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I have been wondering(also searching) for what does the $c$ in eV/$c^2$ stand for? (For example, mass of the electron is $0.511 \, \text{MeV}/c^2$.) I have read that this unit has been derived from Einstein's equation, $E=mc^2$, but how it is possible, We use the symbol $c$ for Coulomb.

Also, tell how to convert this to our normal units of mass ($\mathrm{eV}/c^2 \to \mathrm{kg}$).

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Capital $\mathrm{C}$, in upright font, is the symbol for the coulomb. Lowercase $c$, italicized, is the speed of light in vacuum. Thanks to Einstein's equation, we can switch between mass and energy ($\mathrm{MeV}$ is a unit of energy) by using factors of $c^2$, and sometimes it's more convenient to know the energy equivalent of a particle's mass rather than the mass proper.

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A sample unit conversion for the second half of your question: \begin{alignat}{2} 0.511\,\mathrm{MeV}/c^2 &= 0.511\,\mathrm{MeV}/c^2 \times \frac{10^6\,\mathrm{eV}}{1\,\mathrm{MeV}} \times \frac{1.60\times10^{-19}\,\mathrm{joule}}{1\,\mathrm{eV}} \\ &\quad\qquad \times \frac{1\,\mathrm{kg\cdot m^2/s^2}}{1\,\mathrm{joule}} \times \left( \frac{c}{3.00\times 10^8\,\mathrm{m/s}} \right)^2 \\ &= 9.08 \times 10^{-31}\,\mathrm{kg} \end{alignat} As usual for unit conversion problems, all of the fractions on the right are just clever ways of writing the number one, so multiplying by them doesn't change the value of the quantity at hand.

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For physicists it can be very annoying that our historically evolved units of measurement cause the speed of light $c$ to differ from unity. So physicists often apply a trick to avoid distracting conversion factors corresponding to the numerical value of (powers of) the speed of light popping up in their equations. That trick is simply to define your own units such that length and time units are treated consistently from a relativistic perspective.

So one can, for instance, select an energy unit like the mega-electronvolt ($\text{MeV}$) and use this along with a relativistically consistent unit for mass written as $\text{MeV}/c^2$. Such an expression denotes nothing more than the amount of mass corresponding with one MeV of rest energy.

Similarly, one can use an expression like $\mathrm{s}c$ ($\mathrm{second} \times c$) to denote a unit of distance. This notation, however, is less common, as one usually refers to such a distance unit as light second.

In particle physics it is common to start from the giga-electronvolt ($\text{GeV}$) as energy unit, and and fix all other units such that both the speed of light $c$ and the reduced Planck constant $\hbar$ both have value unity. This leads to units $\text{GeV}/c^2$ for mass, $\hbar / \text{GeV}$ for time, and $\hbar c/ \text{GeV}$ for distance.

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