The capacity of a parallel plate condenser without any dielectric is C. If the distance between the plates is doubled and the space between the plates is filled with a substance of dielectric constant 3, the capacity of the condenser becomes:

Options

A. `(3)/(4) C`
B. `(9)/(2)C`
C. `(2)/(3)C`
D. `(3)/(2)C`

Explanation

To solve the problem, we need to analyze how the capacitance of a parallel plate capacitor changes when the distance between the plates is doubled and a dielectric material is introduced.

### Step-by-Step Solution:

1. **Understand the formula for capacitance**:
The capacitance \( C \) of a parallel plate capacitor without a dielectric is given by the formula:
\[ C = \frac{\varepsilon_0 A}{D} \]
where \( \varepsilon_0 \) is the permittivity of free space, \( A \) is the area of the plates, and \( D \) is the distance between the plates.

2. **Identify the initial conditions**:
The initial capacitance is given as \( C \). Therefore, we can express it as:
\[ C = \frac{\varepsilon_0 A}{D} \]

3. **Change in distance**:
According to the problem, the distance between the plates is doubled. Thus, the new distance \( D' \) becomes:
\[ D' = 2D \]

4. **Introduce the dielectric**:
The space between the plates is filled with a dielectric material that has a dielectric constant \( K = 3 \). The new permittivity \( \varepsilon' \) of the capacitor becomes:
\[ \varepsilon' = K \cdot \varepsilon_0 = 3 \varepsilon_0 \]

5. **Calculate the new capacitance**:
The new capacitance \( C' \) with the dielectric and the new distance can be expressed as:
\[ C' = \frac{\varepsilon' A}{D'} = \frac{3 \varepsilon_0 A}{2D} \]

6. **Relate the new capacitance to the original capacitance**:
We can substitute the original capacitance \( C \) into the equation:
\[ C' = \frac{3 \varepsilon_0 A}{2D} = \frac{3}{2} \cdot \frac{\varepsilon_0 A}{D} = \frac{3}{2} C \]

7. **Final result**:
Thus, the new capacitance \( C' \) of the capacitor is:
\[ C' = \frac{3}{2} C \]

### Conclusion:
The capacity of the condenser after doubling the distance and filling it with a dielectric of constant 3 becomes \( \frac{3}{2} C \).

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