Joint Entrance Examination for Postgraduate Courses in Physics EUF

Joint Entrance Examination for

Postgraduate Courses in Physics

EUF

First Semester 2011

Part 1

—

28 September 2010

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 Statisti-cal Mechanics. All questions have the same weight.

• The duration of this test is4 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, ou 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.

Q1.

Consider a rigid body of mass M and circular cross-section of radius R, which rolls without sliding down a plane surface inclined at angle θto the horizontal, as shown in the figure below. The coefficient of static friction between the body and the surface is µe. The moment of inertia

of the body about its central axis passing through point O isI and the acceleration of gravity is g.

R

h

y x

q

O

(a) Draw the diagram of forces acting on the body. Write the equation that relates the angular velocity, ˙ϕ, of the body about O to the velocity of translation of its axis, ˙x, assuming it rolls without sliding.

(b) Determine the acceleration ¨x, associated with the translation of the axis of the body down the inclined plane, in terms of the variables defined in the problem statement.

(c) Assume that the body starts rolling from rest at the origin of the Cartesian coordinate system indicated in the figure. Calculate its mechanical energy at the start and end of the motion. Is its mechanical energy conserved?

(d) Calculate I, supposing that the body is (i) a ring and (ii) a disk. Assume the mass of the body is distributed uniformly. Now suppose that the angle θ can be gradually increased. At what value of θ does the purely rolling motion cease and the body begin to slide, in cases (i) and (ii)? Express the answer in terms of µe.

Q2.

Consider the inverted pendulum in the figure below, consisting of a rod of massM and moment of inertia I0 about its center of mass, whose coordinates are (X,Y). The rod can turn freely

in the xy plane about an axis of rotation passing through position (xp, yp), which lies at a distance ℓ from the center of mass. The acceleration of gravity is g.

x

y

(X,Y)

(x ,y )

p p

l

q

g

(a) Write equations for the kinetic and potential energy of the system in terms of X, Y andθ.

For items (b), (c) and (d), suppose that an external agent makes the axis of rotation oscillate horizontally at angular frequency ω, such that yp(t) = 0 and xp(t) =Acos(ωt).

(b) Write the Lagrangian of the system in terms of the generalized coordinate θ. (c) Write the equation of motion for the Lagrangian in item (b).

(d) Suppose the system executes small oscillations (θ is small). Show that, in this case,

θ(t) =αcos(ωt) +βsin(ωt) is a solution to the problem. Determine α and β.

Q3.

For items (a), (b) and (c), suppose that, in the Bohr model for a particle of mass m moving in a circular orbit of radius r at velocity v, the Coulomb force be replaced by an attractive central force of magnitude k r (k being a constant). Let the Bohr postulates be valid for the system described. For such a system:

(a) Deduce an expression for the radiirnof the Bohr orbits allowed in this model, as a function of the principal quantum number n and the constants k, ~_{and m. Indicate which values}

of n are possible in this system.

(b) Noting that the potential energy corresponding to the hypothetical central force isV(r) =

kr2

/2, deduce an expression for the energies En of the allowed orbits, as a function of

n, k, ~ _{and m. Determine the frequency of the radiation emitted if the particle makes a}

transition from one orbit to the adjacent one of lower energy.

(c) Calculate the de Broglie wavelength associated with the particle in an energy state cor-responding to the quantum number n = 2, as a function of k, ~ _{and m.}

For item (d), consider a beam of X-rays containing radiation of two distinct wavelengths, diffracted by a crystal in which the spacing between diffracting planes is 1.0 nm (10−9

m). The figure below shows the small-angle region (with respect to the incident beam direction) of the resulting X-ray diffraction spectrum.

Q4.

In an experiment on the photoelectric effect, monochromatic light of wavelength 414 nm and intensity I0 is incident on a clean surface of a metal whose work function is φ= 2.5 eV.

(a) Calculate the maximum kinetic energy of the photoelectrons.

(b) If the intensity of the incident light were doubled, what would happen to the kinetic energy of the photoelectrons?

Consider now a Compton scattering event in which an electron of mass m0 at rest scatters a

photon of wavelength λ = 2λc ≡ 2h/(m0c). In the scattering event, the photon loses half its

energy.

(c) Calculate the wavelength of the scattered photon (express the result as a function of λc alone) and determine its scattering angle.

(d) Calculate the total energy and linear momentum of the electron after the collision (express the result as a function of m0 and c).

Q5.

Imagine that a one-dimensional magnetic material can be modeled as a linear chain of N + 1 spins. Each spin interacts with its nearest neighbors in such a way that the energy of the system is E = n², where n is the number of domain walls separating regions of spin ↑ from regions of spin ↓, as shown in the figure below, where domain walls are represented by dashed lines, and ² is the energy per domain wall. Assume that N À1 and nÀ1.

(a) In how many ways can the n domain walls be arranged?

(b) Calculate the entropy S(E) of the system containing n domain walls.

