In a parallel plate capacitor of capacitance `C`, a metal sheet is inserted between the plates, parallel to them. If the thickness of the sheet is half of the separation between the plates. The capacitance will be

Options

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

Explanation

To solve the problem, we need to analyze the effect of inserting a metal sheet into a parallel plate capacitor. Here’s a step-by-step solution:

### Step 1: Understand the Initial Setup
We have a parallel plate capacitor with capacitance \( C \). The capacitance of a parallel plate capacitor is given by the formula:
\[ C = \frac{A \epsilon_0}{D} \]
where:
- \( A \) is the area of the plates,
- \( \epsilon_0 \) is the permittivity of free space,
- \( D \) is the separation between the plates.

### Step 2: Insert the Metal Sheet
A metal sheet is inserted between the plates, and its thickness is half of the separation between the plates. Therefore, if the original separation between the plates is \( D \), the thickness of the metal sheet is:
\[ \text{Thickness of metal sheet} = \frac{D}{2} \]

### Step 3: Determine the New Effective Distance
After inserting the metal sheet, the effective distance between the plates can be considered as the distance of the air gap remaining. The air gap on either side of the metal sheet will be:
\[ \text{Air gap} = D - \text{Thickness of metal sheet} = D - \frac{D}{2} = \frac{D}{2} \]
Thus, the effective distance \( D_{\text{eff}} \) between the plates becomes:
\[ D_{\text{eff}} = \frac{D}{2} + 0 + \frac{D}{2} = \frac{D}{2} \]
The metal sheet itself does not contribute to the capacitance since it is a conductor, and its effect is to reduce the distance between the plates.

### Step 4: Calculate the New Capacitance
Now, we can calculate the new capacitance \( C' \) using the effective distance:
\[ C' = \frac{A \epsilon_0}{D_{\text{eff}}} = \frac{A \epsilon_0}{\frac{D}{2}} = \frac{2A \epsilon_0}{D} \]
Since the original capacitance \( C \) is given by \( C = \frac{A \epsilon_0}{D} \), we can express the new capacitance as:
\[ C' = 2C \]

### Conclusion
The new capacitance after inserting the metal sheet is:
\[ C' = 2C \]