Energy Density of a Charged Sphere?

[xxacefirexx]

Homework Statement

Since this was on the test, I won't be able to type out the question verbatim, but here it goes:

You have a plastic sphere with charge equally distributed through out the sphere. The radius of the sphere is a. The energy stored in the total sphere is E0 (not to be confused with electric field). Find the fraction of the total energy from r=0 to r=a/2 (half the radius).

Homework Equations

energydensity*volume = energy

The Attempt at a Solution

I simply put a ratio of the two energies, so E(small)/E0. Since charge is equally spread throughout the sphere, I figured that the energy densities would cancel out, leaving the volume(small)/volume(big). Dividing out gives (1/8)E0.

However, this seems unreasonably easy, and I think I'm missing something very important. Did I do it correctly, or am I forgetting something?

 

[Astronuc]

Does it make a difference if the electric field within the sphere is a function of r?

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elesph.html#c4

 

[xxacefirexx]

yes it does...

so it looks like i assumed that because the charge is constant, that the electric field is constant, which is not the case.

I missed 2 big equations in that, so looks like I am not even going to get that much partial credit

at least now i can sleep knowing the answer. btw, that's an excellent site.