E=mc ²

The equation E  =  mc is undoubtedly most famous in the world. Formulated in 1905 by Albert Einstein within the framework of the restricted Relativity, it means that:

the energy ( E ) of a free particle and at rest, is equal to its Masse ( m ) multiplied by the square of the Speed of light ( C ² ) in the Vide.

Its celebrity is probably due to the fact that it made it possible to consider that humanity could extract one day starting from the mass from the matter of gigantic energies, and it is what occurred with the realization of the atomic piles as of the bombs of the same name (that it is advisable to call nuclear ). Let us recall that the France produces in approximately 2006 80  % of sound electricity in nuclear plants whose assessment of energy can be roughly evaluated starting from this formula.

In fact, a thorough study of the theory shows than it is 100  % of the electrical energy (as any other energy) which is appraisable with this formula! Even whenever the nuclear potential does not return directly concerned, and that another potential energy is used.

(See note below.)

One thus has, for the E  equation; =  mc:

• E: the energy, in Joule S in the international system of unit, but often expressed in electronvolt S, more adapted unit;
• m: the Mass in Kilogram S;
• C: the Speed of light in the vacuum: c  = : 299792458  m/s, is c  = : 89875517873681764  m/s very exactly.

• At rest means that the particle, or any other object, has a null speed compared to the selected reference frame.

• Libre means that the particle is subjected to no Potentiel.

Note:

It is appropriate to notice that the field of application of this formula does not stop with the only field of the nuclear reactions. It is in fact a universal relation between the mass and energy: they are like the recto and the back of the same physical entity mass-energy ! Thus, during an oxygenated combustion (use of a chemical potential); for example the combustion of coal, one must expect to observe a variation of the total mass of the molecules entering in reaction. The Carbon dioxide product is very (very) slightly less massive than the coal and the Oxygène from which it is resulting! This variation is actually negligible, because concerned energies are of an order of magnitude of several million times weaker than during the nuclear reactions. This variation (equal to the equivalence of mass of released energy) is about the tenth of billionth of the mass of the chemical reagents, that is to say of 2 orders of magnitude lower than best measurements. This is why one continues to say in chemistry, that the mass is preserved during a reaction.

History

The formula E = mc had been written as of the 19th century. In 1900, Henri Poincaré shows the inertia of the electromagnetic radiation, and proposes to represent the electromagnetic radiation like a fictitious fluid of density e/c ², E being the density in energy. What is equivalent to write E=mc ² in the case of the electromagnetic radiation.

In 1904 and 1905, Friedrich Hasenöhrl obtained an equation which we can rewrite: $E=\; \backslash \; frac\; \{4\}\; \{3\}\; mc^2$ by studying the Pressure of radiation in a closed cavity. Factor 4/3 is due to an error of the physicist. It is interesting to see that one can obtain the relation only by employing the Maxwell's equations (this result is however not astonishing insofar as these equations are most naturally formulated within the framework of the restricted Relativité).

Philipp Lenard presented the equation as being of Hasenöhrl, to make an Aryan creation of it.

Interpretation

This formula, that it is in unit S of mass (E = m) or in conventional units (E = mc), directly binds energy E of a body to its mass m . She says indeed that the mass is a form of energy just like are to it the potential energy or the kinetic energy. The energy of a body is thus not only of the kinetic energy but also of the mass. And just like the kinetic energy can be, for example, exchanged in the form of heat, the mass can also be exchanged in the form of heat, of kinetic energy or in any other form of energy. For this reason, the Conservation of the mass during a reaction is not perfect, in particular during a nuclear reaction.

Because of the importance of the Speed of light, energy corresponding to 1 kg of matter is enormous: $9.10^\; \{16\}$ Joule S, is the energy produced by a Nuclear reactor during approximately two years.

Description by the E/m report/ratio

If one is interested in the report/ratio E/m (energy on mass), as for example in the case of the disintegration of the Positronium which emits two gamma rays of energy (measured) 0,511  MeV = 0,8186×10 −13   J, then one obtains:

$\backslash \; sqrt\; \{\backslash \; frac\; \{E\}\; \{m\}\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{0,8186\; \backslash \; cdot\; 10^\; \{-\; 13\}\}\; \{9,1083\; \backslash \; cdot\; 10^\; \{-\; 31\}\}\}\; =\; \backslash \; sqrt\; \{8,9874\; \backslash \; cdot\; 10^\; \{16\}\}\; =\; 2,997\; \backslash ,\; 56$

\ cdot 10^8 \ \ mathrm {m} \ cdot \ mathrm {S} ^ {- 1}.

That corresponds at the speed C of the light in the vacuum: one finds, in experiments, the equation $E=mc^2$.

Generalization

The equation $E=mc^2$ is valid only at rest, C. - with-D. when the object has a null speed compared to the selected reference frame. It is not valid any more if it acquires a Speed v compared to this reference frame. Its energy observed E then will be given according to the following relation which rises from the restricted Relativité taking account of the kinetic energy thus added by this speed v :

$E=\; \backslash \; gamma\; m\; c^2\; \backslash ,$ with $\backslash \; gamma\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}$ the Factor of Lorentz.

