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

Step-by-step solution

Here the capacitance, (C= A k ₀ d ) (C_ K₁ , C_ K₂ , C_ K₃ ) are in parallel combination and they are connected in series with ( C _ K ₄ ) Here, ( C _ K ₁ = ( A / 3) K ₁ ₀ ~d / 2 = 2 ~K ₁ 3 C ), ( C _ k ₂ = ( A / 3) k ₂ ₀ ~d / 2 = 2 k ₂ 3 C ) (C_ K₃ = ( A / 3) k ₃ ₀ ~d / 2 2 k ₃ C 3 C ) ( C _ K ₄ = ( A ) k ₄ ₀ ~d / 2 =2 k ₄ C ) Now the equivalent capacitance for the combination of four capacitors is ( aligned & 1 C_ c q = 1 (C_ k₁ +C_ k₂ +C_ k₃ ) + 1 C_ k₄ & C_ e q =k C aligned ) ( 1 k C = 3 2 C [ 1 k₁+k₂+k₃ ]+ 1 2