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 figure below. If a single dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectric constants K is given by

Step-by-step solution

The given capacitor can be considered as a combination of four smaller capacitors. The top half consists of three capacitors in parallel, each with area A/3 and thickness d/2 . Their capacitances are: C₁ = k₁ ₀ (A/3) d/2 = 2k₁ ₀ A 3d C₂ = k₂ ₀ (A/3) d/2 = 2k₂ ₀ A 3d C₃ = k₃ ₀ (A/3) d/2 = 2k₃ ₀ A 3d The equivalent capacitance of this top part is: C_ top = C₁ + C₂ + C₃ = 2 ₀ A 3d (k₁ + k₂ + k₃) The bottom half is a single capacitor with area A and thickness d/2 . Its capacitance is: C₄ = k₄ ₀ A d/2 = 2k₄ ₀ A d The to