# Is the Electric field a well defined concept ?

#### DaTario

This question bothers me since ever...

At a given time in a EM course, we stop talking of forces relation between two particles and start conceiving electric interaction as something where one of the charges acts upon the whole space and the other, since it occupies some portion of the space, suffers the consequences of being in an altered space. In this altered space the "who suffers" particle experiences acceleration, which is an evidence that the other particle, the "who acts upon space" particle causes the space to be an out-of-equilibrium ground.

But soon, the teacher comes with the observation that, in order to attach a physically stable meaning to the electric field, one must measure it (the field) with small enough "who suffers" charges (test charges), since larger charge values in the "who suffers" particles will imply forces on the "who acts" charged particles, which will probably change its positions and thus changing the "who acts" action (Electric field).

I have a strange feeling that electric field is not a physically stable concept.

Does any have a comment on this subject ?

Best Wishes

DaTario

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#### Crosson

We get get around this problem by defining the electric field not as the ratio of force to test charge, but as the limit of the ratio of force to test charge, as the magnitude of the test charge approaches zero.

#### DaTario

I wish to add the following considerations:

1) Electric field is offered to us as an alternative way of expressing things in electromagnetism, as one has never talked about "impossible description of some situation in terms of electric forces".

2) As the choice of those "who acts" and those "who suffers" is to be arbitrary, i.e. depends on the student free will, the small test charges criteria implies severe restrictions on the allowed physical situations, using E-field as the descriptive concept.

#### DaTario

Crosson said:
We get get around this problem by defining the electric field not as the ratio of force to test charge, but as the limit of the ratio of force to test charge, as the magnitude of the test charge approaches zero.
But with this in mind, how physically existent is some particular E-field configuration ? Some specific spatial configuration of the E-field is something that has a well definite shape, but exists as long as it is surrounded by "who suffers" charges small enough to label them as: charges approaching zero. Is it?

#### pervect

Staff Emeritus
DaTario said:
I wish to add the following considerations:

1) Electric field is offered to us as an alternative way of expressing things in electromagnetism, as one has never talked about "impossible description of some situation in terms of electric forces".

2) As the choice of those "who acts" and those "who suffers" is to be arbitrary, i.e. depends on the student free will, the small test charges criteria implies severe restrictions on the allowed physical situations, using E-field as the descriptive concept.
I'm not sure what the point of #1 is supposed to be.

As far as #2 goes, there is no limitation on allowed physical situations. If you have two strongly interacting charges, you probe the electric field by using a third, much smaller charge.

This is all speaking classically, of course (after all this is the classical physics forum).

#### DaTario

pervect said:
I'm not sure what the point of #1 is supposed to be.

As far as #2 goes, there is no limitation on allowed physical situations. If you have two strongly interacting charges, you probe the electric field by using a third, much smaller charge.

This is all speaking classically, of course (after all this is the classical physics forum).
sorry for not being clear in #1. Let try again:

I was intending to say that, to the extent of my knowledge, the concept of force is equally valid and makes reference to just the two charges, not mentioning "what would be present at that location if the "who suffers" charge if it wasn't there" .

"As far as #2 goes, there is no limitation on allowed physical situations. If you have two strongly interacting charges, you probe the electric field by using a third, much smaller charge."

But don't you agree that, if you take out the "who suffers" charge from the universe, there will be a rearrengement of charge positions in the "who acts" system of charges ? And if it happens the field then measured with the test "who suffers" charge will nor anymore have correspondence with the field that was at that position in the presence of the "who suffers" charge ?

Best regards,

DaTario

#### robphy

Homework Helper
Gold Member
Some things to keep in mind:

In intro textbooks, the story goes like this.

Using Coulomb's Law, we motivate the concept of the electric field. At one level, one could argue that this is just an alternative way of expressing Coulomb's Law. It also gives us a way to formulate Gauss' Law. At another level, it avoids the "action at a distance" issue and problems with relativity.

Eventually, one learns that the electric field can exist without the presence of charges (e.g., Faraday). In fact, light is an electromagnetic wave.

