In studying magnetism, and magnetic lines of force, it seems clear to me that an error has arisen from misconstruing what a magnetic compass needle is really doing when it is placed near a straight wire direct current. The name magnetism was originally applied to certain objects (including magnetic compass needles) which could display polarized MUTUAL attraction, or repulsion, with a measurable FORCE. Therefore, using sound semantics, you can't have real magnetic lines of force without two such objects. Therefore, to apply this name to something which supposedly encircles a current carrying wire, but doesn't have two objects along the alleged lines of force to refer this alleged force to, redefines the meaning of the phrase "magnetic lines of force" in midstream. This redefinition follows from falsely assuming that any active magnetic compass needle indication always refers to magnetic lines of force. This is definitely not the case.
To explain this inadvertent switch in the meaning of the phrase, "magnetic lines of force", when it is applied to the area around a straight wire current, we have to consider what the true definition of magnetism originally referred to, and therefore, should continue to refer to, unless formally redefined.. The term "magnetism" was originally applied to types of objects which, unknown to everybody at the time, contain multiple looped micro electric currents for there particular force manifestation, which, therefore, have, spatially speaking, at least two opposing parallel currents in them (opposite edges of a loop or coil, or the electrically equivalent situation in magnets). Since a straight wired current in no way has either a loop, or two opposing currents, it does not qualify as a magnetic object which can generate genuine magnetic lines of force. But, next to another parallel wire, or its two equivalents in a pivoting DC powered coil, or compass needle, it does generate radial Ampereic lines of force, either attraction or repulsion.
The situation of a magnetic compass needle near a straight current is a different kind of relationship (easily explained below) than between two genuine magnetic objects (of which the magnetic compass needle can be one of them). The compass needle therefore, near a straight current, falsely projects, in one's imagination, something to which the original meaning of the phrase, "magnetic lines of force", does not apply.
To have a name kicking around in science which has two different meanings is not a helpful thing in my opinion. If the incorrect name, "magnetic lines of force", for this particular situation (magnetic compass indication near a straight current) has been incorporated into so many physics formulas that it can't be extricated, or if it is used in a monitoring/calculating reference system, then I suggest that, at least, the name of the indicated lines be changed to "pseudo magnetic lines".
It would be even better, but a mouthful, to call these compass indicated "north-south" lines around a straight current, "Oersted's right angled offset current direction indicators", because they are simply derived from the physioelectric responses of the two opposite parallel components of a loop current, or their equivalent in a magnetic compass needle, near a straight current, depending on the straight current's direction. This kind of physioelectric effect between current carrying wires, whether there is a passive pivotal loop current device involved or not, should properly be called Ampereism, not magnetism.
The paragraphs following this one are based on the premise that two single turn DC carrying coils placed near to each other, and on the same axis, are merely two rolled up parallel wires which will attract or repel each other depending on whether their parallel currents are in the same or opposite parallel directions. To elaborate a little, if two single turn DC carrying coils are each formed into a square, and the two squares are placed near to each other on the same axis, it will be seen that, in this case, there are 8 parallel wires altogether (16 if opposite sides are also counted), 4 on each square coil. Although within each square coil all the parallel wires will repel each other because of their spatially opposite parallel currents, between the two squares there may be either attraction or repulsion, depending on whether their parallel currents are in the same direction or the opposing direction. Since merely flipping the face of one of the squares, 180 degrees, automatically spatially reverses the current, the previous physical attraction or repulsion between the two squares will then also be reversed.
Since the working force between the two squared coils is obviously the effects of close parallel currents rather than a mysterious force called magnetism, which apparent force is really only a vector resultant of the forces between the shared parallel multiple sections of the two coils, I, therefore, believe this net attractive or repulsive force between DC carrying coils should be called "multiple ampereism", rather than magnetism, in honor of the person who first discovered these physical actions between close parallel current carrying wires. The term "magnetism", I think, is best considered as a nick name. Since physical magnets show the same physical reactions between them as DC carrying coils show, I assume a similar process is at work between magnets, but relying on the net effect of many separate micro electric currents within the magnets rather than on easily locatable electric currents in wires coils.
