UPDATE September 7, 2024: We have a conceptual sign error here. The value of the action in the second section below does rise if we move -Q closer to Q. To cancel the change in the value of the action, some other potential included in V must rise in the action, which is of the usual form
T - V.
That is, when we move -Q closer to Q, we can harvest energy to some other potential in V. Then the value of the action is stationary. This means that Q attracts -Q. Everything is sensible and right.
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The electromagnetic lagrangian seems to be malformed in a way that it cannot handle electric charges of different signs.
Using the electromagnetic action to derive the field of a charged shell Q
E = C / r²
be around it, so that the sum
potential energy of Q + field energy of E
is minimized?
That is, we assume that E is of the form a constant C / r². What should C be?
The potential energy of Q in the field E is
-C Q / R.
The field energy of E is
∞
∫ 1/2 ε₀ C² / r⁴ * 4 π r² dr
R
∞
= ∫ 2 π ε₀ C² / r² dr
R
= 2 π ε₀ C² / R.
The sum of the energies is
1 / R * (-C Q + 2 π ε₀ C²).
The derivative with respect to C is
1 / R * (-Q + 4 π ε₀ C),
and it is zero at
C = 1 / (4 π ε₀) * Q,
which gives Coulomb's law. We see that one can derive the field of Q correctly from the electromagnetic action.
The potential energy of Q at the minimum is
-1 / (4 π ε₀) * Q² / R,
and the field energy is
2 π ε₀ * 1 / (4 π ε₀)² * Q² / R
= 1/2 * 1 / (4 π ε₀) * Q² / R.
This is an example of the virial theorem. The "resisting energy" of the built field E is 1/2 of the "gained energy" in the potential.
The electromagnetic action gets a reasonable solution for the field of a single spherical charged shell. Unfortunately, for two charges, we will get nonsensical results.
Deriving the energy change when a shell of a charge -Q descends on a shell of a charge Q
Let us assume that Q is positive. The field E points to the radial direction. The electric potential is negative at Q. This is just like in the previous section.
R R₂
× | ----> E |
center Q -Q
Let the shell of negative charge be at a radius R₂. How much does the electromagnetic action change if we reduce R₂ by dr?
The potential of the shell Q rises
1 / (4 π ε₀) * Q² / R₂² * dr.
The field energy of E is reduced by
1/2 ε₀ * (1 / (4 π ε₀) * Q / R₂²)²
* 4 π R₂² dr
= 1/2 * 1 / (4 π ε₀) * Q² / R₂² * dr.
The energy of the system rises if we move the outer shell inward. This means that Q repels -Q.
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UPDATE September 7, 2024: No, it does not repel. It attracts, if we think through this carefully.
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Conclusions
The electromagnetic action seems to work ok only in the case where we have a uniform charge distribution on a single spherical shell. If we have two shells, it does not work at all.
How to repair the action for different signs?
It may be necessary to introduce the concept of a private field of a charge, as we have suggested many times in our blog. The electron (or positron) would create its own electric field according to the usual electromagnetic action. Once we have the private fields, we can define the potential of each particle in the field of other particles.
The action should be written for N private fields, instead of one global electromagnetic field.
The interaction between an electron and a positron for very slow movements can be derived from the energy of their combined field (?). Or the action can be written based on the Coulomb force between charges, and we can then forget about the electromagnetic field.
For very quick movements, the interaction is from the Coulomb force. Field energy does not have time to react. How to reconcile these two very different approaches? Either use the Coulomb potential of the other charge, or integrate the field energy over the entire space? We cannot do both at the same time. We have earlier written about the fact that the Coulomb force seems to "know" beforehand what the combined field energy will be, once the fields are updated.
It looks like that a genuine, reasonable action for electromagnetism is not really known at the present. The problems which we uncovered in this blog post are showstoppers.
Quantum field theory may help us?
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