• Author / Uploaded
• kajari

Citation preview

Physics Graduate School Qualifying Examination Spring 2005 Part I ________________________________________________________________________ Instructions: Work all problems. This is a closed book examination. Calculators may not be used. Start each problem on a new pack of correspondingly numbered paper and use only one side of each sheet. Place your 3-digit identification number in the upper right hand corner of each and every answer sheet. All sheets, which you receive, should be handed in, even if blank. Your 3-digit ID number is located on your envelope. All problems carry the same weight. ________________________________________________________________________

1. A uniform flexible rope is suspended above a scale, with the bottom of the rope just touching the scale (gravity points downward). The rope has a length L and a total mass of M. The mass is uniformly distributed along its length.

gravity

L

Scale The rope is released. After a length x>mc2).

5

6. A source illuminates equally two slits (1) and (2). The source can emit either “classical” or quantum particles. Two detectors, (A) and (B), are located directly behind slits (A) and (B) in the viewing screen, respectively (the two detectors are separated by a wall so no particle detected at one detector can also be detected at the other). Finally, a barrier screen is placed so that particles entering slit (1) cannot enter detector (A). barrier screen (A)

(A) detector

(1)

source

(2) (B)

(B) detector

The wavefunctions at the detector locations for particles going through slit (1) and slit (2) are φ1 and φ2. These wavefunctions are orthogonal and when normalized individually they have the values shown in the table below. Location (A) (B) 1 0 φ1 e − iπ / 2 10

φ2

1 10

e − iπ / 2

1 10

e + iπ / 2

a) Write the correctly normalized wavefunction for the two-slit system. b) If 20 “classical” particles (non-interfering) are passed through the slits (equally likely through (1) or (2)), what is the expected number of events observed in each detector (using the above table)? c) If 20 quantum particles are passed through the slits (again, equally likely through (1) or (2)), what is the expected number of events observed in each detector (using the above table)? d) For quantum particles, is interference observed in either detector? From the detection of a quantum particle in detector (A) can you tell whether the particle went through slit (1) or slit (2)? If it were detected in (B)?

6

7. One mole of a monatomic perfect gas initially at temperature T0 expands from volume V0 to 3V0 at constant temperature. Calculate the work of expansion and the heat absorbed by the gas.

7

8. Consider a system with a large number N of distinguishable, noninteracting particles at rest. Each particle has 2 nondegenerate energy levels with energies of 0 and E >0, respectively. Let E/N be the mean energy per particle as N ⇒ ∞ . Energy Levels

E

0

(a) What is the maximum possible value of E/N if the system is not necessarily in thermodynamic equilibrium? (b) What is the maximum possible value of E/N if the system is in thermodynamic equilibrium? (c) Compute the entropy per atom S/N as a function of E/N.

8

Physics Graduate School Qualifying Examination Spring 2005 Part II ________________________________________________________________________ Instructions: Work all problems. This is a closed book examination. Calculators may not be used. Start each problem on a new pack of correspondingly numbered paper and use only one side of each sheet. Place your 3-digit identification number in the upper right hand corner of each and every answer sheet. All sheets, which you receive, should be handed in, even if blank. Your 3-digit ID number is located on your envelope. All problems carry the same weight.

________________________________________________________________________

9. A small mass m is located at a radius b from the center of a turn-table that is initially at rest. The turn-table is horizontally-oriented on the surface of the earth (i.e. the force of gravity points directly down, − zˆ , perpendicular to the surface of the turn-table). At the time t=0, the turn-table starts to accelerate with constant angular acceleration α. The coefficient of static friction between the mass and the surface of the turn-table is μ .

z

α b m

a) At the moment the mass starts to slip, in the rotating frame, write down all of the forces on the mass. b) At what time will the mass start to slip? r v r r r r r r a = ω& × r '+2ω × v '+ω × (ω × r ' ) (Hint: Where the primed system is the accelerated or rotating system. The dot as usual denotes a time derivative.)

9

10. A binary star system (consisting of two stars with masses m1 and m2) has a total mass M=m1+m2. The two stars execute circular orbits about the system’s center of mass with a period of T= 3years. The radii of the two orbits are l1 and l2, respectively. The total distance between the two stars is 2 astronomical units. (one astronomical unit is the distance from the earth to the Sun).

m2 l2

m1

l1

a) What are the orbital velocities of each star? b) What are the centripetal forces on each star? c) Express the mass M of the binary star system in terms of the mass of our Sun, Ms.

10

11. An infinite plane with uniform surface charge density of σ, is lying in the z=0 plane. This infinite plane is moving parallel to itself in the x-direction with r velocity v .

z y

σ

r v

x r a. Find the surface current density K (units of C/m/s).

r b. Find the B field above and below the plane. Now add a second plane with the same charge density σ. It is parallel to the first but at a distance s below the first. It moves with same magnitude of velocity |v| but in the negative x-direction. r c. Find the B field between the two planes. r d. Find the B field outside the two planes. e. What is the magnetic force per unit area on the upper plate?

