# Is the Configuration of Magnetic Fields Only a Convention?

cg0303
If I understand correctly, the concept of electric and magnetic fields originated with Faraday and was developed by reconceptualizing forces acting at-a-distance.

For example, the electric field concept was developed by looking at the force on a test charge in the presence of a source charge rather than looking at the force between two charges (i.e. Coulomb’s law). The magnetic field was similarly developed by looking at the force on iron filings or compass needles in the presence of a magnet or electric current.

But could Faraday have equally developed the concept of magnetic fields by looking at the force on one (test) electric current in the presence of another (source) electric current, per Ampere’s discovery of the attraction/repulsion between currents? If so, wouldn’t that make the configuration of magnetic fields only a convention, in somewhat the same way that positive/negative signs are just a historical convention due to Franklin?

## Answers and Replies

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You may change convention of magnetic field to inverse one if you change also B to -B in Maxwell's equations. If you want to keep Maxwell equations as they are, you should keep current convention.

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If so, wouldn’t that make the configuration of magnetic fields only a convention, in somewhat the same way that positive/negative signs are just a historical convention due to Franklin?
The magnetic field direction is only a convention anyway. You can flip the direction of all magnetic fields either by changing the right hand rule to a left hand rule, or by flipping the order of the cross products.

vanhees71 and Delta2
cg0303
The magnetic field direction is only a convention anyway. You can flip the direction of all magnetic fields either by changing the right hand rule to a left hand rule, or by flipping the order of the cross products.
True, but that's only with respect to whether the direction of the field line is positive or negative. But if the field lines were based on the force between currents, the field lines would diverge from currents rather than curl around them.

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But if the field lines were based on the force between currents, the field lines would diverge from currents rather than curl around them.
Oh, I see what you are saying.

I have not considered this in detail, but I don’t think it would work. If the field lines radiate out from the current then you will have the same field regardless of which direction the current flows.

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I am having a hard time visualizing the direction you want this to point, other than "not in the direction of the B field". Well, there are really only two choices.

One is the magnetic vector potential. One can always use this, since ∇×A= B. However, I suspect that had things evolved this way, it would not be long before we introduced an auxiliary field that was the curl of the A field to ease calculations.

If that's not it, it would be some field like B×A so that it was perpendicular to both. I am pretty sure that this would work, but it would be computationally horrible.

cg0303
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If the field lines radiate out from the current then you will have the same field regardless of which direction the current flows.
Why? We could choose one direction to be 'outwards' and one 'inward', the same way we chose the positive charge to be outward and negative to be inward. The only difference is that that with electric charges opposites attract, while with currents opposites repel.

cg0303
I am having a hard time visualizing the direction you want this to point, other than "not in the direction of the B field". Well, there are really only two choices.

One is the magnetic vector potential. One can always use this, since ∇×A= B. However, I suspect that had things evolved this way, it would not be long before we introduced an auxiliary field that was the curl of the A field to ease calculations.

If that's not it, it would be some field like B×A so that it was perpendicular to both. I am pretty sure that this would work, but it would be computationally horrible.
The field line would point away from an electric current.

I'm not sure what the magnetic vector potential of a current would look like. Could you post a diagram that would help?

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Why? We could choose one direction to be 'outwards' and one 'inward', the same way we chose the positive charge to be outward and negative to be inward.
By symmetry. The laws of physics are invariant under rotation, including specifically a 180 degree rotation. Start with a current ##\vec J## that produces an outward field. By symmetry a 180 degree rotated current ##R_{180}(\vec J)## must also produce an outward field. But ##R_{180}(\vec J)=-\vec J##. So ##\vec J## and ##-\vec J## produce the same field.

cg0303
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Could you post a diagram that would help?

I could, but I posted a link with diagrams.

Dale
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By symmetry. The laws of physics are invariant under rotation, including specifically a 180 degree rotation.

