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. `3C_(1)=5C_(2)`
B. `3C_(1)+5C_(2)=0`
C. `9C_(1)=4C_(2)`
D. `5C_(1)=3C_(2)`

Answer

To solve the problem, we need to analyze the situation with the two capacitors charged to different voltages and then connected in parallel. Here’s a step-by-step solution:

### Step 1: Understand the initial conditions
We have two capacitors, \( C_1 \) and \( C_2 \), charged to 120 V and 200 V, respectively. When connected in parallel, we need to find out under what conditions the potential across both capacitors becomes zero.

**Hint:** Consider how the charges on the capacitors behave when connected in parallel.

### Step 2: Calculate the charges on each capacitor
The charge on a capacitor is given by the formula:
\[ Q = C \times V \]
For capacitor \( C_1 \):
\[ Q_1 = C_1 \times 120 \]
For capacitor \( C_2 \):
\[ Q_2 = C_2 \times 200 \]

**Hint:** Remember that the charge is dependent on both the capacitance and the voltage.

### Step 3: Analyze the connection of the capacitors
When the capacitors are connected in parallel, if we connect them with opposite polarities (the positive terminal of one to the negative terminal of the other), the charges will oppose each other. For the potential across both capacitors to be zero, the total charge must also be zero:
\[ Q_1 - Q_2 = 0 \]
This implies:
\[ Q_1 = Q_2 \]

**Hint:** Think about how opposing charges can cancel each other out.

### Step 4: Set up the equation
From the previous step, we have:
\[ C_1 \times 120 = C_2 \times 200 \]

**Hint:** This equation relates the charges based on the capacitances and voltages.

### Step 5: Rearrange the equation
Rearranging the equation gives us:
\[ 120 C_1 = 200 C_2 \]
Dividing both sides by 40:
\[ 3 C_1 = 5 C_2 \]

**Hint:** Simplifying the equation can help you find a relationship between the two capacitances.

### Step 6: Conclusion
The relationship between the capacitances of the two capacitors is:
\[ \frac{C_1}{C_2} = \frac{5}{3} \]

Thus, the answer is that the ratio of the capacitances \( C_1 \) and \( C_2 \) is \( \frac{5}{3} \).

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