# The laws of Coulomb versus Ampere and the electromagnetic Machian paradox

## Main Question or Discussion Point

The laws of Coulomb versus Ampere and the electromagnetic Machian "paradox"

Here's an apparent paradox that has been tunneling about in my little mind lately. Maybe someone out there can help me with it.

Imagine if you will, a vast empty region of space devoid of any visible distant stars. In this "laboratory" there exists a very long, very thin rigid rod. The rod is electrically charged with a uniform charge density (dQ/dx) along its length.

An observer next to the rod having no relative motion to the rod measures an electric field from the rod due to the electric charges along the rod. He looks for and finds no magnetic field from the rod which he attributes to the fact that there is no relative motion to the charge along the rod and therefor no electric current density so no induced magnetic field. (Ampere's law of current or Maxwell's 4th eq.)

However when the rod moves along side of the observer (at velocity dx/dt), he does measure a magnetic field since now the charges along the rod are moving relative to him and creating an electric current and consequently a magnetic field. His measurements of the field agree with all known parameters such as the charge density/distribution of the rod, its relative motion to him and the magnetic free space permeability constant. If the observer accelerates to the same velocity as the rod, he sees the magnetic field disappear although the electric field remains. All is well in the observers mind.

Now imagine a second identical rod parallel to the first. It has the same charge density/distribution as the first rod. When both rods move with the same velocity relative to the "static" observer, he measures a magnetic field roughly twice as large as the field from one moving rod which he attributes to the fact that there is twice the current density with two rods and so twice the magnetic field.

The observer realizes that there should be a repulsive force between the two rods created by Coulomb's law relating the force to the square of the ratio of the total charge of the rods and the distance between them. He also realizes that there should be an attractive force between the two rods created by the total magnetic field which is described by Ampere's law relating force to the square of the currents (created by the moving charges: i = dQ/dt = (dQ/dx)(dx/dt)) to the distance between the rods.

The observer calculates that the two forces should be in equilibrium and the rods should remain at a fixed distance from each other when the rods are moving at some relative velocity to him.

The "paradox": But if the observer accelerates to the same velocity as the two rods (dx/dt = 0), he should measure no magnetic field from the rods. Will the two rods now separate since there is no attractive force between them? If so, this seems to violate physics' requirement for an unabsolute or non-fixed reference frame. If the rods don't separate, then why?

Question: If the universe is electrically neutral, is there relativity of charge in motion as there is with mass/energy in motion? By which I mean, would an observer moving along with a charged particle see a relative magnetic field of the universe as he sees relative motion of the universe's mass/energy with respect to him?

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Well, they do separate... the electric force is much larger than the magnetic force for low velocities. Som,, the electric force in the first frame dominates the magnetic attraction, and they repel.

Since the electric force is independent of velocity whereas the magnetic force increases with velocity, the distance at which the two forces balance would be dependent on velocity.

If my figurin' is correct, the distance apart where they should balance is:

d = lo((1-(v/c)^2)^.5)/(2 x (v/c)^2)

where: lo is the initial rest length of the rods, v is the velocity of the rods relative to the "stationary" observer and c is the speed of light.

As you can see: as v approaches c, d approaches 0. As v approaches 0, d approaches infinity.

Velocity is relative however the separation distance is not; thus the apparent paradox.

Since the electric force is independent of velocity whereas the magnetic force increases with velocity, the distance at which the two forces balance would be dependent on velocity.

If my figurin' is correct, the distance apart where they should balance is:

d = lo((1-(v/c)^2)^.5)/(2 x (v/c)^2)

where: lo is the initial rest length of the rods, v is the velocity of the rods relative to the "stationary" observer and c is the speed of light.

As you can see: as v approaches c, d approaches 0. As v approaches 0, d approaches infinity.

Velocity is relative however the separation distance is not; thus the apparent paradox.

If the net force on one rod is zero in one frame, it should be in every other frame too.

That means, in the rest frame, although both rods are stationary and electrically charged, they don't exert any force on each other. That's not possible. Thus, for every velocity less than c, there must be a net force. The electric force must overwhelm the magnetic force at every velocity less than c.

If the net force on one rod is zero in one frame, it should be in every other frame too.

... Thus, for every velocity less than c, there must be a net force. The electric force must overwhelm the magnetic force at every velocity less than c.

Except that it doesn't. Hans Christian Oersted discovered in 1820 that two parallel wires carrying current in the same direction attract each other.

The charges in the wires are always moving less than the speed of light. If the current in the wires were zero, there would be no magnetic force attracting the wires. If the speed of the moving charges were c (if possible), the magnetic force attracting the two wires would be infinite.

Ampere was inspired by Oersted's experimental result and developed Ampere's law.

Resolved: The laws of Coulomb versus Ampere and the electromagnetic Machian "paradox"

The apparent paradox of the parallel moving charged particles (rods) being attracted to each other from a force related to some specific velocity (violating common sense requirement of non specific reference frames) was based on an extension of the Oerstead experiment showing that two parallel wires carrying current attracted each other.

Ampere used that experiment to create Ampere's "Law" relating the moving charged particles' velocity and density to an induced magnetic field that pulled the charges in the two wires together.

