Field due to an Infinitely Long Straight Uniformly Charged Wire
This section examines the electric field created by an infinitely long straight wire that carries a linear charge density, denoted as λ. By utilizing Gauss's law, the section calculates the electric field, demonstrating that it is radial and does not vary with the position along the wire’s length.
The setup considers points around the wire, illustrating how they possess mutual symmetry in relation to the charged wire. Consequently, the direction of the electric field is always radial — outward if the wire is positively charged and inward if it is negatively charged.
To determine the field strength, a cylindrical Gaussian surface is employed. Since the electric field is uniform across the curved surface of the cylinder, the flux through the closed surface is given as the product of the electric field and the curved surface area of the cylinder. The charge enclosed by the Gaussian surface is equal to the linear charge density multiplied by the length of the wire that the surface encloses. Then, by applying Gauss's law, the electric field can be expressed mathematically as:
E = λ / (2πε₀r)
where E represents the electric field, ε₀ the permittivity of free space, and r the radial distance from the wire. This formula highlights that the electric field strength decreases inversely with distance from the wire.