The radii of a spherical capacitor are 0.5 m. and 0.6 m. If the empty space is completely filled by a medium of dielectric constant 6, then the capacity of the capacitor will be :–

Options

A. `3.3 xx 10^(-10)F`
B. `2 xx 10^(-9)F`
C. 2F
D. 18F

Answer

To find the capacitance of a spherical capacitor filled with a dielectric medium, we can use the formula for the capacitance of a spherical capacitor, which is given by:

\[ C = \frac{4 \pi \epsilon_0 k a b}{b - a} \]

where:
- \( C \) is the capacitance,
- \( \epsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \, \text{F/m} \)),
- \( k \) is the dielectric constant,
- \( a \) is the inner radius,
- \( b \) is the outer radius.

### Step-by-step Solution:

1. **Identify the given values:**
- Inner radius \( a = 0.5 \, \text{m} \)
- Outer radius \( b = 0.6 \, \text{m} \)
- Dielectric constant \( k = 6 \)

2. **Substitute the values into the formula:**
\[ C = \frac{4 \pi (8.85 \times 10^{-12} \, \text{F/m}) (6) (0.5) (0.6)}{0.6 - 0.5} \]

3. **Calculate the denominator:**
\[ b - a = 0.6 - 0.5 = 0.1 \, \text{m} \]

4. **Calculate the numerator:**
\[ 4 \pi (8.85 \times 10^{-12}) (6) (0.5) (0.6) \]
- First calculate \( 4 \pi \):
\[ 4 \pi \approx 12.5664 \]
- Now calculate the entire numerator:
\[ 12.5664 \times 8.85 \times 10^{-12} \times 6 \times 0.5 \times 0.6 \]
\[ = 12.5664 \times 8.85 \times 10^{-12} \times 1.8 \]
\[ = 12.5664 \times 15.88 \times 10^{-12} \]
\[ \approx 2.000 \times 10^{-10} \, \text{F} \]

5. **Now substitute back into the capacitance formula:**
\[ C = \frac{2.000 \times 10^{-10}}{0.1} \]
\[ C = 2.000 \times 10^{-9} \, \text{F} \]

6. **Final result:**
\[ C = 2 \times 10^{-9} \, \text{F} \, \text{or} \, 2 \, \text{nF} \]

### Final Answer:
The capacitance of the spherical capacitor is \( 2 \times 10^{-9} \, \text{F} \) or \( 2 \, \text{nF} \).