Question

Two parallel plate air capacitors of same capacity '\(C\)' are connected in series to a battery of emf '\(E\)'. Then one of the capacitors is completely filled with dielectric material of constant '\(K\)'. The change in the effective capacity of the series combination is

• A. \(\frac{C}{2}\left[\frac{K-1}{K+1}\right]^{2}\)
• B. \(\frac{C}{2}\left[\frac{K+1}{K-1}\right]^{2}\)
• C. \(\frac{2}{C}\left[\frac{K-1}{K+1}\right]\)
• D. \(\frac{C}{2}\left[\frac{K-1}{K+1}\right]\)

Hint:

Find the series capacitance before and after filling one capacitor with a dielectric and take the difference.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
For two capacitors in series, \(C_{eq}=\dfrac{C_1C_2}{C_1+C_2}\). Filling with a dielectric multiplies that capacitor's capacity by \(K\).

Step 2: Before
\(C_{eq}=\dfrac{C\cdot C}{2C}=\dfrac{C}{2}\)

Step 3: After
\(C_{eq}'=\dfrac{C\cdot KC}{C+KC}=\dfrac{KC}{K+1}\)

Step 4: Calculation
\(\Delta C=\dfrac{KC}{K+1}-\dfrac{C}{2}=\dfrac{C(2K-K-1)}{2(K+1)}=\dfrac{C}{2}\dfrac{K-1}{K+1}\)

Step 5: Compare the options
This is the last option.

Final Answer:
The effective capacity increases by (C/2)(K - 1)/(K + 1). \[ \boxed{\text{(D) }\dfrac{C}{2}\left[\dfrac{K-1}{K+1}\right]} \]