11

12. A metal bar of mass M and length L slides down between two vertical metal posts. It slides with negligible friction but with good electrical contact. The bar falls in the gravitational field of the Earth at a constant speed of v. The falling bar and the vertical posts have negligible electrical resistance, but the base connecting the two posts is an ohmic resistor with resistance R. Throughout the entire region there is a uniform magnetic field of magnitude B coming straight out of the page.

L

v Bar of mass M

B ( out of page)

Resistance R

a. Calculate the amount of current I running through the resistor. Indicate the direction of the conventional current. b. Calculate the constant speed v of the falling bar.

c. Show that the rate of change of the gravitational energy of this system is equal to the rate of energy dissipated in the resistor.

12

13. Consider a simple 1D harmonic oscillator. Its Hamiltonian is

p 2 mω 2 2 H0 = + x 2m 2

a. The eigenfunction of the ground state wave function can be written as

ψ 0 ( x) = Ne −α

2 2

x /2

Determine N and α. b. What is the eigenvalue of the ground state? r c. At t=0, an electric field E is switched adding a perturbation whose Hamiltonian has the form r H' = e E x What is the new ground state energy? d. Assuming that the field is switched on in a time much shorter than 1/ω, what is the probability that the particle stays in the ground state? ∞

(A useful integral:

∫ dxe

−∞

−a 2 x 2

=

π a

)

13

14. Compute the energy levels of the hydrogen atom using the Bohr model. Use this formula to calculate the Kα spectra for atoms with many electrons. That is, use this formula to calculate the energy of the photon released when an electron from the n=1 state is removed and an electron from the n=2 state makes the transition down to the n=1 state. Calculate your answer for Z=2 and for Z=90. Express your answer in terms of ε 0 the electric permitivity, e the charge of the electron, and h Planck’s constant.

14

7. A photon of energy Eγ is incident along the positive x-axis on a stationary electron of mass m at the origin in the lab frame. The goal of this problem is to find the velocity of the center of mass in the lab frame, vcm. This will be accomplished in the three following steps:

a. Give the Lorentz transformation for the linear momentum px of the photon and the electron from the lab frame to the center of mass (cm) frame symbolically in terms of the velocity of the center of mass vcm in the lab frame. b. Find p x (γ ) and p x (e) , the momenta for the photon and the electron, respectively, in the cm frame in terms of vcm and Eγ . c. Solve for vcm.

15

8. Two identical finite bodies of constant heat capacity Cv are at temperatures T1 and T2, respectively. These two bodies are used to run a heat engine. If these bodies remain at constant volume and they undergo no phase change: a) Show that the amount of work obtainable is

W = C v (T1 + T2 − 2T f ) where Tf is the final temperature attained by both bodies. b) Show that when W is a maximum,

T f = T1T2

16

Physics Graduate School Qualifying Examination Spring 2005 Part II ________________________________________________________________________ Instructions: Work all problems. This is a closed book examination. Calculators may not be used. Start each problem on a new pack of correspondingly numbered paper and use only one side of each sheet. Place your 3-digit identification number in the upper right hand corner of each and every answer sheet. All sheets, which you receive, should be handed in, even if blank. Your 3-digit ID number is located on your envelope. All problems carry the same weight. ________________________________________________________________________

1. A small mass m is located at a radius b from the center of a turn-table that is initially at rest. The turn-table is horizontally-oriented on the surface of the earth (i.e. the force of gravity points directly down, − zˆ , perpendicular to the surface of the turn-table). At the time t=0, the turn-table starts to accelerate with constant angular acceleration α. The coefficient of static friction between the mass and the surface of the turn-table is μ .

z

α b m

a) At the moment the mass starts to slip, in the rotating frame, write down all of the forces on the mass. b) At what time will the mass start to slip? r v r r r r r r (Hint: a = ω& × r '+2ω × v '+ω × (ω × r ' ) Where the primed system is the accelerated or rotating system. The dot as usual denotes a time derivative.)

2. A binary star system (consisting of two stars with masses m1 and m2) has a total mass M=m1+m2. The two stars execute circular orbits about the system’s center of mass with a period of T= 3years. The radii of the two orbits are l1 and l2, respectively. The total distance between the two stars is 2 astronomical units. (one astronomical unit is the distance from the earth to the Sun). m2 l2

m1

l1

a) What are the orbital velocities of each star? b) What are the centripetal forces on each star? c) Express the mass M of the binary star system in terms of the mass of our Sun, Ms.

