capacitors

Question

A parallel plate capacitor has capacitance C. If a dielectric slab of dielectric constant K is inserted filling half the space between plates parallel to plates, the new capacitance is:

RankGuru · Accepted Answer

1. **Analyze the Series Combination**: A parallel plate capacitor with a dielectric slab filling half the thickness acts like two capacitors in series: one with air ($d_1 = d/2$) and one with the dielectric ($d_2 = d/2$).
2. **Individual Capacitances**:
   - $C_1 = \frac{\epsilon_0 A}{d/2} = 2C$
   - $C_2 = \frac{K\epsilon_0 A}{d/2} = 2KC$
3. **Equivalent Capacitance ($C_{eq}$)**: Using the series formula $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2}$:
   $$C_{eq} = \frac{C_1 C_2}{C_1 + C_2} = \frac{(2C)(2KC)}{2C + 2KC} = \frac{4KC^2}{2C(1 + K)} = \frac{2KC}{K + 1}$$

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