A parallel plate capacitor with air between the plates has a capacitance of 8pF. The separation between the plates is now reduced by half and the space between them is filled with medium of dielectric constant 5. The value of capacitance of a capacitor in the second case is

Options

A. 0.8 pF
B. 3.2 pF
C. 80 pF
D. 40 pF

Answer

To solve the problem step by step, we will use the formula for the capacitance of a parallel plate capacitor and apply the changes mentioned in the question.

### Step 1: Understand the initial capacitance
The initial capacitance \( C \) of the capacitor with air between the plates is given as:
\[ C = 8 \, \text{pF} = 8 \times 10^{-12} \, \text{F} \]

### Step 2: Identify the formula for capacitance
The formula for the capacitance of a parallel plate capacitor is given by:
\[ C = \frac{A \epsilon}{d} \]
where:
- \( A \) is the area of the plates,
- \( \epsilon \) is the permittivity of the medium between the plates,
- \( d \) is the separation between the plates.

### Step 3: Modify the parameters based on the problem
In the second case:
- The separation \( d \) is reduced by half, so the new separation \( d' = \frac{d}{2} \).
- The medium between the plates is changed to a dielectric with a dielectric constant \( k = 5 \).

### Step 4: Calculate the new permittivity
The new permittivity \( \epsilon' \) when the dielectric is introduced is:
\[ \epsilon' = k \epsilon_0 = 5 \epsilon_0 \]
where \( \epsilon_0 \) is the permittivity of free space (air).

### Step 5: Write the new capacitance formula
The new capacitance \( C' \) can be expressed as:
\[ C' = \frac{A \epsilon'}{d'} = \frac{A (5 \epsilon_0)}{\frac{d}{2}} = \frac{A (5 \epsilon_0) \cdot 2}{d} = 10 \frac{A \epsilon_0}{d} \]
This shows that the new capacitance \( C' \) is 10 times the original capacitance \( C \).

### Step 6: Calculate the new capacitance
Now substituting the initial capacitance:
\[ C' = 10 C = 10 \times 8 \, \text{pF} = 80 \, \text{pF} \]

### Final Answer
The value of capacitance of the capacitor in the second case is:
\[ C' = 80 \, \text{pF} \]

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