Question

Assuming Earth as a spherical conductor of diameter 12,800 km, calculate its capacity.

Hint:

Use \( C = 4\pi\varepsilon_0 R = R/(9\times10^{9}) \) with radius \( R = 6.4\times10^{6}\ \text{m} \) (half the given diameter).

Answer 1

Step 1: Formula for capacitance of an isolated sphere.
The capacitance of an isolated spherical conductor of radius \( R \) is
\[ C = 4\pi\varepsilon_0 R = \frac{R}{\dfrac{1}{4\pi\varepsilon_0}} \]
where \( \dfrac{1}{4\pi\varepsilon_0} = 9\times10^{9}\ \text{N·m}^2/\text{C}^2 \).

Step 2: Find the radius.
Diameter \( = 12800\ \text{km} = 12800\times10^{3}\ \text{m} = 1.28\times10^{7}\ \text{m} \).
\[ R = \frac{1.28\times10^{7}}{2} = 6.4\times10^{6}\ \text{m} \]

Step 3: Substitute.
\[ C = \frac{R}{9\times10^{9}} = \frac{6.4\times10^{6}}{9\times10^{9}} \]

Step 4: Arithmetic.
\[ C = \frac{6.4}{9}\times10^{6-9} = 0.711\times10^{-3}\ \text{F} = 7.11\times10^{-4}\ \text{F} \]
\[ C \approx 711\ \mu\text{F} \]

\[\boxed{C \approx 711\ \mu\text{F} = 7.11\times10^{-4}\ \text{F}}\]