## Dimensions of energy

Dimensions of force = Mass x acceleration = [M][L][T]-2

Dimensions of work = Dimensions of energy = force x distance = [M][L]2[T]-2

Dimensions of rest energy = m0c2

As one would expect, the dimensions of classical energy and of mass-energy are identical.

SI units (based on the mks system)

In using SI units, relativis-tic energy will come out in joules and the units and dimensions of momentum, speed, kinetic energy and mass will be consistent.

The electron-volt system

It is more usual and numerically less cumbersome to use units based on the electron-volt.

An electron-volt (eV) is the kinetic energy gained or lost by an electron (or any other particle with the same size charge) when it moves through a potential difference of 1 volt.

Example:

Mass of proton mp = 1.6726 x 10-27 kg Rest energy of proton E = m0c2 = 1.5053 J

Since moving a charge of 1 coulomb through a potential of 1 volt takes 1 joule of work and the magnitude of the charge of electron is e = 1.602 coulombs, fi 1 eV = 1.602 x 10-19 joules

(which will give it a speed of 593 km/s, which is 0.2% of the speed of light).

103 eV = 1 keV (kilo electron-volt) 103 keV = 1 MeV (mega electron-volt) 103 MeV = 1 GeV (giga electron-volt)

We can now use electron-volts/c2 as a unit of mass. (In practice bigger units such as MeV/c2 and GeV/c2 are more commonly used in high energy particle physics.)

Expressed in these units the mass and rest mass energy of a particle are numerically equal.

Similarly, since the dimensions of momentum are mass x velocity, it is convenient to express momentum in units of eV/c [in this way the units are self-consistent in Equation (A.2)].

A summary of units of mass, energy and momentum in the electron-volt system:

Physical quantity |
Units |

Mass, m0 |
MeV/c2 |

Rest energy, E = m0c2 |
MeV |

Momentum, p |
Mass of proton mp = 938.3 MeV/c2 Rest energy of proton E = m0c2 = 938.3 MeV The advantage of using these units is that the factor c effectively disappears in relativistic equations such as (A.2), as is shown in the example in Appendix 16.3. |

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