Question

In the following circuit, the charge stored in capacitors \( C_1 \), \( C_2 \), and \( C_3 \) are

• A. 18 C, 36 C, 54 C
• B. 24 C, 48 C, 216 C
• C. 2.7 C, 1.33 C, 0.30 C
• D. 12 C, 24 C, 36 C

Hint:

When dealing with circuits involving capacitors in series and parallel, always start by reducing the series capacitors first to find the equivalent capacitance, and then combine with parallel capacitors for the final total capacitance.

Answer 1

The Correct Option is B

We are given the following circuit with capacitors \( C_1 = 3 \, \text{F} \), \( C_2 = 6 \, \text{F} \), and \( C_3 = 27 \, \text{F} \) with a total applied voltage \( V = 8 \, \text{V} \).
Step 1: Simplifying the circuit - First, we need to combine the capacitors in series and parallel. - The first two capacitors \( C_1 \) and \( C_2 \) are in series, so their equivalent capacitance \( C_{\text{eq1}} \) is given by the formula for capacitors in series: \[ \frac{1}{C_{\text{eq1}}} = \frac{1}{C_1} + \frac{1}{C_2} \] Substituting the values of \( C_1 = 3 \, \text{F} \) and \( C_2 = 6 \, \text{F} \): \[ \frac{1}{C_{\text{eq1}}} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2} \] So, \( C_{\text{eq1}} = 2 \, \text{F} \).
Step 2: Combine the equivalent capacitor \( C_{\text{eq1}} \) with \( C_3 \) - Now, \( C_{\text{eq1}} \) is in parallel with \( C_3 \), so the total equivalent capacitance \( C_{\text{eq}} \) of the entire system is: \[ C_{\text{eq}} = C_{\text{eq1}} + C_3 = 2 \, \text{F} + 27 \, \text{F} = 29 \, \text{F} \]
Step 3: Calculate the total charge stored - The total charge \( Q_{\text{total}} \) stored in the system is given by the formula: \[ Q_{\text{total}} = C_{\text{eq}} \times V \] Substituting the values \( C_{\text{eq}} = 29 \, \text{F} \) and \( V = 8 \, \text{V} \): \[ Q_{\text{total}} = 29 \, \text{F} \times 8 \, \text{V} = 232 \, \text{C} \]
Step 4: Calculate the individual charges - The charge stored in the capacitors depends on the voltage across them. - The voltage across \( C_1 \) and \( C_2 \) is the same since they are in series. - The voltage across the parallel combination is the same, so: \[ Q_1 = C_1 \times V_1, \quad Q_2 = C_2 \times V_1, \quad Q_3 = C_3 \times V_3 \] Thus, the charges are 24 C, 48 C, and 216 C respectively.