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Figure shows a long thread along the axis of a long hollow cylinder. The uniform charge per unit length of thread is $λ$ while uniform surface charge density of the cylinder is $- σ$ . The radius of the cylinder is $R$ . Total flux associated with a coaxial cylinder of finite length and radius twice that of the charged cylinder is zero, if the magnitude of $σ$ is

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$\frac{λ}{π R}$

$\frac{λ}{8 π R}$

$\frac{λ}{4 π R}$

$\frac{λ}{2 π R}$

answer is B.

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Detailed Solution

Flux through a Gaussian surfaces (cylindrical shape) of length $l$ and radius $2 R$

$\oint \vec{E} . d \vec{s} = \frac{\sum q_{i n}}{ε_{0}} \oint \vec{E} . d \vec{s} + 2 \oint \vec{E} . d \vec{s} = \frac{λ l + (- σ) 2 π R l}{ε_{0}}$

Total flux $= \frac{l}{ε_{0}} [λ - σ 2 π R] = 0$

$σ = \frac{λ}{2 π R}$

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