But then Einstien showed that the magnetic field is simply a relative manifestation of an electric field in motion. Oerstead's expt. can be explained simply with special relativity and how the moving charges in one wire "see" a higher charge density in the oppositely charged ionic lattice in the other wire since the wire is contracted (relative to the moving charged particles). The higher charge density of the wire (as seen by the moving charges in the other wire) makes the wire, neutral in rest frame, appear oppositely charged to the moving charges in the other wire and so one wire is attracted to the moving charges in the opposite wire and vise versa.

The bottom line is that no magnetic field is required to explain Oerstead's experiment so one should not expect charged rods flying about in space to be contrained by any such magnetic fields (in any reference frame) and should expect them to be pushed apart by the coulomb force to infinity and at the same acceleration regardless of the reference frame of the observer.

Proving once again, that no law is law.

The apparent paradox of the parallel moving charged particles (rods) being attracted to each other from a force related to some specific velocity (violating common sense requirement of non specific reference frames) was based on an extension of the Oerstead experiment showing that two parallel wires carrying current attracted each other.

Ampere used that experiment to create Ampere's "Law" relating the moving charged particles' velocity and density to an induced magnetic field that pulled the charges in the two wires together.

But then Einstien showed that the magnetic field is simply a relative manifestation of an electric field in motion. Oerstead's expt. can be explained simply with special relativity and how the moving charges in one wire "see" a higher charge density in the oppositely charged ionic lattice in the other wire since the wire is contracted (relative to the moving charged particles). The higher charge density of the wire (as seen by the moving charges in the other wire) makes the wire, neutral in rest frame, appear oppositely charged to the moving charges in the other wire and so one wire is attracted to the moving charges in the opposite wire and vise versa.

The bottom line is that no magnetic field is required to explain Oerstead's experiment so one should not expect charged rods flying about in space to be contrained by any such magnetic fields (in any reference frame) and should expect them to be pushed apart by the coulomb force to infinity and at the same acceleration regardless of the reference frame of the observer.

Proving once again, that no law is law.
I have Einstein's 1905 paper "On The Electrodynamics Of Moving Bodies" and never does he say that. He places electric and magnetic fields on equal footing and specifically stated that questions as to which one is the "seat" (primary, root, main, fundamental) "no longer have ant point". He specifically acknowledges forces due to both electric and magnetic fields with no inference as to which one is more basic. I would recheck your reference as to what Einstein said. Are you referring to a different paper? I'm not aware of AE ever changing his position on this matter. I'm just wondering. BR.

Claude

I have Einstein's 1905 paper "On The Electrodynamics Of Moving Bodies" and never does he say that. He places electric and magnetic fields on equal footing and specifically stated that questions as to which one is the "seat" (primary, root, main, fundamental) "no longer have ant point". He specifically acknowledges forces due to both electric and magnetic fields with no inference as to which one is more basic. I would recheck your reference as to what Einstein said. Are you referring to a different paper? I'm not aware of AE ever changing his position on this matter. I'm just wondering. BR.

Claude
Hi Claude,

Do you mean this paper?:
http://www.fourmilab.ch/etexts/einstein/specrel/www/

Philosophically this may be true but since magnetism relies on relative motion (a changing E field) whereas charge is invariant one could easily argue between the two, that electricity is more "fundamental" than magnetism.

Additionally, until someone finds find a magnetic monopole, one could say that a universe devoid of motions or changes (hard to imagine but theoretically possible) would have no magnetism but could still have electrical charge.

Hurkyl
Staff Emeritus
Gold Member

Philosophically this may be true but since magnetism relies on relative motion (a changing E field)
The notion of 'static E field' is not absolute. There is always a reference frame where the B field is nonzero.

whereas charge is invariant
So is the current density (3-form).

one could easily argue between the two, that electricity is more "fundamental" than magnetism.
But one can more easily argue that they are just two facets of the same field.

Additionally, until someone finds find a magnetic monopole
You seem to be forgetting magnetic dipoles, quadrupoles, et cetera. A neutron is magnetic, for example.

one could say that a universe devoid of motions or changes (hard to imagine but theoretically possible) would have no magnetism but could still have electrical charge.
Similarly, there are models of SR where you have magnetic currents, but no charge.

....

You seem to be forgetting magnetic dipoles, quadrupoles, et cetera. A neutron is magnetic, for example.

....
I didn't forget them. They are just not monopoles. There is no analog of a point charge for magnetism. There are analogs between magnetic and electric "loops" though.

The neutron is not an elementary particle. I believe that its the constituent quarks that are interacting with a magnetic field giving the neutron a non zero magnetic moment. I suppose one would have to get an electron to stop spinning to find a charged particle without a magnetic field.

Maybe that would cause its charge to go to zero too. I don't know. Maxwell's eqs and SR don't seem to indicate that to me (at least).

Hi Claude,

Do you mean this paper?:
http://www.fourmilab.ch/etexts/einstein/specrel/www/

Philosophically this may be true but since magnetism relies on relative motion (a changing E field) whereas charge is invariant one could easily argue between the two, that electricity is more "fundamental" than magnetism.

