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Q.

Two point charges q and -q are separated by a distance 2L. Find the flux of the electric field vector across the circle of radius R is shown.

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a

$\frac{2 q}{ϵ_{0}} \{1 - \frac{L}{\sqrt{L^{2} + r^{2}}}\}$

b

$\frac{q}{2 \in_{0}} \{1 - \frac{L}{\sqrt{L^{2} + r^{2}}}\}$

c

$\frac{q}{ϵ_{0}} \{1 - \frac{L}{\sqrt{L^{2} + r^{2}}}\}$

d

Zero

answer is B.

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Detailed Solution

$\begin{array}{l} dϕ = 2 Ecos θ 2 πydy \\ = 2 \cdot \frac{1}{4 {πϵ}_{0}} \frac{q}{(L^{2} + y^{2})} \frac{L}{{(L^{2} + y^{2})}^{1 / 2}} 2 πydy \\ ϕ = \int dϕ = \frac{qL}{ϵ_{0}} \int_{0}^{R} \frac{ydy}{{(L^{2} + y^{2})}^{3 / 2}} \\ = \frac{q}{ϵ_{o}} [1 - \frac{L}{\sqrt{L^{2} + R^{2}}}] \end{array}$

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Two point charges q and -q are separated by a distance 2L. Find the flux of the electric field vector across the circle of radius R is shown.