the three atoms on

the x, y and z axes of its cell. The lattice constant a is equal to 3.92 Å (i.e..the length

of each side of the cube is 3.92 Å).

(1) The crystal is cut in such a way that the plane containing ABCD becomes a

boundary surface and is chosen for doing low-energy electron diffraction

experiments. A collimated beam of electrons with kinetic energy of 64.0 eV is

incident on this surface plane at an incident angle of 15.0o . Note that is

the angle between the incident electron beam and the normal of the surface plane.

The plane containing and the normal of the surface plane is the plane of

incidence. For simplicity, we assume that all incident electrons are back scattered

only by the surface atoms on the topmost layer.

(a) What is the wavelength of the matter waves of the incident electrons?

(b) If a detector is set up to detect electrons that do not leave the plane of incidence

after being diffracted, at what angles with the normal of the surface will these

diffracted electrons be observable?

(2) Assume that the thermal vibrational motions of the surface atoms are simple


2

50 100 150 200 250 300 350 400

0.2

0.4

0.6

28

88

1

Fig. 3b 2221-2

T (K)

0

-0.2

-0.4

-0.6

-0.8

-1.0 0

harmonic. The amplitude of vibration increases as the temperature rises.

Low-energy electron diffraction provides a way to measure the average

amplitude of vibration. The intensity I of the diffracted beam is proportional to

the number of scattered electrons per second. The relation between the intensity I

and the displacement (t) of the surface atoms is given by

(1)

In Eq.(1), I and I0 are the intensities at temperature T and absolute zero,

respectively. and are wave vectors of incident electron and diffracted

electron, respectively. The angle brackets < > is used to denote average over

time. Note that the relation between the wave vector and the momentum

of a particle is = 2 / h, where h is the Planck constant.

To measure vibration amplitudes of surface atoms of a metallic crystal, a

collimated electron beam with kinetic energy of 64.0 eV is incident on a crystal

surface at an incident angle of 15.0o. The detector is set up for measuring

specularly reflected electrons. Only elastically scattered electrons are detected. A

plot of ln ( I / I0 ) versus temperature T is shown in Fig. 3b.

Assume the total energy of an atom vibrating in the direction of the surface

normal is given by kBT , where kB is the Boltzmann constant.

(a) Calculate the frequency of vibration in the direction of the surface normal for the

surface atoms.

(b) Calculate the root-mean-square displacement, i. e. the value of ( < ux2> )1/2, in the

direction of the surface normal for the surface atoms at 300 K.


3

The following data are given:

Atomic weight of the metal M = 195.1

Boltzmann constant kB = 1.38 10 J/K

Mass of electron = 9.11 10-31 kg

Charge of electron = 1.60 10-19 C

Planck constant h = 6.63 10-34 J-s

Theoretical Question 1 (vibrations of a linear crystal lattice)

A very large number N of movable identical point particles (N >>1),each with mass m, are set in a straight chain with N + 1 identical masslesssprings, each with stiffness (spring constant) S, linking them to each otherand the ends attached to two additional immovable particles. See figure.This chain will serve as a model of the vibration modes of a one-dimensional crystal. When the chain is set in motion, the longitudinalvibrations of the chain can be looked upon as a superposition of simpleoscillations (called modes) each with its own characteristic modefrequency.

(a) Write down the equation of motion of the nth particle. [0.7 marks]

(b) To attempt to solve the equation of motion of part (a) use the trial solution

Xn(ω) = A sin nka cos (ωt + α),

where Xn(ω) is the displacement of the nth particle from equilibrium, ω the angularfrequency of the vibration mode and A, k and α are constants; k and ω are thewave numbers and mode frequencies respectively. For each k, there will be acorresponding frequency ω. Find the dependence of ω on k, the allowed values ofk, and the maximum value of ω. The chain’s vibration is thus a superposition ofall these vibration modes. Useful formulas:(d/dx) cos αx = - α sin αx, (d/dx) sin αx = α cos αx, α = constant.

sin(A + B) = sin A cos B + cos A sin B, cos(A + B) = cos A cos B – sinA sin B[2.2 marks]

According to Planck the energy of a photon with a frequency of ω is hω,

where h is the Planck constant divided by 2π. Einstein made a leap from

this by assuming that a given crystal vibration mode with frequency ωalso has this energy. Note that a vibration mode is not a particle, but asimple oscillation configuration of the entire chain. This vibration modeis analogous to the photon and is called a phonon. We will follow up theconsequences of this idea in the rest of the problem. Suppose a crystal ismade up of a very large (∼ 1023) number of particles in a straight chain.

0 a 2 a 3a na (N - 1)a Na (N + 1)a = L

S SSm m mm

S

(c) For a given allowed ω (or k) there may be no phonons; or there may be one; ortwo; or any number of phonons. Hence it makes sense to try to calculate theaverage energy )(ωE of a particular mode with a frequency ω. Let Pp(ω)

represent the probability that there are p phonons with this frequency ω. Then therequired average is

∑

∑∞

=

∞

==

0

0

)(

)(

)(

pp

pp

P

Pp

Eω

ωωω

h

.

Although the phonons are discrete, the fact that there are so many of them (and thePp becomes tiny for large p) allows us to extend the sum to p = ∞, with negligibleerror. Now the probability Pp is given by Boltzmann’s formula

Pp(ω) ∝ exp (– phω/kBT),

where kB is Boltzmann’s constant and T is the absolute temperature of the crystal,assumed constant. The constant of proportionality does not depend on p.Calculate the average energy for phonons of frequency ω. Possibly usefulformula: (d/dx) ef(x) = (df / dx) ef(x).

[2 marks]

(d) We would like next to compute the total energy ET of the crystal. In part (c) wefound the average energy )(ωE for the vibration mode ω. To find ET we must

multiply )(ωE by the number of modes of the crystal per unit of frequency ωand then sum up all these for the entire range from ω = 0 to ωmax. Take an interval∆k in the range of wave numbers. For very large N and for ∆k much larger thanthe spacing between successive (allowed) k values, how many modes can befound in the interval ∆k?

[1 mark]

(e) To make use of the results of (a) and (b), approximate ∆k by (dk/dω)dω and

replace any sum by an integral over ω. (It is more convenient to use the variableω in place of k at this point.) State the total number of modes of the crystal in thisapproximation. Also derive an expression ET but do not evaluate it. The following

integral may be useful: .2/1/1

0

2 π=−∫ xdx [2.2 marks]

(f) The molar heat capacity CV of a crystal at constant volume is experimentallyaccessible: C V = dET/dT (T = absolute temperature). For the crystal underdiscussion determine the dependence of CV on T for very large and very lowtemperatures (i.e., is it constant, linear or power dependent for an interval of thetemperature?). Sketch a qualitative graph of CV versus T, indicating the trendspredicted for very low and very high T.

[1.9 marks]

Theoretical Question 2 (the rail gun)

A young man at P and a young lady at Q were deeply in love. These two places areseparated by a strait of width w = 1000 m. After learning about the theory of rail gunin class, the young man could not wait to construct such a device to launch himselfacross the strait. He constructed a ramp of adjustable elevation of angle θ on which he

laid two metal rails (the length of each rail is D = 35.0 m) in parallel, separated by L =2.00 m. He managed to connect a 2424 V DC power supply to the ends of the rails. Aconducting bar can slide freely on the metal rails such that he could hang on to itsafely as it slides.

A skilled engineer, moved by all these efforts, designed a system that can produce aB = 10.0 T magnetic field that can be directed perpendicular to the plane of the rails.The mass of the young man is 70 kg. The mass of the conducting bar is 10 kg and itsresistance is R = 1.0 Ω.

Just after he had completed the construction and checked that it worked perfectly, hereceived a call from the young lady, sobbing and telling him that her father was goingto marry her off to a rich man unless he can arrive at Q within 11 seconds after thecall, and having said that she hang up.

The young man immediately got into action and launched himself across the strait toQ.

Show, using the steps listed below, whether it is possible for him to make it in time,and if so, what is the range of θ he must set the ramp?

(a) Derive an expression for the acceleration of the young man parallel to the rail. [3 marks]

(b) Obtain an expression in terms of θ for the time spent

i. on the rails, ts andii. in flight, tf.

[3 marks](c) Plot a graph of the total time T = ts + tf against the angle of inclination θ.

[1.5 marks](d) By considering the relevant parameters of this device, obtain the range of

angles that he should set. Plot another graph if necessary. [2.5 marks]

Make the following assumptions:

1) The time between the end of the call and all preparations (such as setting θ to

the appropriate angle) for the launch is negligible. This is to say, the launch isconsidered to start at time t = 0 when the bar (with the young man hanging toit) is starting to move.

2) The young man may start his motion from any point along the metal rails.

P Q

θ

35m

1000mB

3) The higher end of the ramp and Q is at the same level, and the distancebetween them is w = 1000 m.

4) There is no question about safety such as when landing, electric shocks, etc.5) The resistance of the metal rails, the internal resistance of the power supply,

the friction between the conducting bar and the rails and the air resistance areall negligible.

6) Take acceleration due to gravity as g = 10 m/s2.

Some Mathematical notes:

1. ∫−

− −= .a

edxe

axax

2. The solution to bxadt

dx += is given by

btbt exeb

atx )0()1()( +−= .

Theoretical Question 3 (wafer fabrication)

Wafer fabrication refers to the production of semiconductor chips from silicon. Inmodern technologies there are more than 20 processes; we are going to concentrate onthin films deposition.In wafer fabrication process, thin films of various materials are deposited on thesurface of the silicon wafer. The surface of the substrate must be extremely cleanbefore the process of deposition. The presence of traces of oxygen or other elementswill result in the formation of a contamination layer. The rate of formation of thislayer is determined by the impingement rate of the gas molecules hitting the substratesurface. Assuming the number of molecules per unit volume is n , the impingementrate on a unit area of the substrate from the gas is given by

vnJ4

1=

where v is the average or mean speed of the gas molecules.

(a) Assuming that the gas molecules obey a Maxwell-Boltzmann distribution,

,2

4)( )2(/2

2/32 RTvMev

TR

MvW −

=

ππ

where dvvW )( is the fraction of molecules whose speed lie between v and vdv + ,M is the molar mass of the gas, T is the gas temperature and R is the gas constant,show that the average or mean speed of the gas molecules is given by

M∂

TRdvvWvv

8)(

0∫∞

==

[1.5 marks]

(b) Assuming that the gases behave as an ideal gas at low pressure, P , show thatthe rate of impingement is given by

Tmk

PJ

π2=

where m is the mass of the molecule and T is the temperature of the gas.[1.5 marks]

(c) If the residual pressure of oxygen in a vacuum system is 133 Pa, and bymodelling the oxygen molecule as a sphere of radius approximately 10106.3 −× m,estimate how long it takes to deposit a molecule-thick layer of oxygen on the wafer at300o Celsius, assuming that all the oxygen molecules which strike the silicon wafersurface are deposited. Assume also that oxygen molecules in the layer are arrangedside by side.

[1.7 marks]

(d) In reality, not all molecules of oxygen react with the silicon. This can bemodeled by the concept of activation energy where the reacting molecules shouldhave total energy greater than the activation energy before it can react. Physically thisactivation energy describes the fact that chemical bonds between the silicon atomshave to be broken before a new bond between silicon and oxygen atoms is formed.Assuming an activation energy for the reaction to be 1 eV, estimate again how long itwould take to deposit one atomic layer of oxygen at the above temperature. You mayassume that the area under the Maxwell distribution in part (a) is unity.

[2.8 marks](e) For lithography processes, the clean silicon wafer is coated evenly with a layerof transparent polymer (photo-resist) of refractive index µ = 1.40. To measure the

thickness of this photo-resist, the wafer is illuminated with collimated monochromaticbeam of light of wavelength λ = 589 nm. For a certain minimum thickness of photo-resist, d , there is a destructive interference of reflected light, assuming normalincidence on the coating. Derive an expression for relation between d, µ and λ .

Calculate d using the given data. In this point you may assume that silicon behaves asa medium with a refractive index greater than 1.40 and you may ignore multiplereflections.

[2.5 marks]

The following data may be helpful:

Molar mass of oxygen is 32 g mol -1.Boltzmann constant, 231038.1 −×=k J K-1.

Avogadro number, 231002.6 ×=AN mol -1

Useful formula:

∫

+−= −−

k

x

kedxex xkxk

2

23 1

2

1 22

0 500 1000 1500 2000 2500 3000velocity, v

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

borPW(v)

Velocity, v (m/s)

4th Asian Physics Olympiad Theoretical Competition 23 April 2003

page 1

Theoretical Competition

I. Satellite’s orbit transfer

In the near future we ourselves may take part in launching of a satellite which, inpoint of view of physics, requires only the use of simple mechanics.

a) A satellite of mass m is presently circling the Earth of mass M in a circular orbit ofradius 0R . What is the speed )( 0u of mass m in terms of 0, RM and the universalgravitation constant G ? (1 point)

b) We are to put this satellite into a trajectory that will take it to point P at distance 1Rfrom the centre of the Earth by increasing (almost instantaneously) its velocity atpoint Q from 0u to 1u . What is the value of 1u in terms of 100 ,, RRu ?

(2 points)

c) Deduce the minimum value of 1u in term of 0u that will allow the satellite to leavethe Earth’s influence completely. (1 point)

d) (Referring to part b.) What is the velocity )( 2u of the satellite at point P in terms of

100 ,, RRu ? (1 point)

e) Now, we want to change the orbit of the satellite at point P into a circular orbit ofradius 1R by raising the value of 2u (almost instantaneously) to 3u .What is the magnitude of 3u in terms of 102 ,, RRu ? (1 point)

•

u1

u0

u2

R1

R0m

M•

P

Q

•

•


page 2

f)

If the satellite is slightly and instantaneously perturbed in the radial direction so thatit deviates from its previously perfectly circular orbit of radius 1R , derive the periodof its oscillation T of r about the mean distance 1R .

Hint: Students may make use (if necessary) of the equation of motion of a satellitein orbit:

2

2

2

2

rMmGr

dtdr

dtdm −=

− θ ………………………… (1)

and the conservation of angular momentum:

constant2 =θdtdmr ……………………….… (2)

(3 points)

g) Give a rough sketch of the whole perturbed orbit together with the unperturbed one.

(1 point)

mean orbit ofradius R1

X

Y

m

r

θ

•

M •


II. Optical GyroscopeIn 1913 Georges Sagnac (1869-1926) considered the use of a ring resonator to

search for the aether drift relative to a rotating frame. However, as often happen, hisresults turned out to be useful ways that Sagnac himself never dreamt of. One of thoseapplications is the Fibre-Optic Gyroscope (FOG) which is based upon a simplephenomena, first observed by Sagnac. The essential physics associated with the Sagnaceffect is due to the phase shift caused by two coherent beams of light being sent around arotating ring of optical fibre in the opposite directions. This phase shift is also used todetermine the angular speed of the ring.

As shown in a schematic diagram in Fig. 1, a light wave enters a circular opticalfibre light path of radius R at point P on the rotating platform with a uniform angularspeed Ω, in the clockwise direction. Here the light wave is split into two waves whichtravels in the opposite directions, clockwise (CW) and counter clockwise (CCW), throughthe ring. The refractive index of optical fibre material is µ . Assuming the light travelinginside the fibre-optic cable is a smooth circular path of radius R.

a) Practically, the orbital speed of the ring is much less than the speed of light suchthat ( ) 22 cR <<Ω , find the time difference −+ −=∆ ttt where +t and −t denotethe round-trip transit time of the CW and CCW beam respectively. Give youranswer in term of area A enclosed by the ring. (2 points)

b) Find the optical path difference, L∆ , for the CW and CCW beams to complete oneround-trip of the light within the rotating ring. (2 points)

c) For a circular fibre-optic of radius, R = 1 m, what is the maximum value of ∆L forthe rotation of the earth? Given µ = 1.5 . (1 point)

d) In part b), the measurement could be amplified by increasing number of turns infibre-optic coil, N , find the phase difference, θ∆ , for lights to complete the turns.

(1 point)

P • Ω

R

•

Fig.1

Optical fibrelight path


The second scheme of the Optical Gyroscope is Ring Laser Gyroscope (RLG). This couldbe accomplished by the inclusion of active laser cavity into an equilateral triangular ring,of total length L, as illustrated in Fig.2. The laser source here will generate two amplifiedcoherent light sources propagating in the opposite directions. In order to sustain the laseroscillation in this triangular ring resonator, the perimeter of the ring must be equal to theinteger multiple of wavelength λ. Etalon, additional component inserted into the ring, ispossible to cause frequency selective losses in the ring resonator, so that the undesiredmodes can be damped and suppressed.

Fig.2: Schematic illustration of the Ring LaserGyroscope

Fig. 3: Illustration of the Ring LaserGyroscope discussed in this problem

e) Find the time difference of the transit in clockwise and counterclockwise, ∆t, forthe case of the triangular ring as shown in the figure 2. Give the answer in terms ofΩ and the area A enclosed by the ring.Show that this result is the same as that ofthe circular ring. (2 points)

f) If the ring is rotating with an angular frequency Ω as shown in Fig. 2, there will befrequency difference between CW and CCW measurements. What is the observedbeat frequency, ν∆ , between the CW and CCW beams in terms of λ,,ΩL .

(2 points)

Etalon

Detector

Ω

50% coated mirrorM1

M2 M3

60°

60° 60°

Laser tube

L/3

L/3

L/3


III. Plasma Lens

The physics of the intense particle beams has a great impact not only on the basicresearch but also the applications in medicine and industry. The plasma lens is a device toprovide an ultra-strong final focus at the end of the linear colliders. To appreciate thepossibilities of the plasma lens it is quite natural to compare this with the usual magneticand electrostatic lenses. In magnetic lenses, focusing capability is proportional to themagnetic field gradient. The practical upper limit of the quadrupole focusing lens is in theorder of 102 T/m, while for plasma lens having density of 1017 cm-3, its focusingcapability is equivalent to a magnetic field gradient of 3×106 T/m (about four orders ofmagnitude more than that of a magnetic quadrupole lens).

