Problem 1:

A object of mass m orbits a planet of mass M.  The total energy of the object is E.  When its radial distance from the center of the planet is GMm/(8|E|), the radial component of the velocity of the object is zero.  What is the eccentricity e of the orbit?

Hint:  For Kepler orbits 1/r ∝ 1 + e cos(φ - φ0).


Problem 2:

A static charge distribution produces a spherically symmetric, radial electric field E  = A exp(-br)/r2(r/r), where A and b > 0 are constants.
(a)  What is the volume charge density ρ(r)?
(b)  What is the total charge?


Problem 3:

The kinetic energy of a non-relativistic mass point equals T = p2/(2m), with p = mv the momentum of the particle.  Find a formally similar expression for the relativistic kinetic energy in terms of the relativistic momentum p = γmv.


Problem 4:


Two positive and two negative charges are arranged on the corners of a square whose sides have length L.  Each charge has magnitude Q.  The signs of the charges are as indicated.  Point c is at the center of the square.
(a)  What is the electrostatic potential at point c?
(b)  What is the potential energy associated with the charge at point b due to the other three charges?
(c)  How much work must be done to assemble the charges from ∞?  Clearly explain your reasoning, indicate whether the result is + or -, and explain why.


Problem 5:

A positron can be made by bombarding a stationary electron with a photon.
γ + e- --> e- + e+ + e-
What is the minimum photon energy?


Problem 6:

Assume that a star with a radius of 107 km has a purely dipolar magnetic field and that the magnetic field strength in its interior is 100 Gauss.  The star collapses to a neutron star with a radius of 10 km.  Assume that even though mass is lost during this transformation, magnetic flux through an area bounded by the equator is conserved.
(a)  What is the field strength in the interior of the neutron star after the collapse?
(b)  What is the magnetic energy density in the interior of the neutron star?
How does this compare with the energy density from the gravitational field, 2 GeV/fm3 and what does that mean in terms of the ability of the magnetic field to modify the neutron star's structure?