# Is the Configuration of Magnetic Fields Only a Convention?

• cg0303
In summary, Faraday developed the concept of electric and magnetic fields by reconceptualizing forces acting at-a-distance. The electric field was developed by looking at the force on a test charge in the presence of a source charge, while the magnetic field was developed by looking at the force on iron filings or compass needles in the presence of a magnet or electric current. The configuration of magnetic fields is based on a convention, similar to the positive/negative sign convention, and can be changed by flipping the direction of the field lines or using a different field like B×A. However, this would not work for currents as they produce a symmetrical field, making the direction of the field lines arbitrary.
vanhees71 said:
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.
Sorry, you're right, it was Vanadium50. The reasons motivating it are in the OP.

If you want to make a field point in that direction, you need something like B x A to point in that direction. Replacing B with A gives the expression in Post 22.

I don't know why this thread was restarted after more than a month, but I see nothing that suggests the original idea is superior in any way whatsoever to the conventional formulation. It adds a lot of mathematical complexity for no good reason.

cg0303 said:
But if the field lines were based on the force between currents, the field lines would diverge from currents rather than curl around them.
Even if you could get such a scheme to work, it wouldn't be general enough to explain the way a magnet interacts with a current-carrying wire, or the way magnets interact with each other.

Faraday was well aware of the way magnets interact with each other before he ever started to develop an explanation for the way they interact with current-carrying wires as it had been known for about two centuries before Faraday did his work.

One way to reverse magnetic field direction is to change convention of charge sign. Say electron has positive charge and proton has negative charge, current in circuit reverse, so the generated magnetic field reverse also.　Electric field also.

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Delta2
I see nothing that suggests the original idea is superior in any way whatsoever to the conventional formulation. It adds a lot of mathematical complexity for no good reason.
An argument can be made that while there is additional mathematical complexity, the fundamental idea behind it is simpler. This is because it makes magnetism more symmetrical with electricity because the magnetic field would diverge in the same way the electric field would (see post #4). The sources of the fields would also be more similar, with the source of electric fields being charges and the source of magnetic fields being moving charges.

Delta2
cg0303 said:
An argument can be made that while there is additional mathematical complexity, the fundamental idea behind it is simpler. This is because it makes magnetism more symmetrical with electricity because the magnetic field would diverge in the same way the electric field would (see post #4).
I like this idea. However here in physics forums most mentors and moderators are not friendly to new things, they want to discuss only mainstream or should i say well established physics. Your ideas might be revolutionary but are not discussed in any well known book, or a peer reviewed paper. However i like it as i said. I still consider physics forums one of the best site in the internet to discuss about mainstream science (mainly physics and math), but they have the disadvantage i said above.

The sources of the fields would also be more similar, with the source of electric fields being charges and the source of magnetic fields being moving charges.
Nothing new here, this is well established in classical electromagnetism since Maxwell's era 1850s-1860s. I would also add that according to classical electromagnetism, sources of electric field are time varying currents(accelerating charges).

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cg0303
If you want to make a field point in that direction, you need something like B x A to point in that direction. Replacing B with A gives the expression in Post 22.

I don't know why this thread was restarted after more than a month, but I see nothing that suggests the original idea is superior in any way whatsoever to the conventional formulation. It adds a lot of mathematical complexity for no good reason.
Well, obviously it hasn't been answered, whatever was asked. Which direction want you to point the field to and for what purpose?

cg0303
vanhees71 said:
Well, obviously it hasn't been answered, whatever was asked. Which direction want you to point the field to and for what purpose?
It would point in the direction of the force on the test moving charge. Although it would also be a matter of convention in the same way that electric field lines point away from positive charges by convention.

The Lorentz force is part of the usual Maxwell electrodynamics (in SI units),
$$\vec{F}=q (\vec{E}+\vec{v} \times \vec{B}).$$
Of course, there are also some conventions in the definitions, but you don't have too much freedom. That's because Maxwell's theory is a relativistic field theory of a massless spin-1 field, which is necessarily a gauge field (because you don't have continuous polarization-like degrees of freedom), and this fixes to a great extent the dynamics of the system of charges and the field.

