Question

Four identical rectangular plates with length, $l=2\text{cm}$ and breadth, $b=3/2\text{cm}$ are arranged as shown in the figure. The equivalent capacitance between $P$ and $R$ is $\frac{x\times {10}^{-2}{ϵ}_0}{d}$ where $d$ is the distance between the plates in $\text{cm}$. The value of $x$ is ______.

$($Round off to the nearest integer$)$

Answer 1

The equivalent circuit can be drawn as,


Here,
$C=\frac{{ϵ}_{0}A}{d}$

Equivalent capacitance between $P$ and $R$ is given by,

$C_{eq}=\frac{2C\times C}{2C+C}=\frac{2}{3}C=\frac{2}{3}\times \frac{{ϵ}_{0}A}{d}$

$\Rightarrow C_{eq}=\frac{2}{3}\times \frac{{ϵ}_0}{d\times {10}^{-2}}\times (2\times \frac{3}{2})\times {10}^{-4}$
($\because$ Area of the plate = length $\times$ breadth)

$\Rightarrow C_{eq}=2\times {10}^{-2}\times \frac{{ϵ}_0}{d}=\frac{x\times {10}^{-2}{ϵ}_0}{d}$

$\Rightarrow x=2$