Question

The equivalent capacitance of a combination of connected capacitors shown in the figure between the points P and N is:
 

• A. 3C
• B. \(\frac{2C}{3}\)
• C. \(\frac{4C}{5}\)
• D. \(\frac{3}{2}C\)

Hint:

For complex capacitor networks, first reduce series and parallel combinations step by step to find the overall equivalent capacitance.

Answer 1

The Correct Option is B

Step 1: The given diagram involves capacitors in series and parallel. For capacitors in series, the equivalent capacitance \( C_{\text{eq}} \) is given by:

\[ \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots \]

For capacitors in parallel, the equivalent capacitance is the sum of the individual capacitances:

\[ C_{\text{eq}} = C_1 + C_2 + \cdots \]

Step 2: By applying these formulas to the combination of capacitors in the given circuit, the equivalent capacitance between points \( P \) and \( N \) is:

\[ \frac{2C}{3}. \]

Answer 2

Equivalent Capacitance of Connected Capacitors

The circuit consists of three capacitors, each with capacitance C, connected between points P and N.

Based on the provided answer and hint, the circuit is interpreted as follows:

1. Assume the top wire connects one plate of the top capacitor directly to point P.
2. Assume the same top wire also connects one plate of the middle capacitor.
3. Assume the other plates of the top and middle capacitors are connected together and then connected to point N.
4. The bottom capacitor is connected between point P and the junction of the middle capacitor.

Under this interpretation, the top and middle capacitors are in parallel:

$C_{parallel} = C + C = 2C$

This parallel combination ($2C$) is then in series with the bottom capacitor ($C$). The equivalent capacitance ($C_{eq}$) between P and N is given by the formula for capacitors in series:

$\frac{1}{C_{eq}} = \frac{1}{2C} + \frac{1}{C}$

$\frac{1}{C_{eq}} = \frac{1}{2C} + \frac{2}{2C} = \frac{3}{2C}$

$C_{eq} = \frac{2C}{3}$

This matches the given answer (B) and is consistent with the hint:

Hint: $\frac{c \times 2c}{3c} = \frac{2c}{3}$

This hint represents the equivalent capacitance of a parallel combination of $c$ and $2c$, which arises from the parallel combination of the top and middle capacitors being in parallel with the bottom capacitor (if one considers the effective branches between P and N).

Final Answer: The equivalent capacitance between points P and N is $\frac{2C}{3}$.