Question 287 of 311

A long cylindrical volume contains a uniformly distributed charge of density $\rho C{m}^{-3}$. The electric field inside the cylindrical volume at a distance $x=\dfrac{2{\epsilon }_{0}}{\rho }m$ from its axis is _____ $V{m}^{-1}$.

Your answerCheck answer

Show answer and solution▾

Answer: 1

Given here, Volume charge density $=\rho C{m}^{-3}$.
Consider a Gaussian cylindrical surface of height $h$ and radius $x$.
Let $q$ charge is enclosed by the cylinder of radius $x$.

From Gauss's law,
$\int EdS\cos 0=\dfrac{q}{{\epsilon }_{0}}$
Here, $q=$volume charge density$\times$volume of imaginary cylinder.
$\Rightarrow E\left(2\pi xh\right)=\dfrac{\rho \times \pi {x}^{2}h}{{\epsilon }_{0}}$
$\Rightarrow E=\dfrac{\rho x}{2{\epsilon }_{0}}$
$\Rightarrow E=\dfrac{\rho }{2{\epsilon }_{0}}\times \dfrac{2{\epsilon }_{0}}{\rho }=1$
Thus, electric field is $1V{m}^{-1}$.