# How much energy is stored in the charge on each electron?

The energy of electron formation becomes mass, charge, spin and momentum. How much becomes the charge? We know how much becomes mass: E = mc^2. What is the equation for the charge?

• The charge on an electron is measured as $1.60217662 × 10^{-19}$ coulombs. AFAIK, it has no equation either explaining that amount in fundamental terms or by linking it to $E =mc^2$. Sorry if I have misunderstood you or am wrong in my answer. – user108787 Oct 29 '16 at 13:27

We should start by understanding the concept of electric energy. The electric potential energy at some point in space $$\vec{x}$$ is given by $$\Phi(x)=\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{x}')}{|\vec{x}-\vec{x}'|}\mathrm{d}^3x'$$ where the integration is over all chargers in the universe, $$\epsilon_0$$ is the vacuum permittivity and $$\rho(\vec{x})$$ is the charge density. The energy of an electron depends on the specific location. To be precise, you should consider the work done by moving a charge from one point to other. In that case the work done is: $$W=e(\Phi(\vec{x}_A)-\Phi(\vec{x}_B))$$ where $$e$$ is the electron charge. Perhaps the best example to clarify your question is the Electron–positron annihilation where you have a collision of an electron and a positron (anti-electron). The result is the radiation of two gamma rays: $$e^{-}+e^{+}\rightarrow\gamma+\gamma$$

Lets assume that we start an ideal experiment with only two particles $$e^{-}$$ and $$e^+$$, initially at rest an with a separation $$r$$. Without any other force (i.e., gravitation), we will have the total energy given by: $$E_i=2m_ec^2-\frac{1}{4\pi\epsilon_0}\frac{e^2}{r}$$ After the annihilation, the energy of each gamma rays is given by: $$E_f=h\nu$$ where $$h$$ is Planck's constant and $$\nu$$ is the photon's frequency. The total energy during the annihilation is conserved. However, before that the particles will be accelerated. Charged particles radiate energy on electromagnetic waves, therefore some energy is dissipated. Regarding your questions:

How much becomes the charge? We know how much becomes mass: E = mc^2. What is the equation for the charge?

The charge is not part of the rest energy. An electron alone can't be transformed into energy (photons). It is necessary to have particle's interactions to create or destroy electrons. Therefore, you have to consider the whole energy of the system, where the rest mass is part of the total energy. We know that there is electric charge conservation. Therefore, in order to "create" a charged particles, you need the total initial charge to be equal to the total final charge. In other words, the charge is no created by any type of energy.

Any energy we claim exists must be measured between two or more objects. Its obvious for potential energy, but less obvious for kinetic energy: Consider that the kinetic energy we claim is in a moving object can only be detected/extracted by mechanically connecting it to a mass stationary to our person.

Likewise the 'energy' associated with a field can only be detected or extracted if there is another object nearby that also has that kind of field. For example, there is a certain potential energy associated with two charged particles near each other.

The bottom line is that energy (and thus mass, as well) can only be claimed for a closed system (and practically speaking, it can be nearly closed). That's why we cannot say things like, "a particle moving fast enough, will collapse into a black hole". (BTW, it doesn't). This is because that's an open system; specifically, we've neglected to include the mass in our pocket that would be needed to extract such energy. If, however, we say there is a closed box and in that box there are particles moving relative to each other, then we can say that the kinetic energy of those particles contributes to the total energy (and thus mass) of that box system.

So we can't define the energy in a charged particle unless we include other charged particles in the system.

• Positronium is an electron and positron bound pair, with the lowest energy state of -6.8 eV relative to the unbound pair. So what is the energy in an electron in positronium? en.wikipedia.org/wiki/Positronium – user93146 Aug 28 '18 at 0:17