Additionally, until someone finds find a magnetic monopole, one could say that a universe devoid of motions or changes (hard to imagine but theoretically possible) would have no magnetism but could still have electrical charge.
So what you're saying is that since isolated electric charges, i.e. "monopoles" are known to exist, and that the magnetic counterpart has never been observed, that electric charge and fields are more "fundamental" than magnetic flux and fields.

But, per AE's 1905 paper, neither is more fundamental. Just as a static electric field has no associated magnetic field, so does a static magnetic field have no associated electric field. The main difference is that a static E field due to electric monopoles have sources and sinks. An E field vector starts on a positive charge and ends on a negative charge. An H field vector, however, is like the wedding ring I wear on my left hand 3rd finger. It has no source or sink, no start or end. It is solenoidal, a closed loop since no magnetic monopoles are known to exist.

The fact that E can be start-end, vs. H which can only be a closed loop does not make E more fundamental than H, nor vice-versa. The non-existence of point charge magnetic particles, or "monopoles" if you prefer, simply means that the fields are solenoidal. An H field possesses energy just like its E field counterpart. The fact that H is a dipole, quadrapole, octapole, etc. does not change that fact. Energy is energy whether monopole, di-, quadra-, etc.

AE knew that, I know that, and the scientific community knows that. E and H both carry energy. Under static conditions, that said energy can lie in the H field alone, or in the E field alone. But, under time-varying or "dynamic" conditions, "ac" if you prefer, the energy is changing wrt time. Power is dw/dt where w = work or energy. But dw/dt is non-zero under ac conditions. Power is E times H* (complex conjuate of H). A non-zero power exists only if E and H are both non-zero. You can't have one without the other. Neither is more fundamental. In the static case, you can have *either one* without the other. Neither is more fundamental.

Conclusion - neither is more fundamental. Neither is the "cause" nor the "effect" of the other.

I put the word "fundamental" in quotes because is really a matter of symmantics not some absolute truth. Remember, in 1905 the concept of electrons as we know them today didn't exist.

Thompson received the Nobel prize in 1906 for his lecture describing cathode "rays" as point like charged "corpuscles". Magnetic monopoles have been theorized for years but have never been found and probably won't be since they would have to be relatively large with a value of e/(2x alpha). alpha = hyperfine constant

But who really knows? It could be that superstring theory's model of electrons et. al as loops makes the point charge concept of an electric monopole moot.

Do you know of any papers AE wrote, say in the 1920's, that assert such a position on "fundamentality"? They were still worried about the ether in 1905.

Whether or not its "fundamental" the fact is that until an actual magnetic monopole is found there is an fundamental asymmetry in the universe regarding E & M. IMHO

The quantity E^2 - (cB)^2 is Lorentz invariant (the same in all inertial reference frames). For a charged particle like an electron, that can only include the reference frame v=0, when the B field is zero . If some analogous case were true (i.e. E field zero -- B field not), it would require the invariant quantity to be negative which would require an imaginary solution.

"If some analogious case?!" A superconducting loop with static current has "E field zero -- B field not". H (or B if you prefer) can exist without E, but only under *static* conditions. The fact that B/H are soleniodal fields is irrelevant. No monopole needed as energy is still present magnetically but not electricaly.

Also worth noting, under time-varying conditions the E field due to induction is also **solenoidal**, i.e. closed loops with no start or end. Yet energy is present in such an E field. The monopole concept with E fields is only valid with charged particles. The E field due to induction is not monopolar, but dipolar like an H or B field. Induced E fields are closed loops. Dipole, monopole, whatever, it doesn't matter because energy is still present either way.

I'm not sure what you mean by "static" current. Are electrons moving in the loop?

Static current is non-changing wrt time. In other words, the energy is non-changing or static. A loop with inductance L, and steady current (dc) I, has a steady energy equal to (L * (I^2))/2. Under this condition, i.e. superconducting loop, the current will sustain indefinitely with constant energy. This is a static condition with zero E and non-zero H.

The counterpart would be, for example, a charged capacitor with static voltage. If the dielectric is superinsulating, i.e. no leakage, then we have a non-zero E with a zero H. The energy, once again is constant, or non-changing wrt time, W = (C * (V^2))/2.

Thus there are two static conditions where one is zero with the other non-zero. Hence neither can be regarded as the basis or cause. As Einstein said, questions as to which is the "seat" no longer have any point.

Under time-changing conditions, E and H are both non-zero. Which one comes first, i.e. "cause and effect", is chickens and eggs.

Best regards to all.

Claude

Staff Emeritus
2019 Award

I think you are making this more complicated than it needs to be.

Suppose there exists a frame where the net force between two electrically charged current carrying wires is zero: the electrostatic repulsion is exactly balanced by the magnetic attraction. What does this look like in some other frame?

If you boost to a frame where the current is larger, the magnetic force is also larger - but the charge density increases because the distance between the charges is Lorentz contracted. These exactly balance - zero net force in one frame means zero net force in all frames.

Note that in the OP's original post, the force is purely electrostatic in one frame. In that case, there is no frame in which the magnetic force even balances it, much less overcomes it. So the system as described cannot exist.