Question

Calculate the capacitance of the Earth. Treat the Earth as an isolated spherical conductor of radius \(6371 \mathrm{~km}\).

Short answer

Answer: The approximate capacitance of the Earth is 715.4 pF (picoFarads).

Step-by-step solution

Convert radius to meters

First, we need to convert the given radius from kilometers to meters: $$ 6371 \mathrm{~km} \times \frac{1000 \mathrm{~m}}{1 \mathrm{~km}} = 6,371,000 \mathrm{~m} $$ The radius of the Earth in meters is now \(6,371,000 \mathrm{~m}\).

Calculate capacitance

Now, we can plug the values into the capacitance formula: $$ C = 4\pi \epsilon_0 R $$ Where \(\epsilon_0 = 8.854\times 10^{-12} \mathrm{F/m}\) is the vacuum permittivity. Substituting the values, we get: $$ C = 4\pi (8.854\times 10^{-12} \mathrm{F/m}) (6,371,000 \mathrm{~m}) $$

Compute the result

Now, we can perform the calculations to find the capacitance: $$ C \approx 4 \times 3.14 \times (8.854\times 10^{-12} \mathrm{F/m}) \times (6,371,000 \mathrm{~m}) $$ $$ C \approx 715.4 \times 10^{-12} \mathrm{F} $$ Thus, the capacitance of the Earth can be approximated as \(715.4 \times 10^{-12} \mathrm{F}\) or \(715.4 \mathrm{pF}\) (picoFarads).