1. Concept: The capacitance of a parallel plate setup relies on the ratio of its surface area to its gap distance. Manipulating both variables simultaneously multiplies their respective effects.
2. Formula: $$ C = \frac{\epsilon_0 A}{d} $$
3. Solution: - Let initial capacitance be \( C = \frac{\epsilon_0 A}{d} \).
- The area is doubled: \( A' = 2A \).
- The distance is halved: \( d' = d / 2 \).
- Substitute into formula: \( C' = \frac{\epsilon_0 (2A)}{d/2} \).
- The denominator's fraction flips up: \( C' = 2 \times 2 \left( \frac{\epsilon_0 A}{d} \right) = 4C \).
4. Why other options are incorrect: Option D occurs if Area is halved and Distance is halved (canceling out). Option C occurs if only Area is doubled, or only Distance is halved. Combining them yields a 4x multiplier.
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