Field inside uniformly charged cylinder

Statement. Use Gauss's law, ∮E·dA = q_enc/ε0, with a coaxial cylindrical Gaussian surface.

Cylinder radius a, uniform volume charge density ρ, infinitely long. Consider a point at distance r < a from the axis.

      ______________
     /   Gaussian   /|
    |  surface, r  | |   <- length L
    |   (inside)   | /
     \_____________|/
   cylinder of radius a (charged throughout)

Steps

1. By symmetry, E is radial and has the same magnitude at every point on a coaxial cylinder of radius r.
2. Take a Gaussian cylinder of radius r and length L. No flux passes through the end caps, as E is parallel to them. Flux through the curved surface = E (2πrL).
3. Charge enclosed: q_enc = ρ × volume = ρ (πr²L).
4. Apply Gauss's law:

E (2πrL) = ρ π r² L / ε0

5. Cancel πrL:

E = ρ r / (2ε0) (for r < a)

So inside, E grows linearly with r, from zero on the axis. Outside (r > a), q_enc = ρπa²L and E = ρa²/(2ε0 r), and the two expressions agree at r = a.