In what follows, we will illuminate why the intense relativistic particle beams couldproduce self-focusing beams and do not blow themselves apart in free space.

a) Consider a long cylindrical electron beam of uniform number density n andaverage speed v (both quantities in laboratory frame). Derive the expression for theelectric field at a point at distance r from the central axis of the beam usingclassical electromagnetics. (1 point)

b) Derive the expression for the magnetic field at the same point as in a).(2 points)

c) What is then the net outward force on the electron in the electron beam passing thatpoint? (1 point)

d) Assuming that the expression obtained in c) is applicable at relativistic velocities,what will be the force on the electron as v approaches speed of light c , where

00

1µε

=c ? (1 point)

e) If the electron beam of radius R enters into a plasma of uniform density nn <0 (the

plasma is an ionized gas of ions and electrons with equal charge density), what willbe the net force on the stationary plasma ion at distance r′ outside the electronbeam long after the beam entering the plasma. You may assume that the density ofthe plasma ions remains constant and the cylindrical symmetry is maintained.

(3 points)

f) After long enough time, what is the net force on an electron at distance r from thecentral axis of the beam in the plasma, assuming cv → provided that the density ofthe plasma ions remains constant and the cylindrical symmetry is maintained?

(2 points)

Theoretical competition – Problem No.1 Measure the mass in the weightless state

In the spacecraft orbiting the Earth, there is weightless state, so that one cannot use ordinary instruments to measure the weight and then to deduce the mass of the astronaut. Skylab 2 and some other spacecrafts are supplied with a Body Mass Measurement Device which consists of a chair attached to one end of a spring. The other end of the spring is attached to a fixed point of the spacecraft. The axis of the spring passes through the center of mass of the craft. The force constant (the hardness) of the spring is k = 605.6 N/m. 1. When the craft is fixed on the pad, the chair (without person) oscillates with the period T0 = 1.28195 s.

Calculate the mass m0 of the chair [2 pts].

2. When the craft orbits the Earth the astronaut straps himself into the chair and measures the period T’ of the chair oscillations. He obtains T’ = 2.33044 s, then calculates roughly his mass. He feels some doubt and tries to find the true value of his mass. He measures again the period of oscillation of the chair (without person), and find T0’ = 1.27395 s.

What is the true value of the astronaut’s mass and the craft’s mass? [4pts] Note: The mass of the spring is negligible and the astronaut is floating.

Theoretical Competition-Problem No. 2 Optical fiber

O n1

n2

n0=1

θi

x

z

x

z

a

O

An optical fiber consists of a cylindrical core of radius a, made of a transparent material with refraction index varying gradually from the value n=n1 on the axis to n = n2 (with 1<n2<n1) at a distance a from the axis, according to the formula

2 21 1= −α= n(x) n xn

where x is the distance from the core axis and α is a constant. The core is surrounded by a cladding made of a material with constant refraction index n2. Outside the fiber is air, of refractive index n0.

Let Oz be the axis of the fiber, with O - the center of the fiber end. Given n0=1.000; n1 =1.500; n2 = 1.460, a = 25 µm.

1. A monochromatic light ray enters the fiber at point O under an incident angle iθ , the

incident plane being the plane xOz. a. Show that at each point on the trajectory of the light in the fiber, the refractive index n and the angle θ between the light ray and the Oz axis satisfy the relationship n Cθ =cos where C is a constant. Find the expression for C in terms of n1 and θi. [1.0 points]

b. Use the result found in 1.a. and the trigonometric relation ( )1

2 21 tanθ−

= +cos , where θ

tan 'dx xdz

θ = =

'

is the slope of the tangent to the trajectory at point (x, z), derive an equation

for x . Find the full expression for α in terms of n1, n2 and a. By differentiating the two sides of this equation versus z, find the equation for the second derivative ''x . [1.0 points] c. Find the expression of x as a function of z , that is ( )x f z= , which satisfies the above

equation. This is the equation of the trajectory of light in the fiber. [1.0 points] d. Sketch one full period of the trajectories of the light rays entering the fiber under two different incident angles iθ . [1.0 points] 2. Light propagates in the optical fiber. a. Find the maximum incident angle iMθ , under which the light ray still can propagate

inside the core of the fiber. [1.5 points] b. Determine the expression for coordinate z of the crossing points of a light ray with Oz axis for i 0θ ≠ . [1.5 points]

3. The light is used to transmit signals in the form of very short light pulses (of negligible pulse width). a. Determine the time τ it takes the light to travel from point O to the first crossing point with Oz for incident angle θi ≠ 0 and i iMθ θ≤ .

The ratio of the coordinate z of the first crossing point and τ is called the propagation speed of the light signal along the fiber. Assume that this speed varies monotonously with θi.

Find this speed (called ) for Mv i iMθ θ= .

Find also the propagation speed (called ) of the light along the axis Oz. 0vCompare the two speeds. [3.25 points]

b. The light beam bearing the signals is a converging beam entering the fiber at O under different incident angles iθ with 0 iMiθ θ≤ ≤ . Calculate the highest repetition frequency f of

the signal pulses, so that at a distance z = 1000 m two consecutive pulses are still separated (that is, the pulses do not overlap). [1.75 points] Attention 1. The wave properties of the light are not considered in this problem. 2. Neglect any chromatic dispersion in the fiber. 3. The speed of light in vacuum is c = 2.998×108 m/s

4. You may use the following formulae: • The length of a small arc element ds in the xOz plane is

2

1 dxds dzdz

= +

dx ds

dz

•2 2 2

1 sindx bxArcb aa b x

=−

∫

•2

2 2 2 2

2 32 2 2 2 2

sin−= − +

−∫

bxa Arcx dx x a b x ab ba b x

• Arc sin x is the inverse function of the sine function. Its value equals the less angle the sine of which is x. In other words, if Arc siny x= then sin y x= .

Theoretical Competition-Problem No. 3 Compression and expansion of a two gases system

A cylinder is divided in two compartments with a mobile partition NM. The compartment

in the left is limited by the fond of the cylinder and the partition NM (Figure 1). This compartment contains one mole of water vapor. The compartment in the right is limited by the partition NM and a mobile piston AB. This compartment contains one mole of nitrogen gas (N2).

Figure 1

M B

V0 N V0 A

p1 p1 T1 T1

At first, the volumes and temperatures of the gases in two compartments are equal. The partition NM is well heat conductive. His heat capacity is very small and can be neglected.

The specific volume of liquid water is negligible in comparison with the specific volume of water vapor at the same temperature.

The specific latent heat of vaporization L is defined as the amount of heat that must be delivered to one unit of mass of substance to convert it from liquid state to vapor at the same temperature. For water at T0 = 373 K, L = 2250 kJ/kg. 1. Suppose that the piston and the wall of the cylinder are heat conductive, the partition NM can slide freely without friction. The initial state of the gases in the cylinder is defined as follows:

Pressure p1 = 0.5 atm.; total volume V1 = 2V0; temperature T1 = 373 K. The piston AB slowly compresses the gases in a quasi-static (quasi-equilibrium) and

isothermal process to the final volume VF = V0/4 a. Draw the p(V) curve, that is the curve representing the dependence of pressure p on the

total volume V of both gases in the cylinder at temperature T1. Calculate the coordinates of important points of the curve. [1.5 pts]

Gas constant: R = 8.31 J/mol.K or R = 0.0820 L.atm./mol.K 1 atm. =101.3 kPa; Under the pressure p0 =1 atm., water boils at the temperature T0 = 373 K. b. Calculate the work done by the piston in the process of gases compressing. [1.0 pts]

lndV VV

=∫

c. Calculate the heat delivered to outside in the process. [1.5 pts]

2. All conditions as in 1. except that there is friction between partition NM and the wall of the cylinder so that NM displaces only when the difference of the pressures acting on its two opposed faces attains 0.5 atm. and over (assuming that the coefficients of static and kinetic friction are equal).

a. Draw the p(V) curve representing the pressure p in the right compartment as a function of the total volume V of both gases in the cylinder at a constant temperature T1. [1.5 pts]

b. Calculate the work done by the piston in compressing the gases. [0.5 pts]

1

c. After the volume of gases reaches the value VF = V0/4, piston AB displaces slowly to the right and makes a quasi-static and isothermal process of expansion of both substances (water and nitrogen) to the initial total volume 2V0. Continue to draw in the diagram in question 2.a. the curve representing this process [2.0 pts]

Hint for 2. Create a table like the one shown here and use it to draw the curves as required in 2.a.

and 2.c. State Left compartment

Volume | Pressure Right compartment Volume | Pressure

Total volume

Pressure on piston AB

initial V0 | 0.5 atm. V0 | 0.5 atm. 2V0 0.5 atm. 2 | | 3 | | . | | . | | . | | . | | . | |

final | | 2V0

3. Suppose that the wall and the fond of the cylinder and the piston are heat insulator, the partition NM is fixed and heat conductive, the initial state of gases is as in 1. Piston AB moves slowly toward the right side and increases the volume of the right compartment until the water vapor begins to condense in the left compartment.

a. Calculate the final volume of the right compartment. [3 pts] b. Calculate the work done by the gas in this expansion. [1 pts]

The ratio of isobaric heat capacity to isochoric one V

p

CC

=γ for nitrogen is 57

1 =γ and

for water vapor 68

2 =γ .

In the interval of temperature from 353 K to 393 K one can use the following approximate formula:

p = p0 exp

−− )11(

0TTRLµ

where T is boiling temperature of water under pressure p, µ - its molar mass. p0, L and T0 are given above.

2

THEORETICAL COMPETITION

FINAL PROBLEM

2/14

Question 1 1A. SPRING CYLINDER WITH MASSIVE PISTON Consider n=2 moles of ideal Helium gas at a pressure P0, volume V0 and temperature T0 = 300 K placed in a vertical cylindrical container (see Figure 1.1). A moveable frictionless horizontal piston of mass m = 10 kg (assume g = 9.8 m/s2) and cross section A = 500 cm2 compresses the gas leaving the upper section of the container void. There is a vertical spring attached to the piston and the upper wall of the container. Disregard any gas leakage through their surface contact, and neglect the specific thermal capacities of the container, piston and spring. Initially the system is in equilibrium and the spring is unstretched. Neglect the spring’s mass.

a. Calculate the frequency f of small oscillation of the piston, when it is slightly displaced from equilibrium position. (2 points)

b. Then the piston is pushed down until the gas volume is halved, and released with zero velocity. calculate the value(s) of the gas volume when the piston speed is

045gVA

(3 points)

Let the spring constant k = mgA/V0. All the processes in gas are adiabatic. Gas constant R = 8.314 JK-1mol-1. For mono-atomic gas (Helium) use Laplace constant γ = 5/3.

SPRING

PISTON

GAS

Figure 1.1


FINAL PROBLEM

3/14

1B. THE PARAMETRIC SWING (5 points) A child builds up the motion of a swing by standing and squatting. The trajectory followed by the center of mass of the child is illustrated in Fig. 1.2. Let ru be the radial distance from the swing pivot to the child’s center of mass when the child is standing, while rd is the radial distance from the swing pivot to the child’s center of mass when the child is squatting. Let the ratio of rd to ru be 21/10 = 1.072, that is the child moves its center of mass by roughly 7% compared to its average radial distance from the swing pivot. To keep the analysis simple it is assumed that the swing be mass-less, the swing amplitude is sufficiently small and that the mass of the child resides at its center of mass. It is also assumed that the transitions from squatting to standing (the A to B and the E to F transitions) are fast compared to the swing cycle and can be taken to be instantaneous. It is similarly assumed that the squatting transitions (the C to D and the G to H transitions) can also be regarded as occurring instantaneously.

Figure 1.2 How many cycles of this maneuver does it take for the child to build up the amplitude (or the maximum angular velocity) of the swing by a factor of two?

rd ru

StandingSquating


FINAL PROBLEM

5/14

Question 2 MAGNETIC FOCUSING There exist many devices that utilize fine beams of charged particles. The cathode ray tube used in oscilloscopes, in television receivers or in electron microscopes. In these devices the particle beam is focused and deflected in much the same manner as a light beam is in an optical instrument. Beams of particles can be focused by electric fields or by magnetic fields. In problem 2A and 2B we are going to see how the beam can be focused by a magnetic field. 2A. MAGNETIC FOCUSING SOLENOID (4 points) Figure 2.1 shows an electron gun situated inside (near the middle) a long solenoid. The electrons emerging from the hole on the anode have a small transverse velocity component. The electron will follow a helical path. After one complete turn, the electron will return to the axis connecting the hole and point F. By adjusting the magnetic field B inside the solenoid correctly, all the electrons will converge at the same point F after one complete turn. Use the following data:

• The voltage difference that accelerates the electrons V = 10 kV • The distance between the anode and the focus point F, L = 0.50 m • The mass of an electron m = 9.11x 10-31 kg • The charge of an electron e = 1.60 x 10-19 C • 7

0 4 10 H/mµ π −= × • Treat the problem non-relativistically

a) Calculate B so that the electron returns to the axis at point F after one complete

turn. (3 points) b) Find the current in the solenoid if the latter has 500 turns per meter. (1 point)

L

Anode

Figure 2.1

F


FINAL PROBLEM

6/14

2B. MAGNETIC FOCUSING (FRINGING FIELD) (6 points) Two pole magnets positioned on horizontal planes are separated by a certain distance such that the magnetic field between them be B in vertical direction (see Figure 2.2). The poles faces are rectangular with length l and width w. Consider the fringe field near the edges of the poles (fringe field is field particularly associated to the edge effects). Suppose the extent of the fringe field is b (see Fig. 2.3). The fringe field has two components Bx i and Bz k. For simplicity assume that |Bx|= B|z|/b where z=0 is the mid plane of the gap, explicitly:

when the particle enters the fringe field Bx = +B z /b, when the particle enters the fringe field after traveling through

the magnet, Bx = /Bz b−

x

XZ Plane

Narrow Beam of Particles

z

y

B

Fringe Field

l w

θ

Fig.2.2: Overall view (note that θ is very small).


FINAL PROBLEM

7/14

A parallel narrow beam of particles, each of mass m and positive charge q enters the magnet (near the center) with a high velocity v parallel to the horizontal plane. The vertical size of the beam is comparable to the distance between the magnet poles. A certain beam enters the magnet at an angle θ from the center line of the magnet and leaves the magnet at an angle -θ (see Figure 2.4. Assume θ is very small). Assume that the angle θ with which the particle enters the fringe field is the same as the angle θ when it enters the uniform field.

x

y

w v

θ

l

v

eBθ

Bx

Bz

x

z

x=b

Figure 2.3. Fringe field

Figure 2.4. Top view


FINAL PROBLEM

8/14

The beam will be focused due to the fringe field. Calculate the approximate focal length if we define the focal length as illustrated in Figure 2.5 (assume b<<l and assume that the z-component of the deflection in the uniform magnetic field B is very small).

x

f

Figure 2.5. Side view


FINAL PROBLEM

10/14

Question 3 LIGHT DEFLECTION BY A MOVING MIRROR

Reflection of light by a relativistically moving mirror is not theoretically new. Einstein discussed the possibility or worked out the process using the Lorentz transformation to get the reflection formula due to a mirror moving with a velocity vρ. This formula, however, could also be derived by using a relatively simpler method. Consider the reflection process as shown in Fig. 3.1, where a plane mirror M moves with a velocity xevv ˆ=ϖ (where xe is a unit vector in the x-direction) observed from the lab frame F. The mirror forms an angle φ with respect to the velocity (note that

090φ ≤ , see figure 3.1). The plane of the mirror has n as its normal. The light beam has an incident angle α and reflection angle β which are the angles between nϖ and the incident beam 1 and reflection beam 1' , respectively in the laboratory frame F. It can be shown that,

)(sinsinsinsin βαφβα +=−cv (1)

x

y

v

n

a

αβ 1’

1

M

φ

Figure 3.1. Reflection of light by a relativistically moving mirror


FINAL PROBLEM

11/14

3A. Einstein’s Mirror (2.5 points) About a century ago Einstein derived the law of reflection of an electromagnetic wave by a mirror moving with a constant velocity xevv ˆ−=ϖ (see Fig. 3.2). By applying the Lorentz transformation to the result obtained in the rest frame of the mirror, Einstein found that:

2

2

1 cos 2cos

1 2 cos

v vc c

v vc c

αβ

α

⎛ ⎞⎛ ⎞+ −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠=⎛ ⎞− + ⎜ ⎟⎝ ⎠

(2)

Derive this formula using Equation (1) without Lorentz transformation!

Figure 3.2. Einstein mirror moving to the left with a velocity v. 3B. Frequency Shift (2 points)

In the same situation as in 3A, if the incident light is a monochromatic beam hitting M with a frequency f, find the new frequency 'f after it is reflected from the surface of the moving mirror. If 030α = and 0.6 v c= in figure 3.2, find frequency shift f∆ in percentage of f .

αβv n

x

y

φ = 900


FINAL PROBLEM

12/14

3C. Moving Mirror Equation (5.5 Points)

Figure 3.3.

Figure 3.3 shows the positions of the mirror at time 0t and t . Since the observer is moving to the left, the mirror moves relatively to the right. Light beam 1 falls on point a at 0t and is reflected as beam 1' . Light beam 2 falls on point d at t and is reflected

as beam 2 ' . Therefore, ab is the wave front of the incoming light at time 0t . The

atoms at point are disturbed by the incident wave front ab and begin to radiate a wavelet. The disturbance due to the wave front ab stops at time t when the wavefront strikes point d. The semicircle in the figure represents wave-front of the wavelet at time t.

