Question

Three identical parallel plate air capacitors are connected in series to a battery of e.m.f. V. If two capacitors are completely filled with dielectric material of constant k, then potential difference across the third capacitor will become

• A. \(\frac{k}{V(k+2)}\)
• B. \(\frac{kV}{(k+2)}\)
• C. \(\frac{k-2}{kV}\)
• D. \(\frac{V}{k(k+2)}\)

Hint:

In series all capacitors carry the same charge, so voltage is inversely proportional to capacitance.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Each capacitor has capacitance \(C\). Filling two with dielectric makes them \(kC\) each, and the third stays \(C\).

Step 2: Series combination
\(\dfrac{1}{C_{eq}}=\dfrac{2}{kC}+\dfrac{1}{C}=\dfrac{k+2}{kC}\), so \(C_{eq}=\dfrac{kC}{k+2}\).
Charge \(q=C_{eq}V=\dfrac{kCV}{k+2}\).

Step 3: Voltage across the third
\(V_{3}=\dfrac{q}{C}=\dfrac{kV}{k+2}\)

Final Answer:
The potential difference across the third capacitor is \(\frac{kV}{k+2}\). \[ \boxed{\text{(B) }\dfrac{kV}{k+2}} \]