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An infinite plane sheet of charge having uniform surface charge density $+\sigma_5 \mathrm{C} / \mathrm{m}^2$ is placed on $\mathrm{x}-\mathrm{y}$ plane. Another infinitely long line charge having uniform linear charge density $+\lambda_e \mathrm{C} / \mathrm{m}$ is placed at $z=4 \mathrm{~m}$ plane and parallel to $y$-axis. If the magnitude values $\left|\sigma_s\right|=2\left|\lambda_{\mathrm{e}}\right|$ then at point $(0,0,2)$, the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $\pi \sqrt{\mathrm{n}}: 1$. The value of $n$ is
$16$
$20$
$23$
$30$
Solution
$\frac{\mathrm{E}_S}{\mathrm{E}_{\ell}}=\frac{\sigma}{2 \epsilon_0} \times \frac{2 \pi \epsilon_0 \mathrm{r}}{\lambda}$
$=\frac{\pi \times \sigma \mathrm{r}}{\lambda}$
$=\frac{\pi \times 2 \lambda \times 2}{\lambda}=\frac{4 \pi}{1}$
$\therefore \mathrm{n}=16$
