Question
Two uncharged capacitors have capacitance C0 and 2C0 respectively, and are connected in series. This series combination is then connected across the terminals a battery. Which of the following is true?
The capacitor 2C0 has twice the charge of the other capacitor.
The equivalent capacitance is 3C0.
The equivalent capacitance is 2C0/3.
The voltage across each capacitor is the same.
The energy stored by each capacitor is the same.
The capacitor 2C0 has twice the charge of the other capacitor.
The equivalent capacitance is 3C0.
The equivalent capacitance is 2C0/3.
The voltage across each capacitor is the same.
The energy stored by each capacitor is the same.
Answers
GPT-4o mini
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.
1. **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,
\[
C_{\text{eq}} = \frac{2C_0}{3}
\]
2. **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 \( Q \).
3. **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.
4. **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 \( \frac{2C_0}{3} \)**.
1. **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,
\[
C_{\text{eq}} = \frac{2C_0}{3}
\]
2. **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 \( Q \).
3. **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.
4. **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 \( \frac{2C_0}{3} \)**.