Compute voltage inside sphere of uniform charge

[Gary Roach]

Homework Statement

Problem 2.21 from Introduction to Electrodynamics, David J. Griffiths, Third Edition.

Find the potential inside and outside a uniformly charged solid sphere who's radius is R and whose total charge is q. Use infinity as your reference point.

Homework Equations

Given: q, R, r'(r inside sphere)
Variable: r
∫E da =q/ε₀
V = -∫$^{r}_{∞}$E dl

The Attempt at a Solution

Outside of sphere:
|E|∫_s r²sinΘdΘdø = (1/ε₀)q
E = (q)$\widehat{r}$/(4π ε_or²)
V = -∫$^{r}_{∞}$E dr =q/4πε₀r

The above is pretty straight forward. On the other hand, the Voltage inside the sphere completely illudes me. From the answer book, I know that is an A - B type problem but can't seem to get my mind around the concept. I know that the Voltage is a function of the radius but the E field is not zero as in a hollow sphere. How is the basic equation for the E field inside the sphere derived?

 

[lightgrav]

charge inside the gaussian surface at r<R is proportional to the Volume enclosed: rho V_inside
where rho = q/(4/3 pi R^3)

 

[mfb]

For a radius r>R, consider the sphere with a smaller radius (smaller than r) and the hollow sphere with a larger radius (larger than r) as separate objects. How can you calculate their field contributions?