The University
of Georgia
Department of Physics and Astronomy
Graduate Qualifying Exam –– Part I
Instructions: Attempt all problems. Start each problem on a new sheet of paper and use one side only. Print your name on each piece of paper that you submit. This is a closed-book and closed-notes exam. You may use a calculator, but only for arithmetic functions (i.e., not for referring to notes stored in memory, doing symbolic algebra, etc.). For full credit, you must show your work and/or explain your answers. Part I has six problems, numbered 1–6.
Problem 1: (2 parts)
Suppose the electric field in a certain region of space is given in spherical coordinates by:
,
where and
are the r
and
spherical unit vectors, respectively.
(a) Is there a time-varying magnetic field in this region of space? Explain.
(b) Find the work done in moving a 5-C point charge from the origin to the point given by (2 meters, π/4, π/2) in spherical coordinates.
Problem 2: (1 part)
A standard result of Thermodynamics tells us that an equilibrium state of a system in thermal contact with a reservoir is one that minimizes the Helmholtz free energy, F, at constant temperature and constant volume. Under what conditions should one consider the Gibbs free energy, G, instead of the Helmholtz free energy? Explain your answer.
Problem 3: (1 part)
A spacecraft is being designed to dispose of nuclear waste
either (a) by carrying it out of the solar system or (b) by crashing into the
sun. In both cases, assume that the
spacecraft starts in a circular orbit about the sun at the Earth-sun
radius.
Problem 4: (2 parts)
The following constants may be useful in this problem: and
.
(a)
A meson is an unstable particle that is produced
in high-energy particle collisions. It
has a mass-energy equivalent of about 135 MeV, and it exists for an average
lifetime of only
before decaying into two gamma rays. Using the uncertainty principle, estimate the
fractional uncertainty,
,
in its mass determination.
(b)
The neutron has a mass of . Neutrons emitted in nuclear reactions can be
slowed down via collisions with matter.
They are referred to as thermal neutrons once they come into thermal
equilibrium with their surroundings. The
average kinetic energy (3kBT/2)
of a thermal neutron is approximately 0.04 eV.
i. Calculate the de Broglie wavelength of a thermal neutron with a kinetic energy of 0.04 eV.
ii. How does your answer compare to the characteristic atomic spacing in a crystal? Would you expect thermal neutrons to exhibit diffraction effects when scattered by a crystal?
Problem 5: (1 part)
Suppose you wish to test the circuit breakers in your house. Clever physicist that you are, you decide to build a device that, when plugged into any wall outlet, will draw more than 20 A of current, which is the maximum that the wiring in your house can safely carry. If the circuit breakers are working properly, then they will shut off the power when the device is plugged in. But if they are faulty, then the overloaded circuit will start a fire, and your house will burn down. The wall outlets in your house have a voltage of 120 V. The materials at your disposal for building this device include an economy-size box of several dozen 100-W light bulbs and a large spool of zero-resistance wire. Design a circuit capable of drawing more current than your house’s wiring is rated for. However, you must include at least one light bulb in every leg of the circuit; i.e., no short circuits. You may use as many light bulbs as you like. Ignore the fact that the electricity in the house is AC.
Problem 6: (5 parts)
A particle of mass m
moves in a central force field. Its
orbit is a spiral given in polar coordinates by ,
where c is a constant.
(a)
Start with and write down the two equations of motion.
(b) Show that the angular momentum, L, is a constant of the motion.
(c)
Determine the force field, F(r), in terms of m, L,
and c. (Hint:
Make the substitution .)
(d) Determine how θ varies with time t.
(e)
Determine the (constant) total energy, E, of this orbit. Let the zero of potential energy be at .