Two co-axial conducting cylinders of same length with radii and are kept, as shown in Fig. 1. The charge on the inner cylinder is and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric cons

Question

Two co-axial conducting cylinders of same length $ℓ$ with radii $\sqrt{2} R$ and $2 R$ are kept, as shown in Fig. 1. The charge on the inner cylinder is $Q$ and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric constant $k = 5$ . Consider an imaginary plane of the same length $ℓ$ at a distance $R$ from the common axis of the cylinders. This plane is parallel to the axis of the cylinders. The cross-sectional view of this arrangement is shown in Fig. 2. Ignoring edge effects, the flux of the electric field through the plane is ( $ε_{0}$ is the permittivity of free space): [2025]

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(3)

For symmetry, assume $\sqrt{2} R$ is very large.

Outside cylinder will have zero electric field, so the flux generated on the plate will be due to the inner cylinder only in sections AB and CD.

And $Q$ and $ε_{0}$

Flux through an element will be $d ϕ = \vec{E} \cdot \vec{d S}$

$d ϕ = \frac{2 k λ}{r} d y \cdot ℓ \cdot \cos θ$ ...(i)

From figure

$\cos θ = \frac{R}{r} \Rightarrow r = R \sec θ$

$\tan θ = \frac{y}{R} \Rightarrow y = R \tan θ$ $\Rightarrow d y = R \sec^{2} θ d θ$

$d ϕ = \frac{2 k λ}{R \sec θ} \cdot R \sec^{2} θ \cdot ℓ \cdot \cos θ d θ$

$d ϕ = 2 k λ ℓ d θ$

Therefore, $\int_{0}^{ϕ A B} d ϕ = 2 k λ ℓ \int_{π / 4}^{π / 3} d θ$ $= 2 k λ ℓ [\frac{π}{3} - \frac{π}{4}]$

$ϕ_{A B} = 2 k Q [\frac{π}{12}]$

$ϕ_{A B} = 2 (\frac{1}{4 π ε_{0} ε_{r}}) Q [\frac{π}{12}]$

$ϕ_{A B} = \frac{Q}{120 ε_{0}}$

$ϕ_{plate} = ϕ_{A B} + ϕ_{B C} + ϕ_{C D}$

$ϕ_{plate} = \frac{Q}{120 ε_{0}} + 0 + \frac{Q}{120 ε_{0}} = \frac{Q}{60 ε_{0}}$

Solution

1 $\frac{Q}{30 ε_{0}}$

2 $\frac{Q}{15 ε_{0}}$

3 $\frac{Q}{60 ε_{0}}$

4 $\frac{Q}{120 ε_{0}}$

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Solution

1 Q30ε0

2 Q15ε0

3 Q60ε0

4 Q120ε0

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1 $\frac{Q}{30 ε_{0}}$

2 $\frac{Q}{15 ε_{0}}$

3 $\frac{Q}{60 ε_{0}}$

4 $\frac{Q}{120 ε_{0}}$