While what he proposing is still pretty vague, I think he means that clockwise current gives an outward field and counterclockwise gives an inward field. (Or the reverse)

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While what he proposing is still pretty vague, I think he means that clockwise current gives an outward field and counterclockwise gives an inward field. (Or the reverse)
Which is the same issue since a clockwise current rotated 180 degrees becomes counterclockwise.

cg0303
I could, but I posted a link with diagrams.
I was after something different than what was contained in the link, but I think I was able to get what I needed from it.

cg0303
If that's not it, it would be some field like B×A so that it was perpendicular to both. I am pretty sure that this would work, but it would be computationally horrible.
If I understand you correctly, it would be this.

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Then it would be computationally horrible.

cg0303
Then it would be computationally horrible.
Fair enough

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Which is the same issue since a clockwise current rotated 180 degrees becomes counterclockwise.

Take a look at the Wikipedia page, which has a toroid. B is drawn, A is drawn, and their cross-product is outward from the current. So I think the direction is - or at least can be - well-defined. The OP's description still kind of confuses me (sometimes it sounds like A, sometimes perpendicular to A) so I am not sure that it is well-defined. But I think something like he wants could be well-defined.

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B is drawn, A is drawn, and their cross-product is outward from the current. So I think the direction is - or at least can be - well-defined.
Agreed. The direction can be well defined and with your formula it is always outward.

The problem is here:
could Faraday have equally developed the concept of magnetic fields by looking at the force on one (test) electric current in the presence of another (source) electric current
So we have a test current and a source current. The field always points outward from the source current regardless of which direction the current is flowing. So the field at the test current is the same regardless of which direction the current flows. But the force on the test current is not the same. So this field is not and cannot be directly related to the force on the test current.

In order to distinguish the direction of the force would indeed be computationally horrible and would probably wind up solving for B at some point anyway.

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I am having a hard time visualizing the direction you want this to point, other than "not in the direction of the B field". Well, there are really only two choices.

One is the magnetic vector potential. One can always use this, since ∇×A= B. However, I suspect that had things evolved this way, it would not be long before we introduced an auxiliary field that was the curl of the A field to ease calculations.

If that's not it, it would be some field like B×A so that it was perpendicular to both. I am pretty sure that this would work, but it would be computationally horrible.
Well, in fact I think historically there was a lot of confusion about the potentials in the beginning. It's a tricky concept as we know today, because it's not the potentials which uniquely describe an electromagnetic field but only an equivalence class, i.e., "the potentials modulo a gauge transformation". That is, because the Maxwell equations for the potentials do not uniquely determine these potentials given some physical initial (and maybe boundary) conditions, while of course the Maxwell equations for the field(s) ##(\vec{E},\vec{B})## have unique solutions under given initial/boundary conditions, and indeed ##\vec{E}## and ##\vec{B}## are what's operationally defined via the forces on test charges.

Delta2
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computationally horrible and would probably wind up solving for B at some point anyway.

I agree. I believe that the thing that is most likely being discussed works out to be ∇(A2/2)−(A⋅∇)A. One would need to solve for A and take it's curl to get anything that looks like a force.

@vanhees71 is right that A is determined by B only up to a gauge transformation. I think the monster above is also determined by A up to a gauge transformation, and some care might be needed to make sure that one chooses a consistent set. I haven't worked this out, but can see how one might screw this up.

vanhees71 and Dale
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can see how one might screw this up.
Excellent understatement!

vanhees71
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My beautiful Latex was destroyed. (Greg, is there a Purple Heart badge for this?) I tried to write:

$$\nabla (A^2/2) - ({\bf A} \cdot \nabla){\bf A}$$

What's a vector and what is not is important.

All you have to do is invert this.

cg0303
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What do you want invert here? It's not an equation. So far the great feature of classical electromagnetism was that you get very far with linear laws. What should this non-linear expression help (whatever the missing right-hand side of the equation reads).

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I have no idea why anybody would want to add this much complexity.