By referring to figure 3.3 for light wave propagation or using other methods, derive equation (1).

φ φ

7th Asian Physics Olympiad Theoretical Question 1 Page 1 /3


Laser cooling of atoms In this problem you are asked to consider the mechanism of atom cooling with the

help of laser radiation. Investigations in this field led to considerable progress in the understanding of the properties of quantum gases of cold atoms, and were awarded Nobel prizes in 1997 and 2001.

Theoretical Introduction

Consider a simple two-level model of the atom, with ground state energy Eg and

excited state energy Ee. Energy difference is 0ωη=− eg EE , the angular frequency of used laser is ω , and the laser detuning is 00 ωωωδ <<−= . Assume that all atom velocities satisfy c<<υ , where с is the light speed. You can always restrict yourself to first nontrivial orders in small parameters c/υ and 0/ωδ . Natural width of the excited state Ee due to spontaneous decay is 0ωγ << : for an atom in an excited state, the probability to return to a ground state per unit time equalsγ . When an atom returns to a ground state, it emits a photon of a frequency close to 0ω in a random direction.

It can be shown in quantum mechanics, that when an atom is subject to low-intensity laser radiation, the probability to excite the atom per unit time depends on the frequency of radiation in the reference frame of the atom, aω , according to

γγωω

γγ <<−+

= 220

0 /)(412/

ap s ,

where is a parameter, which depends on the properties of atoms and laser intensity.

10 <<s

Ћω0

Eg

ћω

ћδEe ћγ

Fig 1. Note that shown parameters are not in scale.

In this problem properties of the gas of sodium atoms are investigated neglecting the interactions between the atoms. The laser intensity is small enough, so that the number of atoms in the excited state is always much smaller than number of atoms in


the ground state. You can also neglect the effects of the gravitation, which are compensated in real experiments by an additional magnetic field.

Numerical values:

Planck constant η = 1.05·10-34 J s Boltzmann constant = 1.38·10Bk -23 J K-1

Mass of sodium atom m = 3.81·10-26 kg Frequency of used transition ω0 = 2π ·5.08·1014 Hz Excited state linewidth γ = 2π ·9.80·106 Hz Concentration of the atoms n = 1014 cm-3

Questions

а) [1 Point] Suppose the atom is moving in the positive x direction with the velocity

υх, and the laser radiation with frequency ω is propagating in the negative х direction. What is the frequency of radiation in the reference frame of the atom?

b) [2.5 Points] Suppose the atom is moving in the positive x direction with the velocity υх, and two identical laser beams shine along х direction from different sides. Laser frequencies are ω, and intensity parameters are s0. Find the expression for the average force ( )xF υ acting on an atom. For small υх this force can be written as xxF βυυ −=)( . Find the expression for β . What is the sign of

0ωωδ −= , if the absolute value of the velocity of the atom decreases? Assume that momentum of an atom is much larger than the momentum of a photon.

In what follows we will assume that the atom velocity is small enough so that one can use the linear expression for the average force. с) [2.0 Points] If one uses 6 lasers along х, у and z axes in positive and negative

directions, then for β>0 the dissipative force acts on the atoms, and their average energy decreases. This means that the temperature of the gas, which is defined through the average energy, decreases. Using the concentration of the atoms given above, estimate numerically the temperature ТQ, for which one cannot consider atoms as point-like objects because of quantum effects.

In what follows we will assume that the temperature is much larger than ТQ and six lasers along х, у and z directions are used, as was explained in part с). In part b) you calculated the average force acting on the atom. However, because of the quantum nature of photons, in each absorption or emission process the momentum of the atom changes by some discrete value and in random direction, due to the recoil processes.

d) [0.5 Points] Determine numerically the square value of the change of the momentum of the atom, (∆p)2, as the result of one absorption or emission event.


e) [3.5 Points] Because of the recoil effect, average temperature of the gas after long time doesn’t become an absolute zero, but reaches some finite value. The evolution of the momentum of the atom can be represented as a random walk in the momentum space with an average step >∆< 2p , and a cooling due to the dissipative force. The steady-state temperature is determined by the combined effect of these two different processes. Show that the steady state temperature

is of the form: dT

)4/()1( Bd kx

xT += γη . Determine x. Assume that is much larger

than <∆pdT

2>/(2kB m). Note: If vectors P1, P2, … , Pn are mutually statistically uncorrelated, mean square value of their sum is

<( P1 + P2 + … +Pn)2>=P12 + P2 2+…+Pn2

f) [0.5 Point] Find numerically the minimal possible value of the temperature due to recoil effect. For what ratio γδ / is it achieved?



Oscillator damped by sliding friction

Theoretical Introduction

In mechanics, one often uses so called phase space, an imaginary space with the axes comprising of coordinates and moments (or velocities) of all the material points of the system. Points of the phase space are called imaging points. Every imaging point determines some state of the system.

When the mechanical system evolves, the corresponding imaging point follows a trajectory in the phase space which is called phase trajectory. One puts an arrow along the phase trajectory to show direction of the evolution. A set of all possible phase trajectories of a given mechanical system is called a phase portrait of the system. Analysis of this phase portrait allows one to unravel important qualitative properties of dynamics of the system, without solving equations of motion of the system in an explicit form. In many cases, the use of the phase space is the most appropriate method to solve problems in mechanics.

In this problem, we suggest you to use phase space in analyzing some mechanical systems with one degree of freedom, i.e., systems which are described by only one coordinate. In this case, the phase space is a two-dimensional plane. The phase trajectory is a curve on this plane given by a dependence of the momentum on the coordinate of the point, or vice versa, by a dependence of the coordinate of the point on the momentum.

As an example we present a phase trajectory of a free particle moving along x axis in positive direction (Fig.1). p x 0

Fig. 1. Phase trajectory of a free particle.


Questions

A. Phase portraits (3.0)

A1. [0.5 Points] Make a draw of the phase trajectory of a free material point moving between two parallel absolutely reflective walls located at x = - L/2 and x = L/2.

A2. Investigate the phase trajectory of the harmonic oscillator, i.e., of the material point of mass m affected by Hook’s force F = - k x:

a) [0.5 Points] Find the equation of the phase trajectory and its parameters. b) [0.5 Points] Make a draw of the phase trajectory of the harmonic oscillator.

A3. [1.5 Points] Consider a material point of mass m on the end of weightless solid rod of length L, another end of which is fixed (strength of gravitational field is g). It is convenient to use the angle α between the rod and vertical line as a coordinate of the system. The phase plane is the plane with coordinates ( dtd /, αα ). Study and make a draw of the phase portrait of this pendulum at arbitrary angle α. How many qualitatively different types of phase trajectories K does this system have? Draw at least one typical trajectory of each type. Find the conditions which determine these different types of phase trajectories. (Do not take the equilibrium points as phase trajectories). Neglect air resistance.

B. The oscillator damped by sliding friction (7.0)

When considering resistance to a motion, we usually deal with two types of friction

forces. The first type is the friction force, which depends on the velocity (viscous friction), and is defined by F = -γv. An example is given by a motion of a solid body in gases or liquids. The second type is the friction force, which does not depend on the magnitude of velocity. It is defined by the value F = µN and direction opposite to the relative velocity of contacting bodies (sliding friction). An example is given by a motion of a solid body on the surface of another solid body.

As a specific example of the second type, consider a solid body on a horizontal surface at the end of a spring, another end of which is fixed. The mass of the body is m, the elasticity coefficient of the spring is k, the friction coefficient between the body and the surface is µ . Assume that the body moves along the straight line with the coordinate x (x = 0 corresponds to the spring which is not stretched). Assume that static and dynamical friction coefficients are the same. At initial moment the body has a position x=A0 (A0>0) and zero velocity.

B1. [1.0 Points] Write down equation of motion of the harmonic oscillator damped by the sliding friction.

B2. [2.0 Points] Make a draw of the phase trajectory of this oscillator and find the equilibrium points.


B3. [1.0 Points] Does the body completely stop at the position where the string is not stretched? If not, determine the length of the region where the body can come to a complete stop.

B4. [2.0 Points] Find the decrease of the maximal deviation of the oscillator in positive x direction during one oscillation ∆A. What is the time between two consequent maximal deviations in positive direction? Find the dependence of this maximal deviation A(tn) where tn is the time of the n-th maximal deviation in positive direction.

B5. [1.0 Points] Make a draw of the dependence of coordinate on time, x(t), and estimate the number N of oscillations of the body?

Note:

Equation of the ellipse with semi-axes a and b and centre at the origin has the following form:

122

22

=+by

ax



This problem consists of four not related parts.

A. [2.5 points] The Mariana Abyss in the Pacific Ocean has a depth of mH 10920= .

Density of salted water at the surface of the ocean is , bulk modulus is , acceleration of gravity is g=9.81m/s

30 /1025 mkg=ρ

PaК 9101,2 ⋅= 2. Neglect the change in the temperature and in the acceleration of gravity with the depth, and also neglect the atmospheric pressure.

А1) Find the relation between the density )(xρ and pressure at the depth of х.

)(xP

А2) Find the numerical value of the pressure at the bottom of the Mariana Abyss. You may use iterative methods to solve this part.

)(HP

Note: The fluids have very small compressibility. Compressibility coefficient is defined as

constTdPdV

V =

⎟⎠⎞

⎜⎝⎛−=

1κ

Bulk modulus K is the inverse of κ: K=1/ κ. B. [2.5 points] Light mobile piston separates the vessel into two parts. The vessel is

isolated from the environment. One part of the vessel contains m1 = 3g of hydrogen at the temperature of Т10 = 300 К, and the other part contains т2 = 16 g of oxygen at the temperature of Т20 = 400 К. Molar masses of hydrogen and oxygen are

moleg /21 =µ and moleg /322 =µ respectively, and )./(31.8 moleKJR ⋅= The piston weakly conducts heat between oxygen and hydrogen, and eventually the temperature in the system equilibrates. All the processes are quasi stationary.

В1) What is the final temperature of the system T? В2) What is the ratio between final pressure and initial pressure ? fP iP

В3) What is the total amount of heat Q, transferred from oxygen to hydrogen? C. [2.5 points] Two identical conducting plates α and β with charges –Q and +q

respectively (Q > q > 0) are located parallel to each other at a small distance. Another identical plate γ with mass m and charge +Q is situated parallel to the original plates at distance d from the plate β (see fig 1). Surface area of the plates is S. The plate γ is released and can move freely, while the plates α and β are kept fixed. Assume that the collision between the plates β and γ is elastic, and neglect the gravitational force and the boundary effects. Assume that the charge has enough time to redistribute between plates β and γ during the collision.


С1) What is the electric field E1 acting on the plate γ before the collision with the plate β ?

С2) What are the charges of the plates and after the collision? βQ γQ

С3) What is the velocity υ of the plate γ after the collision at the distance d from the plate β ?

d +Q+q-Q

γ β α

Fig. 1

D. [2.5 points] Two thin lenses with lens powers D1 and D2 are located at distance from each other, and their main optical axes coincide. This system creates a

direct real image of the object, located at the main optical axis closer to lens DcmL 25=

1, with the magnification 1'=Γ . If the positions of the two lenses are exchanged, the system again produces a direct real image, with the magnification .4'' =Γ

D1) What are the types of the lenses? On the answer sheet you should mark the gathering lens as «+», and the diverging lens as «-» .

D2) What is the difference between the lens powers 21 DDD −=∆ ?

Theoretical problem 1

Back-and-Forth Rolling of a Liquid-Filled Sphere (10 points) Consider a sphere filled with liquid inside rolling back and forth at the bottom

of a spherical bowl. That is, the sphere is periodically changing its translational and

rotational direction. Due to the viscosity of the liquid inside, the movement of the

sphere would be very complicated and hard to deal with. However, a simplified model

presented here would be beneficial to the solution of such a problem.

Assume that a rigid thin spherical shell of radius r and mass m is fully filled with

some liquid substance of mass M, denoted as W. W has such a unique property that

usually it behaves like an ideal liquid (i.e. without any viscosity), while in response to

some special external influence (such as electric field) it transits to solid state

immediately with the same volume; and once the applied influence removed, the

liquid state recovers immediately. Besides, this influence does not give rise to any

force or torque exerting on the sphere. This liquid-filled spherical shell (for

convenience, called ‘the sphere’ hereafter) is supposed to roll back and forth at the

bottom of a spherical bowl of radius R ( ) without any relative slipping, as shown

in the figure. Assume the sphere moves only in the vertical plane (namely, the plane

of the figure), please study the movement of the sphere for the following three cases:

1. W behaves as in ideal solid state, meanwhile W contacts the inner wall of the

spherical shell so closely that they can be taken as solid sphere as a whole of

radius r with an abrupt density change across the interface between the inside wall

of the shell and W.

(1) Calculate the rotational inertia I of the sphere with respect to the axis passing

through its center C. (You are asked to show detailed steps.)

(1.0 points)

(2) Calculate the period of the sphere rolling back and forth with a small

amplitude without slipping at the bottom of the spherical bowl. (2.5points)

D

R

O A1 A2

A0 A1

θ 0 θ 0 ’

‘ ‘

C

liquid

液态 liquid

液态

solid

固态 solid

固态

A0

2. W behaves as an ideal liquid with no friction between W and the spherical shell.

Calculate the period of the sphere rolling back and forth with a small

amplitude without slipping at the bottom of the spherical bowl. (2.5 points)

3. W transits between ideal solid state and ideal liquid state.

Assume at time , the sphere is kept at rest, the line CD makes an angle

( rad）with the plumb line OD, where D is the center of the spherical bowl.

The sphere contacts the inner wall of the bowl at point , as shown in the figure.

Release the sphere, it starts to roll left from rest. During the motion of the sphere

from to its equilibrium position O, W behaves as ideal liquid. At the moment

that the sphere passes through point O, W changes suddenly into solid state and

sticks itself firmly on the inside wall of the sphere shell until the sphere reaches

the left highest position . Once the sphere reaches , W changes suddenly

back into the liquid state. Then, the sphere rolls right; and W changes suddenly

into solid state and sticks itself firmly on the inside wall of the spherical shell

again when the sphere passes through the equilibrium position O. When the

sphere reaches the right highest position , W changes into liquid state once

again. Then the whole circle repeats time after time. The sphere rolls right and left

periodically but with the angular amplitude decreased time after time. The motion

direction of the sphere is shown by curved arrows in the figure, together with the

words “solid” and “liquid” showing corresponding state of W. It is assumed that

during such process of rolling back and forth, no any relative slide happens

between the sphere and the inside wall of the bowl (or, alternatively, the bottom of

the bowl can supply as enough friction as needed). Calculate the period of the

sphere rolling right and left, and the angular amplitude of the center of the

sphere, namely, the angle that the line CD makes with the vertical line OD when

the sphere reaches the right highest position for the n-th time (only is

shown in the figure). （4.0 points）

Theoretical problem 2

2A. Optical properties of an unusual material (7 points) The optical properties of a medium are governed by its relative permittivity ( )

and relative permeability ( ). For conventional materials like water or glass， which

are usually optically transparent, both of their and are positive, and refraction

phenomenon meeting Snell’s law occurs when light from air strikes obliquely on the

surface of such kind of substances. In 1964, a Russia scientist V. Veselago rigorously

proved that a material with simultaneously negative and would exhibit many

amazing and even unbelievable optical properties. In early 21st century, such unusual

optical materials were successfully demonstrated in some laboratories. Nowadays

study on such unusual optical materials has become a frontier scientific research field.

Through solving several problems in what follows, you can gain some basic

understanding of the fundamental optical properties of such unusual materials. It

should be noticed that a material with simultaneously negative and

possesses the following important property. When a light wave propagates

forward inside such a medium for a distance , the phase of the light wave will

decrease, rather than increase an amount of as what happens in a

conventional medium with simultaneously positive and . Here, positive root

is always taken when we apply the square-root calculation, while is the wave

vector of the light. In the questions listed below, we assume that both the relative

permittivity and permeability of air are equal to 1.

1. (1)According to the property described above, assuming that a light beam strikes

from air on the surface of such an unusual material with relative permittivity

and relative permeability , prove that the direction of the refracted

light beam depicted in Fig.2-1 is reasonable. (1.2 points)

Fig. 2-1

(2) For Fig. 2-1, show the relationship between refraction angle (the angle that

refracted beam makes with the normal of the interface between air and the

material) and incidence angle . (0.8 points)

(3) Assuming that a light beam strikes from the unusual material on the interface

between it and air, prove that the direction of the refracted light beam depicted in

Fig.2-2 is reasonable. (1.2 points)

Fig. 2-2

(4) For Fig. 2-2, show the relationship between the refraction angle (the angle

that refracted beam makes with the normal of the interface between two media)

and the incidence angle . (0.8 points)

2. As shown in Fig. 2-3, a slab of thickness d, which is made of an unusual optical

material with , is placed in air, with a point light source located in

front of the slab separated by a distance of . Accurately draw the ray

diagrams for the three light rays radiated from the point source. (Hints: under

the conditions given in this problem, no reflection would happen at the interface

between air and the unusual material). (1.0 points)

Fig. 2-3

3. As shown in Fig. 2-4, a parallel-plate resonance cavity is formed by two plates

parallel with each other and separated by a distant d. Optically one of the plates,

denoted as Plate 1 in Fig.2-4 , is ideally reflective (reflectance equals to 100%),

and the other one, denoted by Plate 2, is partially reflective ( but with a high

reflectance). Suppose plane light waves are radiated from a source located near

Plate 1, then such light waves are multiply reflected by the two plates inside the

cavity. Since optically the Plate 2 is not ideally reflective, some light waves will

leak out of Plate 2 each time the light beam reaches it (ray 1, 2, 3, as shown in Fig.

