The electric field due to an infinite line charge with linear density λ at distance r is given by which expression?

The electric field due to an infinite line charge with linear density λ at distance r is given by which expression?

• A. E = λ / (2π ε0 r) directed radially away from the line
• B. E = -λ / (2π ε0 r) directed radially toward the line
• C. E = λ / (2π ε0 r^2)
• D. E = λ / (4π ε0 r)

Answer: (B)

The field from an infinite line charge depends only on the distance r from the line and points radially. Use Gauss’s law with a cylinder coaxial to the line. The curved surface area is 2π r L, so the flux is E times that area. The enclosed charge is λL, giving E(2π r L) = λL/ε0, hence E = λ/(2π ε0 r). The direction is radial: away from the line if the line’s charge is positive, toward the line if it is negative. If you take the outward radial unit vector as the reference direction, the vector form is E = (λ/(2π ε0 r)) r_hat. If you prefer inward direction as positive, you can write E = -λ/(2π ε0 r) with the inward direction specified. This matches the given expression when the direction is toward the line.

The field from an infinite line charge depends only on the distance r from the line and points radially. Use Gauss’s law with a cylinder coaxial to the line. The curved surface area is 2π r L, so the flux is E times that area. The enclosed charge is λL, giving E(2π r L) = λL/ε0, hence E = λ/(2π ε0 r). The direction is radial: away from the line if the line’s charge is positive, toward the line if it is negative. If you take the outward radial unit vector as the reference direction, the vector form is E = (λ/(2π ε0 r)) r_hat. If you prefer inward direction as positive, you can write E = -λ/(2π ε0 r) with the inward direction specified. This matches the given expression when the direction is toward the line.