Inside a long straight cylinder with uniform volume charge density, what is E at the axis r = 0?

Inside a long straight cylinder with uniform volume charge density, what is E at the axis r = 0?

• A. 0
• B. ρ/(2 ε0)
• C. ρ R /(2 ε0)
• D. ∞

Answer: (A)

The main idea is symmetry and Gauss’s law inside a uniformly charged cylinder. Because the charge density is the same in all directions around the axis, the electric field must point radially outward and its magnitude grows linearly with distance from the axis. Using a coaxial Gaussian surface of radius r inside the cylinder, the flux is E(r) times the curved surface area 2πrL, and the enclosed charge is ρ times the volume πr²L. Gauss’s law gives E(r) = ρ r / (2ε0). At the axis, where r = 0, the field is zero. So the field at r = 0 is zero. The other values would correspond to nonzero radii (for example, the field at the surface or elsewhere), not at the axis.

The main idea is symmetry and Gauss’s law inside a uniformly charged cylinder. Because the charge density is the same in all directions around the axis, the electric field must point radially outward and its magnitude grows linearly with distance from the axis. Using a coaxial Gaussian surface of radius r inside the cylinder, the flux is E(r) times the curved surface area 2πrL, and the enclosed charge is ρ times the volume πr²L. Gauss’s law gives E(r) = ρ r / (2ε0). At the axis, where r = 0, the field is zero. So the field at r = 0 is zero. The other values would correspond to nonzero radii (for example, the field at the surface or elsewhere), not at the axis.