Calculate the electrostatic energy of a homogeneously charged sphere

[Quelsita]

QUESTION:

Calculate the electrostatic energy of a homogeneously charged sphere of Volume V and
compare the result with 2 times the electrostatic energy of a homogeneously charged
sphere of V/2.

SOLUTION:

OK, so we have a charge Q which is uniformly distributed within a sphere of radius, R.

We know:
q(r) is the charge in the sphere when it has
attained radius, r.
q(r)= $$\rho$$(4/3)$$\Pi$$r³

the work done in bringing a charge dq to it is dW
dW= (q(r) dq)/(4$$\Pi$$r$$\epsilon$$₀

dq= $$\rho$$4$$\Pi$$r²dr

so, we plug in the knowns:

dW= ($$\rho$$(4/3)$$\Pi$$r³ * $$\rho$$4$$\Pi$$r²dr )/ (4$$\Pi$$r$$\epsilon$$₀)

and integrate.

Is this correct thus far?
What are the limits?

Thanks!

 

[G01]

Your bounds would be 0 to R as that covers the whole sphere.