Question

Five capacitors, each of capacitance C, are connected as shown. The ratio of equivalent capacitance between P and R ($C_{PR}$) and the equivalent capacitance between P and Q ($C_{PQ}$) is

• A. 1:4
• B. 2:3
• C. 3:1
• D. 5:2

Hint:

In symmetric capacitor networks, look for series-parallel reductions or bridge balance.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find equivalent capacitances for a cyclic arrangement of capacitors for two different pairs of points.

Step 2: Key Formula or Approach:
For series: \( \frac{1}{C_s} = \sum \frac{1}{C_i} \). For parallel: \( C_p = \sum C_i \).

Step 3: Detailed Explanation:
There are 5 capacitors in a loop.
1. Between P and R:
Path 1: P-Q-R has 2 capacitors in series. \( C_1 = C/2 \).
Path 2: P-T-S-R has 3 capacitors in series. \( C_2 = C/3 \).
These two paths are in parallel. \( C_{PR} = \frac{C}{2} + \frac{C}{3} = \frac{5C}{6} \).
2. Between P and Q:
Path 1: P-Q has 1 capacitor. \( C_3 = C \).
Path 2: P-T-S-R-Q has 4 capacitors in series. \( C_4 = C/4 \).
These two paths are in parallel. \( C_{PQ} = C + \frac{C}{4} = \frac{5C}{4} \).
Ratio:
\[ \frac{C_{PR}}{C_{PQ}} = \frac{5C/6}{5C/4} = \frac{4}{6} = \frac{2}{3} \]

Step 4: Final Answer:
The required ratio is \( 2 : 3 \).