Problem

Find the charge on the $5 \, \mathrm{\mu F}$ capacitor in the circuit shown below:

Data

• Capacitor $C_1 = 3 \, \mathrm{\mu F}$
• Capacitor $C_2 = 2 \, \mathrm{\mu F}$
• Capacitor $C_3 = 5 \, \mathrm{\mu F}$
• Voltage $V = 6 \, \mathrm{V}$.

Prerequisite Concepts

• Capacitors in Parallel:

$$C_{\text{eq}}^{\prime} = C_2 + C_3$$

• Capacitors in Series:

$$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_{\text{eq}}^{\prime}}$$

• Charge on Each Capacitor:
  $Q = C_{\text{eq}} \cdot V$.

Answer

Step 1: Calculate Equivalent Capacitance for Parallel Combination

For $C_2$ and $C_3$:

$$C_{\text{eq}}^{\prime} = C_2 + C_3 = 2 + 5 = 7 \, \mathrm{\mu F}$$

Step 2: Calculate Total Equivalent Capacitance for Series

$$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_{\text{eq}}^{\prime}}$$ $$\frac{1}{C_{\text{eq}}} = \frac{1}{3} + \frac{1}{7}$$

Take LCM:

$$\frac{1}{C_{\text{eq}}} = \frac{7 + 3}{21} = \frac{10}{21}$$ $$C_{\text{eq}} = \frac{21}{10} = 2.1 \, \mathrm{\mu F}$$

Step 3: Calculate Total Charge

$$Q = C_{\text{eq}} \cdot V$$ $$Q = 2.1 \times 10^{-6} \cdot 6 = 12.6 \, \mathrm{\mu C}$$

Step 4: Calculate Voltage Across Parallel Combination

$$V_{\text{parallel}} = \frac{Q}{C_{\text{eq}}^{\prime}}$$ $$V_{\text{parallel}} = \frac{12.6}{7} = 1.8 \, \mathrm{V}$$

Step 5: Calculate Charge on $C_3$

$$Q_3 = C_3 \cdot V_{\text{parallel}}$$ $$Q_3 = 5 \cdot 1.8 = 9 \, \mathrm{\mu C}$$

Final Answer

The charge on the $5 \, \mathrm{\mu F}$ capacitor is $9 \, \mathrm{\mu C}$.

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