Two capacitors having capacitances `C_(1)` and `C_(2)` are charged with 120 V and 200 V batteries respectively. When they are connected in parallel now, it is found that the potential on each one of them is zero. Then,

Options

A. `5 C_(1)=3 C_(2)`
B. `8 C_(1)=5 C_(2)`
C. `9 C_(1)=5 C_(2)`
D. `3 C_(1)=5 C_(2)`

Answer

To solve the problem step-by-step, we will analyze the situation involving the two capacitors charged with different voltages and then connected in parallel.

### Step 1: Understand the Initial Charges on the Capacitors
When the capacitors are charged with their respective batteries, the charge on each capacitor can be calculated using the formula:
\[ Q = C \times V \]
For Capacitor 1 (C1) charged with 120 V:
\[ Q_1 = C_1 \times 120 \]
For Capacitor 2 (C2) charged with 200 V:
\[ Q_2 = C_2 \times 200 \]

### Step 2: Analyze the Connection in Parallel
When the two capacitors are connected in parallel, they are connected such that the positive terminal of one is connected to the negative terminal of the other. This results in the charges on the capacitors opposing each other.

### Step 3: Set Up the Equation for Zero Potential
Since it is given that the potential on each capacitor is zero after they are connected in parallel, this implies that the net charge on the system is zero. Therefore, the charges must be equal in magnitude but opposite in sign:
\[ Q_1 - Q_2 = 0 \]
This means:
\[ Q_1 = Q_2 \]

### Step 4: Substitute the Charges
Substituting the expressions for \( Q_1 \) and \( Q_2 \):
\[ C_1 \times 120 = C_2 \times 200 \]

### Step 5: Rearranging the Equation
Rearranging the equation gives us:
\[ 120 C_1 = 200 C_2 \]
Dividing both sides by 40, we simplify this to:
\[ 3 C_1 = 5 C_2 \]

### Step 6: Final Relation
Thus, we have established the relationship between the capacitances:
\[ 3 C_1 = 5 C_2 \]

### Conclusion
The final relationship between the capacitances of the two capacitors is:
\[ 3 C_1 = 5 C_2 \]