Consider a parallel plate capacitor with plates in the shape of square and in X Y -plane. The gap between the plates is filled with dielectric material. The dielectric constant k varies with X -axis as k(x)= [1+ ( x L )^ ] , where is a constant. Let C_d and C_a are capacitance in the presence of dielectric and air, respectively. If the ratio C_d C_a = 7 6 , then the value of must be

Step-by-step solution

Area of square plate, A=L^2 Consider an elemental capacitor at distance x from above of thickness d x . Capacitance of elemental capacitor, d C= k ₀ d A d = k ₀(L d x) d All such elemental capacitors will be in parallel and equivalent capacitance will be sum of aII, so aligned & C_d= ₀ L d ₀^L 1+ ( x L )^a d x & C_d= ₀ L d [x+ ( x L )^ 1+ (1+ ) 1 L ]₀^L & C_d= ₀ L d [L+ L 1+ ] C_d= ₀ L^2 d ( 2+ 1+ ) aligned It is given, C_d C_a = 7 6 where, C_a= capacitance in absence of dielectric. array ccc ₀ L^2 d ( 2+ 1+ ) ₀ L^