Question

Five capacitors each of capacitance \(C\) are connected as shown in the figure. The ratio of equivalent capacitance between \(P\) and \(R\) and the equivalent capacitance between \(P\) and \(Q\) is

• A. \(2:3\)
• B. \(1:1\)
• C. \(3:1\)
• D. \(5:2\)

Hint:

In symmetric capacitor networks, always identify parallel paths between the given terminals.

Answer 1

The Correct Option is A

Step 1: Observe the circuit symmetry.
The five capacitors of equal capacitance \(C\) form a closed pentagon. Each side of the pentagon has one capacitor.

Step 2: Equivalent capacitance between \(P\) and \(R\).
Between points \(P\) and \(R\), there are two parallel paths:
- One path consists of two capacitors in series \(\Rightarrow C_{1} = \dfrac{C}{2}\).
- The other path consists of three capacitors in series \(\Rightarrow C_{2} = \dfrac{C}{3}\).
These two paths are in parallel, so:
\[ C_{PR} = \frac{C}{2} + \frac{C}{3} = \frac{5C}{6} \]
Step 3: Equivalent capacitance between \(P\) and \(Q\).
Between \(P\) and \(Q\), the two paths are:
- One capacitor directly: \(C\).
- Four capacitors in series: \(C' = \dfrac{C}{4}\).
Hence:
\[ C_{PQ} = C + \frac{C}{4} = \frac{5C}{4} \]
Step 4: Find the ratio.
\[ \frac{C_{PR}}{C_{PQ}} = \frac{\frac{5C}{6}}{\frac{5C}{4}} = \frac{4}{6} = \frac{2}{3} \]
Step 5: Conclusion.
The required ratio is \(2:3\).