UNIPOSRIO-FÍSICAExame Unificado das Pós-Graduações

em Física do Rio de Janeiro

02/12/2011, 14:00 - 18:00

This exam contains problems english.Make sure it has 4 pages. All problems have the same value.

ENGLISH VERSION:

Problem 1: A particle of mass m moves along the trajectory given by x = x0 cos(!1t) and y0 = sin(!1t).

a) Find the force acting on the particle. What are the conditions necessary for this force to be a central force?

b) Given the zero potential energy at the origin, find the potential energy of the particle.

c) Determine the kinetic energy of the particle. Show that the total energy of the particle is conserved.

Problem 2: In a lecture presentation, a small laser pointer is used. It emits red light with wavelength � = 600nm with 3mWpower, concentrated in baem with a diameter of 2mm. Calculate:

(data µ0 = 4⇡ ⇥ 10

�7Tm/A "0 = 8.854⇥ 10

�12C

2/(Nm

2))

a) The frequency ⌫, and I intensity of this wave;

b) The amplitudes of the electric field Ermmax

and magnetic field Brmmax

of the light emitted;

c) The energy U contained in a 3m beam length (the typical distance between the presenter and the screen). item If thepresentation screen is a perfect reflector (ie all the incident light is reflected), and the focuses laser light on the screen is atright angles, determine the pressure P to this radiation exerts on the screen.

Problem 3: Water moves at a speed of 5.0 m/s in a straight section of pipe with 4.0 cm 2. The pipe in which water movesgo gradually down 10 m while its cross section increases to 8.0 cm 2. The water pressure before the descent is 1.5⇥ 10

5 Pa.

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What is the pressure after the descent? Consider that the density of water is 1.0⇥ 10

3 kg/m 3 and the acceleration of gravityis 9.8 m/s 2.

Problem 4: Consider two square plates, P1 and P2, both with thickness 2d and area L2, separated by a distance l, as shownin Figure ??. The board P1 is made of dielectric material with dielectric constant K = 1 and has a positive charge +2Q,uniformly distributed. The board P2 is made of conductive material and has a negative total charge �Q. Consider also thatthe plates are in electrostatic equilibrium and that d/L << 1 e l/L << 1.

a) Use the Gauss’s law to determine the electric field ~E in the following regions:

1 x < �d;

2 �d < x < d;

3 d < x < d+ l;

4 d+ l < x < 3d+ l;

5 3d+ l < x.

b) How do the charges distribute themselves in P2?

Figure 1: P1 and P2 representation.

Attention: From the set of four questions below, choose ONLY TWO

Problem 5: A particle is in a state described by the wave function

=

1p4⇡

�ei� sin ✓ + cos ✓

�g(r)

whereZ 1

0drr2|g(r)|2 = 1 (1)

2

and �, ✓ are the azimuth and polar angles respectively.Obs: The first spherical harmonics are:

Y00 =

r1

4⇡; Y10 =

r3

4⇡cos ✓; Y1±1 = ⌥

r3

8⇡e±i�

sin ✓ (2)

a) What are the possible results of a measurement of the z-component Lz

of the angular momentum of the particle in thisstate?

b) What is the probability of obtaining each of the possible results of part (a)?

c) What is the expectation value of Lz

?

Problem 6: A system of two particles, each with spin 12 is described by an effective Hamiltonian

H = A (s1z + s2z) +B~s1 · ~s2where ~s1 e ~s2 are the two spin operators of the particles 1 and 2 respectively, s1z e s2z are their z components and A and

B are constants.

a) Show that H , ~S2 and Sz

can be simultaneously measured, where ~S = ~s1 + ~s2.

b) Find all the energy levels of this Hamiltonian.

Problem 7: Consider the Hamiltonian for the one dimensional oscillator as:

H =

p2

2m+

1

2

m!2x2.

Knowing that we can define the operators a and a† in the form

a =

1p2m~!

(p� im!x)

a† =

1p2m~!

(p+ im!x)

with a|0i = 0. At t = 0 the system is in a state of the form

| (t = 0)i = 1p5

|1i+ 2p5

|2i

a) Obtain | (t > 0)i.

b) Prove that h (t)|H| (t)i is time independent.

c) Find the expectation value of x as a function of time hx(t)i.

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Problem 8: Consider a Rigid body with momenta of inertia I , which is constrained to rotate in the xy plane, and whosemotion is determined by the equation

�~22I

d2

d'2 = E ,

a) Obtain the eigenvalues and eigenfunctions.

b) Assume that the rotator has a fixed dipole moment �!q in the plane. An constant eletric field�!E is applied to the plane.

Determine, if there is, the energy correction to first order in perturbation theory.

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