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.
