A nonuniform, but spherically symmetric, distribution of charge has a charge density ρ (r) given as follows: ρ (r) = ρ0 (1-r/r) for r≤r ρ (r) = 0 for r≥r where ρ0=3q/πr3 is a positive constant.

A)

dQ = ρ (r) * A * dr = ρ0 (1 - r/R) (4πr²) dr = 4π * ρ0 (r² - r³/R) dr

which when integrated from 0 to r is

total charge = 4π * ρ0 (r³/3 + r^4 / (4R))

and when r = R our total charge is

total charge = 4π*ρ0 (R³/3 + R³/4) = 4π*ρ0*R³/12 = π*ρ0*R³ / 3

and after substituting ρ0 = 3Q / πR³ we have

total charge = Q ◄

B) E = kQ/d²

since the distribution is symmetric spherically

C) dE = k*dq/r² = k*4π*ρ0 (r² - r³/R) dr / r² = k*4π*ρ0 (1 - r/R) dr

so

E (r) = k*4π*ρ0 * (r - r² / (2R)) from zero to r is

and after substituting for ρ0 is

E (r) = k*4π*3Q (r - r² / (2R)) / πR³ = 12kQ (r/R³ - r² / (2R^4))

which could be expressed other ways.

D) dE/dr = 0 = 12kQ (1/R³ - r/R^4) means that

r = R for a min/max (and we know it's a max since r = 0 is a min).

E) E = 12kQ (R/R³ - R² / (2R^4)) = 12kQ / 2R² = 6kQ / R²