An infinite cylinder of radius R has a linear charge density λ. The volume charge density (C/m3) within the cylinder (r≤R) is rho(r)=rho0rR, where rho0 is a constant to be determined.

ρ₀ = 3λ/(2πR⁴)

Explanation:

If we take an elemental part of the cross sectional Area of the infinitely long cylinder.

The small elemental part has a small length dr in the cylinder's radial direction such that the cross sectional Area of the elemental part = 2πr dr

The linear density over the entire radial length of the cylinder will be equal to the sum of (volume charge density × elemental cross sectional Area)

That is,

λ = Σ ρ₀rR × 2πr dr

summing from 0 to R

λ = ∫ᴿ₀ (ρ₀rR × 2πr) dr

λ = ∫ᴿ₀ ρ₀R × 2πr² dr

λ = [ρ₀R × 2πr³/3 ]ᴿ₀

λ = (2π ρ₀R/3) [ r³ ]ᴿ₀

λ = (2π ρ₀R/3) [ R³ ]

λ = 2π ρ₀R⁴/3

ρ₀ = 3λ/(2πR⁴)