Question

A capacitor of capacitance \(6\,\text{F}\) is charged by connecting it to a \(12\,\text{V}\) battery. After disconnecting the battery, the capacitor is connected in parallel to an initially uncharged capacitor of capacitance \(18\,\text{F}\). Find the charge on the \(18\,\text{F}\) capacitor after equilibrium is reached.

• A. \(36\,\text{C}\)
• B. \(48\,\text{C}\)
• C. \(54\,\text{C}\)
• D. \(72\,\text{C}\)

Hint:

In charge sharing problems, always conserve total charge and remember that capacitors in parallel attain the same final voltage.

Answer 1

The Correct Option is C

Step 1: Calculate the initial charge on the charged capacitor.
For the \(6\,\text{F}\) capacitor connected to a \(12\,\text{V}\) battery: \[ Q_{\text{initial}} = CV = 6 \times 12 = 72\,\text{C}. \]
Step 2: Apply conservation of charge.
After disconnecting the battery and connecting the capacitors in parallel, the total charge in the system remains conserved: \[ Q_{\text{total}} = 72\,\text{C}. \]
Step 3: Find the equivalent capacitance of the parallel combination.
\[ C_{\text{eq}} = 6 + 18 = 24\,\text{F}. \]
Step 4: Calculate the final common voltage.
\[ V_{\text{final}} = \frac{Q_{\text{total}}}{C_{\text{eq}}} = \frac{72}{24} = 3\,\text{V}. \]
Step 5: Find the charge on the \(18\,\text{F}\) capacitor.
\[ Q_{18} = C V = 18 \times 3 = 54\,\text{C}. \]
Step 6: Final conclusion.
The charge on the \(18\,\text{F}\) capacitor after equilibrium is: \[ \boxed{54\,\text{C}}. \]