Question

Two capacitors \( C \) and \( 2C \) charged to \( V \) and \( 2V \) respectively are connected in parallel with opposite polarity. The common potential is:

• A. \( V \)
• B. \( \dfrac{V}{2} \)
• C. \( \dfrac{V}{3} \)
• D. \( 3V \)

Hint:

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.

Answer 1

The Correct Option is A

To solve the problem of finding the common potential when two capacitors are connected in parallel with opposite polarity, we will follow these steps: 

1. Calculate the initial charge on both capacitors:
   • The charge on the first capacitor, \( C_1 = C \), charged to a potential \( V \), is given by: \(Q_1 = C \times V\)
   • The charge on the second capacitor, \( C_2 = 2C \), charged to a potential \( 2V \), is given by: \(Q_2 = 2C \times 2V = 4CV\)
2. Determine the net charge when connected with opposite polarity:
   • Because they are connected with opposite polarity, the charges subtract: \(Q_{\text{net}} = -(C \times V) + 4CV = 3CV\)
3. Compute the total capacitance when the capacitors are connected in parallel:
   • The total capacitance: \(C_{\text{total}} = C + 2C = 3C\)
4. Find the common potential:
   • The common potential \( V_{\text{common}} \) is given by: \(V_{\text{common}} = \frac{Q_{\text{net}}}{C_{\text{total}}} = \frac{3CV}{3C} = V\)

Thus, the common potential when the capacitors are connected is \(V\). Therefore, the correct answer is option \( V \) .

This solution demonstrates the conservation of charge principle in capacitors connected in parallel with opposite polarities.

Answer 2

Concept:

• Instead of jumping straight to a memorised formula, set up signed charges relative to one fixed reference direction. This shows exactly why the charges subtract when polarities are opposite, rather than just asserting it.

Step 1: Fix a reference direction and write each charge with a sign.
Take the polarity of capacitor $2C$ as the positive reference direction (since it holds the larger charge).
Charge on $2C$: $Q_2 = 2C \times 2V = +4CV$
Charge on $C$: since its plates face the opposite way, $Q_1 = -(C \times V) = -CV$

Step 2: Add the signed charges to get the total charge in the system.
$Q_{total} = Q_2 + Q_1 = 4CV - CV = 3CV$

Step 3: Find the total capacitance once they share a common potential.
When connected in parallel, capacitance adds directly regardless of polarity:
$C_{total} = C + 2C = 3C$

Step 4: Divide total charge by total capacitance.
$V_{common} = \dfrac{Q_{total}}{C_{total}} = \dfrac{3CV}{3C}$
The positive sign confirms the common potential follows the polarity of the larger capacitor.

Final Answer: $V_{common} = V$