3. An infinite plane with uniform surface charge density of σ, is lying in the z=0 plane. This infinite plane is moving parallel to itself in the x-direction with r velocity v .

z y

σ x

r v

r a) Find the surface current density K (units of C/m/s).

r b) Find the B field above and below the plane. Now add a second plane with the same charge density σ. It is parallel to the first but at a distance s below the first. It moves with same magnitude of velocity |v| but in the negative x-direction.

r c) Find the B field between the two planes. r d) Find the B field outside the two planes. e) What is the magnetic force per unit area on the upper plate?

4. A metal bar of mass M and length L slides down between two vertical metal posts. It slides with negligible friction but with good electrical contact. The bar falls in the gravitational field of the Earth at a constant speed of v. The falling bar and the vertical posts have negligible electrical resistance, but the base connecting the two posts is an ohmic resistor with resistance R. Throughout the entire region there is a uniform magnetic field of magnitude B coming straight out of the page.

L

v Bar of mass M

B ( out of page)

Resistance R

a) Calculate the amount of current I running through the resistor. Indicate the direction of the conventional current. b) Calculate the constant speed v of the falling bar.

c) Show that the rate of change of the gravitational energy of this system is equal to the rate of energy dissipated in the resistor.

5. Consider a simple 1D harmonic oscillator. Its Hamiltonian is

p 2 mω 2 2 H0 = + x 2m 2

a) The eigenfunction of the ground state wave function can be written as

ψ 0 ( x) = Ne −α

2 2

x /2

Determine N and α. b) What is the eigenvalue of the ground state?

r c) At t=0, an electric field E is switched adding a perturbation whose Hamiltonian has the form r H' = e E x What is the new ground state energy? d) Assuming that the field is switched on in a time much shorter than 1/ω, what is the probability that the particle stays in the ground state? ∞

(A useful integral:

∫ dxe

−∞

−a 2 x 2

=

π a

)

6. Compute the energy levels of the hydrogen atom using the Bohr model. Use this formula to calculate the Kα spectra for atoms with many electrons. That is, use this formula to calculate the energy of the photon released when an electron from the n=1 state is removed and an electron from the n=2 state makes the transition down to the n=1 state. Calculate your answer for Z=2 and for Z=90. Express your answer in terms of ε 0 the electric permitivity, e the charge of the electron, and h Planck’s constant.

7. A photon of energy Eγ is incident along the positive x-axis on a stationary electron of mass m at the origin in the lab frame. The goal of this problem is to find the velocity of the center of mass in the lab frame, vcm. This will be accomplished in the three following steps:

a) Give the Lorentz transformation for the linear momentum px of the photon and the electron from the lab frame to the center of mass (cm) frame symbolically in terms of the velocity of the center of mass vcm in the lab frame. b) Find p x (γ ) and p x (e) , the momenta for the photon and the electron, respectively, in the cm frame in terms of vcm and Eγ . c) Solve for vcm.

8. Two identical finite bodies of constant heat capacity Cv are at temperatures T1 and T2, respectively. These two bodies are used to run a heat engine. If these bodies remain at constant volume and they undergo no phase change: a) Show that the amount of work obtainable is

W = C v (T1 + T2 − 2T f ) where Tf is the final temperature attained by both bodies. b) Show that when W is a maximum,

T f = T1T2

Physics Graduate School Qualifying Examination Fall 2005 Part I ________________________________________________________________________ Instructions: Work all problems. This is a closed book examination. Calculators may not be used. Start each problem on a new pack of correspondingly numbered paper and use only one side of each sheet. Place your 3-digit identification number in the upper right hand corner of each and every answer sheet. All sheets, which you receive, should be handed in, even if blank. Your 3-digit ID number is located on your envelope. All problems carry the same weight. ________________________________________________________________________

1. A marble of mass m falls under the influence of gravity into a beaker so that it enters the fluid in the beaker with an initial velocity of vo. The fluid in the beaker is viscous and so produces a drag force linear in velocity (characterized by the parameter α) on the marble of

r r F = − mα v Assume that the buoyant force can be neglected and: (a) Write the equation of motion for the marble; (b) Solve the equation of motion for the location of the marble as a function of time.

r v

gravity

1

2. A long thin tube of negligible mass is pivoted so it may rotate freely without friction in a horizontal plane (see side view). A thin rod of mass M and length L lies in the tube and can slide without friction in the tube. Recall that the moment of inertia of a thin rod about an axis perpendicular to rod’s long axis and through its center is ML2/12.

rotation axis

r r

M

M

θ

L Side view Top view a) Find the equations of motion for the thin rod in terms of the angle of the tube θ and the radial position of the center of the rod r. b) Find an expresion for the vertical component (z-component) of the angular momentum. Is it conserved? Justify your answer. c) Find an expression for the radial velocity r& as a function of r .

2

3. A thin coil with radius a, located at z = 0 in the x-y plane, consists of n turns of wire and carries a current I(t) that is initially zero and then increases linearly to the value I0 over a time τ :

⎧0 ⎪ I (t ) = ⎨ I 0 t / τ ⎪I ⎩ 0

t