2-4), while some light waves will be reflected by it. If these light waves are

in-phase, they will interfere with each other constructively, leading to resonance.

We assume that the light wave gains a phase of by reflection at either of the

two plates. Now we insert a slab of thickness (shown as the shaded area in

Fig. 2-4), made of an unusual optical material with , into the cavity

parallel to the two plates. The remaining space is filled with air inside the cavity.

Let us consider only the situation that the light wave travels along the direction

perpendicular to the plates (the ray diagram depicted in Fig.2-4 is only a

schematic one), calculate all the wavelengths that satisfy the resonance condition

of such a cavity. (Hints: under the condition given here, no reflection would occur

at the interfaces between air and the unusual material). (1.0 points)

Fig. 2-4

4. An infinitely long cylinder of radius R, made of an unusual optical material with

, is placed in air, its cross section in XOY plane is shown in Fig. 2-5

with the center located on Y axis. Suppose a laser source located on the X axis

(the position of the source is described by its coordinate x) emits narrow laser light

along the Y direction. Show the range of x, for which the light signal emitted from

the light source can not reach the infinite receiving plane on the other side of the

cylinder. (1.0 points)

Fig. 2-5

2B. Dielectric spheres inside an external electric field (3 points) By immersing a number of small dielectric particles inside a fluid of low-viscosity,

you can get the resulting system as a suspension. When an external electric field is

applied on the system, the suspending dielectric particles will be polarized with

electric dipole moments induced. Within a very short period of time, these polarized

particles aggregate together through dipolar interactions so that the effective viscosity

of the whole system enhances significantly (the resulting system can be

approximately viewed as a solid). This type of phase transition is called

“electrorheological effect”, and such a system is called “electrorheological fluid”

correspondingly. Such an effect can be applied to fabricate braking devices in practice,

since the response time of such a phase transition is shorter than conventional

mechanism by several orders of magnitude. Through solving several problems in the

following, you are given a simplified picture to understand the inherent mechanism of

the electrorheological transition.

1. When there are many identical dielectric spheres of radius a immersed inside the

fluid, we assume that the dipole moment of each sphere , is induced solely by

the external field , independent of any of the other spheres (Note: ).

(1) When two identical small dielectric spheres exist inside the fluid and contact

with each other, while the line connecting their centers makes an angle with

the external field direction (see Fig. 2-6), write the expression of the energy of

dipole-dipole interaction between the two small contacting dielectric spheres, in

terms of p, a and . (Note: In your calculations each polarized dielectric sphere

can be viewed as an electric dipole located at the center of the sphere) (0.5 points)

Fig. 2-6

(2) Calculate the dipole-dipole interaction energies for the three configurations

shown in Fig. 2-7. (0.75 points)

Fig. 2-7

(3) Identify which configuration of the system is the most stable one. (0.25 points)

(Note: In your calculations each polarized dielectric sphere can be viewed as an

electric dipole located at the center of the sphere, and the energy of dipole-dipole

interaction can be expressed in terms of p and a.)

2. In the case that three identical spheres exist inside the fluid, based on the same

assumption as in question 1,

(1) calculate the dipole-dipole interaction energies for the three configurations

shown in Fig. 2-8; (0.9 points)

(2) identify which configuration of the system is the most stable one; (0.3 points)

(3) identify which configuration of the system is the most unstable one. (0.3 points)

(Note: In your calculations each polarized dielectric sphere can be viewed as an

electric dipole located at the center of the sphere, and the energy of dipole-dipole

interaction can be expressed in terms of p and a.)

Fig. 2-8

Theoretical Problem 3

3A. Average contribution of each electron to specific heat of free

electron gas at constant volume (5 points) 1. According to the classical physics the conduction electrons in metals constitute

free electron gas like an ideal gas. In thermal equilibrium their average energy relates

to temperature, therefore they contribute to the specific heat. The average contribution

of each electron to the specific heat of free electron gas at constant volume is defined

as

, （1）

where is the average energy of each electron. However the value of the specific

heat at constant volume is a constant, independent of temperature. Please calculate

and the average contribution of each electron to the specific heat at constant

volume .

(1.0 points)

2．Experimentally it has been shown that the specific heat of the conduction

electrons at constant volume in metals depends on temperature, and the experimental

value at room temperature is about two orders of magnitude lower than its classical

counterpart. This is because the electrons obey the quantum statistics rather than

classical statistics. According to the quantum theory, for a metallic material the

density of states of conduction electrons (the number of electronic states per unit

volume and per unit energy) is proportional to the square root of electron energy ,

then the number of states within energy range for a metal of volume V can be

written as (2)

where C is the normalization constant, determined by the total number of electrons of

the system.

The probability that the state of energy E is occupied by electron is

, (3)

where is the Boltzmann constant and T is the absolute

temperature, while is called Fermi level. Usually at room temperature is

about several eVs for metallic materials (1eV= J). is called Fermi

distribution function shown schematically in the figure below.

(1) Please calculate at room temperature according to (3.5points)

(2) Please give a reasonable explanation for the deviation of the classical result from

that of quantum theory. (0.5 points)

Note: In your calculation the variation of the Fermi level with temperature

could be neglected, i.e. assume , is the Fermi level at 0K. Meanwhile

the Fermi distribution function could be simplified as a linearly descending function

within an energy range of 2 around , otherwise either 0 or 1, i.e.

At room temperature << , therefore calculation can be simplified accordingly.

Meanwhile, the total number of electrons can be calculated at 0K.

3B. The Inverse Compton Scattering (5 points) By collision with relativistic high energy electron, a photon can get energy from

the high energy electron, i.e. the energy and frequency of the photon increases because

of the collision. This is so-called inverse Compton scattering. Such kind of

phenomenon is of great importance in astrophysics, for example, it provides an

important mechanism for producing X rays and γ rays in space.

1. A high energy electron of total energy (its kinetic energy is higher than static

energy) and a low energy photon (its energy is less than the static energy of an

electron) of frequency move in opposite directions, and collide with each other.

As shown in the figure below, the collision scatters the photon, making the scattered

photon move along the direction which makes an angle with its original incident

direction (the scattered electron is not shown in the figure). Calculate the energy of

the scattered photon, expressed in terms of and static energy of the

electron. Show the value of , at which the scattered photon has the maximum

energy, and the value of this maximum energy. (2.4 points)

2. Assume that the energy of the incident electron is much higher than its static

energy , which can be shown as ，and that the energy of the

incident photon is much less than , show the approximate expression of the

maximum energy of the scattered photon. Taking and the wavelength of

the incident visible light photon , calculate the approximate maximum

energy and the corresponding wavelength of the scattered photon.

Parameters: Static energy of the electron , Planck constant

J s, and eV nm, where c is the light speed in the

vacuum. (1.2 points)

3. (1) A relativistic high energy electron of total energy and a photon move in

opposite directions and collide with each other. Show the energy of the incident

photon, of which the photon can gain the maximum energy from the incident

electron. Calculate the energy of the scattered photon in this case. (0.7 points)

(2) A relativistic high energy electron of total energy and a photon, moving in

perpendicular directions respectively, collide with each other. Show the energy of

the incident photon, of which the photon can gain the maximum energy from the

incident electron. Calculate the energy of the scattered photon in this case. (0.7

points)

1

THEORETICAL COMPETITION 9th Asian Physics Olympiad Ulaanbaatar, Mongolia (April 22, 2008 )

----------------------------------------------------------------------------- Theoretical Problem 1. Tea Ceremony and Physics of Bubbles The tea ceremony is traditional in Asia. One of the important steps in preparation of tea is the boiling of fresh water when bubbles appear inside. Bubbles are familiar from daily life and occupy an important role in physics, chemistry, medicine and technology. Nevertheless, their behavior is often surprising and unexpected - and, in many cases, still not understood. At room temperature the pure water is saturated with gas. With increasing temperature the excess pressure of dissolved gas increases, the dissolved air is liberated and air bubbles (ABs)

appear at the bottom and walls of teakettle (Fig.2). For pure water the wettability is sufficient and an AB represents a truncated sphere with radius and with unwetted foundation with radius . At more heating ABs expand and by reaching certain sizes can detach from

the bottom (Fig.3), flow up to the water surface and burst there. The vapor bubbles (VBs) appear when the water temperature at the bottom reaches the critical value at which the pressure of the saturated vapor exceeds the external pressure. The vapor production increases tens times, VBs expand and detach from the bottom. VB may be considered consisting only of vapor. If the water is heated sufficiently, the uprising VB continue to swell, reach the surface and burst. Else, water is not heated enough in the higher layers and there exits a vertical strong temperature gradient. By reaching relatively cold layers of water VB collapse in the volume of water (Fig.4). This causes the induced degassing - strong oscillations and a considerable amount of dissolved air is released in the form of microscopic air bubbles (MAB). This can generate ultrasonic vibrations. The main stages of the bubble evolution during the boiling process are: - the appearance and growth of AB at the bottom and walls, their transmutation into VB; - the detachment and uprising of VB, their disappearance in the water volume or at the surface; - the appearance of MAB in the water volume and their uprising to the surface. This theoretical description is in good agreement with modern experiments. Particularly, an interesting noise analysis experiment (NAE, Ural State University, Ekaterinburg) for the boiling water was performed. Highly sensitive microphones attached to wide-band amplifiers and brought to an electric teakettle have detected three main origins of noises: 1. AB's detachments from the bottom before boiling (generate oscillations with ~ 100 Hz, ); 2. VB's collapses in the volume of water (generate oscillations with ~ 1 kHz ); 3. MAB's appearances under the water surface (generate oscillations with ~35kHz to 60kHz).

2

Hints: 1) It is well known that a small bubble rises along a rectilinear path and a laminar flow is observed - water flows easy and layer-wise (see Fig.1). Then, the Stokes formula describes the dissipative force for a particle moving with slow velocity : In contrast to this picture, when relatively large bubbles lift to the surface, it disturbs the surrounding water, cavitation hollows appear behind and the turbulent flow is observed (see Fig.1). In this case a part of the kinetic energy of an uprising bubble transfers into the dissipative work.

.

Fig 1. Laminar and turbulent types of flow for rising air bubbles in water 2) When the surface of liquid has a convex (concave) form there appears a surface tension force due to molecular interaction near the edge. This pressure can be given by formula

where - is the surface tension coefficient (unit=N/m), the force coming to unit length of surface, R – is the radius of surface curvity. 3) When dealing with a short process with characteristic duration time "t", its inverse value may

be considered as a characteristic frequency . Use this definition for calculating the noise

frequencies.

3

Theoretical Problem 1, 9th Asian Physics Olympiad (Mongolia)

Useful data:

- atmospheric pressure,

- water density,

- vapor density at T=293K; ( at T=373K)

- vapor pressure at T=293K; ( at T=373K)

- acceleration of gravity, - molecular weight of air

- the gas universal constant - surface tension coefficient of water,

- coefficient of viscosity of water H=10cm – Water attitude in teakettle

Fig. 2. Bubbles in teakettle Fig. 3. Air bubble detaching

from the bottom Fig. 4. Vapor bubble collapsing

4

Theoretical Question 1, 9th Asian Physics Olympiad (Mongolia)

Questions (total 10 points): Consider water boiling in a flat-bottomed cylinder glass teakettle at normal atmospheric pressure. The bottom of the kettle heats up uniformly and a vertical temperature gradient exists, bubbles appear and evolute (Fig.2).

Q1. Write the pressure condition of the growth of an AB in the water volume at height h<H, where H is the water surface level in the teakettle. Take into account the inequality . [in terms of ] (1.0 point) Q2. Write for an AB the condition of the detachment from the bottom of the teakettle (Fig.3). Take into account the relation . [in terms of ] (1.5 points) Q3. Consider an AB with radius at the bottom of the teakettle. As water is boiled, the bubble is saturated with vapor and enlarges its radius. Write the ratio of the masses of the air and saturated vapor inside the bubble at given temperature T. Calculate the ratio at room temperature T=20oC ( ) and at boiling point at T=100oC ( ). [in terms of ] (1.5 points) Q4. By using the NAE data and Newton's Law estimate the radius of the AB detached from the bottom and uprised in distance (Fig.3). Assume, that the added-mass (taking into account surrounding water layer) of AB is a half of the analogous water bubble. (1.0 points) Q5. Write the radius of the foundation of an AB just before the uprising, when the connecting "neck" is very narrow (see Fig.3). [in terms of ]. Calculate it by using the radius found in Q4. (1.5 points) Q6. By using the NAE data estimate the radius of collapsing VB (Fig.4) by assuming that the radial pressure is about 3kPa during this process. (1.2 points) Q7. By using previous results for VB calculate the radius of the MAB produced during induced degassing. (0.5 points) Q8. Write the uprising velocity for typical AB by using the Stokes law of a laminar flow. [in terms of ]. Estimate the uprising time for H=10cm. (0.6 points) Q9. Write the average velocity of the elevation of VB with turbulent type of flow [in terms of ]. Estimate the uprising time for H=10cm. (1.2 points)

1


----------------------------------------------------------------------------------------------------------- Problem 2: Ionic Crystal, Yukawa-type Potential and Pauli Principle Atoms of many chemical elements possess very low ionization energy and easily loose the outer electrons. Vice versa, atoms of other elements accept easily the electrons. Taken into one volume, these positive and negative ions can combine into stable ionic structures. Many solids exhibit a crystal structure, in which the atoms are arranged in extremely regular, periodic patterns. In an ideal crystal the same basic structural unit is repeated through the space.

Fig.1

The face-centered cubic lattice of the sodium chloride (NaCl). The lattice spacing between the atomar centres is constant and is given by parameter .

The main contribution to the binding energy of an ionic crystal is given by the electrostatic potential energy of ions. The electric interaction acting between two point charges q1 and q2 standing in distance R is well defined by Coulomb's potential:

where is the Coulomb constant. A negative force implies an attractive force. The force is directed along the line joining the two charges. For the case of NaCl crystals both types of ions has the unit charge ±e and one should also take into account many other neighbors acting on the chosen ion. Taking into account all positive and negative ions in a crystal of the infinite sizes results in the attractive potential energy , where r is the distance between nearest neighbors and α=1.74756 is the Madelung constant [E.Madelung, Phys. Zs, 19 (1918) p542] and used in determining the energy of a single ion in a crystal. Along the attractive potential energy there should to be a repulsive potential energy due to the Pauli Exclusion Principle and the overlap of electron shells in a crystal lattice. In contrast to the Coulomb-type attractive part, the repulsive potential energy is very short range.

2

Theoretical Problem 2, 9th Asian Physics Olympiad (Mongolia) There are two different models to describe the repulsive potential. Model #1. A reasonable approximation to the repulsive potential is an exponential function: which describes the repulsion interaction of selected ion with the entire crystal lattice. Here λ is coupling strength and ρ stands for the range parameter. Model #2. Another good approximation to the repulsive potential is an inverse power where b is coupling strength and n is integer greater than 2 (the Born exponent). These parameters take into account the repulsion with entire crystals. Obviously, the physical parameters and model potentials depend on the type of crystal lattice. Experimental data for the lattice constant and the dissociation energy (needed to break the lattice into separate ions) are given in Table 1 for some ionic crystals at normal temperature and pressure.

Table 1 Properties of Salt Crystals with the NaCl Structure [C.Kittel, "Introduction to Solid State Physics", N.Y., Wiley (1976) p.92] (in one mole there are the Avogadro’s number of pairs of ions or atoms).

Crystal [nm] NaCl 0.282 +764.4 LiF 0.214 +1014.0 RbBr 0.345 +638.8

3

Theoretical Question 2, 9th Asian Physics Olympiad (Mongolia)

Questions (total 10 points): Q1. Write down the Coulomb potential for an ion located at the centre of cubic lattice in

Fig 1. Let it interact only with the nearest neighbors (in distance up to and including ) of a crystal lattice. Find the Madelung constant corresponding to this approximation. (1.5 points) Q2. By using Model #1 write down the net potential energy per ion . Determine its equilibrium equation for r= and write down the net potential energy [in terms of ]. Use exact Madelung constant α. (1.5 points) Q3. By using experimental data estimate the range parameter ρ . Use . (2.0 points) Q4. By using Model #2 write down the net potential energy per ion. Determine its equilibrium position r= and write down the net potential energy . Use exact Madelung constant α. [in terms of ] (2.0 points) Q5. By using experimental data (from Table 1) estimate the Born exponent n for NaCl. Estimate the proportions of the Coulomb interaction and the Pauli exclusion (repulsive part) in the entire net potential energy in the equilibrium state? (1.5 points) Q6. The ionization energy (required to extract an electron from an atom) of the Na atom is +5.14 eV, the electron affinity (required to receive an electron to an atom) of the Cl atom is -3.61 eV. Estimate the total binding energy (holds an atom inside lattice) per atom in the NaCl crystal. The experimental result is . [in terms of eV]

Use the conversion of units: (1.5 points)


-----------------------------------------------------------------------------

Problem 3. How does a superluminal object look like? Can a body move faster than the speed of light? The answer is “No” if the object is moving in the vacuum. But the answer can be “Yes”, if we deal with the phase speed of light in an optically dense medium with refractive index of ( , where is the speed of light in the medium, and is the speed of light in the vacuum).

We say a body is superluminal, if , where is the velocity of the body. One of the well known examples of the superluminal body is a charged particle generating Cherenkov radiation.

Throughout the problem we will deal with a superluminal body of constant velocity in an optical medium without dispersion. u is the velocity of light in the medium.

For the simplicity, we introduce a notation and an angle given by

and .

1. Radiating superluminal particle

As shown in Fig.1, a radiating particle is moving along the -axis with a constant velocity ( ).

An observer M is located at the distance from -axis.

