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

Question

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

k = k₁ + k₂ + k₃ + 3k₄

k = ${2 \over 3}$ (k₁ + k₂ + k₃) + 2k₄

${2 \over k}$ = ${3 \over {{k_1} + {k_2} + {k_3}}} + {1 \over {{k_4}}}$

${1 \over k}$ = ${1 \over {{k_1}}} + {1 \over {{k_2}}} + {1 \over {{k_3}}} + {3 \over {2{k_4}}}$

Solution

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