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From your point of view, two electrons are initially at rest. In time, they repel one another, leading to an increase in both of their kinetic energies. If they are isolated from the rest of the universe, how is the kinetic energy of the entire system increasing? Asked another way: where is this energy coming from?

I really don’t know how to proceed. Any feedback or sketches of math for me to further investigate are well appreciated.

Thank you for your input.

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    $\begingroup$ Potential energy decreasing as kinetic energy increases? $\endgroup$
    – Claudio Saspinski
    Commented Jun 21, 2020 at 21:48

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The energy is coming from the electromagnetic field.

Electromagnetic fields carry an energy. The energy per unit volume is $$ u = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2 \mu_0} B^2 $$ where $E$ is the electric field and $B$ is the magnetic field. For two charges at rest $B=0$ but $E \ne 0$.

To get the total energy in the field you have to integrate this $u$ over all space. This cannot be done if you think of the charges as point-like (because the answer comes out as infinity), but for the purpose of doing this calculation you can just model the charges as if they were each a very tiny ball, and then you get sensible results. When the charges move apart the field very near each charge does not change much, but the field everywhere else does. One finds that, overall in the case you asked about, the final field energy is smaller than the initial energy, and this is where the energy given to the kinetic energy of the charges came from.

When we teach this at high-school level, we don't normally talk about field energy, but instead we talk about 'potential energy' of the charges. This is two ways of talking about the same thing.

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  • $\begingroup$ Thank you for your answer. I’ll write a computer program to convince myself of this conservation of energy when considering the field as well as the (tiny) particles. I want to upvote your answer but don’t have enough reputation. Please accept this written thank you 🙏🏽 $\endgroup$
    – McMath
    Commented Jun 21, 2020 at 22:07
  • $\begingroup$ If the combined energy is $(E_1+E_2)^2$, and the individual energies are $E_i^2$, then the interaction energy is $2E_1\cdot E_2$, which might be finite even for a point charge. $\endgroup$
    – JEB
    Commented Jun 22, 2020 at 5:15
  • $\begingroup$ What happened to the good old Coulomb potential in this answer? $\endgroup$
    – my2cts
    Commented Jun 22, 2020 at 10:08
  • $\begingroup$ @my2cts The Coulomb potential is a way of tracking field energy. The energy itself is a property of the field; that is where it resides. When we say that something "has potential energy" we are really saying that we think the field will give the energy back to the particle if the particle is released, which is true, except that in practice energy can also be radiated away as e.m. waves and then the concept of potential energy breaks down. But you don't need to know that at high school level. $\endgroup$
    – Andrew Steane
    Commented Jun 22, 2020 at 11:16
  • $\begingroup$ I disagree. In QM the potential is used, as in QFT. $\endgroup$
    – my2cts
    Commented Jun 22, 2020 at 11:20
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Two electrons repel each other by the Coulomb potential and the potential energy is $e^2/4\pi\epsilon_0 r$. The sum of potential and kinetic energy is constant, so the increase in kinetic energy comes from a decrease in potential energy.

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One of the result of the Heisenberg Uncertainty principle states that you can't determine the position or momentum of the electrons can't be determined to 100% accuracy.

$$\Delta x\Delta p = \frac{\hbar}{2\pi}.$$

This means the electrons were never really at rest Where does this energy comes from? Very little is known till now.

But as the other answers have pointed out Some potential energy was needed to bring them together which is lost as the electrons move away due to repelling between them. As the Law of Conservation of Energy states this potential energy gets converted to kinetic energy

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  • $\begingroup$ There is no "conservation of kinetic energy" in this case. $\endgroup$
    – my2cts
    Commented Jun 22, 2020 at 10:02
  • $\begingroup$ This is not an answer to the questio but a comment. Besides, the two electrons could be together in a trap, so localised and close to one another. If the trap is switched off, you arrive at the scenario in the post. $\endgroup$
    – my2cts
    Commented Jun 22, 2020 at 10:06
  • $\begingroup$ My fault, will update the answer accordingly $\endgroup$
    – Tim Crosby
    Commented Jun 22, 2020 at 10:09

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