Two identical parallel plate capacitors are connected in series and then joined in series with a battery of `100 V`. A slab of dielectric constant `K=3` is inserted between the plates of the first capacitor. Then, the potential difference across the capacitor will be, respectively.

Options

A. `25 V,75 V`
B. `75 V,25 V`
C. `20 V,80V`
D. `50 V, 50 V`

Explanation

To solve the problem step by step, we will analyze the circuit with two identical parallel plate capacitors connected in series, where one capacitor has a dielectric slab inserted.

### Step 1: Understand the System
We have two identical capacitors, C1 and C2, connected in series to a 100 V battery. A dielectric slab with a dielectric constant \( K = 3 \) is inserted into the first capacitor (C1).

### Step 2: Determine the Capacitance of the Capacitors
The capacitance of a capacitor without a dielectric is given by:
\[ C = \frac{\varepsilon_0 A}{d} \]
When a dielectric is inserted, the capacitance of the first capacitor (C1) becomes:
\[ C_1 = K \cdot C = 3C \]
The second capacitor (C2) remains unchanged:
\[ C_2 = C \]

### Step 3: Calculate the Total Capacitance in Series
For capacitors in series, the total capacitance \( C_{total} \) is given by:
\[ \frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} \]
Substituting the values:
\[ \frac{1}{C_{total}} = \frac{1}{3C} + \frac{1}{C} \]
Finding a common denominator:
\[ \frac{1}{C_{total}} = \frac{1 + 3}{3C} = \frac{4}{3C} \]
Thus, the total capacitance is:
\[ C_{total} = \frac{3C}{4} \]

### Step 4: Calculate the Total Charge in the Circuit
The total charge \( Q \) stored in the capacitors when connected to the battery is:
\[ Q = C_{total} \cdot V = \frac{3C}{4} \cdot 100 \]
\[ Q = 75C \]

### Step 5: Calculate the Voltage Across Each Capacitor
The voltage across each capacitor can be calculated using the formula \( V = \frac{Q}{C} \).

For the first capacitor (C1):
\[ V_{C1} = \frac{Q}{C_1} = \frac{75C}{3C} = 25 \, \text{V} \]

For the second capacitor (C2):
\[ V_{C2} = \frac{Q}{C_2} = \frac{75C}{C} = 75 \, \text{V} \]

### Step 6: Conclusion
The potential difference across the first capacitor (with dielectric) is 25 V, and across the second capacitor is 75 V. Therefore, the final answer is:
\[ V_{C1} = 75 \, \text{V}, \quad V_{C2} = 25 \, \text{V} \]

### Final Answer
The potential difference across the capacitors will be, respectively, \( 75 \, \text{V} \) and \( 25 \, \text{V} \).

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