An unchanged parallel plate capacitor filled wit a dielectric constant K is connect to an air filled identical parallel capacitor charged to potential `V_(1)`. It the common potential is `V_(2)`, the value of K is

Options

A. `(V_(1)-V_(2))/(V_(1))`
B. `(V_(1))/(V_(1)-V_(2))`
C. `(V_(2))/(V_(1)-V_(2))`
D. `(V_(1)-V_(2))/(V_(2))`

Explanation

To solve the problem, we will follow these steps systematically:

### Step 1: Understand the Setup
We have two identical parallel plate capacitors:
1. Capacitor 1 (C1) is filled with a dielectric of constant \( K \) and is initially uncharged.
2. Capacitor 2 (C2) is filled with air and is charged to a potential \( V_1 \).

### Step 2: Identify the Charges
- The charge on Capacitor 2 (C2) when charged to potential \( V_1 \) can be calculated using the formula:
\[ Q_1 = C_2 \cdot V_1 \]
where \( C_2 \) is the capacitance of the air-filled capacitor.

- The uncharged Capacitor 1 (C1) has no initial charge:
\[ Q_2 = 0 \]

### Step 3: Calculate the Total Charge
The total charge \( Q_{total} \) when both capacitors are connected together is:
\[ Q_{total} = Q_1 + Q_2 = C_2 \cdot V_1 + 0 = C_2 \cdot V_1 \]

### Step 4: Determine the Capacitance of Each Capacitor
- The capacitance of the air-filled capacitor (C2) is \( C \).
- The capacitance of the capacitor filled with dielectric (C1) is:
\[ C_1 = K \cdot C \]

### Step 5: Calculate the Common Potential
When the two capacitors are connected, they will reach a common potential \( V_2 \). The formula for the common potential is given by:
\[ V_2 = \frac{Q_{total}}{C_{total}} \]
where \( C_{total} = C_1 + C_2 = K \cdot C + C \).

### Step 6: Substitute Values
Substituting the values we have:
\[ V_2 = \frac{C \cdot V_1}{K \cdot C + C} \]
This simplifies to:
\[ V_2 = \frac{C \cdot V_1}{C(K + 1)} \]
Cancelling \( C \) from numerator and denominator gives:
\[ V_2 = \frac{V_1}{K + 1} \]

### Step 7: Rearranging to Find K
Now, we can rearrange this equation to solve for \( K \):
\[ V_2 (K + 1) = V_1 \]
\[ K + 1 = \frac{V_1}{V_2} \]
\[ K = \frac{V_1}{V_2} - 1 \]

### Step 8: Final Expression for K
Thus, we can express \( K \) as:
\[ K = \frac{V_1 - V_2}{V_2} \]

### Final Answer
The value of \( K \) is:
\[ K = \frac{V_1 - V_2}{V_2} \]

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