Between the plates of a parallel-plate capacitor there is a metallic plate whose thickness takes up `eta = 0.60` of the capacitor gap. When that plate is absent the capacitor has a capacity `C = 20 nF`. The capacitor is connected to a `dc` voltage source `V = 100 V`. The metallic plate is slowly extraced from the gap. Find :
(a) the energy increment of the capacitor,
(b) the mechanical work performed in the process of plate extraction.

Explanation

To solve the problem step by step, we will break it down into two parts: (a) finding the energy increment of the capacitor and (b) calculating the mechanical work performed during the extraction of the metallic plate.

### Part (a): Energy Increment of the Capacitor

1. **Understanding the Initial Conditions**:
- The capacitance of the capacitor when the metallic plate is absent is given as \( C = 20 \, \text{nF} = 20 \times 10^{-9} \, \text{F} \).
- The voltage across the capacitor is \( V = 100 \, \text{V} \).

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