#### Meir Achuz

Homework Helper
Gold Member
The electric field does not only act on a test charge.
A changing electric field induces a magnetic field.
The connections between electric and magnetic interactions are made much simpler by thinking in terms of the E and B fields. This was Maxwell's approach.
There have been attempts, particularly by Weber in the 19th century to describe this without the fields, but they were not successful, and make relativity much harder to implement. Attempts are still made to use such "action at a distance" theories, but they are not too popular, becuse they get too complicated with unresolved difficulties.

#### DaTario

robphy said:
Some things to keep in mind:

In intro textbooks, the story goes like this.

Using Coulomb's Law, we motivate the concept of the electric field. At one level, one could argue that this is just an alternative way of expressing Coulomb's Law. It also gives us a way to formulate Gauss' Law. At another level, it avoids the "action at a distance" issue and problems with relativity.

Eventually, one learns that the electric field can exist without the presence of charges (e.g., Faraday). In fact, light is an electromagnetic wave.
To the extent of my knowledge, there is no other way of detecting light (or other EM radiation fields) except by methods which can ultimately be named as charge-charge interaction.

Do you have objections to this statement ?

Best Regards,

DaTario

#### DaTario

Let me complete my last post for it seemed to be to be incomplete.

If there is no way of detecting light without the use of charger-charge interaction. It may well be possible that "THAT THING" which exists in empty space and which we decided to call Electric field does not exist by itself, for there is no independent evidence of its existence. All evidences are mediated by charge-charge force mechanism.

Note: I teach this subject with confidence on its validity, but here I am exercizing the attack that I have always thought could be made to this concept. I would like to acknowledge to those who are participating in this debate.

#### Antiphon

The field is a useful device for making calculations. The force definition relies
on at least two charges being present even if one of them is vanishingly
small. While we beleive it is intuitive that a single charge would "create a field"
it's quite possible that this is in fact meaningless.

#### robphy

Homework Helper
Gold Member
Back to the "action at a distance" issue: If q_1 is suddenly moved, does q_2 suddenly feel a different force (since "r" has changed)? If not, how long does q_2 have to wait? and what determines that wait-time?

The following comments might be overkill... but no simpler examples (with explanations that I am happy with) come to mind right now.
So...

The electric field is an observer-dependent concept, as is the magnetic field. They are both components of the electromagnetic field tensor. So, in the sense that the electric field is not [tensor] field in spacetime, it does not exist by itself.

In general relativity, there is a set of equations, which I believe is, called the [electrovacuum] Einstein-Maxwell equations, where the "matter" in this universe is the pure electromagnetic field. (http://scholar.uwinnipeg.ca/courses/38/4500.6-001/Cosmology/OtherEnergyMomentum_Tensors.htm [Broken]) This affects the spacetime geometry, and thus the trajectories of all particles, charged or not.

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#### Nam_Sapper

First, you are confusing the '3-body' problem of measurement in time with electrostatics. Let me explain. In electrostatics, there is no motion. You nail down a positive charge over here, and another charge over there, and calculate the force between them. If you know their mass, and you know there are no other forces, you can calculate their mutual trajectories if you suddenly release them. Mach had a partial answer for this as a means to establish the 3rd Law of action/reaction (and a valid inertial frame for physical laws).

Now you want to describe the force contribution of a single charge to some physical space. You move an arbitrary test-particle (with a charge of '1') anywhere and calculate the force for each point. Luckily, it's a simple (coordinate-free) algebraic equation for this, because a starting axiom says the force varies in a radially symmetric manner, and is best described by a simple distance in any and every direction.

$$F = \frac{Kqq}{d^2}$$

This potential 'field' is NOT the Electric Field. Its just the contribution to the final field by this one particle.

Now imagine if you will a 'MASTER FIELD' like a giant blueprint of space, which is the result of superimposing ALL the contributions from all the particles in the universe (the Law of Superposition !). Now here is the KEY POINT(!): Nobody gets to experience the MASTER FIELD in all its glory. Any and every particle in the universe only experiences (Master Field - Personal Contribution).

The particle's own contribution to the 'Master field' must be subtracted, and what remains is the sum of all the other particle's forces acting on it. That is, NO PARTICLE can exert a force upon itself. That would be like trying to lift yourself up by pulling on your socks! Now this LAW is not arbitrary: it is the natural and absolute result of the PARALLELOGRAM LAW of Vector Addition of Forces, and allows us to calculate and simply (vectorially) add the forces upon a particle from all the other particles. There is no escaping it. If we want to treat Electric repulsion and attraction as forces, we have to treat them in a way that preserves this law.

