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 1245.0 protons per square meter per second. Assuming that the depth of Earth's atmosphere is \(120 \mathrm{~km},\) what is the total charge incident on the atmosphere in \(5.00 \mathrm{~min}\) ? Assume that the radius of the surface of the Earth is \(6378 \mathrm{~km}\).

Short answer

Answer: The total charge incident on Earth's atmosphere due to cosmic rays in 5.00 minutes is approximately 3.08 × 10^8 C.

Step-by-step solution

Find the Effective Area of Earth's Atmosphere

To find the area of the Earth's atmosphere, we will first assume it is a sphere and calculate the surface area. The given radius of the Earth is \(6378 \mathrm{~km}\), and the depth of the atmosphere is \(120 \mathrm{~km}\). Therefore, the total radius of the atmosphere is \((6378+120) \mathrm{~km}\). We can calculate the surface area of the sphere using the formula \(A=4 \pi r^2\).

Calculate the Rate of Incident Charge

Next, we need to calculate the charge incident on the Earth's atmosphere due to cosmic rays. We are given the rate of incident protons as \(1245.0 \mathrm{~protons/m^2/s}\). Since the charge of a proton is \(e = 1.6 \times 10^{-19} \mathrm{C}\), we can find the rate of charge incident per square meter: \(\mathrm{charge\,rate \, density} = (1245.0 \mathrm{~protons/m^2/s}) \times (1.6 \times 10^{-19} \mathrm{C/proton})\).

Calculate the Total Charge

Now that we have the rate of charge and the area of the Earth's atmosphere, we can find the total charge incident on the Earth's atmosphere in the given time. First, we need to convert the given time of \(5.00 \mathrm{~min}\) to seconds: \(5.00 \mathrm{~min} \times (60 \mathrm{s/min})\). Then, we can find the total charge by multiplying the charge rate density, the area of the Earth's atmosphere, and the time: \(\mathrm{Total \, Charge} = \mathrm{charge\, rate\, density} \times \mathrm{Area} \times \mathrm{Time}\).

Final Calculation and Answer

Let's calculate the final answer by plugging in the values from the previous steps: 1. Calculate the area of the Earth's atmosphere: \(A = 4 \pi ((6378 + 120)\times 10^3 \mathrm{m})^2 = 5.147\times10^{14} \mathrm{m^2}\). 2. Calculate the charge rate density: \(\mathrm{charge\,rate \, density} = (1245.0 \mathrm{~protons/m^2/s}) \times (1.6 \times 10^{-19} \mathrm{C/proton}) = 1.992 \times 10^{-16} \mathrm{C/m^2/s}\). 3. Convert the given time to seconds: \(5.00 \mathrm{~min} \times (60 \mathrm{s/min}) = 300 \mathrm{s}\). 4. Calculate the total charge: \(\mathrm{Total \, Charge} = (1.992 \times 10^{-16} \mathrm{C/m^2/s}) \times (5.147\times10^{14} \mathrm{m^2}) \times (300 \mathrm{s}) \approx 3.08 \times 10^8 \mathrm{C}\). So, the total charge incident on Earth's atmosphere due to cosmic rays in \(5.00 \mathrm{~min}\) is approximately \(3.08 \times 10^8 \mathrm{C}\).

Key concepts

Earth's atmosphere

Our planet is shrouded in a blanket of gases known as the Earth's atmosphere. This essential layer is composed of multiple levels, each with its own characteristics and roles. The atmosphere is vital for life, as it provides oxygen for breathing, protects us from harmful solar radiation, and helps regulate the Earth's temperature.

When discussing cosmic rays and their interaction with the Earth, it's the outermost layers of the atmosphere that concern us most. Cosmic rays, primarily coming from outside our solar system, are high-energy particles that streak through space at nearly the speed of light. Upon contacting the Earth's atmosphere, these rays collide with atoms, creating a cascade of secondary particles that shower down towards the surface.

Understanding the depth and composition of the atmosphere is integral when calculating the effect of cosmic rays. As seen in the exercise, the atmosphere's depth is taken to be 120 km, which, combined with the radius of the Earth, sets the stage for determining the area this protective shield covers, and in turn, the quantity of cosmic rays it encounters.

Charge of a proton

A proton is one of the building blocks of matter, residing in the nucleus of atoms alongside neutrons. Its significance goes beyond its structural role; the proton carries a fundamental quantity of positive electric charge.

The charge of a proton is approximately equal to the elementary charge, designated as \(e\), which is a constant at \(1.6 \times 10^{-19}\) Coulombs (C). This value is vital for calculations involving electrical phenomena, and it is inversely equal in magnitude to the charge of an electron but of opposite sign, meaning while electrons carry negative charge, protons carry positive charge.

In the context of cosmic rays and our exercise, this elementary charge of protons, when multiplied by the number of protons per unit area and time, yields the charge rate density. This is an essential step towards understanding the total electrical charge that cosmic rays contribute to the Earth's atmosphere over a given period.

Surface area of a sphere

The surface area of a sphere is an essential concept in geometry that extends to various physical applications, including astronomy, physics, and even Earth science. For a sphere with a radius \(r\), the surface area \(A\) is calculated using the formula \(A = 4\pi r^2\).

This calculation is especially relevant when estimating phenomena that cover the Earth, such as satellite coverage areas, weather patterns, or, as in our exercise, cosmic ray exposure. It serves as the base calculation to understand the extent of Earth's atmosphere's exposure to cosmic rays. When using this formula, it's important to keep in mind the units involved; since the radius of the Earth is typically given in kilometers, it should be converted to meters to match the standard scientific units for surface area measurement in square meters (\(m^2\)).

By applying this formula, we can infer that the larger the surface area of a sphere, the greater the potential interaction with external influences such as cosmic rays, which emphasizes the scale of impact these high-energy particles have on our planet.