Question

A parallel plate air capacitor has capacitance \(C_P\). It is equally filled with parallel layers of materials of dielectric constants \(\mathrm{K}_1\) and \(\mathrm{K}_2\). Now its capacity becomes \(C_K\). The ratio \(C_P\) to \(C_K\) is

• A. \(\mathrm{K}_{1}+\mathrm{K}_{2}\)
• B. \(\frac{K_{1}+K_{2}}{K_{1}K_{2}}\)
• C. \(\frac{K_{1}+K_{2}}{2K_{1}K_{2}}\)
• D. \(\frac{2K_{1}K_{2}}{K_{1}+K_{2}}\)

Hint:

Two layers of equal thickness stacked between the plates act as two capacitors in series.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Air capacitor: \(C_P=\dfrac{\varepsilon_0A}{d}\).
Each dielectric layer has thickness \(\dfrac{d}{2}\).

Step 2: Layer capacitances
\(C_1=\dfrac{K_1\varepsilon_0A}{d/2}=2K_1C_P\) and \(C_2=2K_2C_P\).

Step 3: Series combination
\(C_K=\dfrac{C_1C_2}{C_1+C_2}=\dfrac{2K_1K_2}{K_1+K_2}C_P\).

Step 4: Ratio
\(\dfrac{C_P}{C_K}=\dfrac{K_1+K_2}{2K_1K_2}\).

Final Answer:
The ratio is (K1 + K2) over 2 K1 K2. \[ \boxed{\text{(C) }\dfrac{K_{1}+K_{2}}{2K_{1}K_{2}}} \]