Question

Earth’s atmosphere is constantly bombarded by cosmic ray protonsthat 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$\mathbf{1500}$ protons per second. What would be the electric current intercepted by the total surface area of the planet?

Short answer

The electric current intercepted by the total surface area of the planet is$0.122\text{A}$ .

Step-by-step solution

Stating the given data 

1. The average rate of protons per second through the surface is $1500\frac{\text{protons}}{\text{s}\cdot {\text{m}}^{\text{2}}}$.
2. Radius of the Earth, $\mathrm{R}=6.37\times {10}^6\text{m}$.

Understanding the concept of Ampere’s Law

Using the concept of current flow through a given area, we can get the net current intercepted by the spherical surface.

Formulae:

The net current flowing through a given area,

$\mathrm{i}=\mathrm{rate}\mathrm{of}\mathrm{protons}\mathrm{per}\mathrm{area}\times \mathrm{area}\times \mathrm{electronic}\mathrm{charge}\mathrm{of}\mathrm{the}\mathrm{particle}$ (i)

The net charge considering the number of charges present,$\mathrm{q}=\mathrm{ne}$ (ii)

The total surface area of the sphere,$\mathrm{A}=4{\mathrm{\pi r}}^2$ . (iii)

Calculation of the net current through the surface 

The proton flux is given as 1500 protons per square meter per second, where each proton provides a charge of $\mathrm{q}=+\mathrm{e}$.

Now, we can get the surface area of the Earth using equation (iii) as

$\begin{array}{c}\mathrm{A}=4\mathrm{\pi }{(6.37\times {10}^6\text{m})}^2\\ =5.1\times {10}^{14}{\text{m}}^{\text{2}}\end{array}$

Thus, substituting this area and the equation (ii) in equation (i), the net current intercepting the Earth’s surface is

$\begin{array}{c}\mathrm{i}=(5.1\times {10}^{14}{\text{m}}^{\text{2}})\left(1500\frac{\text{protons}}{\text{s}\cdot {\text{m}}^{\text{2}}}\right)(1.6\times {10}^{-19}\text{C/proton})\\ =0.122\text{A}\end{array}$

Hence, the value of the current is$0.122\text{A}$ .