We choose the point nearest to the observer as the point O, the origin on the x-axis. The time when the particle actually passes over the point x=0 is taken to be t=0.

M

x d

O

Figure 1


2

(1)Suppose the light radiated at the given time is observed at time . Express in terms of and (1.0 point)

(2)At time , the observer first sees the particle at position . Find the apparent position and the observed time for this first appearance in terms of and .

(2.0 points)

(3) Find the apparent position(s) of the particle for any given time t. Write your answer in terms of and . (2.0 points)

(4) Find the apparent velocity(s) (t) of the particle for any given time t. Write your answer in terms of , and . (1.0 point)

(5) Find the apparent velocity(s) of the first appearance of the particle. (0.2 points)

(6) Find the apparent velocity(s) of the particle at infinite distances from the origin, O. Write your answer in terms of and . (0.2 points)

(7) Sketch the graph of the apparent velocity versus time t, indicating clearly asymptotic values of the apparent velocity. (1.0 point)

(8) Can an apparent velocity exceed the light speed in the vacuum, i.e. ? (0.2 points)

2.Radiating linear object

Consider a linear object, radiating light and moving along the x-axis. The length of the linear object is in the rest frame of the object.

A. Parallel movement

In this section, we assume that the radiating linear object moves longitudinally along x-axis as shown in Fig.2.

(9) Determine the time interval of complete appearance of the whole linear object from the first appearance of its front point. Write your answer in terms of and . (0.3 points)

O

Figure 2


3

(10) Determine the apparent length(s) of the object at the moment of its complete appearance. Write your answer in terms of and (0.4 points)

B. Perpendicular movement

In this section, we assume that the radiating linear object moves perpendicularly along x-axis as shown in Fig.3. Let the observer be located at the origin of -axis . The object is symmetrical with respect to x-axis.

(11) Show that for a given time , the apparent form of this object is an ellipse or part(s) of an ellipse. (0.7 points)

Find the following quantities and express them in terms of , and .

(12) Find the position of the centre of symmetry of the ellipse for a given time in terms of and . (0.5 points)

(13) Determine the lengths of the semi-major and semi-minor axes of the ellipse for a given time in terms of and . (0.5 points)

O

Figure 3

y

Theoretical competition Question 1 26 April 2009 Page 1 of 2 -------------------------------------------------------------------------------------------------------------------------

Rolling Cylinders

A thin-walled cylinder of mass M and rough inner surface of radius R can rotate about its fixed

central horizontal axis OZ. The Z-axis is perpendicular to and out of the page. Another smaller

uniform solid cylinder of mass m and radius r rolls without slipping (except for question 1.8) on

the inner surface of M about its own central axis which is parallel to OZ.

1.1) The rotation of M is to be started from rest at the instant 0t = when m is resting at the lowest

point. At a later time t the angular position of the centre of mass of m is θ and by then M

has turned through an angle φ radians. How many radians (designated ψ ) would have mass

m turned through about its central axis relative to a fixed line (for example, the negative Y-

axis)? Give your answer in terms of , , Rθ φ and r . (0.8 point)

1.2) What is the angular acceleration of m , 2

2

ddt

ψ , about its own axis through its centre of mass?

Give your answer in terms of , ,R r and derivatives of θ and φ . (0.2 point)

1.3) Derive an equation for the angular acceleration of the centre of mass of m , 2

2

ddt

θ , in terms of

2

2, , , , , dm g R rdt

θ φ , and the moment of inertia CMI of m about its central axis. (1.8 points)

M

m

R X

Y

Oθ

g = acceleration due to gravity


1.4) What is the period of small amplitude oscillation of m when M is constrained to rotate at a

constant angular velocity? Give your answer only in terms of , ,R r and g . (1.3 point)

1.5) What is the value of θ for the equilibrium position of m in question 1.4? (0.2 point)

1.6) What is the equilibrium position of m when M is rotating with a constant angular acceleration

α ? Give your answer in terms of , ,R g and α . (0.7 point)

1.7) Now M is allowed to rotate (oscillate) freely, without constraint, about its central axis OZ

while m is executing a small-amplitude oscillation by pure rolling on the inner surface of M .

Find the period of this oscillation. (2.5 points)

1.8) Consider the situation in which M is rotating steadily at an angular velocity Ω and m is

rotating (rolling) about its stationary centre of mass, at the equilibrium position found in

question 1.5. M is then brought abruptly to a halt. What must be the lowest value of Ω such

that m will roll up and reach the highest point of the cylindrical surface of M ? The

coefficient of friction between m and M is assumed to be sufficiently high that m begins to

roll without slipping soon after a short skidding right after M is stopped. (2.5 points)

*********************


A Self-excited Magnetic Dynamo

A metallic disc of radius a mounted on a slender axle is rotating with a constant angular velocity

ω inside a long solenoid of inductance L whose two ends are connected to the rotating disc by two

brush contacts as shown. The total resistance of the whole circuit is R . A small magnetic

disturbance can initiate the growth of an induced electromotive force across the terminals P, Q.

2.1) Write down the differential equation for ( )i t , the current through the circuit. Express your

answer in terms of , ,L R and the induced e.m.f. ( )E across the terminals P and Q. (1.0 point)

2.2) What is the value of the magnetic flux density ( )B in terms of , , ,i N and the permeability of

free space 0μ ? Ignore the magnetic field generated by the disc and the axle. (1.5 points)

brush

ω

Q•

long uniform solenoid of N turns, length

a P•

Theoretical competition Question 2 26 April 2009 Page 2 of 2 ------------------------------------------------------------------------------------------------------------------------- 2.3) What is the expression for the induced e.m.f. ( )E in terms of 0 , , , , ,N a iμ and the angular

velocity ω ? (2.0 points)

2.4) Solve the equation in question 2.1 for current at any time t in terms of the initial current ( )0i ,

and other parameters. (1.5 points)

2.5) What is the minimum value of the angular velocity that will permit the current to grow? Give

your answers in terms of 0, , , ,R N aμ and . (2.0 points)

2.6) In order to maintain a certain steady angular velocity ω , what must be the value of torque

applied to the axle at the instant t ? (2.0 points)

*********************


The Leidenfrost Phenomenon

The purpose is to estimate the lifetime of a (hemispherical) drop of a liquid sitting on top of a very

thin layer of vapour which is thermally insulating the drop from the very hot plate below.

It will be assumed here that the flow of vapour underneath the drop is streamline and behaves as a

Newtonian fluid of viscosity coefficient η and of thermal conductivity K . The specific latent heat

of vaporization of the liquid is . And for a Newtonian fluid we have the shear stress FA

η= × the

rate of shear dvdz

where v is the flow velocity and z is the perpendicular distance to the direction of

flow, and the direction of F is tangential to the surface area A .

v is the velocity of vapour in the radial direction at the height z above the mid-plane. The pressure

P inside the vapour must be higher towards the centre O. This will result in the out-flowing of

drop

•

thin layer of vapour very hot surface

Figure 1

Z

R

g

b O

r δr

zv

hemispherical drop of radius R at distance b above the hot surface

hot surface

mid-plane of vapour layer

Figure 2


vapour and in force that holds the drop against the pull of gravity. The thickness of vapour layer

under thermal and mechanical equilibria is b .

For a Newtonian flow of vapour we can approximate that

d z dv Pdz drη

=

3.1) Show that ( )2

2z dv z P C

drη= +

where C is an arbitrary constant of integration. (0.5 point)

3.2) Refer to figure 2, find the value of C in terms of , ,η d Pdr

and b using the boundary condition

0v = for 2bz = ± . (0.5 point)

3.3) Calculate the volume rate of flow of vapour through the cylindrical surface defined by r .

(Hint: the cylinder is of radius r and of height b underneath the drop). (1.0 point)

3.4) By assuming that the rate of production of vapour of density Vρ is due to heat flow from the

hot surface to the drop, find the expression for the pressure ( )P r . Use aP to represent the

atmospheric pressure, and use TΔ for the temperature difference between those of the hot

surface and of the drop. Assume that the system has reached the steady state. (2.0 points)

3.5) Calculate the value of b by equating the weight of the drop to the net force due to pressure

difference between the bottom and the top of the drop. The density of the drop is 0ρ .

(2.0 points)

3.6) Now, what is the total rate of vaporization? (2.0 points)

3.7) Assume that the drop maintains a hemispherical shape, what is the life-time of the drop?

(2.0 points)

*********************

Theoretical Competition 25 April 2010 Page 1 of 5 __________________________________________________________________________________________

Question Number 1

Theoretical Question 1 Particles and Waves

This question includes the following three parts dealing with motions of particles and waves: Part A. Inelastic scattering of particles Part B. Waves on a string Part C. Waves in an expanding universe

Part A. Inelastic Scattering and Compositeness of Particles A particle is considered elementary if it has no excitable internal degrees of freedom such

as, for example, rotations and vibrations about its center of mass. Otherwise, it is composite. To determine if a particle is composite, one may set up a scattering experiment with the

particle being the target and allow an elementary particle to scatter off it. In case that the target particle is composite, the scattering experiment may reveal important features such as scaling, i.e. as the forward momentum of the scattered particle increases, the scattering cross section becomes independent of the momentum.

For a scattering system consisting of an elementary particle incident on a target particle, we shall denote by 𝑄𝑄 the total translational kinetic energy loss of the system. Here the translational kinetic energy of a particle, whether elementary or composite, is defined as the kinetic energy associated with the translational motion of its center of mass. Thus we may write

𝑄𝑄 = 𝐾𝐾i − 𝐾𝐾f , where 𝐾𝐾i and 𝐾𝐾f are the total translational kinetic energies of the scattering pair before and after scattering, respectively.

In Part A, use non-relativistic classical mechanics to solve all problems. All effects due to gravity are to be neglected. (a) As shown in Fig. 1, an elementary particle of mass 𝑚𝑚 moves along the 𝑥𝑥 axis with 𝑥𝑥

-component of momentum 𝑝𝑝1 > 0. After being scattered by a stationary target of mass 𝑀𝑀, its momentum becomes 𝑝2.

𝑝𝑝1 𝑀𝑀

Fig. 1

𝑚𝑚

𝑝2 𝑦𝑦

𝑥𝑥

𝑚𝑚


Question Number 1

From data on 𝑝2, one can determine if the target particle is elementary or composite. We shall assume that 𝑝2 lies in the 𝑥𝑥-𝑦𝑦 plane and that the 𝑥𝑥- and 𝑦𝑦-components of 𝑝2 are given, respectively, by 𝑝𝑝2𝑥𝑥 and 𝑝𝑝2𝑦𝑦 .

(i) Find an expression for 𝑄𝑄 in terms of 𝑚𝑚, 𝑀𝑀, 𝑝𝑝1, 𝑝𝑝2𝑥𝑥 , and 𝑝𝑝2𝑦𝑦 . [0.2 point] (ii) If the target particle is elementary, the momenta 𝑝𝑝1, 𝑝𝑝2𝑥𝑥 , and 𝑝𝑝2𝑦𝑦 are related in a

particular way by a condition. For given 𝑝𝑝1, plot the condition as a curve in the 𝑝𝑝2𝑥𝑥 - 𝑝𝑝2𝑦𝑦 plane. Specify the value of 𝑝𝑝2𝑥𝑥 for each intercept of the curve with the 𝑝𝑝2𝑥𝑥-axis. In the same plot, locate regions of points of 𝑝2 corresponding to 𝑄𝑄 < 0, 𝑄𝑄 = 0, 𝑄𝑄 > 0, and label each of them as such.

[0.7 point] For a stationary composite target in its ground state before scattering, which region(s) of 𝑄𝑄 contains those points of 𝑝2 allowed? [0.2 point]

(b) Now, consider a composite target consisting of two elementary particles each with mass 1

2𝑀𝑀. They are connected by a spring of negligible mass. See Fig. 2. The spring has a force constant 𝑘𝑘 and does not bend sideways. Initially, the target is stationary with its center of mass at the origin O, and the spring, inclined at an angle 𝜃𝜃 to the 𝑥𝑥-axis, is at its natural length 𝑑𝑑0. For simplicity, we assume that only vibrational and rotational motions can be excited in the target as a result of scattering.

The incident elementary particle of mass 𝑚𝑚 moves in the 𝑥𝑥-direction both before and after scattering with its momenta given, respectively, by 𝑝𝑝1 and 𝑝𝑝2. Note that 𝑝𝑝2 is negative if the particle recoils and moves backward. A scattering occurs only if the incident particle hits one of the target particles and 𝑝𝑝2 ≠ 𝑝𝑝1. We assume all three particles move in the same plane before and after scattering.

(i) If the maximum length of the spring after scattering is 𝑑𝑑m , find an equation which

relates the ratio 𝑥𝑥 = (𝑑𝑑m − 𝑑𝑑0)/𝑑𝑑0 to the quantities 𝑄𝑄, 𝜃𝜃, 𝑑𝑑0, 𝑚𝑚, 𝑘𝑘, 𝑀𝑀, 𝑝𝑝1 and 𝑝𝑝2. [0.7 point]

𝜃𝜃

Fig. 2

𝑝𝑝1

𝑝𝑝2 𝑥𝑥

𝑚𝑚

𝑂𝑂

𝑑𝑑0

12𝑀𝑀

12𝑀𝑀


Question Number 1

(ii) Let 𝛼𝛼 ≡ sin2 𝜃𝜃. When the angle of orientation 𝜃𝜃 of the target is allowed to vary, the scattering cross section 𝜎𝜎 gives the effective target area, in a plane normal to the direction of incidence, which allows certain outcomes to occur as a result of scattering. It is known that for all outcomes which lead to the same value of 𝑝𝑝2, the value of 𝛼𝛼 must span an interval (𝛼𝛼min ,𝛼𝛼max ) and we may choose the unit of cross section so that 𝜎𝜎 is simply given by the numerical range (𝛼𝛼max − 𝛼𝛼min ) of the interval. Note that 𝛼𝛼min , 𝛼𝛼max , and, consequently, 𝜎𝜎 are dependent on 𝑝𝑝2 . Let 𝑝𝑝c be the threshold value of 𝑝𝑝2 at which 𝜎𝜎 starts to become independent of 𝑝𝑝2. In the limit of large 𝑘𝑘, give an estimate of 𝑝𝑝c . Express your answer in terms of 𝑚𝑚, 𝑀𝑀, and 𝑝𝑝1. [1.1 points] Assume 𝑀𝑀=3m and in the limit of large 𝑘𝑘, plot 𝜎𝜎 as a function of 𝑝𝑝2 for a given 𝑝𝑝1. In the plot, specify the range of σ and p2. [1.1 points]

Part B. Waves on a String Consider an elastic string stretched between two fixed ends A and B, as shown in Fig. 3.

The linear mass density of the string is 𝜇𝜇. The speed of propagation for transverse waves in the string is 𝑐𝑐. Let the length AB be 𝐿𝐿. The string is plucked sideways and held in a triangular form with a maximum height ℎ ≪ 𝐿𝐿 at its middle point. At time 𝑡𝑡 = 0, the plucked string is released from rest. All effects due to gravity may be neglected.

(c) Find the period of vibration 𝑇𝑇 for the string. [0.5 point] Plot the shape of the string at 𝑡𝑡 = 𝑇𝑇/8. In the plot, specify lengths and angles which serve to define the shape of the string. [1.7 points]

(d) Find the total mechanical energy of the vibrating string in terms of 𝜇𝜇, 𝑐𝑐, ℎ, and 𝐿𝐿. [0.8 point]

Fig. 3

A B

ℎ

𝐿𝐿


Question Number 1

Part C. An Expanding Universe Photons in the universe play an important role in delivering information across the cosmos.

However, the fact that the universe is expanding must be taken into account when one tries to extract information from these photons. To this end, we normally express length and distance using a universal scale factor 𝑎𝑎(𝑡𝑡) which depends on time 𝑡𝑡. Thus the distance 𝐿𝐿(𝑡𝑡) between two stars stationary in their respective local frames is proportional to 𝑎𝑎(𝑡𝑡):

𝐿𝐿(𝑡𝑡) = 𝑘𝑘𝑎𝑎(𝑡𝑡), (1) where 𝑘𝑘 is a constant and 𝑎𝑎(t) accounts for the expansion of the universe. We use a dot above a symbol of a variable to denote its time derivative, i.e. 𝑎(t) = 𝑑𝑑𝑎𝑎(𝑡𝑡)/𝑑𝑑𝑡𝑡, and let 𝑣𝑣(𝑡𝑡) ≡ 𝐿(𝑡𝑡). Taking time derivatives of both sides of Eq. (1), one obtains the Hubble law:

𝑣𝑣(𝑡𝑡) = 𝐻𝐻(𝑡𝑡)𝐿𝐿(𝑡𝑡), (2) where 𝐻𝐻(𝑡𝑡) = 𝑎(𝑡𝑡)/𝑎𝑎(𝑡𝑡) is the Hubble parameter at time 𝑡𝑡. At the current time 𝑡𝑡0, we have

𝐻𝐻(𝑡𝑡0) = 72 km s−1 Mpc−1, where 1 Mpc = 3.0857 × 1019 km = 3.2616 × 106 light-year.

Assume the universe to be infinitely large and expanding in such a way that 𝑎𝑎(𝑡𝑡) ∝ exp(𝑏𝑏𝑡𝑡),

where 𝑏𝑏 is a constant. In such a universe, the Hubble parameter is a constant equal to 𝐻𝐻(𝑡𝑡0). Moreover, it can be shown that the wavelength 𝜆𝜆 of photons travelling in the universe will be stretched in proportion to the expansion of the universe, i.e.

