Gauss's Law Applications

Applying Gauss's law to a coaxial cylindrical surface around an infinite line charge yields $E=\frac{\lambda }{2\pi {\epsilon }_{0}r}$, where $\lambda$ is the linear charge density and $r$ the perpendicular distance, a standard derivation in NCERT Class 12, Chapter 1 (Electric Charges and Fields). The curved area of the Gaussian cylinder grows in proportion to $r$, so the field must fall as $1/r$ to keep the enclosed charge constant. Therefore doubling the distance halves the field. The option inverse-square is wrong because that applies to a point charge whose spherical surface area grows as $r^2$, not to a line whose cylindrical area grows as $r$. The option independent of distance is wrong because that is the behaviour of an infinite charged sheet, not a line. The option directly proportional is wrong because moving away from a charge must weaken, not strengthen, its field. Only the curved surface of the chosen cylinder contributes to the flux, because the field runs parallel to the flat end caps and pierces them not at all. A demonstration with a charged wire confirms this behaviour: bringing a detector twice as close roughly doubles the reading, matching the inverse-first-power law and confirming the correct radial dependence expected from the geometry.