A parallel plate capacitor is made of two square plates of side $a,$ separated by a distance $d (d ≪ a) .$ The lower triangular portion filled with a dielectric of dielectric constant $K,$ as shown in the figure. Capacitance of this capacitor is:

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TestHub · Accepted Answer

Consider a small element at a distance x. It is a two capacitors in series.

$d c = \frac{d c_{1} \times d c_{2}}{d c_{1} + d c_{2}}$

$d c = \frac{ε_{0} . a d x}{\frac{x t a n θ}{k} + \frac{d - x t a n θ}{1}}$

$∴ d c = \frac{ε_{0} K . a d x}{- x \tan θ (K - 1) + d . k}$

$∴ C = \int_{0}^{a} \frac{ε_{0} K . a d x}{- x \tan θ (K - 1) + d . k}$

$= \frac{ε_{0} k . a}{\tan θ (K - 1)} l n \frac{- a \tan θ (K - 1) + d . k}{d . K}$

$= \frac{ε_{0} K a^{2}}{d (K - 1)} l n K$

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