How Does Cylindrical Symmetry Affect the Quadrupole Moment Matrix?

[Josephk1508]

Homework Statement

Show that, if a charge distribution  has cylindrical symmetry, then knowing the value
of Qzz allows you to determine the entire quadrupole moment matrix.

Homework Equations

Tr Q=0, ie Qxx+Qyy+Qzz=0

The Attempt at a Solution

Because of the cylindrical symmetry, the charge density changes as a function of ρ, where ρ=(x^2+y^2)^1/2. Then I don't really know where to go from there.

 

[Josephk1508]

I have now worked out all values for the quadrupole matrix, I just need to calculate what Qzz is. If anyone can help that would be really useful. Also I realized I made a mistake above, cylinrical symmetry means that it is dependent only z only.