A nonconducting sphere of radius R has a volume charge density = 0 r/R, where

A nonconducting sphere of radius R has a volume charge density ρ = ρ₀r/R, where ρ₀ is a constant.

(a) Show that the total charge is Q = πR³ρ₀.

(b) Show that the total charge inside a sphere of radius r < R is q = Qr⁴/R⁴.

(c) Use Gauss's law to find the electric field E_r everywhere.

(d) Use dV = –E_r dr to find the potential V everywhere, assuming that V = 0 at r = ∞. (Remember that V is continuous at r = R.)