Question:
Two capacitors \( C \) and \( 2C \) charged to \( V \) and \( 2V \) respectively are connected in parallel with opposite polarity. The common potential is:
Two capacitors \( C \) and \( 2C \) charged to \( V \) and \( 2V \) respectively are connected in parallel with opposite polarity. The common potential is:
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Treat the two initial voltages as signed because their polarities oppose each other. Find the contribution of each capacitor to the final parallel voltage by dividing $C_iV_i$ by the total capacitance, then add the signed contributions.
Updated On: Aug 14, 2026
- \( V \)
- \( \dfrac{V}{2} \)
- \( \dfrac{V}{3} \)
- \( 3V \)
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The Correct Option is A
Approach Solution - 1
Step 1: Find the initial charges on the capacitors.
\[
Q_1 = C \cdot V = CV, \quad Q_2 = 2C \cdot 2V = 4CV
\]
Step 2: Since the capacitors are connected with opposite polarity, the net charge is: \[ Q_{\text{net}} = 4CV - CV = 3CV \]
Step 3: The equivalent capacitance in parallel is: \[ C_{\text{eq}} = C + 2C = 3C \]
Step 4: The common potential is: \[ V_{\text{common}} = \frac{Q_{\text{net}}}{C_{\text{eq}}} = \frac{3CV}{3C} = V \]
Step 2: Since the capacitors are connected with opposite polarity, the net charge is: \[ Q_{\text{net}} = 4CV - CV = 3CV \]
Step 3: The equivalent capacitance in parallel is: \[ C_{\text{eq}} = C + 2C = 3C \]
Step 4: The common potential is: \[ V_{\text{common}} = \frac{Q_{\text{net}}}{C_{\text{eq}}} = \frac{3CV}{3C} = V \]
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Approach Solution -2
Concept:
Step 1: Identify $C_1, V_1, C_2, V_2$.
$C_1 = C,\ V_1 = V$ and $C_2 = 2C,\ V_2 = 2V$.
Step 2: Apply the common potential formula with opposite polarity.
Since the capacitors oppose each other, take the larger charge as positive:
$V_{common} = \dfrac{C_2V_2 - C_1V_1}{C_1+C_2}$
Step 3: Substitute the values.
$V_{common} = \dfrac{(2C)(2V) - (C)(V)}{C+2C} = \dfrac{4CV - CV}{3C} = \dfrac{3CV}{3C}$
Final Answer: $V_{common} = V$
- When two charged capacitors are connected in parallel, there is a direct formula for the common potential: $V_{common} = \dfrac{C_1V_1 \pm C_2V_2}{C_1+C_2}$, with a minus sign when the polarities oppose. Recalling this saves re-deriving charge conservation each time.
Step 1: Identify $C_1, V_1, C_2, V_2$.
$C_1 = C,\ V_1 = V$ and $C_2 = 2C,\ V_2 = 2V$.
Step 2: Apply the common potential formula with opposite polarity.
Since the capacitors oppose each other, take the larger charge as positive:
$V_{common} = \dfrac{C_2V_2 - C_1V_1}{C_1+C_2}$
Step 3: Substitute the values.
$V_{common} = \dfrac{(2C)(2V) - (C)(V)}{C+2C} = \dfrac{4CV - CV}{3C} = \dfrac{3CV}{3C}$
Final Answer: $V_{common} = V$
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Approach Solution -3
Concept:
Step 1: Assign signs to the initial voltages.
Take the polarity of the $2C$ capacitor as positive. Its voltage is $+2V$, while the voltage of the $C$ capacitor is $-V$.
The total parallel capacitance is $C_T=C+2C=3C$.
Step 2: Find the contribution from the $2C$ capacitor.
$V_2=\dfrac{(2C)(2V)}{3C}=\dfrac{4V}{3}$
Step 3: Find the opposing contribution from the $C$ capacitor.
$V_1=\dfrac{C(-V)}{3C}=-\dfrac{V}{3}$
Step 4: Add the signed contributions.
$V_{common}=V_2+V_1=\dfrac{4V}{3}-\dfrac{V}{3}=V$
The positive sign shows that the final polarity follows the initially stronger $2C$ capacitor.
Final Answer: $V_{common}=V$
- A parallel combination can be handled by superposition: find the common-potential contribution produced by each initially charged capacitor, using signed voltages for opposite polarity.
- Each contribution is divided by the total parallel capacitance.
Step 1: Assign signs to the initial voltages.
Take the polarity of the $2C$ capacitor as positive. Its voltage is $+2V$, while the voltage of the $C$ capacitor is $-V$.
The total parallel capacitance is $C_T=C+2C=3C$.
Step 2: Find the contribution from the $2C$ capacitor.
$V_2=\dfrac{(2C)(2V)}{3C}=\dfrac{4V}{3}$
Step 3: Find the opposing contribution from the $C$ capacitor.
$V_1=\dfrac{C(-V)}{3C}=-\dfrac{V}{3}$
Step 4: Add the signed contributions.
$V_{common}=V_2+V_1=\dfrac{4V}{3}-\dfrac{V}{3}=V$
The positive sign shows that the final polarity follows the initially stronger $2C$ capacitor.
Final Answer: $V_{common}=V$
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