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Capacitor with multiple dielectrics
Concepts tested here
- composite-dielectric-capacitor
All Questions
2016 Phase II 1 question
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A parallel-plate capacitor of area A, plate separation d and capacitance C is filled with four dielectric materials having dielectric constants $k_1$, $k_2$, $k_3$ and $k_4$ as shown in the figure below. If a single dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectric constant k is given by:
Three side-by-side dielectrics add in parallel; that block is in series with $k_4$.
$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$
$\frac{d}{A\varepsilon_0k} = \frac{1}{\frac{A}{3}\varepsilon_0\frac{(k_1 + k_2 + k_3)}{d/2}} + \frac{1}{\frac{A\varepsilon_0k_4}{d/2}}$
$\Rightarrow \frac{2}{k} = \frac{3}{k_1 + k_2 + k_3} + \frac{1}{k_4}$