Electric field at distance r from an infinite line charge with linear density λ is given by which expression?

Electric field at distance r from an infinite line charge with linear density λ is given by which expression?

• A. E = λ /(π ε0 r)
• B. E = 2π ε0 r / λ
• C. E = λ r /(2π ε0)
• D. E = λ /(2π ε0 r)

Answer: (D)

Gauss's law with a long straight line of charge shows the field is radial and falls off with distance as 1/r. Picture a cylindrical Gaussian surface of radius r and length L coaxial with the line. The electric field is the same at every point on the curved surface, so the flux through the surface is E times the surface area, which is E(2π r L). The charge enclosed is λL. Applying Gauss's law, E(2π r L) = (λL)/ε0, and the L cancels, giving E = λ/(2π ε0 r). This shows the field scales with the line density λ and decreases inversely with distance r. The expression that matches this result is E = λ/(2π ε0 r). The other forms either miss a factor of 2 or place r in the numerator, which would incorrectly imply the field grows with distance for an infinite line charge.

Gauss's law with a long straight line of charge shows the field is radial and falls off with distance as 1/r. Picture a cylindrical Gaussian surface of radius r and length L coaxial with the line. The electric field is the same at every point on the curved surface, so the flux through the surface is E times the surface area, which is E(2π r L). The charge enclosed is λL. Applying Gauss's law, E(2π r L) = (λL)/ε0, and the L cancels, giving E = λ/(2π ε0 r). This shows the field scales with the line density λ and decreases inversely with distance r. The expression that matches this result is E = λ/(2π ε0 r). The other forms either miss a factor of 2 or place r in the numerator, which would incorrectly imply the field grows with distance for an infinite line charge.