An uncharged parallel plate capacitor having a dielectric of dielectric constant `K` is connected to a similar air core parallel plate capacitor charged to a potential `V_(0)`. The two share the charge, and the common potential becomes `V`. The dielectric constant `K` is`

Options

A. `V_(0)/V-1`
B. `(V_(0))/V+1`
C. `V/V_(0)-1`
D. `V/V_(0)+1`

Explanation

To solve the problem, we need to analyze the situation step by step:

### Step 1: Understand the initial conditions
We have two parallel plate capacitors:
1. Capacitor 1 (air core) is charged to a potential \( V_0 \).
2. Capacitor 2 (with dielectric constant \( K \)) is initially uncharged.

### Step 2: Calculate the initial charge on the air core capacitor
The charge \( Q_i \) on the air core capacitor can be expressed as:
\[ Q_i = C V_0 \]
where \( C \) is the capacitance of the air core capacitor.

### Step 3: Determine the capacitance of the capacitor with dielectric
When the dielectric is inserted into the second capacitor, its capacitance becomes:
\[ C' = K C \]
where \( K \) is the dielectric constant.

### Step 4: Apply conservation of charge
When the two capacitors are connected, they will share the charge until they reach a common potential \( V \). According to the conservation of charge:
\[ Q_i = Q_f \]
where \( Q_f \) is the final charge after they share the charge.

### Step 5: Express the final charge on both capacitors
The final charge on the capacitor with dielectric is:
\[ Q_f = C' V = K C V \]
The final charge on the air core capacitor is:
\[ Q_f = C V \]

### Step 6: Set up the equation using conservation of charge
Equating the initial charge to the final charge gives:
\[ C V_0 = K C V + C V \]

### Step 7: Factor out \( C \)
Since \( C \) is common in all terms, we can factor it out:
\[ V_0 = K V + V \]

### Step 8: Simplify the equation
This can be rewritten as:
\[ V_0 = (K + 1)V \]

### Step 9: Solve for the dielectric constant \( K \)
Rearranging the equation gives:
\[ K + 1 = \frac{V_0}{V} \]
Thus,
\[ K = \frac{V_0}{V} - 1 \]

### Final Result
The dielectric constant \( K \) is:
\[ K = \frac{V_0}{V} - 1 \]