# I Does matter exert a Force on Fields (or Light)?

#### bob012345

Gold Member
Summary
Do we speak of forces on EM fields or light?
In a simple example of two current carrying wires, there are mutual forces. Do we speak of the forces on each wire as action-reaction or as someone I'm debating with, each wire and the fields from the other wire as action-reaction? Or both? Thanks.

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#### kuruman

Science Advisor
Homework Helper
Gold Member
Put two long current-carrying wires parallel to each other. Wire 1 carries current $I_1$ and is in magnetic field $\vec B_2$ generated by wire 2. You can calculate the force on an element of length $\vec L$ on wire 1 from $\vec F_{12}=I_1\vec L\times \vec B_2$. Wire 2 carries current $I_2$ and is in magnetic field $\vec B_1$ generated by wire 1. You can also calculate the force on an element of length $\vec L$ on wire 2 from $\vec F_{21}=I_2\vec L\times \vec B_1$. When you do all this correctly, you will see that these forces are in accordance with Newton's 3rd law, i.e. it will be true that $\vec F_{21}=-\vec F_{12}$. Turning off the current in either wire, causes both forces to disappear.

• bob012345

#### Ibix

Science Advisor
There are circumstances where you can ignore the fields and just pretend that two physical objects are acting on each other without worrying about the details. Notably, one object pushing another is actually a complicated interaction of the EM fields of their atoms, and you can do Newtonian dynamics without knowing that.

But in general you need to consider the momentum carried by the field - a photon rocket being the obvious example. I'd be reluctant to regard that as a force acting on the field, though. It feels wrong to me. You could certainly calculate the time derivative of the momentum carried by the field, but I'm not sure it makes sense to regard that as a force.

My 2p, anyway.

• bob012345

#### jbriggs444

Science Advisor
Homework Helper
time derivative of the momentum carried by the field, but I'm not sure it makes sense to regard that as a force.
The generalization of "force" to include other instances of momentum being transferred in and out of a system seems fairly natural in retrospect. Though I must admit to the same feeling of unease when introduced to the notion.

If you draw the system boundary in a rocket exhaust stream, you have a momentum flow carried as a mass flow. If you draw the system boundary in the housing for the rocket nozzle you have a momentum flow carried as a plain vanilla force. But where you choose to draw the system boundary ought not have much impact on the system being modeled. It is still basically the same system either way. The words we use to denote the momentum flow should not matter.

A momentum flow in a flashlight rocket should work the same way as in a chemical rocket.

• bob012345

#### bob012345

Gold Member
It feels wrong to me too. Under the Newtonian assumptions of instantaneous interactions at a distance, it's easy to understand. When retardation effects are introduced, I'm not so sure.

#### Ibix

Science Advisor
I see @jbriggs444's point here. I'm not convinced I agree with him, but if I don't we're just disagreeing about whether a momentum flux is always called a force or not.

So I suppose the answer is either. It depends on how you generalise the word "force".

• jbriggs444

#### vanhees71

Science Advisor
Gold Member
I would abandon the word "force" as soon as getting to relativistic physics, and the electromagnetic field is utmost relativistic. I'd rather use the word "interaction".

The discovery of the notion of fields has an experimental basis and some strong hindsight from "natural philosophy" of the 19th century. I think it's pretty fair to say that the discoverer of the field concept is Faraday, questioning the Newtonian idea of "forces" as "actions at a distances" just on heuristic grounds but based in an enormous empirical knowledge from accurate experimental studies of electromagnetic phenomena. Then Maxwell took these ideas up and discovered his famous set of equations describing these phenomena accurately. The trouble was that his theory is not Galilei invariant. The solution of the puzzle famously is the (hitherto final) formulation of special relativity by Einstein (based however on previous work by Lorentz, FitzGerald, Poincare, and others, who fell just short to get the right physical interpretation). Since the special-relativistic space-time structure implies a causality concept that forbids "action-at-a-distance forces" in the Newtonian sense, the most simple descriptions occur using the field concept and substituting action-at-a-distance forces with local-interaction-forces.