And now what most Electrostatic Courses and teachers fail to underline: As a consequence of what we have just said, Every particle experiences a different 'Field' from every other particle.

You don't need to resort to infinitesimal 'particles', or limits. You just need to use your head and understand that there is no single 'field'.

You can only speak of a 'field' value at a given point in relation to a specific particle that is sitting at a given location. The value of the field varies depending upon one's point of view. A similar mistake is made by novices when thinking of the gravitational 'field'. No object really has a 'fixed' Centre of Mass. The location of the Centre of Mass is slightly different from every different location in space.

Is the Electric Field a Well-Defined Concept? Yes, if you have a well-defined location (and particle) to measure it from!

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#### reilly

If you really want to understand Electric and magnetic fields then it's best to start with the history of the concept. The idea of a field is central to modern physics, classical and quantum. And, this idea has a long history. If you can, read Sir E.T. Whittaker's History of the Aether and Electricity. You will learn, among other things the dificulty of devloping the notion of a field -- there's a balancing act between intuition, practicality, rigor and applicability. The success of Radar is quite sufficient to confirm that E and B are very solid classical concepts. QM is another matter.

Regards,
Reilly Atkinson

#### DaTario

I agree the concept has some utility. But the impression is that there is something missing in its rigorous definition.

We agree that a conducting positively charged sphere creates a field of some value one meter from its center. Lets call this field E0. If I put a very small, but very very small charged particle at this position, then the force experienced by this particle divided by its charge will be equal to E0. But if I put another particle with some charge Q1 not so small, it will force the distribution on the sphere, modifying it and providing F/Q1 not equal to E0 but to E1. A particle with charge Q2 will experience a field E2 and so on. Thus, why talk about such a particular property of the space ?

Is this reasoning aomewhat provocative ?

#### robphy

Homework Helper
Gold Member
DaTario said:
I agree the concept has some utility. But the impression is that there is something missing in its rigorous definition.

We agree that a conducting positively charged sphere creates a field of some value one meter from its center. Lets call this field E0. If I put a very small, but very very small charged particle at this position, then the force experienced by this particle divided by its charge will be equal to E0. But if I put another particle with some charge Q1 not so small, it will force the distribution on the sphere, modifying it and providing F/Q1 not equal to E0 but to E1. A particle with charge Q2 will experience a field E2 and so on. Thus, why talk about such a particular property of the space ?

Is this reasoning aomewhat provocative ?
By introducing a conductor, the physical situation is made more complicated... especially for large "test" charges. One has to idealize the situation (e.g., limit of small test charges, which does appear in "more rigorous definitions") and abstract from it the concepts we define. It seems to me that these extra complications you introduce (which we try to idealize away) are not too different from friction present in studying freefall in "realistic" situations.

You never addressed my "action at a distance" question in my previous post.

#### DaTario

robphy said:
Back to the "action at a distance" issue: If q_1 is suddenly moved, does q_2 suddenly feel a different force (since "r" has changed)? If not, how long does q_2 have to wait? and what determines that wait-time?

The following comments might be overkill... but no simpler examples (with explanations that I am happy with) come to mind right now.
So...

The electric field is an observer-dependent concept, as is the magnetic field. They are both components of the electromagnetic field tensor. So, in the sense that the electric field is not [tensor] field in spacetime, it does not exist by itself.

In general relativity, there is a set of equations, which I believe is, called the [electrovacuum] Einstein-Maxwell equations, where the "matter" in this universe is the pure electromagnetic field. (http://scholar.uwinnipeg.ca/courses/38/4500.6-001/Cosmology/OtherEnergyMomentum_Tensors.htm [Broken]) This affects the spacetime geometry, and thus the trajectories of all particles, charged or not.

I agree that this propagation suggested by the "time the particle will have to wait" helps giving the E-field a somewhat rigid structure, but I don´t have a real objection to forces that " wait" to act.

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#### robphy

Homework Helper
Gold Member
DaTario said:
I agree that this propagation suggested by the "time the particle will have to wait" helps giving the E-field a somewhat rigid structure, but I don´t have a real objection to forces that " wait" to act.

