Problem

Two capacitors of capacitance $8 \, \mathrm{\mu F}$ and $10 \, \mathrm{\mu F}$ are connected in series across a $180 \, \mathrm{V}$ power supply. They are then disconnected from the supply and reconnected in parallel with each other. Calculate the new potential difference across the combination and the charge on each capacitor.

Data

• Capacitor $C_1 = 8 \, \mathrm{\mu F}$
• Capacitor $C_2 = 10 \, \mathrm{\mu F}$
• Voltage $V = 180 \, \mathrm{V}$.

Prerequisite Concepts

• Equivalent Capacitance in Series:

$$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2}$$

• Charge on Capacitors in Series:

$$Q = C_{\text{eq}} \cdot V$$

• Equivalent Capacitance in Parallel:

$$C_{\text{total}} = C_1 + C_2$$

• Voltage Across Parallel Combination:

$$V_{\text{new}} = \frac{Q_{\text{total}}}{C_{\text{total}}}$$

Answer

Step 1: Calculate Equivalent Capacitance in Series

$$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2}$$ $$\frac{1}{C_{\text{eq}}} = \frac{1}{8} + \frac{1}{10}$$

Take LCM:

$$\frac{1}{C_{\text{eq}}} = \frac{10 + 8}{80} = \frac{18}{80}$$ $$C_{\text{eq}} = \frac{80}{18} = 4.444 \, \mathrm{\mu F}$$

Step 2: Calculate Total Charge in Series

$$Q = C_{\text{eq}} \cdot V$$ $$Q = 4.444 \cdot 180 = 800 \, \mathrm{\mu C}$$

Step 3: Calculate Total Charge in Parallel

When reconnected in parallel, the total charge becomes:

$$Q_{\text{total}} = Q + Q = 2Q$$ $$Q_{\text{total}} = 2 \cdot 800 = 1600 \, \mathrm{\mu C}$$

Step 4: Calculate Total Capacitance in Parallel

$$C_{\text{total}} = C_1 + C_2$$ $$C_{\text{total}} = 8 + 10 = 18 \, \mathrm{\mu F}$$

Step 5: Calculate New Voltage

$$V_{\text{new}} = \frac{Q_{\text{total}}}{C_{\text{total}}}$$ $$V_{\text{new}} = \frac{1600}{18} = 88.89 \, \mathrm{V}$$

Step 6: Calculate Charge on Each Capacitor

For $C_1$:

$$Q_1 = C_1 \cdot V_{\text{new}}$$ $$Q_1 = 8 \cdot 88.89 = 711.12 \, \mathrm{\mu C}$$

For $C_2$:

$$Q_2 = C_2 \cdot V_{\text{new}}$$ $$Q_2 = 10 \cdot 88.89 = 888.9 \, \mathrm{\mu C}$$

Final Answer

• New Potential Difference: $88.89 \, \mathrm{V}$
• Charge on $C_1$: $711.12 \, \mathrm{\mu C}$
• Charge on $C_2$: $888.9 \, \mathrm{\mu C}$.

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