Electromagnetics Minus the Fields

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...)

WannabeNewton
How will you preserve locality without fields? If you don't have the concept of fields to describe locality then I can't see how a pure force law, withing nothing else added, could be compatible with special relativity.

Nice question by the way!

EDIT: Just to be clear, I know you can more or less derive Maxwell's equations starting from just Coloumb's law and special relativity but this naturally leads to the concept of fields (which defeats the purpose of your question because you are talking about a formulation in which the concept of a field is nonexistent if I read your OP correctly).

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Jano L.
Gold Member
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)

More accurately, it is possible to write down equations of motion for point-like particles that describe subset of situations described by the theory with fields and Maxwell's equations.

For example, we can replace the Maxwell equations by their retarded solutions and write down equations of motion for a system of any number of point-like particles.

It is possible to use other solutions, for example advanced solutions. Feynman and Wheeler studied curious combination half retarded + half advanced solutions:

J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433.

http://rmp.aps.org/abstract/RMP/v21/i3/p425_1

In these theories, the independent existence of the EM field loses its importance, but the partial fields of the particles are still present and useful notions.

1) the energy is conserved in such theories irrespective of how we write the equations. If we do not use fields as variables, we can still express their energy in terms of the particle variables. However, the resulting equation will be probably more complicated and less easily interpreted than the common expressions that contain fields.

2) one could try to modify the equation of motion of the charged particle, as Dirac and others after him tried to do, but so far no such attempt has been very convincing. The idea of radiation drag acting on a point particle has so serious problems that it seems best to avoid it.

Or is there something the fields add which can't really be explained using just charges and the forces between them?

In principle, the description via general EM fields is more general, because they can contain part that does not correlate to motion of any particle (free field, being the solution of the homogeneous Maxwell equations).

How will you preserve locality without fields? If you don't have the concept of fields to describe locality then I can't see how a pure force law, withing nothing else added, could be compatible with special relativity.

Nice question by the way!

EDIT: Just to be clear, I know you can more or less derive Maxwell's equations starting from just Coloumb's law and special relativity but this naturally leads to the concept of fields (which defeats the purpose of your question because you are talking about a formulation in which the concept of a field is nonexistent if I read your OP correctly).

I don't really know enough about SR to understand the significance of locality being violated, but if I understand correctly you're saying that if such a particle-force interpretation of electromagnetics existed, it would fall apart as soon as you tried to make it compatible with modern physics? (I guess for the same reason that the action-at-a-distance Newtonian model of gravity was scrapped for general relativity?)

More accurately, it is possible to write down equations of motion for point-like particles that describe subset of situations described by the theory with fields and Maxwell's equations.

For example, we can replace the Maxwell equations by their retarded solutions and write down equations of motion for a system of any number of point-like particles.

It is possible to use other solutions, for example advanced solutions. Feynman and Wheeler studied curious combination half retarded + half advanced solutions:

J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433.

http://rmp.aps.org/abstract/RMP/v21/i3/p425_1

In these theories, the independent existence of the EM field loses its importance, but the partial fields of the particles are still present and useful notions.

1) the energy is conserved in such theories irrespective of how we write the equations. If we do not use fields as variables, we can still express their energy in terms of the particle variables. However, the resulting equation will be probably more complicated and less easily interpreted than the common expressions that contain fields.

2) one could try to modify the equation of motion of the charged particle, as Dirac and others after him tried to do, but so far no such attempt has been very convincing. The idea of radiation drag acting on a point particle has so serious problems that it seems best to avoid it.

In principle, the description via general EM fields is more general, because they can contain part that does not correlate to motion of any particle (free field, being the solution of the homogeneous Maxwell equations).

Thanks for the reply! One thing I'm still wondering, though, is what happens to conservation of momentum under such a theory? The force law Griffiths presents doesn't always obey Newton's third law, and so momentum isn't always conserved (he actually gives an example of this earlier in the book to motivate the idea that EM fields carry momentum). So would we be forced to change the definition of momentum from p = mv to something that actually is conserved? Or maybe we'd have to abandon momentum conservation altogether?

Jano L.
Gold Member
The conservation of momentum is maintained in electromagnetic theory, and can be easily expressed with fields.

Particle theory of the above kind is still consistent with electromagnetic theory, so the total momentum is conserved.

This total momentum has contribution from the particles + contribution from the fields in whole space.

If we could express the field in terms of the particle variables, the momentum could be expressed in such way too and from the above it follows that such total momentum would be conserved.

However, it is quite problematic, because the new expression for momentum would refer not only to quantities at the present time, but to all particle variables at all past times (retarded field), or all future times (advanced fields) or both (half ret + half adv. fields). So expressing momentum in such theory would be very different mathematically from what we are accustomed to see with fields. That is also one advantage of the field concept - one can write equations referring to one instant of time.

BruceW
Homework Helper
Another thing is that (classically) the fields can exist without matter. We could (for example) have a universe with just radiation, and without matter. (And I think at some time in the past, the universe was primarily radiation-dominated). Such a situation is not completely trivial, even though no matter is around.

No, you need the fields for a complete description of electromagnetic. The fields exist in their own right. How can you describe an isolated electromagnetic wave in free space without using fields? The force construction replacing the fields construction only works when you are talking about the interaction of charges. For a EM wave in free space, what is exerting a force on what?

Jano L.
Gold Member
No, you need the fields for a complete description of electromagnetic. The fields exist in their own right.
That is one possible but not necessary viewpoint.

How can you describe an isolated electromagnetic wave in free space without using fields? The force construction replacing the fields construction only works when you are talking about the interaction of charges. For a EM wave in free space, what is exerting a force on what?

An EM wave in free space is an idealization that is introduced for convenience, not for necessity. In practice what is modeled by a plane wave can originate as a spherical wave in a distant charged particle.