The capacitance of the earth, viewed as a spherical conductor of radius 6408 km is

Options

A. `980 mu F`
B. `1424 muF`
C. `712 muF`
D. `356 muF`

Answer

To find the capacitance of the Earth viewed as a spherical conductor, we can follow these steps:

### Step 1: Understand the formula for capacitance
The capacitance \( C \) of a spherical conductor can be expressed using the formula:
\[ C = \frac{Q}{V} \]
where \( Q \) is the charge on the conductor and \( V \) is the potential of the conductor.

### Step 2: Determine the potential \( V \)
For a spherical conductor, the potential \( V \) at its surface due to a charge \( Q \) is given by:
\[ V = \frac{kQ}{R} \]
where \( k \) is Coulomb's constant (\( k = \frac{1}{4\pi \epsilon_0} \)), and \( R \) is the radius of the sphere.

### Step 3: Set up the equation for capacitance
Since the Earth is isolated, we can set the potential \( V \) at infinity to zero. Therefore, the potential difference becomes:
\[ V = \frac{kQ}{R} - 0 = \frac{kQ}{R} \]
Substituting this into the capacitance formula gives:
\[ C = \frac{Q}{\frac{kQ}{R}} = \frac{R}{k} \]

### Step 4: Substitute the values
Given that the radius \( R \) of the Earth is \( 6408 \) km, we convert this to meters:
\[ R = 6408 \times 10^3 \, \text{m} \]
Now, we need to substitute the value of \( k \):
\[ k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \]
Thus, the capacitance \( C \) becomes:
\[ C = \frac{6408 \times 10^3}{9 \times 10^9} \]

### Step 5: Calculate the capacitance
Now, we perform the calculation:
\[ C = \frac{6408 \times 10^3}{9 \times 10^9} = \frac{6408}{9} \times 10^{-6} \, \text{F} \]
Calculating \( \frac{6408}{9} \):
\[ C \approx 712 \times 10^{-6} \, \text{F} \]
This can also be expressed in microfarads:
\[ C \approx 712 \, \mu\text{F} \]

### Final Answer
The capacitance of the Earth viewed as a spherical conductor is approximately \( 712 \, \mu\text{F} \).

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