A parallel-plate capacitor of area A, plate separation d and… | NEET 2016 (Phase 2)

Q102NEET 2016 · PHASE 2Difficult

Dielectrics

A parallel-plate capacitor of area A, plate separation d and capacitance C is filled with four dielectric materials having dielectric constants k1, k2, k3 and k4 as shown in the figure. 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

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Answer: (c)

Top half (thickness $d/2$ ): $k_1, k_2, k_3$ side by side in parallel, each taken as area $A/3$ : $C_{top} = \frac{2\epsilon_0 A}{d} \cdot \frac{k_1 + k_2 + k_3}{3}$ .
Bottom half: $C_4 = \frac{2\epsilon_0 A k_4}{d}$ , in series with the top.
$\frac{1}{C} = \frac{d}{2\epsilon_0 A}\left[\frac{3}{k_1 + k_2 + k_3} + \frac{1}{k_4}\right]$ with $C = \frac{k\epsilon_0 A}{d}$
$\frac{2}{k} = \frac{3}{k_1 + k_2 + k_3} + \frac{1}{k_4}$ , so the answer is (c).

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