Poissons' Equation, Electric Potential

[Bassalisk]

Homework Statement

In an imaginary sphere with radius R, there exist uniformly distributed space charge ρ. Find the potential in all points in space, bounded by this sphere. (r<R). Dielectric is air. ******Must use Poisson or Laplace equation******

Homework Equations

$\nabla\cdot\nabla ={\Large -\frac{\rho}{\epsilon _{0}}}$

The Attempt at a Solution

I did everything, and I found that one of the constants is 0.

But I get stuck when trying to find the second constant.

$\phi (r) ={\Large -\frac{\rho r^{2}}{6\epsilon _{0}}}+C_{2}$

I know that potential is constant on R. But I don't know how to use that. I am basically stuck here. I tried with some derivations, that equal 0 etc. But didn't get me anywhere.

I have the solution for the constant: $C_{2}={\Large \frac{\rho R^{2}}{2\epsilon _{0}}}$

Can somebody help me?

 

[Bassalisk]

Nevermind I got it. I misplaced the la place equation.