Joint Entrance Examination for Postgraduate Courses in Physics

EUF

Second Semester/2011 Part 1 10 May 2011

Instructions:

DO NOT WRITE YOUR NAME ON THE TEST. It should be identified only by your candidate number (EUFxxx).

This test is the first part of the joint entrance exam for Postgraduate Physics. It contains questions on: Classical Mechanics, Modern Physics, Thermodynamics and Statistical Mechanics. All questions have the same weight.

The duration of this test is 4 hours. Candidates must remain in the exam room for a minimum of 90 minutes.

The use of calculators or other electronic instruments is NOT permitted in the exam. ANSWER EACH QUESTION ON THE CORRESPONDING PAGE OF THE

ANSWER BOOKLET. The sheets with answers will be reorganized for marking. If you need more answer space, use the extra sheets in the answer booklet. Remember to write the number of the question (Q1, Q2, or . . . ) and your candidate number (EUFxxx) on each extra sheet. Extra sheets without this information will not be marked. Use separate extra sheets for each question. Do not detach the extra sheets.

If you need spare paper for rough notes or calculations, use the sheets marked SCRATCH at the end of the answer booklet. DO NOT DETACH THEM. The scratch sheets will be discarded and solutions written on them will be ignored.

Do NOT write ANYTHING on the List of Constants and Formulae provided; RETURN IT at the end of the test, as it will be used in the test tomorrow.

Have a good exam!

1

Q1. A bullet of mass m is fired with velocity v and hits a homogeneous disc of mass M and radius R lying flat at rest on a frictionless horizontal surface, getting embedded near its surface as shown in the figure. Assume that the center of mass of the system (disc + bullet) after the collision remains at the center of the disc. Note: use 212CMdiscI MR= .

(a) What is the velocity of the center of the disc after the collision? (b) What is the angular velocity of the system (disc + bullet) after the collision? (c) Determine the change in total kinetic energy of the system in the collision?

Q2. A particle of mass m, subject to constant acceleration due to gravity, g, is constrained to move inside the surface of an inverted cone of aperture 2. The apex of the cone is at the origin and its axis is vertical. Friction is negligible.

(a) Determine the kinetic and potential energy of the particle. Hint: use spherical coordinates.

(b) Obtain the Lagrangian of the system and find the equations of motion. (c) Are there physical quantities that are conserved in this motion? If yes, what are these

quantities and explain how you concluded that they are conserved? (d) Using the definition of the Hamiltonian, obtain its explicit form in terms of generalized

coordinates and momenta. Compare it to the mechanical energy of the particle. (e) Show that the particle may execute small radial oscillations about an equilibrium radius

r0 and find its frequency. Compare this value to the frequency of revolution in the circular motion.

m

v

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Q3. Part I The figure below presents curves of energy versus distance r between the nuclei in two diatomic molecules, named A and B. Each graph displays two states: the fundamental state, ( )0U r , and the first excited electronic state, ( )1U r .

(a) For molecule A, what is the meaning of r0 and r1, indicated in the graph? (b) Suppose molecule B is initially in its fundamental state, but then absorbs a photon and

jumps to the first excited electronic state. What do you expect to happen to this molecule after the absorption?

Part II The electron wave function of a hydrogen atom in the 1s state is given by

( ) 030

1 r ar ea

= ,

where 0a is the Bohr radius and r is the distance between the electron and the nucleus.

(c) Calculate the most probable distance r at which to find the electron in the 1s state. (d) Calculate r , the mean value of r in this state.

Q4. A particle of rest mass m0, moving initially at a speed 45v c= , as measured in the

laboratory reference frame, collides with an identical particle which is initially at rest in the same reference frame. As a result of the collision, the two particles combine to form a single particle of mass M. Assume relativistic mechanics.

(a) What are the total energy and linear momentum of each particle before the collision and of the combined particle after the collision?

(b) What is the speed of the combined particle after the collision? (c) What is the mass M of the combined particle?

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Q5. Consider n moles of an ideal monatomic gas.

(a) Using the first law of thermodynamics, express the entropy of the gas as a function of T, V, and n.

(b) A Carnot cycle consists of: 1) a reversible isothermal expansion at temperature qT ; 2) a reversible adiabatic expansion to a temperature fT ; 3) a reversible isothermal compression at temperature fT ; 4) a reversible adiabatic compression (use the figure labels). Calculate the work done and the heat exchanged in each of the 4 processes of the Carnot cycle for n moles of an ideal gas.

(c) Calculate the efficiency of the cycle.

Joint Entrance Examination for Postgraduate Courses in Physics

EUF

Second Semester/2011 Part 2 11 May 2011

Instructions:

DO NOT WRITE YOUR NAME ON THE TEST. It should be identified only by your candidate number (EUFxxx).

This test is the second part of the joint entrance exam for Postgraduate Physics. It contains questions on: Electromagnetism, Quantum Mechanics, Thermodynamics and Statistical Mechanics. All questions have the same weight.

The duration of this test is 4 hours. Candidates must remain in the exam room for a minimum of 90 minutes.

The use of calculators or other electronic instruments is NOT permitted in the exam. ANSWER EACH QUESTION ON THE CORRESPONDING PAGE OF THE

ANSWER BOOKLET. The sheets with answers will be reorganized for marking. If you need more answer space, use the extra sheets in the answer booklet. Remember to write the number of the question (Q1, Q2, or . . . ) and your candidate number (EUFxxx) on each extra sheet. Extra sheets without this information will not be marked. Use separate extra sheets for each question. Do not detach the extra sheets.

