Extra inertia in an electric field: it cannot be "private"!

Let us calculate the electric potential of an electron next to a kilogram of hydrogen, assuming that the electrons have been removed from hydrogen.

The inertia in an electric field cannot be "private"

A kilogram of hydrogen contains 6 * 10²⁶ atoms, which correspond to a charge of 10⁸ coulombs. If the electron is at a distance of 1 meter, the potential is

       V = 9 * 10⁹ * 1.6 * 10⁻¹⁹ * 10⁸ J

           = 0.14 J.

That is a huge potential compared to the mass-energy of the electron, which is 0.8 * 10⁻¹³ J.

In our March 25, 2023 post we suggested that the inertia of an electron is "private" in the field of each other charge. That is, the amounts of inertia with respect to each other charge have to be summed. Now we see that the hypothesis cannot be true. The inertia of an electron close to any matter would be immense.

We have to return to the hypothesis that the extra inertia has to come from the combined field of charges.

How much energy is shipped in the fields of protons and electrons if we move a test electron?

A ton of matter contains some 10³⁰ electrons or protons. The field energy of these particles around 10⁻¹⁰ m from the particle is

       ~ 10⁻⁵

of the mass-energy of the electron, 511 keV. At longer distances than 10⁻¹⁰ m, the fields of the electrons and the protons cancel each other out almost entirely.

                                                            ^

                                                            |

                     ●                                    •

     proton or electron           test electron

Let us put a ton of matter within a meter from our test electron. The electric field of the test electron close to the other particle is

       ~ 10⁻²⁰

times of the field of the particle. If we move the electron in the diagram one meter vertically, then

       ~ 10⁻²⁰ * 10⁻⁵ * 511 keV

of field energy moves a distance of

       ~ 10⁻¹⁰ m.

That is, the energy shipping in the field of the other particle is 10⁻³⁵ times the shipping in the movement of the test electron.

We conclude that a ton of matter close to the test electron can add a fraction

       ~ 10³⁰ * 10⁻³⁵ = 10⁻⁵

to the inertia of the test electron. Shorter distances than 10⁻¹⁰ m add less.

Let us add R meters of mass around the test electron. The effect for each 1 meter of more mass goes as

       ~ 1 / r,

where r is the distance. The effect of R meters of mass on the inertia of the test electron is

       ~ 10⁻⁵ * ln(R).

It is not negligible for Earth, for which R = 6 * 10⁶ meters.

We see that the energy shipping adds a small, but non-negligible fraction of inertia to the test electron. Polarization of matter adds more inertia. It is probably hard to distinguish these between added inertia from polarization, and added inertia from energy shipping in the electric field.

Conclusions

We must return back to the hypothesis that the extra inertia of a test charge comes from energy flowing in the field. That is case 2 in our March 25, 2023 blog post.

The fact that the Coulomb force can be calculated from the field energy of the combined electric field (if charges move slowly) suggests that the field energy does play some role in the force.

But we face the problem that the field is only updated at the speed of light. We may speculate that the system works as if the field would be updated infinitely fast. Then the Coulomb force does come from the field energy. Maybe the extra inertia, too, is determined by an infinitely fast update of the field?