Question

Uniformly charged long cylinder has volume charge density $\rho$. Find the electric field at a distance x < R from the axis of the cylinder.

• A. $\frac{\rho x}{{\epsilon }_0}$
• B. $\frac{\rho x}{2{\epsilon }_0}$
• C. $\frac{\rho x}{3{\epsilon }_0}$
• D. $\frac{\rho x}{4{\epsilon }_0}$

Answer 1

The correct option is B $\frac{\rho x}{2{\epsilon }_0}$

Assume a hypothetical cylinder of radius $x$ and length $l$. Apply Gauss's law, $\oint E.ds=\frac{Q_{en}}{{ϵ}_o}$ where $Q_{en}$ is the charge enclosed by the hypothetical cylinder. here $Q_{en}=\rho \times$ volume of the hypothetical cylinder.

thus $Q_{en}=\rho \times \pi x^{2}l$

now, $E.\left(2\pi xl\right)=\frac{\rho \times \pi x^{2}l}{{ϵ}_0}$

$E=\frac{\rho x}{2{ϵ}_0}$