If you need spare paper for rough notes or calculations, use the sheets marked SCRATCH at the end of the answer booklet. DO NOT DETACH THEM. The scratch sheets will be discarded and solutions written on them will be ignored.

It is NOT necessary to return the List of Constants and Formulae.

Have a good exam!

1

Q6. In a factory of chocolate powder, pipes with compressed air are used to move the chocolate

between the different sectors. However, the chocolate powder becomes electrically charged through friction, creating a uniform positive volumetric charge density inside the pipe of radius R. Suppose that the pipe is conductive and grounded, and that the dielectric constant of air is not changed by the chocolate powder.

(a) Calculate the electric field inside and outside the pipe. Consider the pipe as a long cylinder.

(b) Calculate the electric potential inside and outside the pipe. Take V = 0 at the tube wall.

(c) Sketch the graph of the electric field and potential as a function of the distance from the axis of the pipe.

(d) If the electric field is greater than a certain value E0, it will exceed the dielectric strength of air, resulting in an electric spark. As the chocolate powder is quite flammable, a spark inside the pipe could result in an explosion. Determine what condition the pipe must satisfy to avoid this risk.

Q7. A plasma can be thought of as a classical gas (not relativistic) formed by positive ions and electrons. We are initially interested in the interaction of an electromagnetic wave with the free electrons present in the plasma, since these have a much smaller mass than the positive ions.

(a) The electric field EG

of a transverse harmonic electromagnetic wave may be written:

( )0 i k r tE E e =G GG G

.

Show that for operations involving , this operator may be replaced by ikG , and the time derivatives t by i . Use this to rewrite Maxwells equations.

Consider a harmonic wave propagating in the z-direction and assume that the average number of electrons per unit volume is n. (b) Show that the current density induced by the electric field of the wave is

2neJ i Em=G G

,

where e and m are the electric charge and mass of the electron, respectively, and is the wave frequency. Explain carefully your reasoning.

(c) Using Maxwells equations, obtain the dispersion relation ( )k for the wave propagation.

(d) Does the plasma allow wave propagation for any frequency? Justify your answer.

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Q8. The wave function for a particle in one dimension is denoted by ( , )x t . The probability density ( , )x t is defined as *( , ) ( , ) ( , )x t x t x t . The value of ( , )x t can change over time because of probability density flow, which can expressed by the continuity equation,

jt x = ,

where ( , )j x t is the probability current density. (a) Given the Schrdinger equation,

2 2

22 ( )i V xt m x = + == ,

express the time derivative of ( , )x t as a function of , * and their space derivatives.

(b) Obtain the explicit expression for ( , )j x t .

(c) Find the equation relating the time derivative of the expected value of position, d xdt , to the expected value of momentum, p . Hint: use integration by parts and assume

that the functions and its derivative, x

, vanish at infinity faster than

1x

.

Q9. The Hamiltonian of a given physical system is:

( )0 1 2 H w a a= += . The eigenstates of this Hamiltonian are denoted by n , are nondegenerate and fulfill the eigenvalue equation N n n n= , where n is an integer and N a a . (a) Assume that the operators a and a obey the commutation relation [ ] 1 ,a a = . Show

that the states a n and a n are also eigenstates of N , using the commutation relation. Find the corresponding eigenvalues for these states, n and n , respectively.

(b) Since all states n are nondegenerate, determine the proportionality constant that relates the states a n to the states n , which were found in item (a) above. Hint: remember that all states are normalized. Assume that the expected value of the Hamiltonian, for any eigenstate, is a positive number, 0H , and also assume that

0 0a = . What can be concluded about the number of states n : is it finite or infinite? (c) Assume now the operators a and a obey the anticommutation relation

{ } 1 ,a a aa a a= + = . Show that the states a n and a n are also eigenstates of N , using the anticommutation relation. Find the corresponding eigenvalues for these states, n and n , respectively. Since all states n are nondegenerate, determine the proportionality constant that relates the states a n to the states n . Hint: remember that all states are normalized.

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(d) Assume, as in item (c), that the operators a and a obey the anticommutation relation, that the expected value of the Hamiltonian for any eigenstate is a positive number,

0H , and that 0 0a = . From this it follows that the number of states n is finite. What are the only non-null states n in this case?

Q10. The Stefan-Boltzmann law states that the total energy density of the electromagnetic field

inside a cavity at thermal equilibrium is given by

( ) 4u T aT= , where a is a constant.

(a) The Stefan-Boltzmann law can be derived by a thermodynamical argument. Knowing that, at thermal equilibrium, the density of electromagnetic energy is independent of the material of the cavity walls, it can be inferred that any extensive variable of the radiation field must be proportional to the volume of the cavity and can only be a function of temperature. In particular, the internal energy and the entropy are

( )U u T V= and ( )S s T V= , respectively. Independently, it is known from classical electromagnetic theory that the radiation pressure on the cavity walls is ( )3

u TP = . Use the above information and the first law of thermodynamics to finish the argument and derive the Stefan-Boltzmann law.

(b) Now you will derive the Stefan-Boltzmann law from statistical physics, by assuming that the electromagnetic radiation is a photon gas.

i. Calculate the partition function, Z, and show that the average number of photons with energy j is

1 11

lnjj j

Zne

= = ,

where 1Bk T

= .

ii. Obtain the Stefan-Boltzmann law. You can use the result that the total number of photons per unit volume with frequency in the interval [ ], d + is given by

( )2

1dg d

e = ,

where is a constant and = = is the photon energy.