Of course, there are certain conventions, which in principle you could change, but at the end the physics stays the same of course. E.g., you can use the east- or west-coast convention of the Minkowski bilinear form. These two possibilities are even used in parallel in the literature for decades now. It's just your personal choice in the beginning of your physics education, depending on the scientific community within you do research.

Other issues are more consistently settled by tradition. E.g., in 3D vector calculus everybody uses the right-hand rule to define the various orientations occurring in the fundamental theorems: The orientation of the boundary of a surface is related to the arbitrary orientation of the surface-normal vectors (important in Stokes's theorem). The boundary surface's normal vectors of a volume are always defined by convention to point out of the volume (the only exception I know of is the very valuable classic textbook on relativity by von Laue) etc. Of course, you could redefine all these conventions, but that's not useful for anything. At the end the physics is the same.

cg0303
anuttarasammyak said:
One way to reverse magnetic field direction is to change convention of charge sign. Say electron has positive charge and proton has negative charge, current in circuit reverse, so the generated magnetic field reverse also.　Electric field also.

Convention of reversing magnetic field should be accompanied by reversing electric field so that Poynting vector which is flow of energy has to be unchanged. Here I do not refer changing rules of mathematics, e.g. vector product A X B.

But what should all this redefinition be good for? I'm glad that at least the signs of the field components are uniquely defined in the literature. It's already a nuissance to still have at least three different systems of units (non-rationalized Gauss, rationalized Gauss (aka Heaviside-Lorentz), SI) and all the different sign conventions in relativity (##\eta_{\mu \nu} = \pm \mathrm{diag}(1,-1,-1,-1)##, ##\epsilon^{\mu \nu \rho \sigma}##,...).

vanhees71 said:
The Lorentz force is part of the usual Maxwell electrodynamics (in SI units),
$$\vec{F}=q (\vec{E}+\vec{v} \times \vec{B}).$$
At the end the physics is the same.
Well, if magnetic fields are defined by the force they apply to moving charges, then it follows that magnetic field would do work. It could be that in the end the physics still remains the same, but I'm not sure.

I'm aware, as pointed out in #41, that this make take the thread outside the forum guidelines. Just flagging it as a possible concern.

weirdoguy
From the Lorentz force it immediately follows that magnetic fields don't do work. The power is
$$P=\dot{\vec{v}} \cdot \vec{F}=q \vec{v} \cdot \vec{E}.$$

vanhees71 said:
But what should all this redefinition be good for?

An excellent question. This dead horse keeps getting flogged, and I don't understand why.

vanhees71
An excellent question. This dead horse keeps getting flogged, and I don't understand why.
I think the OP would like to drop the concept of the magnetic field, instead using some field they (confusingly) keep calling a magnetic field, which points in the direction ##\vec v\times\vec B## ("in the direction of the force on the test moving charge", as in #43). OP seems to feel that this would be more intuitive.

It seems to me that this implies that the force field isn't well defined, since two charges may pass through the same point with different velocities. Unless I, too, fail to understand what the OP wants.

cg0303 said:
Well, if magnetic fields are defined by the force they apply to moving charges, then it follows that magnetic field would do work.

Then show us your calculations that prove that.

Ibix said:
I think the OP would like to drop the concept of the magnetic field, instead using some field they (confusingly) keep calling a magnetic field
My question is whether the magnetic field could have been defined differently. A differently defined magnetic field would still be called a magnetic field.

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cg0303 said:
My question is whether the magnetic field could have been defined differently. A differently defined magnetic field would still be called a magnetic field.
Maybe. It's still confusing to refer to your hypothetical definition by the same term used for a mainstream concept.

I notice you didn't comment on the rest of my post. Am I correct in understanding that you want to define a field that points in the direction ##\vec v\times\vec B##?

By the way, if you are still able to edit your post would you mind removing the "of" you seem to have inserted after "instead" in your quote of my post?

weirdoguy said:
Then show us your calculations that prove that.
I made a mistake. I mistakenly thought that if the magnetic field and the force on the charge were not perpendicular, the magnetic field would do work. In reality, only if the charge's velocity and the force on the charge were not perpendicular, the magnetic field would do work.

An embarrassing mistake on my part. You were right in pushing me to show my calculations, that way I caught it.

