A parallel plate capacitor with air as dielectric is charged to a potential .v. using a battery . Removing the battery. The charged capacitor is then connected across an identical uncharged parallel plate capacitor filled with was of dielectric constant k. The common potential of both the capacitor is

Options

A. V volts
B. kV volts
C. (k+1) V volts
D. `(V)/(k+1) ` volts

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the Initial Conditions
We have a parallel plate capacitor (let's call it Capacitor 1) charged to a potential \( V \) with air as the dielectric. The dielectric constant of air is \( k = 1 \). The capacitance of this capacitor is \( C_1 = C \).

### Step 2: Calculate Initial Charge
The charge \( Q \) on Capacitor 1 when it is charged to potential \( V \) can be calculated using the formula:
\[ Q = C_1 \cdot V = C \cdot V \]

### Step 3: Remove the Battery
After charging, the battery is removed, and the charge \( Q \) remains on Capacitor 1 since it is isolated.

### Step 4: Connect to the Second Capacitor
Now, we connect Capacitor 1 (charged) to an identical uncharged capacitor (let's call it Capacitor 2) which has a dielectric material with dielectric constant \( K \). The capacitance of Capacitor 2 is:
\[ C_2 = K \cdot C \]

### Step 5: Calculate Total Capacitance
When the two capacitors are connected in parallel, the total capacitance \( C_{total} \) becomes:
\[ C_{total} = C_1 + C_2 = C + K \cdot C = C(1 + K) \]

### Step 6: Apply Charge Conservation
Since charge is conserved, the initial charge on Capacitor 1 is equal to the total charge on the combined system:
\[ Q = C \cdot V = C_{total} \cdot V_{common} \]
Where \( V_{common} \) is the common potential across both capacitors after they are connected.

### Step 7: Substitute for \( C_{total} \)
Substituting \( C_{total} \) into the equation gives:
\[ C \cdot V = C(1 + K) \cdot V_{common} \]

### Step 8: Solve for \( V_{common} \)
Now, we can solve for \( V_{common} \):
\[ V_{common} = \frac{C \cdot V}{C(1 + K)} = \frac{V}{1 + K} \]

### Final Answer
Thus, the common potential across both capacitors after they are connected is:
\[ V_{common} = \frac{V}{1 + K} \]

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