To analyze the series connection of the two capacitors \( C_1 = C_0 \) and \( C_2 = 2C_0 \), we need to determine their equivalent capacitance and other properties when connected across a battery.
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Equivalent Capacitance in Series: The formula for the equivalent capacitance \( C_{\text{eq}} \) of capacitors in series is given by: \[ \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} \] Plugging in the values: \[ \frac{1}{C_{\text{eq}}} = \frac{1}{C_0} + \frac{1}{2C_0} = \frac{2}{2C_0} + \frac{1}{2C_0} = \frac{3}{2C_0} \] Therefore, \[ C_{\text{eq}} = \frac{2C_0}{3} \]
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Charge on Capacitors: In a series connection, the charge \( Q \) on each capacitor is the same. The total charge stored is related to voltage \( V \) applied across the equivalent capacitance: \[ Q = C_{\text{eq}} \cdot V \] This means that both capacitors \( C_0 \) and \( 2C_0 \) will store the same charge \( Q \).
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Voltage Across Each Capacitor: The voltage across each capacitor can be calculated using the formula \( V = \frac{Q}{C} \):
- For the capacitor \( C_0 \): \[ V_1 = \frac{Q}{C_0} \]
- For the capacitor \( 2C_0 \): \[ V_2 = \frac{Q}{2C_0} \] Since \( V_1 \) and \( V_2 \) are different, we can conclude that the voltage across the capacitors is not the same.
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Energy Stored in Each Capacitor: The energy stored in a capacitor is given by \( U = \frac{1}{2} C V^2 \). Since the charge is the same, the energy stored will also be different due to the difference in capacitance values.
Now, let's summarize the deductions:
- The capacitor \( 2C_0 \) does not have twice the charge of \( C_0 \); both have the same charge.
- The equivalent capacitance is indeed \( \frac{2C_0}{3} \).
- The voltage across each capacitor is not the same.
- The energy stored by each capacitor is not the same due to differences in capacitance.
Conclusion:
The correct statement is: The equivalent capacitance is \( \frac{2C_0}{3} \).
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