Two identical parallel plate air capacitors are connected in series to a battery of emf V. If one of the capacitor is completely filled with dielectric material of constant K, then potential difference of the other capacitor will become

Options

A. `110 muJ`
B. `112 muJ`
C. `114.5 muJ`
D. `115.2 muJ`

Answer

To solve the problem step by step, we will analyze the configuration of the capacitors and apply the relevant formulas.

### Step 1: Understand the Configuration
We have two identical parallel plate capacitors connected in series to a battery of emf \( V \). One of the capacitors is filled with a dielectric material of constant \( K \).

**Hint:** Remember that in a series circuit, the total voltage is the sum of the voltages across each component.

### Step 2: Define the Variables
Let:
- \( C \) = Capacitance of each air capacitor (before the dielectric is added)
- \( V_1 \) = Potential difference across the capacitor with the dielectric
- \( V_2 \) = Potential difference across the air capacitor
- \( Q \) = Charge on each capacitor (since they are in series, the charge is the same)

**Hint:** The charge \( Q \) on a capacitor is given by the formula \( Q = C \cdot V \).

### Step 3: Write the Voltage Equation
Since the capacitors are in series, the total voltage \( V \) is the sum of the voltages across both capacitors:
\[ V = V_1 + V_2 \]

**Hint:** This relationship is crucial for finding the individual voltages.

### Step 4: Relate Charge and Capacitance
For the capacitor with the dielectric:
\[ Q = C_1 \cdot V_1 = K \cdot C \cdot V_1 \]
For the air capacitor:
\[ Q = C_2 \cdot V_2 = C \cdot V_2 \]

Since \( Q_1 = Q_2 \):
\[ K \cdot C \cdot V_1 = C \cdot V_2 \]

**Hint:** You can cancel \( C \) from both sides since it is common.

### Step 5: Solve for \( V_1 \)
From the equation:
\[ K \cdot V_1 = V_2 \]
We can express \( V_1 \) in terms of \( V_2 \):
\[ V_1 = \frac{V_2}{K} \]

**Hint:** This shows how the potential across the capacitor with the dielectric relates to the potential across the air capacitor.

### Step 6: Substitute into the Voltage Equation
Substituting \( V_1 \) into the total voltage equation:
\[ V = \frac{V_2}{K} + V_2 \]

**Hint:** This step combines the voltages to find a relationship between \( V \) and \( V_2 \).

### Step 7: Factor and Solve for \( V_2 \)
Combine the terms:
\[ V = V_2 \left( \frac{1}{K} + 1 \right) \]
\[ V = V_2 \left( \frac{1 + K}{K} \right) \]
Now, solve for \( V_2 \):
\[ V_2 = \frac{V \cdot K}{1 + K} \]

**Hint:** This final expression gives you the potential difference across the air capacitor.

### Final Answer
The potential difference across the other capacitor (the one without the dielectric) is:
\[ V_2 = \frac{V \cdot K}{1 + K} \]