A parallel plate capacitor of capacitance C has spacing d between...

Let ${m}^{th}$ layer is at a distance $‘x’$ from left plate. Also, let there be a small strip of thickness 'dx' having small capacitance $′′d$\left(\dfrac{1}{C}\right)′′$with medium$′′{K}_{m}′′$in between.$⇒ \dfrac{x}{m} = \dfrac{D}{N}$Also,$d$\left(\dfrac{1}{C}\right) = \dfrac{dx}{{K}_{m}{ε}_{0}A} = \dfrac{dx}{{K}_{m}{ε}_{0}A} = \dfrac{dx}{K{ε}_{0}A\left(1 + \dfrac{m}{N}\right)} = \dfrac{dx}{K{ε}_{0}A\left(1 + \dfrac{x}{D}\right)}$ $⇒ \dfrac{1}{{C}_{eq}} = ∫ d$\left(\dfrac{1}{C}\right) = \mathop{∫}_{0}^{D}\dfrac{Ddx}{K{ε}_{0}A\left(D + x\right)} \dfrac{1}{{C}_{eq}} = \dfrac{D}{K{ε}_{0}A}\ln 2 {C}_{eq} = \dfrac{K{ε}_{0}A}{Dln2} ⋅$Therefore$\alpha = 1