Question

A capacitor, \( C_1 = 6 \, \mu F \), is charged to a potential difference of \( V_1 = 5 \, \text{V} \) using a 5V battery. The battery is removed and another capacitor, \( C_2 = 12 \, \mu F \), is inserted in place of the battery. When the switch 'S' is closed, the charge flows between the capacitors for some time until equilibrium condition is reached. What are the charges \( q_1 \) and \( q_2 \) on the capacitors \( C_1 \) and \( C_2 \) when equilibrium condition is reached?

• A. \( q_1 = 15 \, \mu C, \, q_2 = 30 \, \mu C \)
• B. \( q_1 = 30 \, \mu C, \, q_2 = 15 \, \mu C \)
• C. \( q_1 = 10 \, \mu C, \, q_2 = 20 \, \mu C \)
• D. \( q_1 = 20 \, \mu C, \, q_2 = 10 \, \mu C \)

Hint:

When capacitors are in parallel and the switch is closed, the total charge is conserved, and the potential across all capacitors will be the same at equilibrium. Use charge conservation and the capacitance values to find the final charges on each capacitor.

Answer 1

The Correct Option is C

Step 1: Initial charge on \( C_1 \) at \( t = 0 \) is: \[q_1 = C_1 \cdot V_1 = 6 \, \mu F \cdot 5 \, \text{V} = 30 \, \mu C\]

Step 2: Upon closing switch 'S', charge redistributes between capacitors \( C_1 \) and \( C_2 \) until equilibrium. At equilibrium, the potential difference across both capacitors is equal, denoted as \( V_c \). \[q_1 = C_1 \cdot V_c \quad \text{and} \quad q_2 = C_2 \cdot V_c\] Charge conservation dictates: \[q_1 + q_2 = 30 \, \mu C\] Substituting the expressions for \( q_1 \) and \( q_2 \): \[C_1 \cdot V_c + C_2 \cdot V_c = 30 \, \mu C\] \[V_c \cdot (C_1 + C_2) = 30 \, \mu C\] Solving for \( V_c \): \[V_c = \frac{30 \, \mu C}{C_1 + C_2} = \frac{30 \, \mu C}{6 \, \mu F + 12 \, \mu F} = \frac{30 \, \mu C}{18 \, \mu F} = 1.67 \, \text{V}\]

Step 3: Final charges on the capacitors are: \[q_1 = C_1 \cdot V_c = 6 \, \mu F \cdot 1.67 \, \text{V} = 10 \, \mu C\] \[q_2 = C_2 \cdot V_c = 12 \, \mu F \cdot 1.67 \, \text{V} = 20 \, \mu C\] Therefore, \( q_1 = 10 \, \mu C \) and \( q_2 = 20 \, \mu C \), corresponding to option (3).