A parallel plate capacitor with air between the plates has a capacitance of 10 pF. The capacitance, if the distance bgetween the plates is reduced by half and the space between tehm is filled with a substance of dielectric constant 4 is

Options

A. `80 pF`
B. `96pF`
C. `100 pF`
D. `120 pF`

Explanation

To solve the problem, we need to determine the new capacitance of a parallel plate capacitor when the distance between the plates is halved and the space between them is filled with a dielectric material of dielectric constant \( k = 4 \).

### Step-by-Step Solution:

1. **Understanding the Formula for Capacitance**:
The capacitance \( C \) of a parallel plate capacitor is given by the formula:
\[ C = \frac{k \cdot A \cdot \epsilon_0}{d} \]
where:
- \( C \) is the capacitance,
- \( k \) is the dielectric constant,
- \( A \) is the area of the plates,
- \( \epsilon_0 \) is the permittivity of free space,
- \( d \) is the distance between the plates.

2. **Initial Conditions**:
We know that the initial capacitance \( C \) is 10 pF (picoFarads) with air between the plates, which means \( k = 1 \) (for air). Therefore, we can write:
\[ C = \frac{1 \cdot A \cdot \epsilon_0}{d} = 10 \text{ pF} \]

3. **New Conditions**:
In the new scenario, the distance \( d \) is reduced by half, so the new distance \( d' \) is:
\[ d' = \frac{d}{2} \]
The dielectric constant of the new material is \( k = 4 \).

4. **Calculating the New Capacitance**:
We can now calculate the new capacitance \( C' \) using the modified distance and dielectric constant:
\[ C' = \frac{k \cdot A \cdot \epsilon_0}{d'} \]
Substituting \( d' \):
\[ C' = \frac{4 \cdot A \cdot \epsilon_0}{\frac{d}{2}} = \frac{4 \cdot A \cdot \epsilon_0 \cdot 2}{d} = \frac{8 \cdot A \cdot \epsilon_0}{d} \]

5. **Relating New Capacitance to Initial Capacitance**:
From the initial capacitance equation, we know:
\[ \frac{A \cdot \epsilon_0}{d} = C \]
Therefore, we can express \( C' \) in terms of \( C \):
\[ C' = 8 \cdot \frac{A \cdot \epsilon_0}{d} = 8C \]
Substituting \( C = 10 \text{ pF} \):
\[ C' = 8 \cdot 10 \text{ pF} = 80 \text{ pF} \]

### Final Answer:
The new capacitance \( C' \) is **80 pF**.

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