The radius of the earth is `6400km`, what is its capacitance?

Answer

To find the capacitance of the Earth, we can treat it as a spherical capacitor. The formula for the capacitance \( C \) of a sphere is given by:

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

where:
- \( C \) is the capacitance,
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{F/m} \),
- \( r \) is the radius of the sphere.

### Step 1: Convert the radius of the Earth to meters.
The radius of the Earth is given as \( 6400 \, \text{km} \). We need to convert this to meters:

\[ r = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6.4 \times 10^6 \, \text{m} \]

### Step 2: Substitute the values into the capacitance formula.
Now we can substitute the values of \( \epsilon_0 \) and \( r \) into the capacitance formula:

\[ C = 4 \pi (8.85 \times 10^{-12} \, \text{F/m}) (6.4 \times 10^6 \, \text{m}) \]

### Step 3: Calculate the numerical value.
Calculating the above expression step by step:

1. Calculate \( 4 \pi \):
\[ 4 \pi \approx 12.5664 \]

2. Multiply \( 12.5664 \) by \( 8.85 \times 10^{-12} \):
\[ 12.5664 \times 8.85 \times 10^{-12} \approx 1.113 \times 10^{-10} \, \text{F/m} \]

3. Now multiply by \( 6.4 \times 10^6 \):
\[ C \approx 1.113 \times 10^{-10} \times 6.4 \times 10^6 \approx 7.112 \times 10^{-4} \, \text{F} \]

### Step 4: Convert to microfarads.
To express the capacitance in microfarads (\( \mu F \)):
\[ C \approx 7.112 \times 10^{-4} \, \text{F} = 711.2 \, \mu F \]

### Final Answer:
The capacitance of the Earth is approximately \( 711.2 \, \mu F \).

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