A parallel plate capacitor is made up of stair like structure with a plate area $$A$$ of each stair and that is connected with a wire of length $$b$$, as shown in the figure. The capacitance of the arrangement is $$\frac{x}{15}\frac{\varepsilon_0 A}{b}$$. The value of $$x$$ is ______.

Concept:
All sections are connected between the same two conductors, hence they have the same potential difference ⇒ they are in parallel.

Why distances are b,3b,5b:
Each step increases separation by b:

• First gap = b
• Next gap = b+b+b=3b
• Last gap = b+b+b+b+b=5b

Step 1: Individual capacitances

$$C=\frac{\varepsilon_0A}{d}​$$

$$C_1=\frac{\varepsilon_0A}{b},\quad$$

$$C_2=\frac{\varepsilon_0A}{3b},\quad$$

$$C_3=\frac{\varepsilon_0A}{5b}$$

Step 2: Equivalent capacitance (parallel)

$$C_{\text{eq}}=C_1+C_2+C_3$$

= $$\varepsilon_0A\left(\frac{1}{b}+\frac{1}{3b}+\frac{1}{5b}\right)$$

= $$\frac{\varepsilon_0A}{b}\cdot\frac{23}{15}$$

Step 3: Compare with given form

$$C_{\text{eq}}=\frac{x}{15}\cdot\frac{\varepsilon_0A}{b}$$

$$\Rightarrow x=23$$

Final Answer:

23