Electric potential related to electric field question

[David0709]

Homework Statement

(i) Consider a non-conducting sphere of radius R with non-homogeneous charge density ρ = ρ(r) = r, where r is the radial co-ordinate.

1. (a) Find the electric field inside and outside of the sphere
2. (b) Find and plot the electric potential inside and outside of the sphere

Homework Equations

[/B]
E=−∇V
E = KQ1Q2/r^2
V = KQ1Q2/R

The Attempt at a Solution

For part a) Field inside the sphere = r^2 / 4ε0
Outside the sphere R^4 / 4ε0r^2

With regards to part b)
I am aware that E=−∇V

So applying this formula we have:
Outside sphere:
R^4 /4ε0 r
Inside sphere potential
-r^3 /12ε0

However when we let R = r we see these two equations for potential do not agree suggesting something (am not sure what ) is wrong.

Any help would be appreciated

 

[haruspex]

these two equations for potential do not agree

Potential is always relative to some arbitrary zero, often taken as the potential at infinity.
To put it another way, going from field to potential is an integration, and an integral has a constant of integration, resolved by the bounds. The result of that is just the difference in potential between the endpoints, not an absolute potential.