The picture is that instead of the "action-at-a-distance-force" acting between electric charges (instantaneous Coulomb force), an electric charge distribution, somewhere localized, implies the existence of an electromagnetic field around it, which is a "fundamental dynamical entity" as the "charged matter" itself. Maxwell's equations then describe the very laws connecting the localized charge distributions with the electromagnetic field. A symmetry analysis based on Noether's theorem shows that the system of charge-current distributions and the electromagnetic field has clearly defined contributions to energy and momentum (as well as stress) by the matter and the fields, and the corresponding energy-momentum-conservation laws translate into local conservation laws in terms of corresponding Noether currents (energy and momentum densities and its currents), which provide the local "forces" acting on the charge-current distributions (as well as the back reaction to the field). These "forces" in this way are made local, i.e., the "force" acting on charges and currents is "caused" by the presence of the field, and energy-momentum conservation thus becomes local and thus consistent with the constraints on causality implied by the relativistic space-time description. That makes it well justified to abandon the notion of "forces", reserving this word for the "action-at-a-distance interactions" in Newtonian physics and talk about "local interactions" in the relativistic case.

There were attempts to get rid of the fields again and finding forms of action-at-a-distance forces compatible with relativity. A famous example is the Wheeler-Feynman absorber theory using half-retarded-half-advanced actions between charges eliminating the em. field. This non-local classical theory can be indeed interpreted in the way making it consistent with causality, but as Pauli famously predicted, Wheeler and Feynman could not produce the promised "quantization" of their absorber theory.

Finally it turned out that the up to today only working quantized theory of electromagnetic interactions is quantum electrodynamics, where all entities, "radiation" and "matter", are described by quantized fields (e.g., in the minimal QED model, a quantized Dirac field and the electromagnetic field with an interaction term provided by the minimal coupling ansatz based on the idea of conjecturing a Dyson-renormalizable quantum field theory).

Thus, as it seems today, Faraday's intuition that Newtons "actions at a distance" are fictitious and to be substituted by "local interactions" mediated by fields has prevailed in a surprising day, providing the most accurate theory ever, the Standard Model of elementary particles.

Unfortunately, it's incomplete since no working theory to also implement gravitational interactions. There's of course General Relativity giving an accurate description of gravity on the classical level in terms of a dynamical spacetime structure. Unfortunately all attempts to quantize this theory, most famously string theory and loop-quantum gravity, have not yet lead to a truely satisfactory description. Unfortunately there's no clue from observations, how a future successful quantum theory of gravity might look like :-(.

#### bob012345

Gold Member
Thanks for the interesting and involved answer! I think you are saying only local interactions mean anything. Then Newton's Third Law as applied to the original situation of the two wires only applies under the classical assumption of instantaneous forces which isn't the reality implying it doesn't actually apply between the two wires themselves although for steady state currents or slowly varying currents it appears so practically. What actually applies is the instantaneous interaction between each wire and the local fields. Is that correct?

#### vanhees71

Science Advisor
Gold Member
In the relativistic-field description it's NOT an instantaneous interaction but a local interaction between the field and the matter. Of course, for quasistatic situations, where retardation effects can be neglected, i.e., if the typical timescales of the relevant dynamics of the fields and the matter is very slow compared to the "signal propgation speed" (which is the speed of light), the Newtonian action-at-a-distance description is a very good approximation.

In the so far most comprehensive and successful fundamental theory, which is the standard model of elementary particle physics, both matter and radiation are described by quantized fields only. This solves a lot of problems of a theory where you have "point particles" as fundamental entities.

Not even for classical electrodynamics there's a fully consistent description due to the radiation-reaction/self-interaction problem. It's of course better for macroscopic matter using continuum-mechanics descriptions like relativistic hydro or relativistic transport theory.

#### bob012345

Gold Member
By instantaneous I just meant very local. Whatever the field is at that instant.

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