Question

The Earth is constantly being bombarded by cosmic rays, which consist mostly of protons. These protons are incident on the Earth's atmosphere from all directions at a rate of \(1245 .\) protons per square meter per second. Assuming that the depth of Earth's atmosphere is \(120.0 \mathrm{~km}\), what is the total charge incident on the atmosphere in 5.000 min? Assume that the radius of the surface of the Earth is \(6378 . \mathrm{km}\).

Short answer

Answer: The total charge incident on the Earth's atmosphere in 5 minutes is approximately \(3.05 \times 10^2 \mathrm{C}\).

Step-by-step solution

1. Calculate the surface area of the Earth.

We know that the surface area of a sphere is given by the formula: \(A = 4 \pi r^2\). Using the Earth's radius (\(r = 6378 \mathrm{~km}\)), we can find the surface area of the Earth's atmosphere. Note that the radius should be converted to meters. \(A = 4 \pi (6378 \times 10^3)^2 = 4 \pi \times (4.076 \times 10^{14}) = 5.109 \times 10^{15} \mathrm{m^2}\)

2. Find the number of incident protons per second on the entire Earth's surface.

Now, we are given the rate of protons incident per square meter per second, which is \(1245 \mathrm{~protons/m^2s}\). To find the total number of incident protons per second, we multiply this rate by the surface area of the Earth. Total Incident Protons/second = (Rate of Incident Protons/square meter) x (Surface Area of Earth) Total Incident Protons/second = \(1245 \mathrm{~protons/m^2s} \times 5.109 \times 10^{15} \mathrm{m^2} = 6.359 \times 10^{18} \mathrm{protons/s}\)

3. Calculate the total number of incident protons within 5 minutes.

We want to find the total number of incident protons within 5 minutes. Since we have the number of protons incident per second, we just need to multiply this by the number of seconds in 5 minutes. Total Incident Protons (5 minutes) = (Total Incident Protons/second) × (Number of seconds in 5 minutes) Total Incident Protons (5 minutes) = (\(6.359 \times 10^{18} \mathrm{protons/s}\)) × (5 × 60) = \(1.909 \times 10^{21} \mathrm{protons}\)

4. Find the total charge incident on the atmosphere.

We know that each proton has a charge of \(1.602 \times 10^{-19} \mathrm{C}\). To find the total charge, we multiply this charge by the total number of incident protons. Total Charge = (Charge of a single proton) × (Total Incident Protons) Total Charge = (\(1.602 \times 10^{-19} \mathrm{C}\)) × (\(1.909 \times 10^{21} \mathrm{protons}\)) = \(3.05 \times 10^2 \mathrm{C}\) The total charge incident on the Earth's atmosphere in 5 minutes is approximately \(3.05 \times 10^2 \mathrm{C}\).