Further, by careful definition, you also can't have "lines of magnetic force" encircling the individual wires of a direct current carrying coil, because you only get genuine "magnetism" off the faces of a loop, or off the ends of a tubular coil, or off the ends of a magnet, (or magnetic compass needle) as a MUTUAL VECTOR RESULTANT FORCE between at least two pairs of parallel currents, at least one pair in each magnetic entity, when they are brought near to each other. This mutual vector resultant force varies with different orientations between any two magnetic entities. However, encircling the individual internal wires of a coil, you only have pseudo magnetic lines. These lines of no force, naturally, cannot be added up to create a net magnetic force. However, there are radial lines of Ampereic attractive force around each wire which must be added up altogether. But proper semantics demands that this radial force around each wire of a current carrying loop, or coil, should NOT be called magnetic force.
Since genuine magnetism, according to the original definition, requires at least two parallel opposing currents in each coil, or the electrical equivalent in magnets, it is clear that magnetism and electromagnetism is a more complex arrangement of a more simple force system relating to the physical reactions between close parallel currents. Since it was Andre M. Ampere who first discovered this physical reaction between close parallel currents, it would seem only proper to call this more primary system, Ampereism, and the forces operating there, Ampere's radial lines of mutual physical force.
From the above considerations, it appears to me that the overall problem of properly relating magnetism to electricity is that magnetism is a compound forces relation system built up of a lower order forces relation system, which latter system should properly be called Ampereism. Therefore, magnetism provides only a confusing view of Ampereism.
What both the loop currents and the straight currents do have in common is they both have inductive fields, when these fields are moving, which the working physicist and electrical engineer need to keep track of in order to get a mathematical hold on either field's electrical and physical effects. But it is not helpful to drag around confusing names. A straight wire current's inductive field is just that. It is not a loop current's inductive field, so it should not be called a magnetic inductive field. On the other hand, spatially speaking, a magnetic inductive field is a combination of at least two sub inductive fields. If a common name is going to be used for both types of inductive fields, and the lines for their flux densities, it should obviously be "Faradayic flux density" instead of, "magnetic flux density".
Also, referencing Ampereic lines of force, instead of magnetic lines of force, makes it easier to explain the physical reactive motions of current carrying wires in relation to magnetic fields. For instance, the Lorentz Force resolves into the, less mysterious, slack current carrying wire being attracted toward the side of a current carrying coil which has the same direction current as the slack current carrying wire The same applies to currents in cathode ray tubes when they are manipulated with varying currents in coils. The Hall Effect is easily explained in a similar way.
Finally, if there can't be real mutual magnetic lines of force encircling single straight current carrying wires, then it follows that there can't be real transverse magnetic waves in electromagnetic waves. Instead, the accompanying wave component must necessarily be described as a radially moving, longitudinally alternating, wave of inductive force.
To restate the whole above argument in a different vein:
If it had happened in history that Ampereism had been discovered before magnetism, then whenever we rolled up energized parallel wires into a loop, or coil, we would naturally deem the physical force between any two coils to be emanating straight directly from the wires rather than from in the middle area in between the wires (central area of the loop), as we do now; and two dimensional diagrams of the physical force field between, and around, air core electromagnetic coils would look much different than they are currently depicted. They would look more like thorns on a cactus rather than as an elongated toroid. But if we then measured, or sensed, the vector RESULTANTS of these radial forces, which are coming straight from the wires, we would get the field diagram that we currently get for air core electromagnetic fields. But we would not fail to make the distinction between the basic radial forces, and their resultants.
In my opinion, what has happened is that, because of our prior experience with magnets, which are self powered, electrically fuzzy equivalents of air core electromagnets, we have over prioritized magnetic lines of force as a basic phenomenon, and failed to analyze more deeply, their origin. Consequently, we have also described electromagnetic waves wrongly.