A parallel-plate air-spaced capacitor of capacitance C and a resistor of resistance R are in series with an ac source of frequency $ω$ . The voltage-drop across R is V_R. Half the capacitor is now filled with a material of dielectric constant K but the remainder of the circuit remains unchanged. The voltage-drop across R is now 2V_R. Neglecting edge effects, the dielectric constant K is $(\frac{m}{\sqrt{1 - n R^{2} C^{2} ω^{2}}} - 1)$ . Find the value of m + n.

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Detailed Solution

When half the capacitor is filled with the material, the capacitance of the capacitor (two capacitors in parallel) becomes

$C^{'} = \frac{C}{2} + \frac{K C}{2}$

The voltage across R is $(\frac{V}{Z}) R$ , where V is the voltage of the ac source and Z is the total impedance of the circuit. Thus

$V_{n e w} = V_{o l d}$
$\Rightarrow \frac{R}{\sqrt{R^{2} + \frac{1}{ω^{2} C^{' 2}}}} = 2 \frac{R}{\sqrt{R^{2} + \frac{1}{ω^{2} C^{2}}}}$

$\Rightarrow K = (\frac{4}{\sqrt{1 - 3 R^{2} C^{2} ω^{2}}} - 1)$

$\begin{array}{l} T h u s, m = 4, n = 3 \\ \Rightarrow m + n = 7 \end{array}$

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