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Physics-

General

Easy

Question

In the figure a capacitor is filled with dielectrics. The resultant capacitance is

$\frac{2 ε_{0} A}{d} [\frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}}]$
$\frac{ε_{0} A}{d} [\frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}}]$
$\frac{2 ε_{0} A}{d} [k_{1} + k_{2} + k_{3}]$
None of these

The correct answer is: None of these

$C subscript 1 end subscript equals fraction numerator K subscript 1 end subscript epsilon subscript 0 end subscript fraction numerator A over denominator 2 end fraction over denominator open parentheses fraction numerator d over denominator 2 end fraction close parentheses end fraction equals fraction numerator K subscript 1 end subscript epsilon subscript 0 end subscript A over denominator d end fraction$
$C subscript 2 end subscript equals fraction numerator K subscript 2 end subscript epsilon subscript 0 end subscript open parentheses fraction numerator A over denominator 2 end fraction close parentheses over denominator open parentheses fraction numerator d over denominator 2 end fraction close parentheses end fraction equals fraction numerator K subscript 2 end subscript epsilon subscript 0 end subscript A over denominator d end fraction$ and $C subscript 3 end subscript equals fraction numerator K subscript 3 end subscript epsilon subscript 0 end subscript A over denominator 2 d end fraction equals fraction numerator K subscript 3 end subscript epsilon subscript 0 end subscript A over denominator 2 d end fraction$
Now, $C subscript e q end subscript equals C subscript 3 end subscript plus fraction numerator C subscript 1 end subscript C subscript 2 end subscript over denominator C subscript 1 end subscript plus C subscript 2 end subscript end fraction equals open parentheses fraction numerator K subscript 3 end subscript over denominator 2 end fraction plus fraction numerator K subscript 1 end subscript K subscript 2 end subscript over denominator K subscript 1 end subscript plus K subscript 2 end subscript end fraction close parentheses. fraction numerator epsilon subscript 0 end subscript A over denominator d end fraction$

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