Four identical rectangular plates with length, $$l = 2$$ cm and breadth, $$b = \frac{3}{2}$$ cm are arranged as shown in figure. The equivalent capacitance between $$A$$ and $$C$$ is $$\frac{x\varepsilon_0}{d}$$. The value of $$x$$ is ________. (Round off to the Nearest Integer)

We need to find the equivalent capacitance between terminals $$A$$ and $$C$$ for the given system of four identical rectangular plates.

1. Analyze the Plate Dimensions and Geometry

From the layout shown on the page, the system consists of four parallel plates ($$A$$, $$B$$, $$C$$, and $$D$$) separated by a uniform distance $$d$$. Notice the wire connection: plate $$B$$ is connected directly to plate $$D$$ by a conducting wire.

The area ($$Area$$) of each rectangular plate is calculated from its length ($$l$$) and breadth ($$b$$):

• Length, $$l = 2\text{ cm} = 2 \times 10^{-2}\text{ m}$$
• Breadth, $$b = \frac{3}{2}\text{ cm} = 1.5 \times 10^{-2}\text{ m}$$
• $$Area = l \times b = 2 \times \frac{3}{2} = 3\text{ cm}^2 = 3 \times 10^{-4}\text{ m}^2$$

The capacitance ($$C_0$$) between any two adjacent parallel surfaces is:

$$C_0 = \frac{\varepsilon_0 \cdot Area}{d} = \frac{3 \cdot \varepsilon_0}{d}$$

2. Determine the Equivalent Capacitor Network

The four consecutive plates form three separate parallel-plate capacitor regions in the gaps between them:

1. Capacitor 1 (between plates A and B): Connected between terminal $$A$$ and node $$B$$.
2. Capacitor 2 (between plates B and C): Connected between node $$B$$ and terminal $$C$$.
3. Capacitor 3 (between plates C and D): Connected between terminal $$C$$ and node $$D$$.

Since plates $$B$$ and $$D$$ are joined by a wire, they share the same electrical potential (they form a single common node). Consequently:

• Capacitor 2 is connected between $$C$$ and $$B$$.
• Capacitor 3 is connected between $$C$$ and $$D$$ (which is electrically the same as $$B$$).

This means Capacitor 2 and Capacitor 3 are connected in parallel with each other across the nodes $$C$$ and $$B$$. Their combined capacitance is:

$$C_{23} = C_0 + C_0 = 2C_0$$

Now, this parallel combination