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A capacitor is to be designed to operate, with constant capacitance, even when temperature varies. The distance between plates is adjusted by spacer to compensate temperature effect. Let $α_{1} and α_{2}$ be the coefficient of thermal expansion of plate and spacer respectively. Find the ratio of $\frac{α_{2}}{α_{1}}$ , So that no change in capacitance of capacitor with change in temperature.

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answer is 2.

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Detailed Solution

$\begin{array}{l} C = \frac{ϵ_{0} A}{x}; \frac{dC}{dT} = ϵ_{0} [\frac{1}{x} \frac{dA}{dT} - \frac{A}{x^{2}} \frac{dx}{dT}] \\ dA = A 2 α_{1} dT; dx = {xα}_{2} dT \end{array}$

For no change in capacitance

$dA = dx$

$2 α_{1} = α_{2} \Rightarrow \frac{α_{2}}{α_{1}} = 2$

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