Mentor
Greg, is there a Purple Heart badge for this?)
$$\heartsuit (A^2/2) - ({\bf A} \cdot \heartsuit){\bf A}$$
There, I fixed it for you. And you're welcome.

cg0303 and Vanadium 50
cg0303
I have no idea why anybody would want to add this much complexity.
1. It might only be complex with respect to its definition with respect to the vector potential. However, if we were starting from scratch (not that we would want to) and are defining B for the first time, it may not be as complex. I'm not using the words 'might' amd 'may' because I really don't know. My grasp on the maths here is at at a beginners level.

2.
I’ve read and thought a little more about this question, and I think the question can be strengthened somewhat. However, I think it ultimately can be resolved. Please bear with me.
I came across the following chapter which explains the history: https://link.springer.com/chapter/10.1007/978-1-349-11139-8_6
As a quick digest of the chapter, Faraday and Ampere disagreed about two main issues:
a) Ampere conceived of magnetism in terms of Newtonian forces acting at-a-distance, while Faraday thought in terms of fields.
b) To Ampere, the attractive/repulsive forces between two current carrying wires was the fundamental fact about magnetism, and could explain all other magnetic phenomena. He believed that all magnetic phenomena were caused by miniature electric currents that existed inside magnetic materials. He struggled to find a model that was consistent with experiment, and so Faraday rejected his view.

Faraday’s field view (issue 1) prevailed. Regarding issue (2), it seems that Ampere’s view prevailed to a certain extent, as we now explain the properties of magnetic material as result of the motion of the electrons inside their constituent atoms. This latter point seems to strengthen my original question. Faraday, given the scientific knowledge at the time, would not have developed the concept of magnetic fields by looking at the force on a test electric current (or moving charge) in the presence of a source electric current (or moving charge). He had no reason to see currents as the fundamental fact of magnetism, and therefore constructed the concept of magnetic field by looking at the force on iron filings. However, that is merely a historical artefact, since today we know that the motion of charge within a material is what makes gives it its magnetic properties. If Faraday were to construct the concept of magnetic fields today, he may have based it on the forces between currents.

Now, here’s where I think why it wouldn’t work.

In a uniform magnetic field, a moving charge experiences a force which causes it to move in a circular path.

(taken from https://openstax.org/books/college-...in-a-magnetic-field-examples-and-applications)

A magnetic field constructed from the force between currents would not be able to produce such a force (as far as I can tell).

Thoughts?

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Are you saying that you can't follow our math, but you're sure we're wrong?
Are you sure that's the path you want to go down?

berkeman
cg0303
Are you saying that you can't follow our math, but you're sure we're wrong?
Are you sure that's the path you want to go down?
Not at all!

All I was trying to say is:
a) There seems to be a historical motivation for the suggestion in the OP
b) I don't see how Amperian forces between currents/moving charges could account for the circular path.

Maybe I worded post #26 too strongly (although I kept saying 'I think' and 'may'), but that's all I was trying to say.

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One should note that Ampere hasn't had yet the full dynamical knowledge about electromagnetism. This came later with Maxwell's addition of the socalled "displacement current" making, unknowingly of course, electromagnetism the paradigmatic example for a relativsitic field theory. Ampere's (and Weber's) theoretical framework was based on the action-at-a-distance picture, which you get from assuming magnetostatics and neglect the displacement current and retardation effects.

Given that the electromagnetic field is operationally defined via the forces on charges and currents, one could think that it is just an auxilliary quantity which may be eliminated from the physical laws, leading to a description of systems of charges and currents with direct action-at-a-distance forces.

