E=mc²

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E = mc² er ei likning innan teoretisk fysikk som fastset samanhengen mellom masse (m) og energi (E), i eikvar form. Konstanten ljosfarten (c) er òg med i likninga.

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Innhaldsliste

1 Meanings of the formula
2 Implications
3 Background and consequences
4 Applicability of the equation
- 4.1 Using relativistic mass
- 4.2 Using rest mass
5 Low energy approximation
6 Einstein and his 1905 paper
7 Contributions of others
8 Television biography
9 Sjå òg
10 Bakgrunnsstoff
11 På verdsveven

[endre] Meanings of the formula

This formula proposes that when a body has a mass, it has a certain amount of energy, even if it is at rest, and does not have any form of potential energy, chemical energy, etc, it still has that amount of energy. As opposed to the Newtonian mechanics in which a massive body could have no energy at all. That is why we often call the mass the rest energy of the body. The E of the formula can be seen as the total energy of the body, which is proportional to the mass only when the body is at rest.

Conversely, a cloud of photons travelling in empty space, with each photon having no rest mass, still have a mass, m, due to their kinetic energy.

This formula also gives the quantatative relation of energy and mass in any process when they transform into each other, such as a nuclear explosion. Then this E could be seen as the energy released when a certain amount of mass m is annihilated, or the energy absorbed to create a certain amount of mass m. In those cases, the energy released(absorbed) equals in quantity to the mass annihilated(created) times the speed of light squared.

[endre] Implications

In the context of special relativity theory, the implication is that energy and mass are equivalent, and that, now, mass is considered as a form of energy. In practical terms, it led to the atomic bomb and other applications. It is one of the best-known equations of all time. Even those who may not explicitly know what it means may have some idea of its meaning, through culture.

[endre] Background and consequences

The equation resulted from Albert Einstein's inquiry into the dependence of the inertia of a body on its energy content. The famous result of this inquiry is that the mass of a body is actually a measure of its energy content. To understand the significance of this relationship, compare the electromagnetic force with the gravitational force. In electromagnetism, energy is contained in the fields (electric and magnetic) associated with the force and not in the charges. In gravitation, the energy is contained in the matter itself. It is not a coincidence that mass bends spacetime, while the charges of the other three fundamental forces do not.

$\mbox{Rest Energy} = \mbox{mass}\,\times\,\mbox{(speed of light)}^2$

According to the equation, the maximum amount of energy obtainable from an object is equivalent to the mass of the object multiplied by the square of the speed of light.

This equation was crucial in the development of the atomic bomb. By measuring the mass of different atomic nuclei and substraying from that number the mass of the individual protons and neutrons, one can obtain an estimate of the binding energy available within an atomic nucleus. This not only showed that it was possible to release this binding energy by fusion of light nuclei or fission of heavy nuclei, but also to estimate the amount of binding energy which can be released. Note that the masses of the protons and neutrons are still there, and that they too represent an amount of energy.

It is a little known piece of trivia that Einstein originally wrote the equation in the form dm = L/c² (with an "L", instead of an "E", representing energy, the E being utilised elsewhere in the demonstration to represent energy too).

A kilogram of mass completely converts into

89,875,517,873,681,764 joules or
24,965,421,632 kilowatt-hours or
21.48076431 megatons of TNT
approximately 0.0851900643 Quads (quadrillion British thermal units)

It is important to note that practical conversions of mass to energy are seldom 100 percent efficient. One theoretically perfect conversion would result from a collision of matter and antimatter; for most cases, byproducts are produced instead of energy, and therefore very little mass is actually converted. In the equation, mass is energy, but for the sake of clarity, the word converted is used.

[endre] Applicability of the equation

Mal:Cleanup-date

E=mc² applies to all objects with mass, as it is a statement that mass is derived from energy, or energy from mass, and it is possible to convert between the two. Its applicability to moving objects depends on the definition of mass used in the equation.

Usually, this equation applies to an object that is not moving as seen from a reference point. But this same object can be moving from the standpoint of an other frame of reference, so that, for this latter frame, the equation still stands but the total energy (or equivalantly, mass) differs in amount to the former frame. So unlike Newtonian mechanics, in Special Relativity, mass differs in different reference frames.

