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I have been introduced to $E =mc^2$ and binding energy recently while studying nuclear physics and I have a hard time understanding the following:

Mass-Energy

  • what does $E=mc^2$ really mean, as I have read some posts and from what I understand it is not the fact that "energy is transformed to mass" so what does this equation really show?
  • Is $E=mc^2$ a new type of energy? is it like all the other energies I'm familiar with but only that this one is due to the physical property "mass" of the object if so can I use it in the energy conservation equation for a system $E_{initial} = E_{final}$?
  • What is mass-energy conservation and how can I use it? Is it similar to energy conservation but more fundamental? : $\sum{Energy_{initial}+Mass_{initial}}=\sum{Energy_{final}+Mass_{final}}$?

Binding Energy

Consider the following fusion reaction:

$_1^2H+_1^3H\rightarrow_2^4He+_0^1n$

Let's say we are given binding energy for each of the nuclei and we are given that the mass of the deuterium that reacts is $2.0g$ and that of tritium is $3.0g$.

The difference in binding energy is:

$\sum (E_B)_{final} - \sum(E_B)_{initial} = \Delta E_{Reaction} $

  • Why is this difference the change in energy of the Reaction, what does binding energy have to do with the change in energy of the reaction?
  • how can I find the energy released in this reaction and why does the energy released in this reaction have to do with the binding energies at all?

Conclusion

I have given the following example of the fusion reaction together in this post with the $E=mc^2$ confusion because I believe that they are closely related. I have attempted to try to understand the problems that I have with binding energy by trying to apply the idea of mass-energy and $E= mc^2$ however in the end I came to the conclusion that I do not understand $E=mc^2$ and mass-energy properly thus I think that is the reason I also do not understand the example of binding energy I gave above.

Thank you for any help!

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closed as too broad by Bill N, Gert, rob May 1 at 22:43

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ This question is too broad. I suggest that 1) you search this PSE for each of your topics separately ($E=mc^2$, binding energy, nuclear reaction energy, fission, fusion) and read the questions and answers which are already written. Vote for the ones that help you. THEN if you still have a good conceptual specific question, come back and write it. $\endgroup$ – Bill N May 1 at 21:45
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The equation

$$ E = mc^2 $$

is supposed to mean this: any body of inertial mass $m$ has, in the frame where it is at rest, localized energy associated with it of value $mc^2$.

This formula is an extrapolation of Einstein's observation that in relativistic theory of matter, any change of internal energy of a body must be associated with change of its inertial mass:

$$ \Delta E = \Delta m c^2. $$

There is some confusion in textbooks and literature that makes people think "mass can be transformed into energy" but you are correct in that this is not a good way to explain this relation. It is true that energy can be transformed from dormant form (potential energy) into apparent or useful form (radiation, heat). But energy is always locally conserved, one cannot have disappearance of mass and appearance of energy. When body increases its internal energy, it also increases its mass.

My answers to your questions:

  • The extrapolated equation means there is a belief that sum of energies of matter particles (kinetic and rest energies) + energies of all fields in a material body ($E$) determines inertial mass of the body. Or in the opposite direction: measuring inertial mass means determining total internal energy of the body.

  • $E$ in the equation is total internal energy of the body in its rest frame, stored in the volume of the body (localized in it). If the body has some potential energy due to being in external field, such as potential energy in gravity field, or electric field, this potential energy is not part of $E$. Potential energy that is due to interaction of parts of the system, however, is part of $E$ - this can be usually localized in the space region that the body occupies. There are cases where this interaction energy is not well localized (Coulomb potential energy of electrically charged body); in that case, it needs to be chosen whether we count energy out of the region as part of $E$, or whether we want to count that as separate energy belonging to the space region outside the body. Usually, this contribution is taken as part of $E$, because it is due to charges in the body.

  • It is a strange name for local energy conservation: energy in some region can change, but only via continuous flow of energy from one place to another. The old law of conservation of mass alone is not valid in relativistic physics, because particles can disappear and turn into radiation. Energy is locally conserved, but mass becomes an inappropriate concept in such processes - how would one measure mass of radiation?

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  • $\begingroup$ Might be helpful to be clear that potential energy between the parts of a system does contribute to the mass of the system. And that this contribution can have either sign. $\endgroup$ – dmckee May 1 at 22:02
  • $\begingroup$ So if I would want to take into account the total energy of a system, I would have to take into account the other external energies and the internal energies that are in that system? $E_{total}= \sum E_{external} + E_{internal}$ where $E_{internal}=mc^2$? $\endgroup$ – Luca Ion May 1 at 22:20
  • $\begingroup$ @LucaIon it depends on what "total energy" means in the question at hand. Something external contribution is a part of it, sometimes not. $\endgroup$ – Ján Lalinský May 1 at 22:30
  • $\begingroup$ @dmckee you are right, I made some changes. $\endgroup$ – Ján Lalinský May 1 at 22:34
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    $\begingroup$ @LucaIon If internal energy changes form and as a result, it heats the body (radioactive decay) but still stays in the body, mass does not change. Only if the heat energy leaves the body, mass decreases. $\endgroup$ – Ján Lalinský May 2 at 19:16

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