A parallel plate capacitor with plate area ‘A’ and distance of separation d is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as : ε(x) = ε0 + kx, for ε(x) = ε0 + k(d

Engineering

Physics

Parallel Plate Capacitor

Question

A parallel plate capacitor with plate area ‘A’ and distance of separation d is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as :

ε(x) = ε₀ + kx, for $(0 < x \leq \frac{d}{2})$

ε(x) = ε₀ + k(d – x), for $(\frac{d}{2} \leq x \leq d)$

$\frac{kA}{2 ℓn (\frac{2 ε_{0} + kd}{2 ε_{0}})}$

$\frac{kA}{2} ℓn (\frac{2 ε_{0}}{2 ε_{0} - kd})$

${(ε_{0} + \frac{kd}{2})}^{2 / kA}$

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$\frac{1}{C_{eq}} \Rightarrow \int \frac{dx}{Aε}$

$\Rightarrow \int_{0}^{d / 2} \frac{dx}{A (ε_{0} + kx)} + \int_{d / 2}^{d} \frac{dx}{A [ε_{0} + k (d - x)]}$

$\Rightarrow \frac{1}{A} {\frac{1}{k} {[ℓn (ε_{0} + kx)]}_{0}^{d / 2} + \frac{1}{k} {[ℓn (ε_{0} + k (d - x))]}_{d / 2}^{d}}$

$\Rightarrow \frac{1}{Ak} \times 2 ℓn \frac{ε_{0} + \frac{kd}{2}}{ε_{0}} \Rightarrow C_{eq} = \frac{kA}{2 ℓn (\frac{2 ε_{0} + kd}{2 ε_{0}})}$