Question

The field at a distance r from a long string of charge per unit length $\lambda$ is :

• A. $k\frac{\lambda }{r^2}$
• B. $k\frac{\lambda }{r}$
• C. $k\frac{\lambda }{2r}$
• D. $k\frac{2\lambda }{r}$

Answer 1

The correct option is D $k\frac{2\lambda }{r}$

Consider a surface of a cylinder of radius $r$ and length $l$ as Gaussian surface. Apply Gauss's law, $\overrightarrow{E}.\overrightarrow{ds}=\frac{Q_{en}}{{ϵ}_0}$ where $Q_{en}$ is the charge enclosed by the Gaussian surface. here $Q_{en}=\lambda l$

thus, $E.\left(2\pi rl\right)=\frac{\lambda l}{{ϵ}_0}$

$\therefore E=\frac{\lambda }{2\pi {ϵ}_{0}r}=k\frac{2\lambda }{r}$ where $k=\frac{1}{4\pi {ϵ}_{0}r}$