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+

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Correct answer: B. 5

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 x and C_d=( _0 L)/(d)[x+ ((x)/(L))^ 1+α (1+α) · (1)/(L) ]_0^L and C_d=( _0 L)/(d)[L+(L)/(1+α)] _d=( _0 L^2)/(d)((2+α)/(1+α)) 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+α)/(1+α)))/(( _0 L^2)/(d))=(7)/(6) and (2+α)/(1+α)=(7)/(6) 12+6 α=7+7 α and α=5 array

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