Ranking of rest energies

Problem

Ranking of rest energy. Order from smallest to largest the rest energies $E_0=mc^2$ of: an electron ($m_e\approx 9{,}1\cdot 10^{-31}$ kg), a proton ($m_p\approx 1{,}67\cdot 10^{-27}$ kg), a mosquito ($m_z\approx 2$ mg). Express the values in joules.

Solution

Formula. $E_0 = mc^2$, with $c^2 \approx 9\cdot 10^{16}\ \text{m}^2/\text{s}^2$. Since $c^2$ is common to all, the order of the energies follows the order of the masses.

Electron.

$E_0^{e} = (9{,}1\cdot 10^{-31})(9\cdot 10^{16}) \approx 8{,}2\cdot 10^{-14}\ \text{J}.$

(Equivalent to $\approx 0{,}51$ MeV, the well-known value.)

Proton.

$E_0^{p} = (1{,}67\cdot 10^{-27})(9\cdot 10^{16}) \approx 1{,}5\cdot 10^{-10}\ \text{J}.$

(About $938$ MeV, almost $2000$ times the electron.)

Mosquito ($m_z = 2\ \text{mg} = 2\cdot 10^{-6}$ kg).

$E_0^{z} = (2\cdot 10^{-6})(9\cdot 10^{16}) \approx 1{,}8\cdot 10^{11}\ \text{J}.$

(Tens of thousands of kWh: the rest energy of a single mosquito is enormous.)

Ordering.

$\ev{E_0^{\,e^-} < E_0^{\,p} < E_0^{\,\text{mosquito}}}$

The energies span $\sim 25$ orders of magnitude: even a tiny mass hides colossal quantities of energy.

Links

Topics: Special relativity Concepts: Mass-energy equivalence Skills: Ranking reasoning