Maxwell studied Ampere's Laws more deeply than any of his contemporaries.

In his discussions of electrodynamics, [Treatise On Electricity and Magnetism] he reminded his readers:

"...It must be carefully remembered, that the mechanical force which urges a conductor carrying a current, across the lines of magnetic force, acts, not on the electric current, but on the conductor which carries it."

(Note: The mechanical force which is observed acting on the conductor is known as the "Ponderomotive Force").

" ...The only force which acts on the electric current is the electromotive force, which must be distinguished from mechanical force."

This clear distinction between ponderomotive force and electromotive force was wrongly obliterated by the introduction of the Lorentz force law, on which the obviously flawed theory of relativistic electromagnetism, was erroneously based.

Why does this matter? The distinction between mechanical forces, such as stress, or acceleration, which act on ponderable matter should never be confused with any of the activities of the so-called "fields".       Confusion results in delusion.

Again, quoting Maxwell, "...Electromotive force is always to be understood to act on electricity only, not on the bodies in which the electricity resides. It is never to be confounded with ordinary mechanical force, which acts on bodies only, not on the electricity between them."

Ampere primarily studied ponderomotive [mechanical] interactions of current-carrying objects. Ampere himself said, "...Newton taught us that motion of this kind, like all motions in nature, must be reducible by calculation, to forces acting between two material particles, along the straight line between them, such that, the action of one upon the other is equal and opposite to that which the latter has on the former, and consequently, assuming the two particles to be permanently associated [rigidly linked] so that no motion whatsoever will result from their mutual interaction." (This is just Newton's third law applied to the situations which result in ponderomotive forces.)]

Directly quoting Newton, his third law reads: "To every action there is always opposed an equal reaction: or the mutual actions of the two bodies are always equal and directed to contrary parts."

Newton's law is applicable to situations which involve two, and only two, material entities. It does not apply to interactions between the material body and some "field", such as the electromotive forces, which are NON-RECIPROCAL in behavior. Further, according to the above, neither material entity acts exclusively on itself.

Fundamentally, Ampere's law predicts a longitudinal force will exist along the path of a material conductor, due to the activities of Newton's third law on the material (atomic lattices) which comprises the conductor. In these studies, there is no linkage whatsoever between the "electric fluid" and the conductor material. In our present day knowledge of such systems, any mechanical forces which may arise due to collisions between electron transports and lattice ions are entirely negligible, relative to the amount of heat released by such events, and due to the miniscule mass of the electron, relative to the atomic lattice structures. Fundamentally, Ampere's law describes the responses of the atomic lattices which comprise the conductor, to the application of an electrical current I , such that the magnetic force H which is exerted on a unit magnetic pole is inversely proportional to the shortest distance, r, to the wire. This is known as the Biot-Savart law, and is written as,

H = k I/r,

where k is a dimensional constant.

Today, this is understood as,

dH = k/R ^2 I ds sin Theta,   due to LaPlace.

This is the form where the concept of a "current-carrying element", first became known. This became the "particle" of Ampere-Neumann electrodynamics, by which the current carrying wire is divided into many small individual "elements", each of which carry current and act ponderomotively.

Ampere's law regarding the mechanical force which may be expected to act between any two material current-carrying elements is of the general form,

delta F _m,n = - i_m i_n dm dn/ r^2_m,n f(alpha, beta, gamma)

The delta informs us that we are dealing with an elemental force which cannot be measured directly, simply because the individual mechanical elements which comprise a conductor cannot be examined in isolation from one another. In the above, the current elements carry currents denoted i_m and i_n, where the lengths of these elements are given as dm and dn.

The distance between the center points of these elements is given as r_m,n, while the directional properties of the actions of these elements are given by the angle function, f(alpha, beta, gamma), which activities may, or may not, lie in the same plane.

Finally, restating Ampere, "...all attractions and repulsions of these parts can be regarded as directed along a straight line, so that they are additive. Such action must also be proportional to the intensities of the two individual current elements.", in the manner of Newton.

The point of all this is, for example, by application of Ampere's law, it is easy to see how we may develop a method of propelling a boat along the water by application of a current to the water, which shall result in a ponderomotive force being appreciated by the boat, resulting in the motion thereof.