A parallel plate capacitor C with plates of unit area and...

We have, the time constant, $τ = RC ' \ldots \left(i\right)$

Now, $C ' = \dfrac{{C}_{1}{C}_{2}}{{C}_{1} + {C}_{2}}$

$= \dfrac{\left(\dfrac{A{ε}_{0}}{d - x}\right)\left(\dfrac{KA{ε}_{0}}{x}\right)}{\dfrac{A{ε}_{0}}{d - x} + \dfrac{KA{ε}_{0}}{x}}$

$∴ C ' = \dfrac{KA{ε}_{0}}{x + K ( d - x )}$

Putting the value of $C '$ in Equation. (i), we get

$τ = \dfrac{RKA{ε}_{0}}{\dfrac{d}{3} - vt + k\left(d - \dfrac{d}{3} + vt\right)}\left[∵ x = \dfrac{d}{3} - vt\right]$

Given, $A = 1$ and $K = 2$

$∴ τ = \dfrac{3 \times 2{ε}_{0} \times R}{d - 3vt + 6d - 2d + 6vt} = \dfrac{6R{ε}_{0}}{5d + 3vt}$