The one who is not a MIB fires up the oven and half-bakes some fancy cipherin'...

The textbook descriptions of the magnetic field and the forces on charges moving through it usually involve an observer watching two moving charged particles. The first particle produces a magnetic field as seen by the observer, and the other particle "cuts the lines" of the field and feels a resulting (Lorentz) force. If either particle stationary in the observer's frame, no forces are produced. For particles with charges q1 and q2, velocity vectors v1 and v2, and a distance vector r between them, the force can be written in terms of vector dot products. The result will have terms like (v1.v2)r, (r.v1)v2, and (r.v2)v1, all multiplied by a constant containing (q1 q2 r)/|r|^3 to give an inverse-square fall-off with distance.

Dr. Marinov's "Annus Horribilis" paper [1] uses formulae of this form, and points out some momentum conservation issues with the textbook form. Many texts point out that the equations are only valid for currents in closed loops. All of them still require the two moving charges to have a Lorentz "line cutting" force, trying to use one of the particles as the frame of reference kills the force. This seems wrong, it is not properly frame-invariant.

It should be possible to find a direct formula for the electric field seen by an observer from a moving charged particle, directly including electrostatic and motion-related forces, without needing the idea of the magnetic field. The observer is at the origin, using a tiny test charge to see any electric fields. The moving particle is at some location vector r, is moving at some velocity vector v, and carries charge q. The observer looks for an electric field E caused by the particle motion. The electrostatic part is ignored here.

One might expect the result to contain terms like (r.v)v and (v.v)r, noting (v.v) = |v|^2. Using the USWAG [2] method, the following force equation is proposed. Constants PI=3.14159..., u=magnetic permeability of medium. '^'=raising to a power, '.'=vector dot-product.

E = (uq/8PI|r|^3) [2(r.v)v - |v|^2r]

This is the fundamental equation of NotAMibbean Magneto-crackpottics. [3]

Plain TeX [4] users can use the form below to render it.

\def\Ev{\vec E}

\def\rv{\vec r}

\def\uv{\vec u}

\def\vv{\vec v}

$$\Ev = {\mu q \over 8 \pi |\rv|^3} [2(\rv\cdot\vv)\vv - |\vv|^2\rv]$$

\bye

The first term looks like an odd drag-effect [5]. The second term is a motion-caused change in the electrostatic force. For a single particle, the E field would go up as the square of the velocity, unusual for most of electromagnetism. If one considers neutral "current elements" the squared terms neatly cancel, leaving forms that usually match textbook results.

A "current element" can be modeled as a pair of charges +q and -q at the same initial location, moving in opposite velocities +v and -v. The whole element can the be moved relative to the observer by adding another velocity u, so the positive charges moves at (u+v) and the negative charge at (u-v). Using this in the force equation results in the |u|^2 and |v|^2 terms cancelling, leaving a uv term that resembles the textbook results. The constants in the equation were chosen to match the usual results for long wires and circular current loops, moving relative to the observer.

This equation would suggest testable outcomes, and appears to allow a longitudinal force between moving particles. Highly-charged wires and magnets might also behave differently from neutral ones, exposing the squared terms [6].

This also seems to clarify the issues about "conduction" versus "convection" electric currents, and what it means to move relative to one. For an observer moving parallel to a pure electron beam, the current depends upon the observer's velocity, and it is not clear what part of the motion generates B-field lines and what part cuts them. For a current in a neutral wire, one has fixed positive metal ions, and moving electrons. If the observer moves along the wire, the observed current remains the same, since both kinds of charges move by the observer, and only the difference is observed as current. The same generating versus cutting issue remains.

Using the new equation, one treats each type of charge separately, and the force emerges directly. One is stationary with respect to the neutral current when the positive charge and negative charge currents are equal and opposite. For real-world wire currents, the average electron velocity is some fraction of a millimeter per second, so imbalance effects would be small enough to have missed.

Thing not done yet.

1. Finding a neutral current configuration that would manifest longitudinal magnetic forces. This would yield good experiments for testing the equation. Square coils might show such effects near the corners. The ideal case would be something that accumulates such a force around a closed path, allowing one to turn a wire loop into a negative resistor.

2. Re-integrating the new force law back into Maxwell's equations to see if it gives new wave-propagation modes or other changes.

3. Looking at stability of electron orbits in atoms with this force law. This should probably only be attempted by a licensed quantum mechanic.

4. Looking at the high-speed (relativistic) case.

Footnotes

1:

http://itis.volta.alessandria.it/episteme/ep6/ep6-marin.htm (Not to be confused with "Anus Horribilis", which is an unpleasant medical condition usually caused by eating the deadly bean burritos served at the cafeteria on Thursdays.)

2: UnScientific Wild *ssed Guess and some numerical modeling to match the textbook results for long wires and neutral current loops.

3: This is possibly the greatest advance in electro-crackpottics since the invention of the silly tinfoil hat.

4:

http://www.ctan.org/ (for rendering and printing math)

5: You can almost feel the aether-wind blowing.

6: Such conditions might be found inside Testatika pots or those SM ring-things.