Two identical parallel plate capacitors, of capacitance C each, have plates of area A, separated by a distance d. The space between the plates of the two capacitors, is filled with three dielectrics, of equal thickness and dielectric constants K1, K2 and K3. The first capacitor is filled as shown in fig. I, and the second one is filled as shown in fig. II. If these two modified capacitors are charged by the same potential V, the ratio of the energy stored in the two, would be (E1 refers to capacitor (I) and E2 to capacitor (II)) :

Engineering

Physics

Parallel Plate Capacitor

Question

Two identical parallel plate capacitors, of capacitance C each, have plates of area A, separated by a distance d. The space between the plates of the two capacitors, is filled with three dielectrics, of equal thickness and dielectric constants K₁, K₂ and K₃. The first capacitor is filled as shown in fig. I, and the second one is filled as shown in fig. II. If these two modified capacitors are charged by the same potential V, the ratio of the energy stored in the two, would be (E₁ refers to capacitor (I) and E₂ to capacitor (II)) :

$\frac{E_{1}}{E_{2}} = \frac{(K_{1} + K_{2} + K_{3}) (K_{2} K_{3} + K_{3} K_{1} + K_{1} K_{2})}{K_{1} K_{2} K_{3}}$

$\frac{E_{1}}{E_{2}} = \frac{K_{1} K_{2} K_{3}}{(K_{1} + K_{2} + K_{3}) (K_{2} K_{3} + K_{3} K_{1} + K_{1} K_{2})}$

$\frac{E_{1}}{E_{2}} = \frac{9 K_{1} K_{2} K_{3}}{(K_{1} + K_{2} + K_{3}) (K_{2} K_{3} + K_{3} K_{1} + K_{1} K_{2})}$

$\frac{E_{1}}{E_{2}} = \frac{(K_{1} + K_{2} + K_{3}) (K_{2} K_{3} + K_{3} K_{1} + K_{1} K_{2})}{9 K_{1} K_{2} K_{3}}$

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$C_{1} = \frac{3 ε_{0} {AK}_{1}}{d}$

$C_{2} = \frac{3 ε_{0} {AK}_{2}}{d}$

$C_{3} = \frac{3 ε_{0} {AK}_{3}}{d}$

$\frac{1}{C_{eq}} = \frac{1}{C_{1}} + \frac{1}{C_{2}} + \frac{1}{C_{3}}$

$\Rightarrow E_{eq} = \frac{3 ε_{0} {AK}_{1} K_{2} K_{3}}{d (K_{1} K_{2} + K_{2} K_{3} + K_{3} K_{1})} . ...... (i)$

$c_{1} = \frac{ε_{0} K_{1} A}{3 d}$

$c_{2} = \frac{ε_{0} K_{2} A}{3 d}$

$c_{3} = \frac{ε_{0} K_{3} A}{3 d}$

$C_{eq}^{'} = C_{1} + C_{2} + C_{3}$

$= \frac{ε_{0} A}{3 d} (K_{1} + K_{2} + K_{3}) . ...... (ii)$