The figure shows a parallel-plate capacitor having square plates of edge a and plate-separation d. The gap between the plates is filled with a directrices of dielectric constant k which varies as $K = K_{0} + αx$ parallel to an edge as where K and $α$ are constants and x is the distance from the left end. Calculate the capacitance.

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$\frac{ε_{0} a^{2}}{d} (K_{0} + \frac{aα}{2})$

$\frac{ε_{0} a}{d} (K_{0} + \frac{aα}{2})$

$\frac{ε_{0} a}{d} (K_{0} - \frac{aα}{2})$

$\frac{ε_{0} a^{2}}{d} (K_{0} - \frac{aα}{2})$

answer is A.

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Detailed Solution

$\int d C = \int_{0}^{a} \frac{(K_{0} + αx) ϵ_{0} a}{d} \times dx$

$= \frac{ε_{0} a}{d} (K_{0} a + \frac{α a^{2}}{2})$

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