Delta2
Ibix said:
Maybe. It's still confusing to refer to your hypothetical definition by the same term used for a mainstream concept.

I notice you didn't comment on the rest of my post. Am I correct in understanding that you want to define a field that points in the direction ##\vec v\times\vec B##?

By the way, if you are still able to edit your post would you mind removing the "of" you seem to have inserted after "instead" in your quote of my post?
I don't know what else to call it. I guess I could just say the "hypothetic magnetic field".

Yes, I want to define a field that points in the direction of v × B.

Ibix said:
By the way, if you are still able to edit your post would you mind removing the "of" you seem to have inserted after "instead" in your quote of my post?
Done

cg0303 said:
Yes, I want to define a field that points in the direction of v × B.

But how would it work when there is no moving charge and hence no ##\vec{v}##? How would you describe interaction of two magnets?

vanhees71
cg0303 said:
Yes, I want to define a field that points in the direction of v × B.
In addition to @weirdoguy's point, then, how do you define this field when two or three particles pass through the same point with different velocities? Your field has different strengths and directions from the same source at the same point.
cg0303 said:
Done
Thank you.

cg0303 and vanhees71
Ibix said:
I think the OP would like to drop the concept of the magnetic field, instead using some field they (confusingly) keep calling a magnetic field, which points in the direction ##\vec v\times\vec B## ("in the direction of the force on the test moving charge", as in #43). OP seems to feel that this would be more intuitive.

It seems to me that this implies that the force field isn't well defined, since two charges may pass through the same point with different velocities. Unless I, too, fail to understand what the OP wants.
Ok, but this doesn't make sense from a conceptual point of view. The important paradigm change from Newtonian mechanics, where interactions are treated as action-at-a-distance forces, is the concept of locality, i.e., one introduces fields as "mediators" of interactions. In Newtonian mechanics you can also take this standpoint of the field concept, but there it doesn't really change the qualitative picture, because all that's done is to introduce a static field. In practice that are electrostatic fields and the gravitational field. However, the idea is already clear in these cases: You first calculate a field, independent of the test particle moving "in this field", which mediates the interaction/forces.

E.g., in Newtonian theory of gravity you start with the field equation
$$-\Delta \Phi(\vec{x})=-4 \pi G \rho(\vec{x}),$$
where ##\rho## is the mass distribution of the "source of the gravitational field". The solution of course is
$$\Phi(\vec{x})=-G \int_{\mathbb{R}^3} \mathrm{d}^3x \frac{\rho(\vec{x}')}{|\vec{x}-\vec{x}'|}.$$
Then the gravitational force on a test mass is given by the potential
$$U(\vec{x})=m \Phi(\vec{x}).$$
the important point here is that the field ##\Phi## is independent of the test mass, upon which the force acts.

The same you have with electrostatic fields as well as magnetostatic fields and test charges in Newtonian physics. The important point also here is that the field is independent of the properties of the test charge. So no quantities related to it appear in the fields, neither the charge nor the charge's velocity.

The field concept and the locality concept becomes, however, crucial in the context of relativistic physics, where the fields necessarily becomes a dynamical entity of its own. It's not just a mathematical trick to "mediate forces" but is a fundamental dynamical "thing" for itself, as "point particles" or "fluids" are fundamental dynamical entities for themselves within classical physics. The strict locality in space and (!) time of the successful dynamical models within the theory of relativity are key for the consistency with relativistic spacetime models and particularly causality.

To make a long speach short: It doesn't make sense to intermingle properties of the (test) particles/fluids with the field but keep everything within a strictly local description.

etotheipi, cg0303 and Ibix
weirdoguy said:
But how would it work when there is no moving charge and hence no ##\vec{v}##? How would you describe interaction of two magnets?
Magnets have the properties they do because of the motion of the charges inside them. It would probably take a lot of work to figure out how to apply the hypothetical magnetic field concept to two magnets, but it seems possible in principle.

Ibix said:
In addition to @weirdoguy's point, then, how do you define this field when two or three particles pass through the same point with different velocities? Your field has different strengths and directions from the same source at the same point.

Thank you.
Good point, I didn't think of that.

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