Which statement about the electric field due to an infinite line charge is true?

Which statement about the electric field due to an infinite line charge is true?

• A. E is proportional to 1/r
• B. E is proportional to r
• C. E is proportional to 1/r^2
• D. E is constant with distance

Answer: (A)

Because an infinite line of charge has cylindrical symmetry, the electric field must point radially away from the line and depend only on the distance r from it. Take a cylindrical Gaussian surface of radius r and length L coaxial with the line. The field is uniform on the curved surface and perpendicular to it, so the flux comes only from that curved surface: Φ = E(r) (2π r L). The charge enclosed is Q_enc = λ L, where λ is the line’s charge density. Gauss’s law gives E(r) (2π r L) = Q_enc / ε0 = (λ L)/ε0, which yields E(r) = λ / (2π ε0 r). So the field falls off as 1/r with distance from the line. This 1/r behavior contrasts with 1/r^2 that you’d get for a point charge (spherical symmetry) and with a constant field for an infinite plane (which has a different symmetry). The key reason is that the flux through a cylindrical surface grows in area proportional to r, so the field must decrease inversely with r to keep Gauss’s law balanced.

Because an infinite line of charge has cylindrical symmetry, the electric field must point radially away from the line and depend only on the distance r from it. Take a cylindrical Gaussian surface of radius r and length L coaxial with the line. The field is uniform on the curved surface and perpendicular to it, so the flux comes only from that curved surface: Φ = E(r) (2π r L). The charge enclosed is Q_enc = λ L, where λ is the line’s charge density. Gauss’s law gives E(r) (2π r L) = Q_enc / ε0 = (λ L)/ε0, which yields E(r) = λ / (2π ε0 r). So the field falls off as 1/r with distance from the line.

This 1/r behavior contrasts with 1/r^2 that you’d get for a point charge (spherical symmetry) and with a constant field for an infinite plane (which has a different symmetry). The key reason is that the flux through a cylindrical surface grows in area proportional to r, so the field must decrease inversely with r to keep Gauss’s law balanced.