An infinite plane sheet of charge having uniform surface charge density is placed on the plane. Another infinitely long line charge having uniform linear charge density is placed at m plane and parallel to the y-axis. If the magnitude values then at

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Solution

Q.
An infinite plane sheet of charge having uniform surface charge density $+ σ_{s}$ $C / m^{2}$ is placed on the $x - y$ plane. Another infinitely long line charge having uniform linear charge density $+ λ_{e}$ $C / m$ is placed at $z = 4$ m plane and parallel to the y-axis. If the magnitude values $∣ σ_{s} ∣ = 2 ∣ λ_{e} ∣$ then at point $(0, 0, 2),$ the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $π \sqrt{n} : 1$ . The value of $n$ is ______ . [2024]

Ans.
(16)

${\vec{E}}_{sheet} = \frac{σ_{s}}{2 ε_{0}} \hat{k} = \frac{λ_{e}}{ε_{0}} \hat{k}$

${\vec{E}}_{wire} = \frac{2 k λ_{e}}{r} (- \hat{k}) = \frac{2 k λ_{e}}{2} (- \hat{k}) = k λ (- \hat{k})$

$\frac{| {\vec{E}}_{sheet} |}{| {\vec{E}}_{wire} |} = \frac{λ_{e}}{ε_{0}} \times \frac{1}{k λ_{e}} = \frac{1}{ε_{0} k} = \frac{1}{ε_{0} \times \frac{1}{4 π ε_{0}}} = 4 π$

$\Rightarrow 4 π = π \sqrt{n} \Rightarrow n = 16$

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