A parallel plate capacitor of capacitance C has spacing d between two plates having area A . The region between the plates is filled with N dielectric layers, parallel to its plates, each with thickne

Question

A parallel plate capacitor of capacitance $C$ has spacing d between two plates having area $A .$ The region between the plates is filled with $N$ dielectric layers, parallel to its plates, each with thickness $δ = \frac{d}{N} .$ The dielectric constant of the $m^{t h}$ layer is $K_{m} = K (1 + \frac{m}{N}) .$ For a very large $N (> 10^{3}),$ the capacitance $C$ is $α (\frac{K \in_{0} A}{d l n 2}) .$ The value of $α$ will be____
$[\in_{0}$ is the permittivity of free space]

TestHub · Accepted Answer

Let m t h layer is at a distance ‘ x ’ from left plate. Also, let there be a small strip of thickness 'dx' having small capacitance ′ ′ d 1 C ′ ′ with medium ′ ′ K m ′ ′ in between. ⇒ x m = D N Also, d 1 C = d x K m ε 0 A = d x K m ε 0 A = d x K ε 0 A 1 + m N = d x K ε 0 A 1 + x D ⇒ 1 C e q = ∫ d 1 C = ∫ 0 D D d x K ε 0 A D + x 1 C e q = D K ε 0 A l n 2 C e q = K ε 0 A D l n 2 ⋅ Therefore α = 1

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