So, the rest of the question says
"how long does q_2 have to wait? and what determines that wait-time?"

For such ' forces that " wait" to act ', is there some kind of "message" being sent to the distant charge that says "a force of (say) 2 N" is going to be applied?

#### pervect

Staff Emeritus
DaTario said:
Let me complete my last post for it seemed to be to be incomplete.

If there is no way of detecting light without the use of charger-charge interaction. It may well be possible that "THAT THING" which exists in empty space and which we decided to call Electric field does not exist by itself, for there is no independent evidence of its existence. All evidences are mediated by charge-charge force mechanism.

Note: I teach this subject with confidence on its validity, but here I am exercizing the attack that I have always thought could be made to this concept. I would like to acknowledge to those who are participating in this debate.
This strikes me as a purely philosophical point. One has a wide lattitude of what one regards as "real" and what one regards as a "mathematical abstraction". For instance, is the Hamiltonian of a system "real", or is it just a mathematical abstraction? How about the Lagrangian? It doesn't really matter whether or not one regardes any of these as "real" or as "abstractions", except that it probably helps one to take the ideas more seriously by considering them as "real".

One point in favor of regarding the electric field as "real" is that one usually wishes to view energy as "real". Without the electric field, one has major difficulties in coherently keeping tract of what the energy of a system is and where it is going when one applies forces to charges.

The concept of an electric field is well defined as long as one has the ability to use smaller and smaller test charges, there isn't really any wiggle room.

There is one potential problem here that is not philosophical. This is the problem that arises when one can't make the test charge smaller, because the charge on an electron is the smallest charge that one can obtain.

This problem, though, is what separates classical mechanics from quantum mechanics. Philosophically speaking, one starts to question the "reality" of many more things than the electric field when one starts to address the problems of QM. In fact, one has to question the very philosophy of realism, as Bell's theorem shows that the philosophy of realism is incompatible with QM and locality.

#### DaTario

Quantization of charge may be included in classical physics as a aditional postulate. It doesn't imply, I think, wave-particle duality and the whole quantum phenomenology as I understand it now.

I guess this discussion in my opinion has led me to a conclusion.
E-field is something that can be defined in terms of that small test charge ratio between its force and the charge, but we have to be aware and careful when using a particular field configuration to deduce forces on a real (and not necessarily small charges) charged particle. And the reason for this care is that the particle suffering that field may (and probably will) change the "who acts" charged particle's positions and , therefore, the field configuration itself.

This may be put differently saying that we have to always pay much attention to the role of "boundary condition" played by the charges who experiences the field generated by a predefined configuration of "who acts" charged particles.

Best Regards

DaTario

#### reilly

(Re: Datarios #15)
No. That's how electricity actually works. The standard def. of E is a static one. But, indeed, real physics is dynamic. Turns out that under many sets of circumstances similar to your description, the static def, of E, a la Faraday, Maxwell and Einstein, generalizes seamlessly ino a general definition good for all seasons.

I'm sure the general physics community will rejoice in your acceptance of the field concept, or E or whatever. But, your concerns can easily be mitigated by a study of the history of the concepts of E and B -- the materials for the opportunity for so doing are in rich supply.

regards,
Reilly Atkinson

#### DaTario

reilly said:
(Re: Datarios #15)
No. That's how electricity actually works. The standard def. of E is a static one. But, indeed, real physics is dynamic. Turns out that under many sets of circumstances similar to your description, the static def, of E, a la Faraday, Maxwell and Einstein, generalizes seamlessly ino a general definition good for all seasons.

I'm sure the general physics community will rejoice in your acceptance of the field concept, or E or whatever. But, your concerns can easily be mitigated by a study of the history of the concepts of E and B -- the materials for the opportunity for so doing are in rich supply.

regards,
Reilly Atkinson
Thanks,

Thanks again.

Best Regards,

DaTario

#### Nam_Sapper

DaTario said:
Quantization of charge may be included in classical physics as a aditional postulate. It doesn't imply, I think, wave-particle duality and the whole quantum phenomenology as I understand it now.
My response to this is yes: Quantization of charge doesn't lead to wave-particle duality. You could have many different systems of QM, describing scenario universes, each having quantization of charge, but also having different behaviours, and wave-particle duality effects would be alterable options. However, physics primarily deals with actually observed phenomenae, so at least the effects of 'duality' are here to stay: their interpretation is another question.

