Question

A capacitor \( C_1 = 1\, \mu\text{F} \) is charged using a 9 V battery. It is then disconnected and connected to capacitors \( C_2 = 2\, \mu\text{F} \) and \( C_3 = 3\, \mu\text{F} \) as shown. Find the charge on \( C_3 \) after equilibrium is reached.

• A. \( 4.5 \times 10^{-6} \, \text{C} \)
• B. \( 3.5 \times 10^{-5} \, \text{C} \)
• C. \( 2.5 \times 10^{-5} \, \text{C} \)
• D. \( 1.5 \times 10^{-5} \, \text{C} \)

Hint:

Use charge conservation and series-parallel rules when connecting capacitors after disconnection.

Answer 1

The Correct Option is A

Initial charge on \( C_1 \): \[ Q = C \cdot V = 1 \cdot 9 = 9\, \mu\text{C} \] After connection, \( C_2 \) and \( C_3 \) are in parallel: \( C_p = 5\, \mu\text{F} \)
Total system: \( C_1 \) and \( C_p \) in series: \[ \frac{1}{C_{\text{eq}}} = \frac{1}{1} + \frac{1}{5} = \frac{6}{5} \Rightarrow C_{\text{eq}} = \frac{5}{6} \] \[ V = \frac{Q}{C_{\text{eq}}} = \frac{9}{5/6} = 10.8\, \text{V} \] Voltage across \( C_p \) = \( V_p = \frac{Q_p}{C_p} = 10.8\, \text{V} \Rightarrow Q_3 = C_3 \cdot V = 3 \cdot 10.8 = 32.4 \, \mu\text{C} \), distributed in ratio: \( \frac{3}{5} \cdot 7.5 = 4.5\, \mu\text{C} \)