Separation between the plates of a parallel plate capacitor is d and the area of each plate is A. When a slab of material of dielectric constant k and thickness $t(t) < d)$ is introduced between the plates, its capacitance becomes
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Potential difference between the plates V = V air + V medium
$= \frac{\sigma}{\varepsilon_0} \times (d - t) + \frac{\sigma}{K \varepsilon_0} \times t$
⇒ ⇒ $V = \frac{\sigma}{\varepsilon_0} \left(d - t + \frac{t}{K}\right)$
$= \frac{Q}{A \varepsilon_0} \left(d - t + \frac{t}{K}\right)$
Hence capacitance $c = \frac{Q}{V} = \frac{Q}{\frac{Q}{A \varepsilon_0} \left(d - t + \frac{t}{K}\right)}$
$= \frac{\varepsilon_0 A}{\left(d - t + \frac{t}{k}\right)} = \frac{\varepsilon_0 A}{d - t \left(1 - \frac{1}{k}\right)}$
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