If the body has a nonnull mass, and if its speed v approaches speed of light C , then v/c tends towards 1 , and its energy $E=\; \backslash \; gamma\; mc^2$ becomes extremely large C. - with-D. tends towards the infinite one when v tends towards C . This prevents it from reaching exactly the speed C , which is thus a limit.

Another way of writing this result uses the Quantité of movement $p=\; \backslash \; gamma\; mv$ of the body: $E^2=m^2c^4\; +\; p^2c^2\; \backslash ,$.
And if one wants to establish the link with the traditional Mécanique, one must introduce the kinetic concept of energy $T$: $E=T+mc^2\; \backslash ,$.

One from of deduced the formulas then:
$\backslash \; begin\; \{boxes\}\; p^2c^2=T^2+2Tmc^2\; \backslash \; \backslash \; T=\; (\backslash \; gamma-1)\; mc^2\; \backslash \; end\; \{boxes\}$

Particular case of a body of null mass

If the body has a null mass, the equation is used: $E^2=m^2c^4+p^2c^2\; \backslash ,$, which is written then: $E=pc\; \backslash ,$. Such a body has a speed equal to C and one speaks about Impulsion rather than of kinetic energy, term which would be unsuitable here.

In Physical of the particles, several particles reaches speed C by their null mass: the Photons which transport the electromagnetic Rayonnement, the bosons of gauge which transmit the others fundamental interactions standard model. Within the framework of the General relativity the gravitational waves also move with speed of light and the associated particle, called Graviton should also be of null mass. Nevertheless to date, and contrary to all the other quoted particles, neither the graviton nor associated gravitational radiation were observed in experiments.

E in units of mass

All the formulas used above are in conventional units . However they can also be presented in units of mass.

That means that one regards the kilograms as unit of energy: according to the formula $E=mc^2$, C can be used like a conversion factor of the kilograms into joules. Thus, the formula $E=mc^2$ where E is expressed in conventional units (joules), can be expressed in units of mass: $E\; =\; m\; \backslash ,$.

In the same way, the Temps T can be expressed in meters instead of the being in seconds. That is done by multiplying T by C , and by replacing T by c×t in the equations used. The constant C then becomes a conversion factor of the seconds into meters.

Remark :

C cannot be regarded as conversion factor of the mass to the energy , but only of the kilograms to the joules . The same remark applies to C .

.

Nonrelativistic limit

The equation $E\; =\; T+mc^2\; \backslash ,$ with $T\; =\; (\backslash \; gamma-1)\; mc^2\; \backslash ,$, in which T represents the kinetic energy, can be approximated when the speed of the element is low in front of C .

For a Speed of the element of mass m much smaller than C , the factor γ can be developed mathematically in a series to give us:
$\backslash \; gamma\; =\; 1+\; \backslash \; frac\; \{v^2\}\; \{2c^2\}\; +\; \backslash \; frac\; \{3v^4\}\; \{8c^4\}\; +\dots$.
Of this series, only the first two terms are physically significant, from where:
$(\backslash \; gamma-1)\; =\; \backslash \; frac\; \{v^2\}\; \{2c^2\}$.
By replacing this term in the expression of the kinetic energy, one finds: $T\; =\; \backslash \; frac\; \{1\}\; \{2\}\; mv^2$, which corresponds to the kinetic energy in mechanical not-relativist. Thus, the relativistic theory joined the classical theory when speeds of the solid elements are negligible in front of that of the light. With the same assumption, the total energy of the solid element, can be approximate by the expression: $E\; =\; m\; (\backslash \; frac\; \{1\}\; \{2\}\; v^2+c^2)$.

Other possible notations

In the equation $E=mc^2\; \backslash ,$, E $E\_\; \{is\; not\; total\; energy\; early\}\; \backslash ,$, but only energy at rest $E\_0$. Indeed, total energy includes/understands energy at rest (the mass m), and the kinetic energy T.

For less confusion, it would be necessary to write: $E\_0\; =\; m\_0c^2\; \backslash ,$ where $m\_0\; \backslash ,$ is the rest mass . Thus, while posing:

$E\; =\; E\_0\; +\; T\; \backslash ,$ (total energy)

and $m\; =\; \backslash \; gamma\; m\_0\; \backslash ,$ (mass observed),

one finds $E=mc^2\; \backslash ,$, which is then valid if the body is not at rest.

Remark :

the fact that an object is moving or at rest , depends on the reference frame from which one takes measurements. An object can thus be in rest for a reference frame has and being moving for a reference frame B, and thus to have a total energy $E\_\; \{early\}$ different. But it has a clean mass i.e. the mass $m\_0$, of value independent of the reference frame, i.e. it with the same value in any reference frame. Or in other words, an object at rest compared to any reference frame given (has; B; …) the same mass $m\_0$ in the specified reference frame has, whatever it is.

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