Problem 31 - Fundamentals of Physics

Question

Earth’s atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total surface area of the planet?

Step-by-Step Solution

Answer

The total electric current intercepted by Earth's surface is approximately 0.122 A.

Step 1: Understanding the Problem

To find the electric current, we need to know the charge carried by the protons and how many protons hit the Earth per second. From the problem, each square meter intercepts 1500 protons per second.

Step 2: Finding the Charge of a Proton

Each proton has a charge of approximately \(1.6 \times 10^{-19}\) coulombs.

Step 3: Calculating Current for One Square Meter

The current is calculated as the product of the number of protons and the charge per proton. For one square meter: \[ I = 1500 \times 1.6 \times 10^{-19} = 2.4 \times 10^{-16} \text{ A (amperes)} \]

Step 4: Calculating Earth's Total Surface Area

The surface area \(A\) of Earth is calculated using the formula for the surface area of a sphere: \[ A = 4\pi r^2 \] where \(r\) is Earth's radius approximately \(6.37 \times 10^6\) meters. Thus, \[ A = 4\pi (6.37 \times 10^6)^2 \approx 5.1 \times 10^{14} \text{ m}^2 \]

Step 5: Calculating Total Electric Current Intercepted

Multiply the current per square meter by the total surface area to find the total current intercepted: \[ I_{\text{total}} = 2.4 \times 10^{-16} \times 5.1 \times 10^{14} = 1.224 \times 10^{-1} = 0.122 \text{ A} \]

Key Concepts

Cosmic Rays

Cosmic rays are highly energetic particles that travel through space and frequently reach the Earth's atmosphere. These particles originate from various sources, including the sun, distant stars, and even remnants of supernovae.

• Most cosmic rays are atomic nuclei that have been stripped of their electrons, primarily consisting of protons, which are positively charged hydrogen nuclei.
• Some cosmic rays can also be electrons or heavier atomic nuclei.

When they enter Earth's atmosphere, cosmic rays interact with molecules and atoms, creating showers of secondary particles. This phenomenon is a valuable source of information for scientists studying the universe, as it provides insights into cosmic events and conditions beyond our planet.
Understanding cosmic rays is essential for fields like astrophysics and helps in considering protective measures for satellites and space missions.

Protons

Protons, as a key component of cosmic rays, are subatomic particles with a positive electric charge found in the nucleus of atoms.

• The charge of a single proton is approximately equal to \(1.6 \times 10^{-19}\) coulombs.
• Protons are incredibly stable and along with neutrons, they form the core of atomic matter.

Due to their charge and mass, protons play a critical role in determining the electrical characteristics of atoms and are pivotal in forming molecules through chemical bonds. When we observe protons as cosmic rays, their interaction with the Earth's atmosphere can tell us a lot about both cosmic activities and the processes occurring within atmospheric layers.
Given their charge, protons contribute directly to the calculation of electric currents when they reach Earth's surface.

Earth's Surface

The surface of Earth acts as a vast target area for cosmic ray bombardment. These particles, specifically the protons in cosmic rays, constantly strike every square meter of the Earth's surface at a high rate.

• It is important to recognize that Earth is not a flat surface, but roughly spherical in shape.
• Therefore, the surface area for potential proton impact is calculated using the formula for the surface area of a sphere: \( A = 4\pi r^2 \).

With Earth's radius being approximately \(6.37 \times 10^6\) meters, this results in a total surface area of about \(5.1 \times 10^{14}\) square meters. The sheer magnitude of this area is why cosmic rays, despite having small individual effects, can accumulate to significant levels when considering the entire planet.
The distribution and impact of cosmic rays across the Earth's surface also vary based on altitude and atmospheric conditions, making the study of cosmic rays an essential aspect of environmental and space science.

Charge Calculation

Charge calculation is a fundamental process to determine electric currents, especially when discussing phenomena like cosmic rays striking Earth. The current generated by these charged particles depends on the charge each particle carries and their impact frequency.

• To find the total current a surface area intercepts, we multiply the number of protons by the charge of one proton and further by the total area of impact.
• For our example, each square meter of Earth's surface intercepts 1500 protons per second, each contributing a charge of \(1.6 \times 10^{-19}\) coulombs.

The electric current for one square meter is calculated as \( I = 1500 \times 1.6 \times 10^{-19} = 2.4 \times 10^{-16} \text{ A} \).
The comprehensive scale is achieved by considering the entire surface of the Earth. Multiplying the current for one square meter by Earth's total surface area \(5.1 \times 10^{14}\) results in the total intercepted current: \( I_{\text{total}} = 0.122 \text{ A} \).
Understanding these calculations is vital for applications in electrical science and planetary sciences, providing insights into natural processes involving charged particles.