A parallel plate condenser with air between the plates possesses the capacity of `10^(-12) F` .Now, the plates are removed apart, so that the separation is twice the original value. The space between the plates is filled with a material of dielectric constant 4.0. Then new value of the capacity is (in farad)

Options

A. `4 xx 10^(-12)`
B. `3 xx 10^(-12)`
C. `2 xx 10^(-12)`
D. `0.5 xx 10^(-12)`

Answer

To solve the problem, we need to calculate the new capacitance of a parallel plate capacitor after the plates are separated and filled with a dielectric material. Here are the steps to find the solution:

### Step 1: Understand the initial conditions
The initial capacitance of the parallel plate capacitor is given as:
\[ C = 10^{-12} \, \text{F} \]
The initial separation between the plates is \( d \).

### Step 2: Write the formula for capacitance
The capacitance \( C \) 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 3: Determine the new separation
The plates are moved apart, and the new separation becomes:
\[ d' = 2d \]

### Step 4: Introduce the dielectric constant
When the space between the plates is filled with a dielectric material of dielectric constant \( k = 4 \), the new capacitance \( C' \) can be calculated using the modified formula:
\[ C' = \frac{k A \epsilon_0}{d'} \]
Substituting \( d' = 2d \):
\[ C' = \frac{k A \epsilon_0}{2d} \]

### Step 5: Substitute the values
Now substituting \( k = 4 \) into the equation:
\[ C' = \frac{4 A \epsilon_0}{2d} \]
This simplifies to:
\[ C' = \frac{2 A \epsilon_0}{d} \]

### Step 6: Relate the new capacitance to the initial capacitance
From the initial capacitance formula, we know that:
\[ C = \frac{A \epsilon_0}{d} \]
Thus, we can express \( C' \) in terms of \( C \):
\[ C' = 2 \left( \frac{A \epsilon_0}{d} \right) = 2C \]

### Step 7: Calculate the new capacitance
Now substituting the value of \( C \):
\[ C' = 2 \times 10^{-12} \, \text{F} = 2 \times 10^{-12} \, \text{F} \]

### Final Answer
The new value of the capacitance is:
\[ C' = 2 \times 10^{-12} \, \text{F} \]

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