[endre] Using relativistic mass

Einstein's original papers (e.g. [1]) treated m as what would now be called the relativistic mass. This is related to the rest mass m_o (i.e. the mass of the object in the frame of reference in which it is stationary) in the following way:

$m = \gamma m_0 = \frac{m_0}{\sqrt{1-v^2/c^2}}$

But to obtain the E = mc² equation, we have to start from the equation E² = p²c² + m²c⁴ and put p = 0, which means that we have to put v = 0. That means that we have now a special case where the object is not moving, and where E² is only equal to m²c⁴, or E = mc². It is only in that special case that E = mc² holds. At any other velocity, we have to put back the p²c² in the general equation.
If we now put v = 0 in the equation $m = \gamma m_0 = \frac{m_0}{\sqrt{1-v^2/c^2}}$ we get m = $m o$ . So, at rest i.e. at speed v = 0, rest mass and relativistic mass are the same quantity, and the equation E = mc² can be rewriten as E = $m o c 2$ : there is no difference, except perhaps, that we would have to say that $m o$ is for v = 0.

Then, using the relativistic mass, the equation $E = m c 2$ in the title must be rewritten as 'E = $m o c 2$ for v = 0', which means it does not apply to objects moving at any velocity but only at velocity zero, because the $m o$ here is only for v = 0, and at v = 0, $m o$ = m .

[endre] Using rest mass

Relativistic mass is little used by modern physicists, who use m to denote rest mass so that E = mc² is the rest energy (i.e. energy of the object when at rest) of the object. In this case the equation only applies to stationary objects; the modern form of the equation for an object with any velocity is

$E = \sqrt{p^2c^2+m^2c^4} = \gamma mc^2$ ,

where $p = γ m v$ is the relativistic momentum of the object. This reduces to E = mc² for the zero-velocity case. Notwithstanding the modern usage, for clarity the remainder of this article uses m for relativistic mass and m₀ for rest mass.

[endre] Low energy approximation

Since the rest energy is m₀c², and total energy is kinetic energy plus rest energy, the relativistic kinetic energy is given by

$E_\mathrm{kinetic} = E_\mathrm{total} - E_\mathrm{rest} = \gamma m_0 c^2 - m_0 c^2 = \left(\gamma - 1 \right) m_0 c^2$

and at low velocities this should agree with the classical expression for the kinetic energy,

$E_\mathrm{kinetic}= \frac{1}{2} m_0 v^2$ .

The two can be shown to agree by expanding $γ$ using a Taylor series,

$\gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}} \approx \left( 1+ \frac{1}{2} \left(\frac{v}{c} \right)^2 \right)$ .

Plugging this back into our original equation,

$E_\mathrm{kinetic} \approx \frac{1}{2} \left(\frac{v}{c} \right)^2 m_0 c^2 =\frac{1}{2} m_0 v^2$ ,

we therefore have ½m0v² = E total - E rest or E total = E rest + ½m0v² , the relativistic expression of energy, which is not in agreement with the classical Newtonian expression for the energy which is only kinetic. This shows that relativity is a higher order correction to classical mechanics and that in the low energy or classical regime Newtonian and relativistic mechanics are not equivalent.
Then what is equivalent? It is only the expression of the kinetic energy, not the total energy.

In the case of extrapolating classical mechanics to the very large or very fast, Einstein showed classical mechanics was wrong. In the case of smaller slower objects such as were used in establishing classical mechanics, classical mechanics is a subset of relativistic mechanics. The two theories only contradict each other outside the classical regime.

[endre] Einstein and his 1905 paper

Albert Einstein did not formulate exactly this equation in his 1905 paper "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" ("Does the Inertia of a Body Depend Upon Its Energy Content?", published in Annalen der Physik on September 27), one of the articles now known as his Annus Mirabilis Papers.
What that paper says is exactly this: 'If a body gives off the energy L in the form of radiation, its mass diminishes by L/c².', radiation being, in this case, kinetic energy, and the mass being the ordinary concept of mass used in those times, the same one that we call today, rest energy or invariant mass, depending on the context.
It is the difference in the mass before the ejection of energy and after it, that is equal to L/c², not the entire mass of the object. At this moment it was only theoritical and not proven experimentally.

[endre] Contributions of others

Einstein was not the only one to have related energy with mass, but he was the first to have presented that as a part of a bigger theory, and even more, to have deduced the formula from the premices of this theory. According to Umberto Bartocci (University of Perugia historian of mathematics), the equation was first published two years earlier by Olinto De Pretto, an industrialist from Vicenza, Italy, though this is not generally regarded as true or important by mainstream historians. Even if De Pretto introduced the formula, it was Einstein who connected it with the theory of relativity.

[endre] Television biography

E=mc² was used as the title of a 2005 television biography about Einstein concentrating on the year 1905.

[endre] Sjå òg

Celeritas for the origins of using the c notation in E=mc².
Energy-momentum relation
Mass-energy equivalence
Relativistic mass
mass, momentum, and energy
Inertia

[endre] Bakgrunnsstoff

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Mal:Book reference

[endre] På verdsveven

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