If the circumference of a sphere is 2 m, then capacitance of sphere in water would be

Options

A. 2700 pF
B. 2760 pF
C. 2780 pF
D. 2800 pF

Answer

To find the capacitance of a sphere in water given its circumference, we can follow these steps:

### Step 1: Find the radius of the sphere
Given the circumference \( C \) of the sphere is 2 m, we can use the formula for the circumference of a circle:

\[ C = 2\pi r \]

Substituting the given circumference:

\[ 2 = 2\pi r \]

Dividing both sides by \( 2\pi \):

\[ r = \frac{1}{\pi} \text{ m} \]

### Step 2: Use the formula for capacitance of a sphere
The capacitance \( C \) of a sphere in a medium with relative permittivity \( \epsilon_r \) is given by:

\[ C = 4\pi \epsilon_0 \epsilon_r r \]

Where:
- \( \epsilon_0 \) (permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{F/m} \)
- \( \epsilon_r \) (relative permittivity of water) is approximately 79

### Step 3: Substitute the values into the capacitance formula
Now substituting the values we have:

\[ C = 4\pi (8.85 \times 10^{-12}) (79) \left(\frac{1}{\pi}\right) \]

### Step 4: Simplify the expression
The \( \pi \) in the numerator and denominator cancels out:

\[ C = 4 \times 8.85 \times 10^{-12} \times 79 \]

### Step 5: Calculate the capacitance
Calculating the numerical value:

\[ C = 4 \times 8.85 \times 79 \times 10^{-12} \]

Calculating \( 4 \times 8.85 \times 79 \):

\[ C \approx 2800 \times 10^{-12} \, \text{F} \]

### Step 6: Convert to picoFarads
Since \( 1 \text{ pF} = 10^{-12} \text{ F} \):

\[ C \approx 2800 \text{ pF} \]

### Final Answer
The capacitance of the sphere in water is approximately **2800 picoFarads (pF)**.

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