Basically my question is this: could we, in principle, formulate all of classical electromagnetic theory purely in terms of the force one charge exerts on another, or do we need to include the electromagnetic fields in order to arrive at a complete theory? ("complete" as in equivalent to classical electrodynamics using Maxwell's equations)
Griffiths' EM book got me wondering this. Starting with the Lienard-Wiechert potentials, he derives an equation for the force that one arbitrarily moving point charge exerts on another arbitrarily moving point charge. He then says that "the entire theory of classical electrodynamics is contained in that equation." The expression is really complicated and unwieldy, so it's fairly clear why we would prefer to use Maxwell's equations and quantities like the electric and magnetic field, but I'm wondering if it's really true that we could replace those things with just a force law if we wanted to (in principle).
Two things make me question whether we could actually do everything in terms of this "generalized" force law:
1) If we eliminated the notions of electric and magnetic fields, wouldn't conservation of energy/momentum break down since we can't say that the fields carry energy/momentum to make up the difference?
2) Related to the idea of fields carrying energy... I can't understand how the "radiation drag" of an accelerating charge would appear under this paradigm.
So that's why I'm wondering... is it true to say that we don't really need the electric and magnetic fields, and electromagnetics can, in principle, be boiled down to finding the force between two charges? Or is there something the fields add which can't really be explained using just charges and the forces between them?
(By the way, the expression for the force can be found at the end of Section 10.3 of "Introduction to Electrodynamics" (3rd Ed.) by D. Griffiths. I'll type it out if someone doesn't have access to that book, but that's more TeX coding than I feel like doing right now...)
Griffiths' EM book got me wondering this. Starting with the Lienard-Wiechert potentials, he derives an equation for the force that one arbitrarily moving point charge exerts on another arbitrarily moving point charge. He then says that "the entire theory of classical electrodynamics is contained in that equation." The expression is really complicated and unwieldy, so it's fairly clear why we would prefer to use Maxwell's equations and quantities like the electric and magnetic field, but I'm wondering if it's really true that we could replace those things with just a force law if we wanted to (in principle).
Two things make me question whether we could actually do everything in terms of this "generalized" force law:
1) If we eliminated the notions of electric and magnetic fields, wouldn't conservation of energy/momentum break down since we can't say that the fields carry energy/momentum to make up the difference?
2) Related to the idea of fields carrying energy... I can't understand how the "radiation drag" of an accelerating charge would appear under this paradigm.
So that's why I'm wondering... is it true to say that we don't really need the electric and magnetic fields, and electromagnetics can, in principle, be boiled down to finding the force between two charges? Or is there something the fields add which can't really be explained using just charges and the forces between them?
(By the way, the expression for the force can be found at the end of Section 10.3 of "Introduction to Electrodynamics" (3rd Ed.) by D. Griffiths. I'll type it out if someone doesn't have access to that book, but that's more TeX coding than I feel like doing right now...)
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