𝜆𝜆(𝑡𝑡) ∝ 𝑎𝑎(𝑡𝑡). Now suppose that photons making up a Lyman-alpha emission line were emitted at 𝑡𝑡e by

a star that was stationary in its local frame and that we as observers are stationary in our local frame. When these photons were emitted, their wavelength was 𝜆𝜆(𝑡𝑡e) = 121.5 nm. But when they reach us now at 𝑡𝑡0, their wavelength is red-shifted to 145.8 nm. (e) As these photons traveled, the universe kept expanding so that the star kept receding from

us. Given that the speed of light in vacuum 𝑐𝑐 has never changed, what was the distance 𝐿𝐿(𝑡𝑡e ) of the star from us when these photons were emitted at 𝑡𝑡e? Express the answer in units of Mpc. [2.2 points]

(f) What is the receding velocity 𝑣𝑣(𝑡𝑡0) of the star with respect to us now at 𝑡𝑡0? Express the answer in units of the speed of light in vacuum 𝑐𝑐. [0.8 point]


Question Number 1

Appendix The following formula may be used when needed:

𝑒𝑒𝛽𝛽𝑥𝑥𝑏𝑏

𝑎𝑎𝑑𝑑𝑥𝑥 =

1𝛽𝛽

(𝑒𝑒𝛽𝛽𝑏𝑏 − 𝑒𝑒𝛽𝛽𝑎𝑎 ).


Question Number 2


Strong Resistive Electromagnets Resistive electromagnets are magnets with coils made of a normal metal such as copper or

aluminum. Modern strong resistive electromagnets can provide steady magnetic fields higher than 30 tesla. Their coils are typically built by stacking hundreds of thin circular plates made of copper sheet metal with lots of cooling holes stamped in them; there are also insulators with the same pattern. When voltage is applied across the coil, current flows through the plates along a helical path to generate high magnetic fields in the center of the magnet.

In this question we aim to assess if a cylindrical coil (or solenoid) of many turns can serve as a magnet for generating high magnetic fields. As shown in Fig. 1, the center of the magnet is at O. Its cylindrical coil consists of 𝑁𝑁 turns of copper wire carrying a current 𝐼𝐼 uniformly distributed over the cross section of the wire. The coil’s mean diameter is 𝐷𝐷 and its length along the axial direction 𝑥𝑥 is ℓ. The wire’s cross section is rectangular with width 𝑎𝑎 and height 𝑏𝑏. The turns of the coil are so tightly wound that the plane of each turn may be taken as perpendicular to the 𝑥𝑥 axis and ℓ = 𝑁𝑁𝑎𝑎. In Table 1, data specifying physical dimensions of the coil are listed.

In assessing if such a magnet can serve to provide high magnetic fields, two limiting

factors must not be overlooked. One is the mechanical rigidity of the coil to withstand large Lorentz force on the field-producing current. The other is that the enormous amount of Joule heat generated in the wire must not cause excessive temperature rise. We shall examine these two factors using simplified models.

The Appendix at the end of the question lists some mathematical formulae and physical data which may be used if necessary.

Part A. Magnetic Fields on the Axis of the Coil Assume 𝑏𝑏 ≪ 𝐷𝐷

so that one may regard the wire as a thin strip of width 𝑎𝑎. Let O be the

origin of 𝑥𝑥 coordinates. The direction of the current flow is as shown in Fig. 1.

ℓ = 12.0 cm 𝐷𝐷 = 6.0 cm 𝑎𝑎 = 2.0 mm 𝑏𝑏 = 5.0 mm

Table 1

Figure 1 𝑎𝑎

𝑥𝑥 𝑂𝑂

ℓ/2

I 𝐼𝐼

𝐷𝐷

𝑎𝑎

ℓ/2

𝑏𝑏

𝑏𝑏


Question Number 2

(a) Find the 𝑥𝑥-component 𝐵𝐵(𝑥𝑥) of the magnetic field on the axis of the coil as a function of 𝑥𝑥 when the steady current passing through the coil is 𝐼𝐼. [1.0 point]

(b) Find the steady current 𝐼𝐼0 passing through the coil if 𝐵𝐵(0) i s10.0 T. Use data given in Table 1 when computing numerical values. [0.4 point]

Part B. The Upper Limit of Current In Part B, we assume the length ℓ of the coil is infinite and 𝑏𝑏 ≪ 𝐷𝐷. Consider the turn of

the coil located at 𝑥𝑥 = 0. The magnetic field exerts Lorentz force on the current passing through the turn. Thus, as Fig. 2 shows, a wire segment of length Δ𝑠𝑠 is subject to a normal force Δ𝐹𝐹n which tends to make the turn expand.

(c) Suppose that, when the current is 𝐼𝐼, the mean diameter of the expanded coil remains at a

constant value 𝐷𝐷′ larger than 𝐷𝐷, as shown in Fig. 2. Find the outward normal force per unit length 𝛥𝛥𝐹𝐹n /𝛥𝛥𝑠𝑠. [1.2 point] Find the tension 𝐹𝐹t acting along the wire. [0.6 point]

(d) Neglect the coil’s acceleration during the expansion. Assume the turn will break when the wire’s unit elongation (i.e. tensile strain or fractional change of the length) is 60 % and tensile stress (i.e. tension per unit cross sectional area of the unstrained wire) is 𝜎𝜎b =455 MPa. Let 𝐼𝐼b be the current at which the turn will break and 𝐵𝐵b the corresponding magnetic field at the center O. Find an expression for 𝐼𝐼b and then calculate its value. [0.8 point] Find an expression for 𝐵𝐵b and then calculate its value. [0.4 point]

Part C. The Rate of Temperature Rise When the current 𝐼𝐼 is 10.0 kA and the temperature 𝑇𝑇 of the coil is 293 K, assume that the

resistivity, the specific heat capacity at constant pressure, and the mass density of the wire of the coil are, respectively, given by 𝜌𝜌e = 1.72 × 10−8 Ω ∙ m , cp = 3.85 × 102 J/(kg ∙ K) and 𝜌𝜌𝑚𝑚 = 8.98 × 103 kg ∙ m−3. (e) Find an expression for the power density (i.e. power per unit volume) of heat generation in

the coil and then calculate its value. Use data in Table 1. [0.5 point]

Figure 2 𝑥𝑥

Δ𝑠𝑠

𝐷𝐷′/2

𝑂𝑂

𝐼𝐼 𝑏𝑏

𝐹𝐹t 𝐹𝐹t

Δ𝐹𝐹n


Question Number 2

(f) Let 𝑇 be the time rate of change of temperature in the coil. Find an expression for 𝑇 and then calculate its value. [0.5 point]

Part D. A Pulsed-Field Magnet If the large current needed for a strong magnet lasts only for a short time, the temperature

rise caused by excessive Joule heating may be greatly reduced. This idea is employed in a pulsed-field magnet.

Thus, as shown in Fig. 3, a capacitor bank of capacitance 𝐶𝐶 charged initially to a potential 𝑉𝑉0 is used to drive the current 𝐼𝐼 through the coil. The circuit is equipped with a switch 𝐾𝐾. The inductance 𝐿𝐿

and resistance 𝑅𝑅

of the circuit are assumed to be entirely due to

the coil. The construct and dimensions of the coil are the same as given in Fig. 1 and Table 1. Assume 𝑅𝑅, 𝐿𝐿, and 𝐶𝐶

to be independent of temperature and the magnetic field is the same as

that of an infinite solenoid with ℓ → ∞.

(g) Find expressions for the inductance 𝐿𝐿 and resistance 𝑅𝑅. [0.6 point]

Calculate the values of 𝐿𝐿 and 𝑅𝑅. Use data given in Table 1. [0.4 point] (h) At time 𝑡𝑡 = 0, the switch 𝐾𝐾 is thrown to position 1 and the current starts flowing.

For 𝑡𝑡 ≥ 0, the charge 𝑄𝑄(𝑡𝑡) on the positive plate of the capacitor and the current 𝐼𝐼(𝑡𝑡) entering the positive plate are given by

𝑄𝑄(𝑡𝑡) =𝐶𝐶𝑉𝑉0

sin𝜃𝜃0𝑒𝑒−𝛼𝛼𝑡𝑡 sin(𝜔𝜔𝑡𝑡 +𝜃𝜃0), (1)

𝐼𝐼(𝑡𝑡) =𝑑𝑑𝑄𝑄𝑑𝑑𝑡𝑡

= (−𝛼𝛼

cos𝜃𝜃0)𝐶𝐶𝑉𝑉0

sin𝜃𝜃0𝑒𝑒−𝛼𝛼𝑡𝑡 sin𝜔𝜔𝑡𝑡 , (2)

in which 𝛼𝛼 and 𝜔𝜔 are positive constants and 𝜃𝜃0 is given by

tan𝜃𝜃0 =𝜔𝜔𝛼𝛼

, 0 < 𝜃𝜃0 <𝜋𝜋2

. (3)

Note that, if 𝑄𝑄(𝑡𝑡) is expressed as a function of a new variable 𝑡𝑡′ ≡ (𝑡𝑡 + 𝜃𝜃0/𝜔𝜔), then 𝑄𝑄(𝑡𝑡′) and its time derivative 𝐼𝐼(𝑡𝑡) are identical in form except for an overall constant factor. The time derivative of 𝐼𝐼(𝑡𝑡) may therefore be obtained similarly without further differentiations.

Find 𝛼𝛼 and 𝜔𝜔 in terms of 𝑅𝑅, 𝐿𝐿, and 𝐶𝐶. [0.8 point] Calculate the values of 𝛼𝛼 and 𝜔𝜔 when 𝐶𝐶

is 10.0 mF. [0.4 point]

Figure 3 𝐶𝐶

𝐾𝐾 𝐿𝐿

1 2

𝑉𝑉0

𝑅𝑅


Question Number 2

(i) Let 𝐼𝐼m be the maximum value of |𝐼𝐼(𝑡𝑡)| for 𝑡𝑡 > 0. Find an expression for 𝐼𝐼m . [0.6 point] If 𝐶𝐶 = 10.0 mF, what is the maximum value 𝑉𝑉0b of the initial voltage 𝑉𝑉0 of the capacitor bank for which 𝐼𝐼m

will not exceed 𝐼𝐼b found in Problem (d)? [0.4 point]

(j) Suppose the switch 𝐾𝐾 is moved instantly from position 1 to 2 when the absolute value of the current | 𝐼𝐼(𝑡𝑡)| reaches 𝐼𝐼m . Let ∆𝐸𝐸 be the total amount of heat dissipated in the coil from 𝑡𝑡 = 0 to ∞ and ∆𝑇𝑇 the corresponding temperature increase of the coil. Assume the initial voltage 𝑉𝑉0 takes on the maximum value 𝑉𝑉0b obtained in Problem (i) and the electromagnetic energy loss is only in the form of heat dissipated in the coil. Find an expression for ∆𝐸𝐸 and then calculate its value. [1.0 point] Find an expression for ∆𝑇𝑇 and then calculate its value. Note that the value for ∆𝑇𝑇 must be compatible with the assumption of constant 𝑅𝑅 and 𝐿𝐿. [0.4 point]

Appendix

1. 𝑑𝑑𝑥𝑥

(𝐷𝐷2 + 𝑥𝑥2)3/2

𝐿𝐿

0=

1𝐷𝐷2

𝐿𝐿(𝐷𝐷2 + 𝐿𝐿2)1/2

2. sin(𝛼𝛼 ± 𝛽𝛽) = sin𝛼𝛼 cos𝛽𝛽 ± cos𝛼𝛼 sin𝛽𝛽

3. permeability of free space 𝜇𝜇0 = 4𝜋𝜋 × 10−7 T ∙ m/A

------------------------------------------------------ END ------------------------------------------------


Question Number 3

Theoretical Question 3 Electron and Gas Bubbles in Liquids

This question deals with physics of two bubble-in-liquid systems. It has two parts: Part A. An electron bubble in liquid helium Part B. Single gas bubble in liquid

Part A. An Electron Bubble in Liquid Helium When an electron is planted inside liquid helium, it can repel atoms of liquid helium and

form what is called an electron bubble. The bubble contains nothing but the electron itself. We shall be interested mainly in its size and stability.

We use ∆𝑓𝑓 to denote the uncertainty of a quantity 𝑓𝑓. The components of an electron’s position vector 𝑞 = (𝑥𝑥,𝑦𝑦, 𝑧𝑧) and momentum vector 𝑝 = (𝑝𝑝𝑥𝑥 ,𝑝𝑝𝑦𝑦 ,𝑝𝑝𝑧𝑧) must obey Heisenberg’s uncertainty relations ∆𝑞𝑞𝛼𝛼∆𝑝𝑝𝛼𝛼 ≥ ℏ/2 , where ℏ is the Planck constant divided by 2 𝜋𝜋 and 𝛼𝛼 = 𝑥𝑥,𝑦𝑦, 𝑧𝑧.

We shall assume the electron bubble to be isotropic and its interface with liquid helium is a sharp spherical surface. The liquid is kept at a constant temperature very close to 0 K with its surface tension 𝜎𝜎 given by 3.75 × 10−4 N ∙ m−1 and its electrostatic responses to the electron bubble may be neglected.

Consider an electron bubble in liquid helium with an equilibrium radius 𝑅𝑅. The electron, of mass 𝑚𝑚, moves freely inside the bubble with kinetic energy 𝐸𝐸k and exerts pressure 𝑃𝑃e on the inner side of the bubble-liquid interface. The pressure exerted by liquid helium on the outer side of the interface is 𝑃𝑃He .

(a) Find a relation between 𝑃𝑃He , 𝑃𝑃e , and 𝜎𝜎. [0.4 point] Find a relation between 𝐸𝐸k and 𝑃𝑃e . [1.0 point]

(b) Denote by 𝐸𝐸0 the smallest possible value of 𝐸𝐸k consistent with Heisenberg’s uncertainty relations when the electron is inside the bubble of radius 𝑅𝑅. Estimate 𝐸𝐸0 as a function of 𝑅𝑅. [0.8 point]

(c) Let 𝑅𝑅e be the equilibrium radius of the bubble when 𝐸𝐸k = 𝐸𝐸0 and 𝑃𝑃He = 0. Obtain an expression for 𝑅𝑅e and calculate its value. [0.6 point]

(d) Find a condition that 𝑅𝑅 and 𝑃𝑃He must satisfy if the equilibrium at radius 𝑅𝑅 is to be locally stable under constant 𝑃𝑃He . Note that 𝑃𝑃He can be negative. [0.6 point]

(e) There exists a threshold pressure 𝑃𝑃th such that equilibrium is not possible for the electron bubble when 𝑃𝑃He is less than 𝑃𝑃th . Find an expression for 𝑃𝑃th . [0.6 point]


Question Number 3

Part B. Single Gas Bubble in Liquid — Collapsing and Radiation In this part of the problem, we consider a normal liquid, such as water. When a gas bubble in a liquid is driven by an oscillating pressure, it can show dramatic

responses. For example, following a large expansion, it can collapse rapidly to a small radius and, near the end of the collapse, emit light almost instantly. In this phenomenon, called single-bubble sonoluminescence, the gas bubble undergoes cyclic motions which typically consist of three stages: expansion, collapse, and multiple after-bounces. In the following we shall focus mainly on the collapsing stage.

We assume that, at all times, the bubble considered is spherical and its center remains stationary in the liquid. See Fig 1. The pressure, temperature, and density are always uniform inside the bubble as its size diminishes. The liquid containing the bubble is assumed to be isotropic, nonviscous, incompressible, and very much larger in extent than the bubble. All effects due to gravity and surface tension are neglected so that pressures on both sides of the bubble-liquid interface are always equal.

Radial motion of the bubble-liquid interface As the bubble’s radius 𝑅𝑅 = 𝑅𝑅(𝑡𝑡) changes with time 𝑡𝑡, the bubble-liquid interface will move

with radial velocity 𝑅 ≡ 𝑑𝑑𝑅𝑅/𝑑𝑑𝑡𝑡. It follows from the equation of continuity of incompressible fluids that the liquid’s radial velocity 𝑟 ≡ 𝑑𝑑𝑟𝑟/𝑑𝑑𝑡𝑡 at distance 𝑟𝑟 from the center of the bubble is related to the rate of change of the bubble’s volume 𝑉𝑉 by

𝑑𝑑𝑉𝑉𝑑𝑑𝑡𝑡

= 4𝜋𝜋𝑅𝑅 2𝑅 = 4𝜋𝜋𝑟𝑟2𝑟. (1)

This implies that the total kinetic energy 𝐸𝐸k of the liquid with mass density 𝜌𝜌0 is

𝐸𝐸k =12 𝜌𝜌0

𝑟𝑟0

𝑅𝑅(4𝜋𝜋𝑟𝑟2𝑑𝑑𝑟𝑟)𝑟2 = 2𝜋𝜋𝜌𝜌0𝑅𝑅4𝑅2

1𝑟𝑟2

𝑟𝑟0

𝑅𝑅𝑑𝑑𝑟𝑟 = 2𝜋𝜋𝜌𝜌0𝑅𝑅4𝑅2

1𝑅𝑅−

1𝑟𝑟0 (2)

where 𝑟𝑟0 is the radius of the outer surface of the liquid.