I guess this discussion in my opinion has led me to a conclusion.
E-field is something that can be defined in terms of that small test charge ratio between its force and the charge, but we have to be aware and careful when using a particular field configuration to deduce forces on a real (and not necessarily small charges) charged particle. And the reason for this care is that the particle suffering that field may (and probably will) change the "who acts" charged particle's positions and , therefore, the field configuration itself.
I think it is a basic mistake speaking of 'who acts' vs 'who is acted upon'. There is no such distinction either in the theory properly understood, or in the analysis or methods of calculation. This was a bad analogy gone to seed. See my previous post #13 and read the second half carefully.

This may be put differently saying that we have to always pay much attention to the role of "boundary condition" played by the charges who experiences the field generated by a predefined configuration of "who acts" charged particles.
Again, I think the whole concept of "who acts" vs "who is acted upon" is wrong-headed, artificial, and misleading. It is like the 'collapse of the wave function" nonsense that Bohr et al spouted about QM in the 50's. It has no actual relevance to QM. Similarly here, the copious talk of 'acted upon vs acting' has the same function as the many muddled descriptions of time paradoxes in the 20th century discussions of relativity. It only confuses, and does nothing to elucidate or assist in application of the methods.

#### DaTario

Nam_Sapper said:
My response to this is yes: Quantization of charge doesn't lead to wave-particle duality. You could have many different systems of QM, describing scenario universes, each having quantization of charge, but also having different behaviours, and wave-particle duality effects would be alterable options. However, physics primarily deals with actually observed phenomenae, so at least the effects of 'duality' are here to stay: their interpretation is another question.

I think it is a basic mistake speaking of 'who acts' vs 'who is acted upon'. There is no such distinction either in the theory properly understood, or in the analysis or methods of calculation. This was a bad analogy gone to seed. See my previous post #13 and read the second half carefully.

Again, I think the whole concept of "who acts" vs "who is acted upon" is wrong-headed, artificial, and misleading. It is like the 'collapse of the wave function" nonsense that Bohr et al spouted about QM in the 50's. It has no actual relevance to QM. Similarly here, the copious talk of 'acted upon vs acting' has the same function as the many muddled descriptions of time paradoxes in the 20th century discussions of relativity. It only confuses, and does nothing to elucidate or assist in application of the methods.

I am sorry to disagree with you, but, by stating that

"when field concept is applied, we are making use of some sort of focused description where some charges are named the 'who acts upon' (or 'who creates field') and the remaining charge is named the 'who feels the field' "

seems to me as a kind of procedure which clearifies the linguistic structure of the field concept, once the complete statement would be EVERY CHARGE ACTS UPON (produce forces on) EVERY OTHER CHARGE. Field concept imply breaking this electric simmetry and offers us a way of treating separatedly one charge, considering all the others as the producers of an electric scenario of influences.

Finally I would add that longer statements does not imply necessarily inefficient explanations.

Best Regards,

DaTario

#### krab

DaTario said:
I guess this discussion in my opinion has led me to a conclusion.
E-field is something that can be defined in terms of that small test charge ratio between its force and the charge, but we have to be aware and careful when using a particular field configuration to deduce forces on a real (and not necessarily small charges) charged particle. And the reason for this care is that the particle suffering that field may (and probably will) change the "who acts" charged particle's positions and , therefore, the field configuration itself.

This may be put differently saying that we have to always pay much attention to the role of "boundary condition" played by the charges who experiences the field generated by a predefined configuration of "who acts" charged particles.
This conclusion is correct.

I might also say that there are very practical consequences of your conclusion. Example: A stream of charged particles from a particle accelerator can be controlled (steered, focused) by electric fields. Such fields are set up by applying voltages to conductors of required shape (parallel plates for steerers, hyperbolic plates for focusing, etc.) For a low current charged particle beam (say less than about a nanoampere), the fields calculated from the voltages and the shapes are sufficient to find the effect on the particles. However, for milliamperes, it is necessary to take into account that the charged particle beam disrupts the charge distributions on the conducting surfaces. This is the same as saying that there are "image charges" that act back on the particle beam.

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