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(C) The electric flux through a surface is related to the solid angle subtended by the surface at the line charge. Alternatively,we can use the concep

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$6$

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(C) The electric flux through a surface is related to the solid angle subtended by the surface at the line charge. Alternatively,we can use the concept of symmetry. The rectangular surface has a width $a$ and length $L$. The distance from the line charge to the center of the rectangle is $d = \frac{\sqrt{3}}{2} a$.The angle $	heta$ subtended by the width $a$ at the line charge is given by $	an(	heta/2) = \frac{a/2}{d} = \frac{a/2}{(\sqrt{3}/2)a} = \frac{1}{\sqrt{3}}$.Thus,$	heta/2 = 30^{\circ}$,which means $	heta = 60^{\circ}$.The total angle around the line charge is $360^{\circ}$. The number of such identical rectangular surfaces required to enclose the line charge completely is $n = \frac{360^{\circ}}{60^{\circ}} = 6$.According to Gauss's Law,the total flux through a closed surface enclosing a charge $q_{enclosed}$ is $\frac{q_{enclosed}}{\varepsilon_0}$. For a length $L$ of the line charge,the enclosed charge is $q = \lambda L$.Since the flux is distributed equally among the $6$ surfaces,the flux through one surface is $\phi = \frac{\lambda L}{6 \varepsilon_0}$.Comparing this with the given expression $\frac{\lambda L}{n \varepsilon_0}$,we get $n = 6$.

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