A charge is kept at the central point of a cylindrical region. The two edges subtend a half-angle at , as shown in the figure. When , then the electric flux through the curved surface of the cylinder is . If , then the electric flux through the curved s

Question

A charge is kept at the central point $P$ of a cylindrical region. The two edges subtend a half-angle $θ$ at $P$ , as shown in the figure. When $θ = 30 °$ , then the electric flux through the curved surface of the cylinder is $Φ$ . If $θ = 60 °$ , then the electric flux through the curved surface becomes $\frac{Φ}{\sqrt{n}},$ where the value of $n$ is _________. [2024]

Smart Achievers · Accepted Answer

(3)

Solid angle subtended at the centre by the plane surface

$Ω = 2 \times 2 π (1 - \cos θ)$

$= 4 π - 4 π \cos θ$

So, solid angle made by the curved surface $= 4 π - Ω$

$= 4 π - (4 π - 4 π \cos θ)$ $= 4 π \cos θ$

Flux through curved surface $= \frac{Q}{ε_{0}} \cos θ$

$ϕ_{30 °} = ϕ = \frac{4 π \cos 30 °}{4 π} \cdot \frac{Q}{ε_{0}} = \cos 30 ° \frac{Q}{ε_{0}}$

$ϕ_{60 °} = \frac{4 π \cos 60 °}{4 π} \cdot \frac{Q}{ε_{0}} = \cos 60 ° \frac{Q}{ε_{0}}$

$\frac{ϕ_{30 °}}{ϕ_{60 °}} = \frac{\cos 30 °}{\cos 60 °} = \sqrt{3}$

$\frac{ϕ}{ϕ_{60 °}} = \sqrt{3}$ $\Rightarrow ϕ_{60 °} = \frac{ϕ}{\sqrt{3}}$

$∴$ $n = 3$

Solution

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