91Ó°ÊÓ

The average number of protons per cubic meter is calculated by,

\(\overline N = \frac{{{\rho _c}\left( {0.10} \right)}}{{{m_p}}}\)

Here\(\overline N \) is the number of protons per cubic meter,\({\rho _c}\)is the critical density,\({m_p}\)is the mass of proton.

With the help of the previous question, infer:

\({\rho _{closure}}{\rm{ }} = {\rm{1}}{{\rm{0}}^{{\rm{ - 26}}}}\,{\rm{kg/}}{{\rm{m}}^{\rm{3}}}\)

Then, the density will be:

\(\begin{align}\rho &= {\rm{0}}{\rm{.1 }}{\rho _{closure}} \\ &= {\rm{1}}{{\rm{0}}^{{\rm{ - 27}}}}{\rm{ kg/}}{{\rm{m}}^3}\end{align}\)

The mass per cubic meter is\(m{\rm{ }} = {\rm{ 1}}{{\rm{0}}^{{\rm{ - 27}}}}\,{\rm{kg/}}{{\rm{m}}^3}\).

The number of protons is then evaluated as:

\(\begin{align}\overline N {\rm{ }} &= {\rm{ }}\frac{{{\rm{1}}{{\rm{0}}^{{\rm{ - 27}}}}{\rm{ kg/}}{{\rm{m}}^3}}}{{{\rm{1}}{\rm{.67 \times 1}}{{\rm{0}}^{{\rm{ - 27}}}}{\rm{kg}}}}{\rm{ }}\\ &= {\rm{ 0}}{\rm{.60 per cubic metre}}\end{align}\)

Therefore, the average number of protons per cubic metre is \({\rm{0}}{\rm{.6 }}{{\rm{m}}^{{\rm{ - 3}}}}\).