The space between the plates of a parallel plate capacitor is filled with a dielectric whose dielectric constant varies with distance as per the relation, K(x) = Ko + λx (λ = a constant). The capacitance C, of this capacitor, would be related to its vacuum capacitance Co as per the relation:

Capacitance of the vacuum capacitor Co = Aεo/d where A is the area of the plates of capacitor. Dielectric constant of the medium at a distance x from first plate is given by K(x) = Ko + λx Capacitance of capacitor of thickness dx, dC = K(x)Aεo dx = Aεo(Ko + λx)dx Total capacitance 1/C = ∫dodx/Aεo(Ko + λx) Or 1/C = 1/Aεo × ln(Ko + λx)/λ |d0 Or 1/C = 1/Aεoλ ln(1 + λd/Ko) Or C = Aεoλ ln(1 + λd/Ko) ⇒ C = λdln(1 + λd/Ko)Co