A long cylindrical shell of radius $a=0.10,\text{m}$ carries a uniform surface charge density $\sigma =5.0,\mu {\text{C/m}}^2$. A coaxial cylindrical Gaussian surface of radius $r=0.20,\text{m}$ is drawn outside the shell. Calculate the electric flux through this Gaussian surface per unit length of the cylinder.

Step-by-step solution

1. Step 1: For a cylindrical shell, the charge per unit length is $lambda=sigma(2pia)$.
2. Step 2: Compute $\lambda$: $2\pi a=2\pi \times 0.10=0.628,\text{m}$; thus $\lambda =5.0\times 10^{-6}\times 0.628=3.14\times 10^{-6},\text{C/m}$.
3. Step 3: By Gauss’s law, the electric flux per unit length through the Gaussian surface is $Phi/L=lambda/varepsilo{n}_0$.
4. Step 4: Substitute ${\epsilon }_0=8.854\times 10^{-12},{\text{C}}^2{\text{/N m}}^2$: $\Phi /L=3.14\times 10^{-6}/8.854\times 10^{-12}\approx 3.55\times 10^5,{\text{N m}}^2\text{/C m}$.
5. Step 5: Express the result in standard form: $3.5\times 10^5,{\text{N m}}^2\text{/C m}$.

Concept tested: Gauss's law for cylindrical shell

Published 7 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked