9. Two equal charges
are located at the points
, and two charges
at
. Find the leading term in the potential at large distances.
24. Assume that the pressure
in a star of radius
and with spherical symmetry is related to the density
by the equation of state
; this equation of state actually quite well approximates the pressure of nucleons in a neutron star. Use the fluid equilibrium equation
to find a relation between
and
, the gravitational potential. Hence show that Poisson's equation yields
Solve this equation with the boundary conditions that
is finite at
and vanishes at the surface of the star. Show that the radius of the star is determined solely by
and is independent of its mass
. Show also that
. Evaluate the mass and radius of the star if
g/cm3 (about ten times nuclear density) and
(the maximum pressure for a given density is
- so this is pretty close).
You will find the expression for the Laplacian in spherical coordinates helpful: