Between the plates of a parallel plate capacitor of capacity C, two parallel plates of the same material and same area as the plates of the original capacitor are placed. If the thickness of each plate `=(1)/(5)` the distance between the plates of the original capacitor, then the capacity of the new capacitor will be

Options

A. a) `3/5` C
B. b) `5/3` C
C. c)`(10)/(3)C`
D. d) `(5)/(4)`C

Explanation

To solve the problem, we need to determine the capacitance of a new capacitor formed by inserting two plates of the same material and area as the original capacitor between the plates of the original capacitor. The thickness of each new plate is given as \( \frac{1}{5} \) of the distance between the original plates.

### Step-by-Step Solution:

1. **Identify the Original Capacitance**:
The capacitance \( C \) of a parallel plate capacitor 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,
- \( D \) is the distance between the plates.

2. **Determine the Thickness of the New Plates**:
Let the distance between the original plates be \( D \). The thickness of each new plate is:
\[ t = \frac{1}{5} D \]
Therefore, the total thickness of the two new plates is:
\[ 2t = 2 \times \frac{1}{5} D = \frac{2}{5} D \]

3. **Calculate the New Distance Between the Plates**:
When the two plates are inserted, they occupy some space between the original plates. The new distance \( D' \) between the plates of the capacitor becomes:
\[ D' = D - 2t = D - \frac{2}{5} D = \frac{3}{5} D \]

4. **Calculate the New Capacitance**:
The capacitance of the new configuration \( C' \) can be calculated using the same formula for capacitance, but with the new distance \( D' \):
\[ C' = \frac{\varepsilon_0 A}{D'} = \frac{\varepsilon_0 A}{\frac{3}{5} D} = \frac{5 \varepsilon_0 A}{3D} \]

5. **Relate the New Capacitance to the Original Capacitance**:
We know that the original capacitance \( C \) is:
\[ C = \frac{\varepsilon_0 A}{D} \]
Thus, we can express \( C' \) in terms of \( C \):
\[ C' = \frac{5}{3} \cdot \frac{\varepsilon_0 A}{D} = \frac{5}{3} C \]

### Final Result:
The capacitance of the new capacitor is:
\[ C' = \frac{5}{3} C \]