𝑃𝑃0 0T

𝑟𝑟0

𝑅𝑅

𝑟𝑟

𝜌𝜌0

Fig. 1 𝑃𝑃

𝑇𝑇


Question Number 3

(f) Assume the ambient pressure 𝑃𝑃0 acting on the outer surface 𝑟𝑟 = 𝑟𝑟0 of the liquid is constant. Let 𝑃𝑃 = 𝑃𝑃(𝑅𝑅) be the gas pressure when the radius of the bubble is 𝑅𝑅. Find the amount of work 𝑑𝑑𝑑𝑑 done on the liquid when the radius of the bubble changes from 𝑅𝑅 to 𝑅𝑅 + 𝑑𝑑𝑅𝑅. Use 𝑃𝑃0 and 𝑃𝑃 to express 𝑑𝑑𝑑𝑑. [0.4 point] The work 𝑑𝑑𝑑𝑑 must be equal to the corresponding change in the total kinetic energy of the liquid. In the limit 𝑟𝑟0 → ∞, it follows that we have Bernoulli’s equation in the form

12 𝜌𝜌0 𝑑𝑑𝑅𝑅m 𝑅2 = (𝑃𝑃 − 𝑃𝑃0)𝑅𝑅n𝑑𝑑𝑅𝑅. (3)

Find the exponents m and n in Eq. (3). Use dimensional arguments if necessary. [0.4 point]

Collapsing of the gas bubble From here on, we consider only the collapsing stage of the bubble. The mass density of the

liquid is 𝜌𝜌0 = 1.0 × 103 kg ∙ m−3, the temperature 𝑇𝑇0 of the liquid is 300 K and the ambient pressure 𝑃𝑃0 is 1.01 × 105 Pa. We assume that 𝜌𝜌0, 𝑇𝑇0, and 𝑃𝑃0 remain constant at all times and the bubble collapses adiabatically without any exchange of mass across the bubble-liquid interface.

The bubble considered is filled with an ideal gas. The ratio of specific heat at constant pressure to that at constant volume for the gas is 𝛾𝛾 = 5/3. When under temperature 𝑇𝑇0 and pressure 𝑃𝑃0, the equilibrium radius of the bubble is 𝑅𝑅0 = 5.00 μm.

Now, this bubble begins its collapsing stage at time 𝑡𝑡 = 0 with 𝑅𝑅(0) = 𝑅𝑅i = 7𝑅𝑅0, 𝑅(0) = 0, and the gas temperature 𝑇𝑇i = 𝑇𝑇0. Note that, because of the bubble’s expansion in the preceding stage, 𝑅𝑅i is considerably larger than 𝑅𝑅0 and this is necessary if sonoluminescence is to occur.

(g) Express the pressure 𝑃𝑃 ≡ 𝑃𝑃(𝑅𝑅) and temperature 𝑇𝑇 ≡ 𝑇𝑇(𝑅𝑅) of the ideal gas in the bubble as a function of 𝑅𝑅 during the collapsing stage, assuming quasi-equilibrium conditions hold. [0.6 point]

(h) Let 𝛽𝛽 ≡ 𝑅𝑅/𝑅𝑅i and 𝛽 = 𝑑𝑑𝛽𝛽/𝑑𝑑𝑡𝑡 . Eq. (3) implies a conservation law which takes the following form

12 𝜌𝜌0 𝛽2 + 𝑈𝑈(𝛽𝛽) = 0. (4)

Let 𝑃𝑃i ≡ 𝑃𝑃(𝑅𝑅i) be the gas pressure of the bubble when 𝑅𝑅 = 𝑅𝑅i . If we introduce the ratio 𝑄𝑄 ≡ 𝑃𝑃i/[(𝛾𝛾 − 1)𝑃𝑃0] , the function 𝑈𝑈(𝛽𝛽) may be expressed as

𝑈𝑈(𝛽𝛽) = 𝜇𝜇𝛽𝛽−5[𝑄𝑄(1 − 𝛽𝛽2) − 𝛽𝛽2(1− 𝛽𝛽3)]. (5) Find the coefficient 𝜇𝜇 in terms of 𝑅𝑅i and 𝑃𝑃0. [0.6 point]

(i) Let 𝑅𝑅m be the minimum radius of the bubble during the collapsing stage and define 𝛽𝛽m ≡ 𝑅𝑅m /𝑅𝑅i. For 𝑄𝑄 ≪ 1, we have 𝛽𝛽m ≈ 𝐶𝐶m𝑄𝑄 . Find the constant 𝐶𝐶m . [0.4 point]


Question Number 3

Evaluate 𝑅𝑅m for 𝑅𝑅i = 7𝑅𝑅0. [0.3 point] Evaluate the temperature 𝑇𝑇m of the gas at 𝛽𝛽 = 𝛽𝛽m . [0.3 point]

(j) Assume 𝑅𝑅i = 7𝑅𝑅0. Let 𝛽𝛽𝑢𝑢 be the value of 𝛽𝛽 at which the dimensionless radial speed 𝑢𝑢 ≡|𝛽| reaches its maximum value. The gas temperature rises rapidly for values of 𝛽𝛽 near 𝛽𝛽𝑢𝑢 . Give an expression and then estimate the value of 𝛽𝛽𝑢𝑢 . [0.6 point] Let 𝑢𝑢 be the value of 𝑢𝑢 at 𝛽𝛽 = 𝛽 ≡ (𝛽𝛽m + 𝛽𝛽𝑢𝑢)/2. Evaluate 𝑢𝑢. [0.4 point] Give an expression and then estimate the duration ∆𝑡𝑡m of time needed for 𝛽𝛽 to diminish from 𝛽𝛽𝑢𝑢 to the minimum value 𝛽𝛽m . [0.6 point]

Sonoluminescence of the collapsing bubble Consider the bubble to be a surface black-body radiator of constant emissivity 𝑎𝑎 so that

the effective Stefan-Boltzmann’s constant 𝜎𝜎eff = 𝑎𝑎𝜎𝜎SB . If the collapsing stage is to be approximated as adiabatic, the emissivity must be small enough so that the power radiated by the bubble at 𝛽𝛽 = 𝛽 is no more than a fraction, say 20 %, of the power 𝐸 supplied to it by the driving liquid pressure.

(k) Find the power 𝐸 supplied to the bubble as a function of 𝛽𝛽. [0.6 point] Give an expression and then estimate the value for an upper bound of 𝑎𝑎. [0.8 point]

Appendix

1. 𝑑𝑑𝑑𝑑𝑥𝑥

𝑥𝑥𝑛𝑛 = 𝑛𝑛𝑥𝑥𝑛𝑛−1

2. Electron mass 𝑚𝑚 = 9.11 × 10−31 kg

3. Planck constant ℎ = 2𝜋𝜋 ℏ = 2𝜋𝜋 × 1.055 × 10−34 J ∙ s

4. Stefan-Boltzmann’s constant 𝜎𝜎SB = 5.67 × 10−8 W ∙ m−2 ∙ K−4

END ----------------------------------------------------------------------------------------------------------------

Page 1 of 3

Theoretical Question 1: The Shockley-James Paradox

In the year 1905, Albert Einstein proposed the special theory of relativity to resolve the inconsistency between

Newton’s mechanics and Maxwell’s electromagnetism. Proper understanding of the theory led to the resolution of

many apparent paradoxes. At the time, the discussion focused mostly on the propagation of electromagnetic waves.

In this question, we solve a paradox of a different type. For a fairly simple system of charges proposed by W.

Shockley and R. P. James in 1967, understanding the conservation of linear momentum requires careful relativistic

analysis. If a point charge is located near a magnet of changing magnetization, there's an induced electric force on the

charge, but no apparent reaction on the magnet. The process may be slow enough that any electromagnetic radiation

(and any momentum carried away by it) is negligible. Thus, apparently we get a cannon without recoil.

In our analysis of this system, we will demonstrate that in relativistic mechanics, a composite body may hold a non-

zero mechanical momentum while remaining stationary.

Part I: Understanding the impulse on the point charge (3.3 points)

Consider a circular current loop of radius carrying a current , and a second, larger current loop of radius ,

concentric with the first and lying in the same plane.

a. (1 pt.) A current passing through loop 2 (the larger loop) generates a magnetic flux through loop 1. Find

the ratio . It is called the mutual inductance coefficient.

b. (0.8 pts.) Given that , obtain the total induced EMF in the larger loop as a result of a

variation of the current in the smaller loop. Neglect the current in the larger loop. Hint: the induced

EMF is equal to the rate of change of the magnetic flux through the loop.

c. (0.5 pts.) The EMF you found in part (b) is due to the tangential component of an induced electric field. Obtain an

expression for the tangential electric field at radius as a function of the rate of change of the current.

Figure 1: A circular current loop and a point charge .

We now remove the larger current loop, and instead put a massive point charge at radius , as shown in Figure 1. It

may be assumed that the charge moves very little during the relevant time periods.

d. (1 pt.) Find the total tangential impulse received by the point charge as the current in the small loop changes

from an initial value to the final value .

Part II: Understanding the recoil of the current loop (4.4 points)

We will now understand the origin of the recoil of the loop, using a loop of different geometry.

Page 2 of 3

e. (1.1 pts.) Consider a hollow tube with walls made of a neutral insulating material of length and cross section

carrying an electric current . The current is due to charged particles of rest mass and charge distributed

homogenously inside the tube with number density . Assume that the charged particles are all moving along the

tube with the same velocity. Find the total momentum of the charged particles in the tube, taking Special

Relativity effects into account.

f. (3.3 pts.) Consider a square current loop with side . At a distance from the loop, there is a point charge ;

see Figure 2. The loop carries current . We will model the current loop as a neutral tube, as in part (e). The charge

carriers can move freely along the loop, colliding elastically with the walls and making elastic right turns at the

corners. Neglect all interactions among the charge carriers. Assume also that all the charge carriers at a given

section along the tube always move with the same velocity. Assume that the loop is heavy and that its motion can

be neglected. Calculate the total linear momentum of the charge carriers in the loop. It is called "hidden

momentum".

Figure 2: A square current loop and a point charge .

When the current stops, this linear momentum is transferred to the loop, and it gets an impulse equal to minus the

impulse received by the point charge . This is the missing recoil that we were looking for (note that in the initial state

there is also momentum in the electromagnetic field; this is important for conservation of the total momentum of the

entire system).

Part III: Summarizing the results (2.3 points)

g. (0.8 pts.) Current loops are often characterized by their magnetic moment , where is the current and is

the loop’s area. Express the answer to part (d) in terms of , , and . Likewise, express the answer to part (f) in

terms of , , and . Note that the electric and magnetic constants are related by:

where is the speed of light.

h. (1.5 pts.) In a more realistic model, the current loop is a conducting wire, and the field of the point charge does

not penetrate into the conductor. We assume that the current is still conducted by charge carriers inside the wire.

Decide whether each of the following statements is true or false, and circle the correct option in the Answer Form.

Note: You may leave a statement undecided, but if you decide incorrectly, you will not get credit at all for part

(h).

A. (0.5 pts.) The linear momentum of the current loop is zero.

B. (0.5 pts.) As the total current in the loop changes from to zero, the charge carriers decelerate, causing induced

currents in the wire’s conducting material. Because of these induced currents, the point charge will not get a net

impulse.

Page 3 of 3

C. (0.5 pts.) The surface charges on the wire, induced by the presence of the external charge, will experience an

electric force as the current changes from to zero. This way, the loop will get the same impulse as found in part

(f).

Page 1 of 3

Theoretical Question 2: Creaking Door

The phenomenon of creaking is very common, and can be found in doors, closets, chalk squeaking on a blackboard,

playing a violin, new shoes, car brakes and other systems from everyday life. Here in Israel, a similar phenomenon

causes violent earthquakes with a period of several decades. These originate in the Dead Sea rift, located right above

the deepest known break in the earth's crust.

The physical mechanism for creaking is based on elasticity combined with the difference between the static and the

kinetic friction coefficients. In this question, we will study this mechanism and its application to the case of an

opening door.

Part I: General model (7.5 points)

Consider the following system (see Figure 1):

A box with mass is attached to a long ideal spring with spring constant , whose other end is driven at a constant

velocity . The static and the kinetic friction coefficients between the box and the floor are given respectively by

and , where .

We would like to understand why this setup supports two different forms of motion:

1. The friction is always kinetic. This is known as a pure slip mode.

2. Kinetic and static friction alternate. This is known as a stick-slip mode. Stick-slip motion is the source of

the creaking sound commonly encountered.

a. (1 pt.) Consider the case where at the initial time , the box slides on the floor with velocity , and the

spring’s tension exactly balances the kinetic friction. Assume . The spring’s elongation will

oscillate as a function of .

a1. (0.6 points) Find the period and the amplitude of these oscillations.

a2. (0.4 points) Sketch a qualitative graph of the spring’s elongation for .

b. (1.2 pts.) Now, consider the case where at the box is at rest, while the initial spring elongation is the same

as in part (a). Sketch a qualitative graph of the velocity of the box with respect to the floor for ,

where is the (new) period of the oscillations . Motion to the right corresponds to a positive sign of .

Indicate approximately on your graph the horizontal line .

c. (0.5 pts.) For the initial conditions of part (b), find the time-averaged value of the spring’s elongation after a

sufficiently long time has passed.

Figure 1: A general model for creaking

Page 2 of 3

d. (2.4 pts.) For the conditions of part (b), find the period of the oscillations .

Generically, stick-slip motion stops at high driving velocities . We will now discuss one of the possible mechanisms

behind this effect.

e. (2.4 pts.) Suppose that during each period , a small amount of energy is dissipated into heat in the spring, via an

additional mechanism. Let be the fractional amplitude loss per period due to dissipation in pure-slip

motion. For , find the critical driving velocity above which periodic stick-slip becomes impossible.

The results of part (e) are not required for part II.

Part II: Application to creaking door (2.5 points)

A door hinge is a hollow, open-ended metal cylinder with radius , height and thickness . The lower end of the

cylinder lies on a metal base attached to the wall (the area of contact is a ring of radius ); see Figure 2. The static and

the kinetic friction coefficients between the cylinder and its base are and respectively, with . The upper

end of the cylinder is attached to the door, which can be regarded as perfectly rigid. A typical door hangs on two or

three such hinges, but its weight is concentrated on only one of them – this is the hinge that will creak. The cylinder of

that hinge presses down on its metal base with the weight of the entire door, whose mass is .

The cylinder is not a perfectly rigid body – it can twist tangentially without changing its overall form, so that vertical

line segments become tilted with a small angle ; see Figure 3. The elastic force on a small area element of the

base due to this deformation is given by:

,

where is a material property known as the shear modulus. Use the values , , ,

, , , , . You may use the approximation .

f. (1 pt.) We start rotating the door very slowly from equilibrium (zero torque). For small rotation angles, obtain an

expression for the torsion coefficient , where is the torque which must be applied to rotate the door by

an angle .

g. (1.5 pts.) At very low angular velocity, when a transition from stick to slip occurs, a sound pulse is emitted. Find

the angular velocity of the door for which the frequency of these pulses enters the audible range at .

Figure 2: Schematic drawing of a door

hinge

Figure 3: The twisted hinge cylinder

Wall Door

Page 3 of 3

Assume that the frequency of pure-slip oscillations in the hinge is much higher: . Provide an expression

and a numerical result.

Page 1 of 3

Figure 1: A partially

inflated birthday balloon.

Theoretical Question 3: Birthday Balloon

The picture shows a long rubber balloon, the kind that is popular at birthday parties.

A partially inflated balloon usually splits into two domains of different radii. In this

question, we consider a simplified model to help us understand this phenomenon.

Consider a balloon with the shape of a long homogeneous cylinder (except for the

ends), with a mouthpiece through which the balloon can be inflated. All processes

will be considered isothermal at room temperature. At all times, the pressure inside

the balloon exceeds the atmospheric pressure by a small fraction, so the air may

be treated as an incompressible fluid. Gravity and the balloon's weight may also be neglected. The inflation is slow

and quasistatic. In parts (a)-(d), the balloon is inflated uniformly throughout its length. We denote by and the

radius and length of the balloon before it was inflated.

a. (1.8 pts.) The balloon is held by the mouthpiece, while its other parts hang freely. Find the ratio between the

longitudinal surface tension (in the direction parallel to the balloon’s axis) and the transverse surface tension

(in the direction tangent to the balloon’s circular cross-section).

The surface tension of a rubber film is the force that adjacent parts exert on each other, per unit length of the

boundary.

Hooke's Law is a linear approximation of real-world elasticity for small tensions. Assume that the balloon’s length

remains constant at , while the surface tension depends linearly on the inflation ratio :

(

)

b. (1 pt.) With these assumptions, obtain an expression for the dependence of the pressure inside the balloon on the

balloon's volume . Sketch a plot of as a function of . What is the maximal inflation pressure

resulting from Hooke's elasticity approximation?

In reality, because the inflation ratio is large (in Figure 1, typical values of about 5 can be observed), one must

consider non-linear behavior of the rubber and changes in the balloon’s length. These effects allow higher inflation

pressures than predicted by part (b). In a typical balloon, the graph of is composed of three pieces:

1. For small inflation ratios, grows in a Hooke-like manner.

2. At the balloon’s length begins to increase, and reaches a long plateau where it grows very

slowly.

3. At some large inflation ratio, the rubber starts strongly resisting any further stretch, which leads to a sharp rise

in .

This behavior is depicted in Figure 2.

c. (1.3 pts.) Sketch a qualitative plot of the pressure difference as a function of for a uniformly inflated

balloon that behaves according to Figure 2. Indicate any local extremum points on your plot. Indicate also the

Page 2 of 3

points corresponding to and . Find the values of at these two points with 10%

accuracy.

To explore the consequences of the behavior you found in part (c), we approximate for a uniformly inflated

balloon with a cubic function:

where and are positive constants. Assume that the volume is larger than the balloon’s uninflated volume ,

and is large enough so that the function (2) is positive in the entire physical range . See Figure (3).

The balloon is attached to a large air reservoir maintained at a controllable pressure . It may happen that some values

of are consistent with more than one value of the volume . If the balloon suffers occasional perturbations (such as

local stretching by external forces) while held at such inflation pressure, it may jump to a state of different volume.

