Question

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

Short answer

Question: Calculate the Earth's capacitance, given that its radius is 6371 km. Answer: The Earth's capacitance is approximately \(1.8 \times 10^{-3} \mathrm{~F}\).

Step-by-step solution

Determine the value for vacuum permittivity.

The vacuum permittivity is a constant value, denoted by \(\epsilon_0\). The value of vacuum permittivity is given by \(\epsilon_0 = 8.854 \times 10^{-12} \mathrm{ F/m}\) (farads per meter).

Convert the Earth's radius from kilometers to meters.

The Earth's radius is given as \(6371\) km. To convert this value to meters, we multiply it by \(1000\), since there are \(1000\) meters in a kilometer. Thus, we have \(R = 6371 \times 10^3 \mathrm{~m}\).

Calculate the capacitance using the formula.

Now we have everything we need to calculate the Earth's capacitance. Using the formula \(C = 4\pi\epsilon_0 R\), we plug in the values for \(\epsilon_0\) and \(R\) as follows: \(C = 4\pi(8.854 \times 10^{-12} \mathrm{~F/m})(6371 \times 10^3 \mathrm{~m})\) Now, multiply these values together: \(C = 4\pi(8.854 \times 10^{-12})(6371 \times 10^3)\) \(C = 1.8 \times 10^{-3} \mathrm{~F}\) (approximately) So, the capacitance of the Earth is approximately \(1.8 \times 10^{-3} \mathrm{~F}\).