Let's do this for two currents in thin two wire loops (of arbitrary shape) at rest carrying somehow currents ##I_1## and ##I_2##. Let the closed curves describing the wires be ##C_1## and ##C_2##. Let's start from the modern field view, which simplifies the analysis considerably. There the idea is that ##C_1## leads to a magnetic field, which can be described (in the static approximation) by Biot-Savart's Law,
$$\vec{B}(\vec{x})=\mu_0 I_1 \int_{C_1} \mathrm{d} x_1 \times \frac{\vec{x}-\vec{x}_1}{4 \pi |\vec{x}-\vec{x}_1|^3}.$$
The total force on ##C_2## then is
$$\vec{F}=I_2 \int_{C_2} \mathrm{d} \vec{x}_2 \times \vec{B}(\vec{x}_2).$$
Plugging the above formula for ##\vec{B}## in this you get
$$\vec{F}=\frac{\mu_0 I_1 I_2}{4 \pi} \int_{C_2} \mathrm{d} \vec{x}_2 \times \int_{C_1} \mathrm{d} \vec{x}_1 \times \frac{\vec{x}_2-\vec{x}_1}{|\vec{x}_1-\vec{x}_2|^3}.$$
Here you have a form, where the magnetic field is eliminated at the prize of a complicated non-local law.

It's also possible to do something similar for the general case of the full electromagnetic theory with taking into account the full relativistic effects of retardation. The elimination of the electromagnetic field then leads to quite complicated non-local dynamical laws. A famous example is Feynman's and Wheelers "absorber theory". It's called absorber theory because you must use an reinterpretation of the field picture to make "causal sense" of the necessity to introduce "half retarded and half advanced Green's functions". The original idea was to eliminate the fields and then make a pure "particle quantum theory" for point particles out of it by somehow "quantizing" this non-local classical particle symmetry.

This, however, never worked out, and the key of electromagnetism in quantum theory (both non-relativsitic quantum mechanics of point particles in classically treated electromagnetic fields, the socalled semiclassical approximation, as well as the full relativistic quantum field theory , quantum electrodynamics) are to use the potentials and gauge invariance and staying with strictly local laws. From the modern point of view this lesson teaches us that the most fundamental description of Nature seems to be closer to a local field-theoretical point of view than a point-particle action-at-a-distance point of view.

etotheipi, Delta2 and cg0303
cg0303
Here you have a form, where the magnetic field is eliminated at the prize of a complicated non-local law.
Just to clarify, the suggestion in the OP was not that we could eliminate fields in favour of forces acting-at-a-distance. Rather, it was that the definition of the magnetic field could be based on the force betwen currents, much like the definition of the elctric field is based on the forces bewteen charges.

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Ok, yes. I think that was the case in the 19th century, where no charged particles and the em. forces on them were observed yet, but only after Oersted discovered the close relation between currents and magnetism, which was only possible since physicists could have steady currents through the discovery of Volta and Galvani before. The first charged particles at disposal for experiment were of course electrons (discovered by Thomson in 1897) and the related "electron theory" by Lorentz.

Before all that it wasn't even known that electricity and magnetism are related, and the physicists considered only permanent magnets discribing them by "poles", being well aware phenomenologically that single magnetic poles are not observed but only in terms of "dipoles".

cg0303
cg0303
My beautiful Latex was destroyed. (Greg, is there a Purple Heart badge for this?) I tried to write:

$$\nabla (A^2/2) - ({\bf A} \cdot \nabla){\bf A}$$

What's a vector and what is not is important.

All you have to do is invert this.
Would you be able to show me how you derived this?

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Could you remind me, what this term (there's no equation) was supposed to be used for? I'm a bit lost about what the actual mathematical or physical question is

cg0303
Could you remind me, what this term (there's no equation) was supposed to be used for? I'm a bit lost about what the actual mathematical or physical question is
In post #6 you described the field I was trying to describe in the OP as: "it would be some field like B×A so that it was perpendicular to both". So the other side of the equation would be B×A (if I understand correctly). Thank you!

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That wasn't me. I've no clue, where the gauge-dependent expression ##\vec{B} \times \vec{A}## might come from. AFAIK it has no immediate physical meaning to begin with.