Consider an infinite perfect conductor that occupies all space for z < 0. The
surface of this conductor coincides with the x?y plane. Although we
don?t know what exactly exists on top of the conductor, we know that
the space z > 0 is occupied by a simple dielectric with a dielectric
constant ?r. A total charge Q is placed on this conductor, which
results in a surface charge density of
?s =?0/((a2 + x2 + y2)^(3/2)) .
where ?0 and a are positive constants.

1. Find an equation that relates Q and ?0. Your equation may contain an
integral as long as the integral is clearly defined so a computer may
calculate it without any additional information.

2. Find the polarization of the dielectric on the surface of the conductor
(i.e. at z = 0+).

3. Relate the voltage at (0,0,-5000) with the voltage at (0,0,-1). Justify your
answer.

"here ?0 and a are positive constants..."

should be 
"here ?0 and a2 are positive constants.."

??

Charge would re-distribute itself to become more uniform.
Roughly with speed of light (depending on permitivity of conductor)

Since conductor is infinite, it will take infinite time for system to
reach equilibrium.