This will happen when it becomes energetically favorable for the entire system, consisting of the balloon, the

atmosphere and the machinery maintaining the pressure . If the pressure is slowly increased from , and sufficient

perturbations exist at every step, this explosive volume jump will happen at a certain pressure where the energy

required to move between the two states is zero. Above this pressure, going from the smaller volume to the larger

volume branch releases energy, and vice versa. This type of discontinuity is often found in nature, and is sometimes

referred to as a “phase transition”.

d. (2.3 pts.) By considering equation (2), obtain the value of , the volume of the balloon before the jump, and

the volume after the jump. Express your answers using and .

A more realistic inflating agent, such as a birthday boy, is unable to supply enough air for the instantaneous volume

change described above. Instead, air is pumped gradually into the balloon, effectively controlling the balloon’s volume

rather than the pressure. In this case, a new type of behavior becomes possible. If it helps to minimize the total energy

of the system, the balloon will split (given sufficient perturbations) into two cylindrical domains of different radii,

Figure 2: for a realistic party balloon.

Figure 3: A plot of equation (2).

Page 3 of 3

whose lengths will gradually change. The splitting boundary itself requires energy, which you may neglect. We shall

also neglect the length of the boundary layer (these assumptions are valid for a very long balloon.)

e. (1 pt.) Sketch a qualitative graph of the pressure difference as a function of , taking the split into account.

Indicate on your axes the pressure and the volumes and .

f. (1.4 pts.) The balloon is in the volume range that supports two coexisting domains. Find the length of the

thinner domain as a function of the total air volume . Express your answer in terms of , and the radius of

the thinner domain.

g. (1.2 pts.) The balloon is in the volume range that supports two coexisting domains. Find the latent work

that must be performed on the balloon to convert a unit length of the thin domain into the thick domain.

Express your answer in terms of , , and the radius of the thinner domain.

Theory Question-I Page 1 of 3

13th

Asian Physics Olympiad

May 01-07, 2012, New Delhi

Figure 2

The Drag on a Falling Magnet

A clear and detailed discussion on eddy currents was first provided by the British

physicist Sir James H. Jeans (1877-1946) in his celebrated book The mathematical

theory of electricity and magnetism (1925). The present problem is based on electricity

and magnetism.

A small size magnet with dipole moment of

magnitude 𝑝 and mass 𝑚 is dropped through a very

long vertically held non-magnetic metallic tube as

shown in Fig. (1) (figure is not to scale). In general

the fall is governed by

Here 𝑔 is the acceleration due to gravity. Note

that the damping parameter 𝑘 is due to the

generation of eddy currents in the tube.

I.1 Obtain the terminal velocity (𝑣𝑇) of the magnet. [0.5 point]

I.2 Obtain 𝑧(𝑡) , i.e. position of the magnet at time 𝑡 . Take 𝑣(𝑡 = 0) = 0 and

𝑧(𝑡 = 0) = 0. [1.0 point]

We shall attempt to understand the dynamics of the fall. In order

to do this we consider in part (I.3) – part (I.8) a simplified problem

of the magnet falling axially towards a fixed non-magnetic

metallic ring of radius 𝑎, resistance 𝑅 and inductance 𝐿 as

shown in Fig. (2). In this problem, we shall ignore radiation

effects.

In our case it is convenient to change the reference coordinates to

a set of cylindrical ones (𝜌, 𝜑, 𝑧) as shown in Fig. (2) where 𝑧-axis

is the ring axis, the magnet is initially at rest at the origin and the

center of the ring is at distance 𝑧0 from the origin. Cartesian axes

(𝑥, 𝑦, 𝑧) are also shown in the figure. The magnet has dipole

moment 𝑝 in the positive 𝑧 direction (𝑝 = 𝑝𝑘 ) where 𝑘 is unit

vector in 𝑧 direction. We will assume that during the fall,

magnetic moment remains in the same direction. The axial

component (𝐵𝑧) and radial component (𝐵𝜌 ) of the magnetic

field at an arbitrary point (𝜌, 𝜑, 𝑧) when the magnet is at the

origin are given by

𝑚𝑧 = 𝑚𝑔 − 𝑘𝑧 …………………(1)

James H. Jeans

(1877-1946)

Figure 1


13th



𝐵𝑧 =𝜇0

4𝜋

𝑝

𝜌2 + 𝑧2 32

3𝑧2

𝜌2 + 𝑧2− 1

𝐵𝜌 =𝜇0

4𝜋

3𝑝𝑧𝜌

(𝜌2 + 𝑧2)5/2

where 𝜇0 is the permeability of free space.

I.3 Let the instantaneous speed of the magnet be 𝑣. Obtain the magnitude of the

induced emf (𝑒𝑖) in the ring. [1.5 points]

I.4 This emf will give rise to an induced current (𝑖) in the ring. Obtain the magnitude

of the instantaneous electromagnetic force (𝑓𝑒𝑚 ) on the ring in terms of 𝑖.

[1.0 point]

I.5 What is the magnitude of the force on the magnet due to this ring?

[0.5 point]

I.6 Express the emf in the ring in terms of L, R and i. Do not solve for 𝑖. [0.5 point]

I.7 As the magnet falls it loses gravitational potential energy. Identify the three main

forms of energy into which the gravitational potential energy is converted and

write down the expressions you would use to calculate each of the three

contributions. [1.0 point]

I.8 Does the magnetic field of the magnet do any work in this process? Tick in the

appropriate box. [0.5 point]

Next we will estimate the damping parameter 𝑘 due to the

pipe (see Eq. (1)). Take an infinitely long pipe with radius 𝑎,

small thickness 𝑤, and electrical conductivity 𝜍. For this and

later part, we take inductance of the pipe to be negligible. It

would help if you considered the pipe to be made of many

rings each of height ∆𝑧′, radius 𝑎 , small thickness 𝑤 and

electrical conductivity 𝜍 (see Fig. (3)). For simplicity, the two

ends of the pipe are at 𝑧 = −∞ and at 𝑧 = ∞, respectively.

I.9 Obtain the resistance of an individual ring.

[0.5 point]

Figure 3


13th



I.10 Obtain the damping parameter 𝑘 due to the entire pipe in terms of 𝑝, 𝜍 and

geometrical parameters of the ring. Since each ring is very thin, you may take

magnetic field to be constant over the thickness of the ring and equal to 𝐵𝜌(𝜌 =

𝑎) . Assume that at an instant 𝑡 , the magnet has a coordinate 𝑧(𝑡) with an

instantaneous speed 𝑧 . You should leave your answer in terms of a dimensionless

integral 𝐼, involving a dimensionless variable 𝑢 = (𝑧 − 𝑧′)/𝑎. [2.0 points]

I.11 Assume that the damping constant 𝑘 depends on the following

𝑘 = 𝑓(𝜇0, 𝑝, 𝑅0, 𝑎)

where 𝑅0 is the effective resistance of a long pipe. Use dimensional analysis to

obtain an expression for 𝑘. Take the dimensionless constant to be unity.[1.0 point]

The following integral may be useful:

𝑢𝑑𝑢

𝑢2 + 𝑎2 𝑛=

1

2

𝑎2 + 𝑢2 1−𝑛

1 − 𝑛+ Constant (n > 1)

Theory Question Page 1 of 2

13th


May 01-07, 2012, New Delhi II

Chandrasekhar Limit

In a famous work carried out in 1930, the Indian Physicist Prof Subrahmanyan

Chandrasekhar (1910-1995) studied the stability of stars. The problem will help you

to construct a simplified version of his analysis.

You may find the following symbols and values useful.

Speed of light in vacuum 𝑐 = 3.00 × 108m. s−1 Planck’s constant ℎ = 6.63 × 10−34J. s

Universal constant of Gravitation 𝐺 = 6.67 × 10−11N. m2 . kg−2

Rest mass of electron 𝑚𝑒 = 9.11 × 10−31kg

Rest mass of proton 𝑚𝑝 = 1.67 × 10−27kg

II.1. Consider a spherical star of uniform density, radius 𝑅 and mass 𝑀. Derive an

expression for its gravitational potential energy (𝐸𝐺) due to its own gravitational

field (gravitational self energy). [1.0 point]

II.2. We assume that the star is made up of only hydrogen and that all the hydrogen is in

ionized form. We consider the situation when the star’s energy production due to

nuclear fusion has stopped. Electrons obey the Pauli exclusion principle and their

total energy can be computed using quantum statistics. You may take this total

electronic energy (ignoring the protonic energy) to be

𝐸𝑒 =ℏ2𝜋3

10𝑚𝑒42/3

3

𝜋

7/3 𝑁𝑒5/3

𝑅2

where 𝑁𝑒 is the total number of electrons and ℏ = ℎ/2𝜋. Obtain the equilibrium

condition of the star relating its radius (𝑅𝑤𝑑 ) to its mass. This radius is called the

‘White Dwarf’ radius. [2.0 points]

II.3. Numerically evaluate 𝑅𝑤𝑑 given that mass of the star is the same as the solar

mass (𝑀S = 2.00 × 1030kg ). [1.5 points]

II.4. Assuming that the electron distribution is homogeneous, obtain an order of

magnitude estimation of the average separation (𝑟𝑠𝑒𝑝 ) between electrons if the

radius of the star is 𝑅𝑤𝑑 as obtained in part (II.3). [1.0 point]

II.5. Let us estimate the speed of electrons. For this purpose, assume each electron to

form a standing wave in a one-dimensional box of length 𝑟𝑠𝑒𝑝 . Estimate the speed of

electron (𝑣) in the lowest energy state using de-Broglie hypothesis [1.0 point]

II.6. Consider now a modification of the analysis in part (II.2). If we take electrons in the

ultrarelativistic limit (𝐸 = 𝑝𝑐), a similar analysis yields

S. Chandrasekhar

(1910-1995)


13th


May 01-07, 2012, New Delhi II

𝐸𝑒𝑟𝑒𝑙 =

𝜋2

44/3

3

𝜋

5/3 ℏ𝑐

𝑅𝑁𝑒

4/3

Obtain the expression for the mass for which, the star can be in equilibrium in terms

of the constants provided at the beginning of the question. We call this the critical

mass (𝑀𝑐). [1.5 points]

II.7. If the mass 𝑀 of the star is greater than the critical mass 𝑀𝑐 obtained in part (II.6),

state whether the star will expand or contract. Tick in appropriate box. [0.5 point]

II.8. Calculate a numerical estimate of this critical mass in units of solar mass (𝑀𝑆).

(Note: Your answer may differ from Chandrasekhar’s famous result because of the approximations made in this analysis)

[1.5 points]


13th


May 01-07, 2012, New Delhi III

Pancharatnam Phase

This problem deals with the two beam phenomena associated with light, its

interference, polarization and superposition. The particular context of the problem

was studied by the Indian physicist S. Pancharatnam (1934–1969).

Consider the experimental set up as shown in Fig. (1). Two coherent monochromatic light

beams (marked as beam 1 and 2), travelling in the 𝑧 direction, are incident on two narrow

slits and separated by a distance 𝑑 (𝑆1𝑆2 = 𝑑). After passing through the slits the two

beams interfere and the pattern is observed on the screen 𝑆. The distance between the slits

and the screen is 𝐷 and 𝐷 ≫ 𝑑. Assume that the width of each slit 𝑆1 and 𝑆2 is much

smaller than the wavelength of light.

III.1. Let the beams 1 and 2 be linearly polarized at 𝑧 = 0. The corresponding

electric field vectors are given by

where 𝑖 is the unit vector along the 𝑥-axis, 𝜔 is angular frequency of light and 𝐸0 is

the amplitude. Find the expression for the intensity of the light 𝐼 𝜃 , that will be

observed on the screen where 𝜃 is the angle shown in Fig. (1). Express your answer

in terms of 𝜃, 𝑑, 𝐸0, 𝑐 and 𝜔 where 𝑐 is the speed of light. Also, note that the

intensity is proportional to the time average of the square of the electric field. Here

you make take the proportionality constant to be 𝛽. You may ignore the attenuation

in the magnitude of the electric fields with distance from the slits to any point on

the screen.

[1.0 point]

III.2. A perfectly transparent glass slab of thickness 𝑤 and refractive index 𝜇 is

𝐸 1 = 𝑖 𝐸0 cos(𝜔𝑡) …………………(1a)

𝐸 2 = 𝑖 𝐸0 cos(𝜔𝑡) …………………(1b)

Figure 1

S. Pancharatnam

(1934–1969)


13th



introduced in the path of beam 1 before the slits. Find the expression for the

intensity of the light 𝐼 𝜃 that will be observed on the screen. Express your answer

in terms of 𝜃, 𝑑, 𝐸0, 𝑐, 𝜔, 𝜇 and 𝑤.

[1.0 point] III.3. An optical device (known as quarter wave plate (QWP)) is introduced in the

path of beam 1, before the slits, replacing the glass slab. This device changes the

polarization of the beam from the linear polarization state

to a circular polarization state which is given by

where 𝑗 is the unit vector along the 𝑦-axis.

Assume that the device does not introduce any additional path difference and that

it is perfectly transparent. Note that the tip of the electric field vector traces a

circle as time elapses and hence, the beam is said to be circularly polarized. We

assume that the angle 𝜃 is small enough so that intensity from slit one does not

depend on the angle 𝜃 even for 𝑗 polarization.

III.3.a. Find the expression for the intensity 𝐼 𝜃 of the light that will be observed

on the screen. Express your answer in terms of 𝜃, 𝑑, 𝐸0, 𝑐 and 𝜔.

III.3.b. What is the maximum intensity (𝐼𝑚𝑎𝑥 )?

III.3.c. What is the minimum intensity (𝐼𝑚𝑖𝑛 )?

[2.0 points]

III.4.

Now, consider the experimental setup (see Fig. (2)) in which the beam 1 is subjected to

the device (QWP) described in part 3 and,

a linear polarizer (marked as I), between 𝑧 = 𝑎 and 𝑧 = 𝑏 which allows only

the component of the electric field parallel to an axis (𝑖 ′) to pass through. The

𝐸 1 = 𝑖 𝐸0 cos(𝜔𝑡)

𝐸 1 = 1

2 𝑖 𝐸0 cos(𝜔𝑡) + 𝑗 𝐸0 sin(𝜔𝑡)

…………………(2)

Figure 2


13th



unit vector 𝑖 ′ is defined as

𝑖 ′ = 𝑖 cos γ + 𝑗 sin 𝛾

and,

another linear polarizer (marked as II) between 𝑧 = 𝑏 and 𝑧 = 𝑐 which

polarizes the beam back to 𝑖 direction.

Thus the beam 1 is back to its original state of polarization. Assume that the

polarizers do not introduce any path difference and are perfectly transparent.

III.4.a. Write down the expression for the electric field of beam 1 after the first

polarizer at 𝑧 = 𝑏 [𝐸 1(𝑧 = 𝑏)]. III.4.b. Write down the expression for the electric field of beam 1 after the

second polarizer at 𝑧 = 𝑐 [𝐸 1(𝑧 = 𝑐)].

III.4.c. What is the phase difference (𝛼) between the two beams at the slits?

[2.0 points]

The most general type of polarization is elliptical polarization. A convenient way of

expressing elliptical polarization is to consider it as a superposition of two orthogonal

linearly polarized components i.e.

where 𝑖 ′ and 𝑗 ′ and this state of polarization are depicted in Fig. 3.

The tip of the electric field vector traces an ellipse

as time elapses. Here 𝑒 represents the ellipticity

and is given by

tan 𝑒 =Semi-minor axis of the ellipse

Semi-major axis of the ellipse

Linear polarization (Eqs. (1)) and circular

polarization (Eq. (2)) are special cases of elliptical

polarization (Eq. (3)). The two parameters

𝛾(∈ [0, 𝜋])and 𝑒(∈ [−𝜋/4, 𝜋/4]) completely

describe the state of polarization.

𝐸 = 𝑖 ′𝐸0 cos 𝑒 cos(𝜔𝑡) + 𝑗 ′𝐸0 sin 𝑒 sin(𝜔𝑡) …………………(3)

Figure 3


13th



The polarization state can also be represented by

a point on a sphere of unit radius called the

Poincare sphere. The polarization of the beam

described in Eq. (3) is represented by a point 𝑃

on the Poincare sphere (see Fig. 4), then latitude

∠𝑃𝐶𝐷 = 2𝑒 and longitude ∠𝐴𝐶𝐷 = 2𝛾. Here 𝐶

is the center.

III.5. Consider a point on the equator of the Poincare sphere.

III.5.a. Write down the electric field (𝐸 Eq ) corresponding to this point.

III.5.b. What is its state of polarization?

[0.5 point]

III.6. Consider a point at the north pole of the Poincare sphere.

III.6.a Write down the electric field (𝐸 NP ) corresponding to this point.

III.6.b What is its state of polarization?

[0.5 point]

III.7. Now, consider the three polarization states of beam 1 as given in part 4. Let

the initial polarization (at 𝑧 = 0) be represented by a point 𝐴1 on the Poincare

sphere; after the optical device, let the state (at 𝑧 = 𝑎) be represented by point 𝐴2

and after the first polarizer (say, at 𝑧 = 𝑏), the state be represented by point 𝐴3. At

𝑧 = 𝑐, the polarization returns to its original state which is represented by 𝐴1.

Locate these points (𝐴1, 𝐴2, and 𝐴3) on the Poincare sphere.

[1.5 points]

III.8. If these three points (𝐴1, 𝐴2 , and 𝐴3 from the part (III.7)) are joined by great

circles on the sphere, a triangle on the surface of the sphere is obtained (Note: A

great circle is a circle on the sphere whose center coincides with the center of the

sphere). The phase difference 𝛼 obtained in part 4 and the area 𝑆 of the curved

surface enclosed by the triangle are related to each other. Relate 𝑆 to 𝛼.

This relationship is general and was obtained by Pancharatnam and the phase difference is called

the Pancharatnam phase. [1.5 points]

Figure 4

Documents

apho (asian physics olympiad)