(c) Express the internal energy of the system as a function of temperature, E(T). The result should be given in terms of N,²,T and physical constants alone.

(d) Sketch the function E(T), indicating the values of E at T = 0 and as T → ∞.

Joint Entrance Examination for

Postgraduate Courses in Physics

EUF

First Semester 2011

Part 2

—

29 September 2010

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: Electromagnetism, Quantum Mechanics, Thermodynamics and Sta-tistical Mechanics. All questions have the same weight.

• The duration of this test is4 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, ou 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 Constant and Formulae.

Q6.

An uncharged metal sphere of radius R is placed in a region of space initially filled with an electric field given by E~i =E0 ˆk . Set the origin of the coordinate system at the center of the

sphere.

(a) Sketch the electric field lines in the entire region outside the sphere. Justify your sketch on the basis of physical arguments.

(b) Find the electric field in the presence of the sphere,E~f(~r) , throughout the region of space. In particular, deduce the fields at points where |~r| ÀR and |~r| ≈ R and check whether they are consistent with the sketch in item (a).

(c) Find the charge density on the surface of the sphere. If the radius of the sphere Ris 10 cm and E0 = 100 N/C, calculate the charges that accumulate on the northern and southern

hemispheres of the sphere.

(d) Suppose that the metal sphere were replaced by a dielectric sphere. Discuss inqualitative

terms what would happen in this case and sketch the electric field lines throughout the region of space.

Q7.

Consider the hypothetical arrangement illustrated in the figure below, in which a solid wire of radius a, stretched along the z-axis, conducts an electric current I (which is maintained constant) uniformly distributed over its cross-section. A small gap in the wire, of widthw¿a, forms a parallel-plate capacitor. The charge on the capacitor is zero at time t= 0.

(a) Find the electric field vector in the gap as a function of the distanceρfrom the z-axis and time t, as well as of the parameters I, w and a. Ignore edge effects.

(b) Find the magnetic field vector in the gap as a function of ρ, t and the parameters I, w

and a.

(c) Calculate the electromagnetic energy density uem and the Poynting vector in the gap,

indicating explicitly the direction and sense of the vector.

(d) Determine the total electromagnetic energyUem in the gap as a function of time. Compare

the rate of change of Uem with respect to time with the energy flux per unit time (power

flux), calculated as a surface integral of the Poynting vector.

• • • • • •

w

z

I I

a +σ _−σ

Q8.

Consider a particle of massmsubject to a one-dimensional harmonic potentialV(x) = 1 2mω

2

x2

, where ω is the angular frequency of the oscillator andx is the coordinate of the particle.

(a) The stationary wave functions of the ground state ψ0 and the first excited state ψ1 are

given by:

ψ0(x) = A exp ³

−mω

2~ x

2´

, ψ1(x) =B x exp

³

−mω

2~ x

2´

,

whereA andB are normalization constants. Calculate Aand B, assuming that the wave functions are real.

(b) Let E0 be the energy of the ground state. We know that the energy of the first excited

state is E1 = E0 +~ω, since the quantum of energy of the oscillator is ~ω. Use the

Schr¨odinger equation to find the value of E0.

(c) For stationary states, the mean position of the particle hxi is always nil. Construct a non-stationary wave function as a linear combination of ψ0 e ψ1 with real coefficients,

such that the mean position hxi has the highest possible value. In other words, take the normalized quantum state:

ψ(x) =p1−β2

ψ0(x) +β ψ1(x) ,

where 0≤β2

≤1, and determine the coefficient β that maximizes the value of hxi. (d) Let the wave function constructed in the previous item describe the state of the harmonic

oscillator at time t = 0. Write the wave function for the state at an arbitrary timet >0, assuming that no measurement has been made on the system. For this state, give an expression for the mean position as a function of time, hxi(t).

Q9.

Consider a particle with angular momentum l = 1.

(a) In the representation in which the matrices L2

e Lz are diagonal, obtain the matrix of

component Lx. Remember that this matrix must represent a Hermitian operator. We

suggest you use the ladder operators L±.

(b) Calculate the eigenvalues of Lx.

(c) Find the eigenvector of Lx with the highest eigenvalue.

(d) Suppose now that the result of a measurement of Lx is the highest eigenvalue. Calculate

the probabilities of measuring, respectively, +~_{, 0 and −}~ _{in a succeeding measurement}

of Lz.

Q10.

One mole of an ideal monatomic gas is at a temperature T and occupies a volume V. The internal energy per mole of an ideal gas is given by u = cVT, where cV is the molar heat capacity, which we assume to be constant. Answer the following questions:

(a) Suppose the gas is in continuous contact with a large heat reservoir at temperatureT and undergoes a reversible quasi-static expansion, during which its volume increases from V

to 2V. Calculate the work done by the gas during its expansion.

(b) Again with reference to the physical process described in (a), determine the quantity of heat transferred between the gas and the heat reservoir.

(c) Calculate the entropy changes in the gas and in the heat reservoir, in the process described in (a).