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Figure shows a system of inductor and parallel plate capacitor made of 2 parallel circular plates of area A and filled with dielectric liquid of dielectric constant K as shown. A small leak develops in capacitor and liquid starts to fill the inductor of same dimensions having n turns per unit length. Find the ratio of magnitude of initial to final reactance of circuit after liquid fills the inductor completely.
$G i v e n : ω^{2} A^{2} n^{2} = c^{2}$
$ω \to angular frequency of AC$
$c \to speed of light$
$and μ_{r} \to relative permeability of liquid$

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$\frac{1}{K} \frac{(1 - K)}{(1 - μ_{r})}$

$\frac{(1 + μ_{r}) K}{(1 + K)}$

$K \frac{(K - 1)}{(μ_{r} + 1)}$

$\frac{1}{K} \frac{(K + 1)}{(1 - μ_{r})}$

answer is B.

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Detailed Solution

$\frac{X_{i}}{X_{f}} = |\frac{\frac{1}{ω C_{i}} - ω L_{i}}{\frac{1}{ω C_{F}} - ω L_{f}}| \frac{= \frac{l}{ω K ε_{0} A} - ω μ_{0} n^{2} A l}{\frac{l}{ω ε_{0} A} - ω μ_{0} μ_{r} n^{2} A l} = \frac{1}{K} (\frac{1 - K}{1 - μ_{r}})$

$[U \sin g ε_{0} μ_{0} = \frac{1}{C^{2}}] a n d [ω^{2} A^{2} n^{2} = C^{2}]$

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