The intensity of cosmic ray radiation decreases rapidly with increasing energy, but there are occasionally extremely energetic cosmic rays that create a shower of radiation from all the particles they create by striking a nucleus in the atmosphere. Suppose a cosmic ray particle having an energy of \(10^{10} \mathrm{GeV}\) converts its energy into particles with masses averaging \(200 \mathrm{MeV} / \mathrm{c}^{2}\) (a) How many particles are created? (b) If the particles rain down on a \(1.00-\mathrm{km}^{2}\) area, how many particles are there per square meter?

First, we'll convert the given energy units into SI units: Joules (J). 1 GeV = \(1\times10^9\) eV, and 1 eV = \(1.6\times10^{-19}\) J. So, the initial energy of the cosmic ray particle, E_initial, is: \(E_{initial} = 10^{10}\,\mathrm{GeV} \times\frac {1\times10^9\,\mathrm{eV}}{1\,\mathrm{GeV}}\times\frac {1.6\times10^{-19}\,\mathrm{J}}{1\,\mathrm{eV}}\) The average mass of the particles created can also be converted into energy using E=mc^2. Let's convert the mass to energy: \(E_{particle} = (200\,\frac{\mathrm{MeV}}{\mathrm{c}^2})\times\frac {1\times10^6\,\mathrm{eV}}{1\,\mathrm{MeV}}\times\frac {1.6\times10^{-19}\,\mathrm{J}}{1\,\mathrm{eV}}\times\frac {c}{c^2}\) Where c is the speed of light \(3\times10^8\) m/s.

To find the number of particles created, divide the initial energy by the energy per particle: \(N = \frac{E_{initial}}{E_{particle}}\)

The given area is 1.00 km^2, and we are asked to determine the particles per square meter: Area = \(1.00 \,\mathrm{km}^2 \times\frac{1\times10^6\,\mathrm{m}^2}{1\,\mathrm{km}^2}\) Now, we can calculate the number of particles per square meter: \(N_{\mathrm{square\,meter}} = \frac{N}{\mathrm{Area}}\)