The separation between the plates of a parallel-plate capacitor is 0.500 cm and its plate area is `100 cm^2.`A 0.400cm thick metal plate is inserted into the gap with its faces parallel to the plates .Show that the capacitance of the assembly is independent of the position of the metal plate within the gap and find its value.

Explanation

To solve the problem, we need to find the capacitance of a parallel-plate capacitor with a metal plate inserted between its plates. The capacitance will be shown to be independent of the position of the metal plate within the gap. Let's break down the solution step by step.

### Step 1: Understand the Configuration
We have a parallel-plate capacitor with:
- Plate separation \( D = 0.500 \, \text{cm} = 0.005 \, \text{m} \)
- Plate area \( A = 100 \, \text{cm}^2 = 100 \times 10^{-4} \, \text{m}^2 = 0.01 \, \text{m}^2 \)
- A metal plate of thickness \( t = 0.400 \, \text{cm} = 0.004 \, \text{m} \) inserted between the plates.

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