A solid uniformly charged insulating sphere has uniform volume charge density p and radius R. Apply Gauss's law to determine an expression for the magnitude of the electric field at an arbitrary distance r from the center of the sphere, such that r < R, in terms of rho and r

electric field E = (1 / 3 e₀) ρ r

Explanation:

For the application of the law of Gauss we must build a surface with a simple symmetry, in this case we build a spherical surface within the charged sphere and analyze the amount of charge by this surface.

The charge within our surface is

ρ = Q / V

Q ' = ρ V '

The volume of the sphere is V = 4/3 π r³

Q ' = ρ 4/3 π r³

The symmetry of the sphere gives us which field is perpendicular to the surface, so the integral is reduced to the value of the electric field by the area

I E da = Q ' / ε₀

E A = E 4 πi r² = Q ' / ε₀

E = (1/4 π ε₀) Q ' / r²

Now you relate the fraction of load Q 'with the total load, for this we use that the density is constant

R = Q ' / V' = Q / V

How you want the solution depending on the density (ρ) and the inner radius (r)

Q ' = R V'

Q ' = ρ 4/3 π r³

E = (1 / 4π ε₀) (1 / r²) ρ 4/3 π r³

E = (1 / 3 e₀) ρ r