Air filled capacitor of capacitance `2 muF` is filled with three dielectric material of dielectric constant `K_(1) = 4, K _(2) = 4 ` and `K_(3) = 6` as shown in the figure . The new capacitance of the capacitor is

Options

A. `50 mu F`
B. `2 muF`
C. ` 20 mu F`
D. ` 10mu F `

Answer

To find the new capacitance of the capacitor filled with three dielectric materials, we can follow these steps:

### Step 1: Understand the Initial Conditions
The initial capacitance of the air-filled capacitor is given as:
\[ C_0 = 2 \, \mu F \]

### Step 2: Identify the Dielectric Constants
The dielectric constants for the three materials are:
- \( K_1 = 4 \)
- \( K_2 = 4 \)
- \( K_3 = 6 \)

### Step 3: Calculate the Equivalent Dielectric Constant
When multiple dielectrics are present, the equivalent dielectric constant \( K_{eq} \) can be calculated based on how the dielectrics are arranged (series or parallel). Assuming they are arranged in series, the formula for the equivalent dielectric constant is:
\[ \frac{1}{K_{eq}} = \frac{1}{K_1} + \frac{1}{K_2} + \frac{1}{K_3} \]

Substituting the values:
\[ \frac{1}{K_{eq}} = \frac{1}{4} + \frac{1}{4} + \frac{1}{6} \]

### Step 4: Calculate Each Term
Calculating each term:
\[ \frac{1}{4} = 0.25, \quad \frac{1}{4} = 0.25, \quad \frac{1}{6} \approx 0.1667 \]

Adding these values together:
\[ \frac{1}{K_{eq}} = 0.25 + 0.25 + 0.1667 = 0.6667 \]

### Step 5: Find the Equivalent Dielectric Constant
Taking the reciprocal to find \( K_{eq} \):
\[ K_{eq} \approx \frac{1}{0.6667} \approx 1.5 \]

### Step 6: Calculate the New Capacitance
The new capacitance \( C \) can be calculated using the formula:
\[ C = K_{eq} \cdot C_0 \]
Substituting the values:
\[ C = 1.5 \cdot 2 \, \mu F = 3 \, \mu F \]

### Step 7: Conclusion
The new capacitance of the capacitor after filling it with the three dielectric materials is:
\[ C = 3 \, \mu F \]