Field inside a long straight cylinder of radius R with uniform volume charge density ρ for r < R. Which expression gives E(r) inside?

Field inside a long straight cylinder of radius R with uniform volume charge density ρ for r < R. Which expression gives E(r) inside?

• A. E(r) = ρ r / (2 ε0)
• B. E(r) = ρ / (2 ε0)
• C. E(r) = ρ R / (2 ε0)
• D. E(r) = ρ r / (2 ε0)

Answer: (D)

This question tests Gauss's law for a long cylinder with cylindrical symmetry. Inside the cylinder, the electric field grows linearly with the distance from the axis. Take a Gaussian surface that is a coaxial cylinder of radius r (less than R) and length L. The electric field is radial and uniform over this curved surface, and there is no flux through the end caps because the field is perpendicular to them. So the total flux is E(r) multiplied by the curved surface area, 2π r L. The charge enclosed by this Gaussian surface is the volume charge density times the inner cylinder’s volume: q_enc = ρ × (π r^2 L). Gauss’s law gives E(r) × (2π r L) = q_enc / ε0 = (ρ π r^2 L) / ε0. Solving for E(r) yields E(r) = (ρ r) / (2 ε0). So the electric field inside grows linearly with r, and at the outer surface r = R it would be E(R) = ρ R / (2 ε0).

This question tests Gauss's law for a long cylinder with cylindrical symmetry. Inside the cylinder, the electric field grows linearly with the distance from the axis.

Take a Gaussian surface that is a coaxial cylinder of radius r (less than R) and length L. The electric field is radial and uniform over this curved surface, and there is no flux through the end caps because the field is perpendicular to them. So the total flux is E(r) multiplied by the curved surface area, 2π r L.

The charge enclosed by this Gaussian surface is the volume charge density times the inner cylinder’s volume: q_enc = ρ × (π r^2 L).

Gauss’s law gives E(r) × (2π r L) = q_enc / ε0 = (ρ π r^2 L) / ε0. Solving for E(r) yields E(r) = (ρ r) / (2 ε0).

So the electric field inside grows linearly with r, and at the outer surface r = R it would be E(R) = ρ R / (2 ε0).