An infinitely long uniform line charge distribution of charge per unit length lies parallel to the -axis in the plane at (see figure). If the magnitude of the flux of the electric field through the rectangular surface lying in the plane with its cent

Question

An infinitely long uniform line charge distribution of charge per unit length $λ$ lies parallel to the $y$ -axis in the $y - z$ plane at $z = \frac{\sqrt{3}}{2} a$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $A B C D$ lying in the $x - y$ plane with its centre at the origin is $\frac{λ L}{n ε_{0}}$ ( $ε_{0}$ = permittivity of free space), then the value of $n$ is [2015]

Smart Achievers · Accepted Answer

(6)

From the figure $\tan θ = \frac{B C}{O B} = \frac{a / 2}{\sqrt{3} a / 2} = \frac{1}{\sqrt{3}}$

$∴$ $θ = 30 °$

Electric flux through the complete cylinder by Gauss's theorem

$ϕ_{cylinder} = \frac{q_{in}}{ε_{0}} = \frac{λ L}{ε_{0}}$

(where L = length of cylinder)

$∴$ Electric flux passing through the cylindrical surface i.e., for $60^{o}$ angle $= \frac{λ L}{6 ε_{0}}$

Hence, $n = 6$

Solution

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Solution

Ans. (6) From the figure tanθ=BCOB=a/23a/2=13 ∴ θ=30° Electric flux through the complete cylinder by Gauss's theorem ϕcylinder=qinε0=λLε0 (where L = length of cylinder) ∴ Electric flux passing through the cylindrical surface i.e., for 60o angle=λL6ε0 Hence, n=6

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