Two metal spheres of capacitances `C_1 and C_2`carry some charges . They are put in contact and then seperated.The final charges `Q_1 and Q_2` on them will satisfy

Options

A. `(Q_1)/(Q_2)lt(C_1)/(C_2)`
B. `(Q_1)/(Q_2) = (C_1)/(C_2) `
C. `(Q_1)/(Q_2) gt(C_1)/(C_2) `
D. `(Q_1)/(Q_2) = (C_2)/(C_1)`

Answer

To solve the problem step by step, we need to analyze the situation of two metal spheres with capacitances \( C_1 \) and \( C_2 \) that are put in contact and then separated.

### Step 1: Understanding the Initial Conditions
Initially, the two metal spheres have charges \( Q_1 \) and \( Q_2 \) and their respective capacitances are \( C_1 \) and \( C_2 \). The potentials of the spheres can be expressed as:
\[ V_1 = \frac{Q_1}{C_1} \quad \text{and} \quad V_2 = \frac{Q_2}{C_2} \]

### Step 2: Contacting the Spheres
When the spheres are brought into contact, charge will flow between them until they reach the same potential. This is because charge flows from a region of higher potential to a region of lower potential. Therefore, we set the potentials equal:
\[ V_1 = V_2 \]

### Step 3: Setting the Potentials Equal
From the expressions for the potentials, we have:
\[ \frac{Q_1}{C_1} = \frac{Q_2}{C_2} \]

### Step 4: Rearranging the Equation
Cross-multiplying gives us:
\[ Q_1 \cdot C_2 = Q_2 \cdot C_1 \]

### Step 5: Expressing the Charge Ratio
We can rearrange this equation to express the ratio of the charges:
\[ \frac{Q_1}{Q_2} = \frac{C_1}{C_2} \]

### Conclusion
Thus, the final charges \( Q_1 \) and \( Q_2 \) on the spheres after they are separated will satisfy the relationship:
\[ \frac{Q_1}{Q_2} = \frac{C_1}{C_2} \]

### Final Answer
The final charges \( Q_1 \) and \( Q_2 \) will satisfy the equation \( \frac{Q_1}{Q_2} = \frac{C_1}{C_2} \).

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