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

To find the final charge on the \(6\,\text{F}\) capacitor, we need to understand the process of charge redistribution that occurs when capacitors are connected in parallel. Here is the step-by-step solution:

1. The \(12\,\text{F}\) capacitor is initially connected to a \(5\,\text{V}\) battery and fully charged. The charge \(Q_1\) on the capacitor is calculated using the formula \(Q = C \cdot V\), where \(C\) is the capacitance and \(V\) is the voltage.
2. \(Q_1 = 12\,\text{F} \times 5\,\text{V} = 60\,\text{C}\) 
3. After being charged, the \(12\,\text{F}\) capacitor is disconnected from the battery and connected in parallel with an uncharged \(6\,\text{F}\) capacitor. When connected in parallel, the total charge is redistributed between the two capacitors.
4. Since the capacitors are in parallel, they will have the same voltage across them. Let \(V_f\) be the final common voltage across both capacitors after redistribution.
5. The total initial charge is conserved, so the equation for charge conservation is: \(Q_{\text{total}} = Q_1 + Q_2 = (C_1 + C_2) \cdot V_f\) where \(C_1 = 12\,\text{F}\) and \(C_2 = 6\,\text{F}\) and \(Q_2 = 0\) initially.
6. \(60\,\text{C} = (12\,\text{F} + 6\,\text{F}) \cdot V_f\)
7. Simplifying gives: \(V_f = \frac{60\,\text{C}}{18\,\text{F}} = \frac{10}{3}\,\text{V}\)
8. Now, we can find the final charge on the \(6\,\text{F}\) capacitor using the formula \(Q = C \cdot V\): \(Q_{\text{final, 6F}} = 6\,\text{F} \times \frac{10}{3}\,\text{V} = 20\,\text{C}\)

Thus, the final charge on the \(6\,\text{F}\) capacitor is \(20\,\text{C}\). The correct answer is \(20\,\text{C}\).