Question

A \(12\,\text{F}\) capacitor is connected to a \(5\,\text{V}\) battery and fully charged. After disconnecting the battery, it is connected in parallel to an uncharged \(6\,\text{F}\) capacitor. Find the final charge on the \(6\,\text{F}\) capacitor.

• A. \(10\,\text{C}\)
• B. \(15\,\text{C}\)
• C. \(20\,\text{C}\)
• D. \(30\,\text{C}\)

Hint:

When capacitors are connected in parallel after charging, total charge is conserved and the final voltage becomes common.

Answer 1

The Correct Option is C

Step 1: Calculate the initial charge on the charged capacitor.
For the \(12\,\text{F}\) capacitor: \[ Q_{\text{initial}} = CV = 12 \times 5 = 60\,\text{C}. \]
Step 2: Apply conservation of charge.
After disconnection from the battery and connection in parallel, the total charge is conserved.
Total charge in the system: \[ Q_{\text{total}} = 60\,\text{C}. \]
Step 3: Find the equivalent capacitance.
\[ C_{\text{eq}} = 12 + 6 = 18\,\text{F}. \]
Step 4: Calculate the final common voltage.
\[ V_{\text{final}} = \frac{Q_{\text{total}}}{C_{\text{eq}}} = \frac{60}{18} = \frac{10}{3}\,\text{V}. \]
Step 5: Find the final charge on the \(6\,\text{F}\) capacitor.
\[ Q_{6} = C V = 6 \times \frac{10}{3} = 20\,\text{C}. \]
Step 6: Final conclusion.
The final charge on the \(6\,\text{F}\) capacitor is: \[ \boxed{20\,\text{C}}. \]