Hi A. I. Arbab ,

I feel this question includes a paradoxon:

On one hand, a conductor is present, and this introduces immediately a large number of movable electrons (plus a lot of bound charges). On the other hand, a single electron is moving which is certainly accompanied by an electric and a magnetic field.

Now, if we neglect the conductor then there is no difference to an electron moving in vacuum which isn't the topic of your question. Or if we take the conductor into account then we have the birectional coupling of the one moving electron and all the other charges which again isn't the gist of your question (if I understand it correctly).

Additionally, there is a more philosophical problem: There seems to be no feasible way to prove or disprove the content of any answer experimentally.

The best I can imagine is: Take a conducting pipe (with extremely small inner diameter), and cool it to nearly 0°K (= nearly no moving electrons). Accelerate a single electron (outside the pipe), and guide it into the pipe. Measure the magnetic field inside and outside the pipe.

My guess is: The kinetic energy of the electron is rapidly dissipated into thermal energy of a lot of other electrons, and the resulting magnetic field looks like noise at all locations.

Dear Fricke,

Thank you very much for your nice and prompt reply. In fact I derived a formula for this magnetic field but wanted to know if it is derived before. The magnetic field is found to be given by B=m L v/(e A), where m and e are the electron mass and charge, v the velocity, L is the conductor length and A is its cross-sectional area. This arises from the motion of the electron mass. Does this sound real to you?

Dear A. I. Arbab ,

I'm sorry, but I don't think this can be true. To begin with, your formula doesn't contain any spatial information which would mean that the magnetic flux density is the same everywhere and always (as long as the electron keeps moving without leaving the conductor).

Secondly, it would follow that an electron with a certain velocity inside a conductor of a certain length and cross-section would cause a field strength B, and if we double the length of the conductor, the field strength would double, too.

Dear Fricke,

Recall that the electric flux is also constant and is equal to q/epsilon_0, why not the same happens to the magnetic flux which is m v L/q. Why should the magnetic flux be a function of position?

Dear A. I. Arbab ,

I agree about the electric flux but B denotes usually the magnetic flux density.

To look at it from another point: In the equation you provided in your question, there is the radius (or distance from the axis of the conductor) r part of the denominator which is a spatial quantity. I agree with this equation.

If your equation for one electron would be correct, and if the current I in the first equation consists of many moving electrons why isn't the first equation just

B = n m L v/(e A), where n is the number of moving electrons?

If we proceed from one electron to many electrons, how is r introduced?

Dear Fricke,

I wonder which way do electrons move across the conductor. Is it along the length L or circling across the face? This could be the magnetic field density emanating from the two faces of the conductor perpendicular to the cross-sectional area A, and not across the sides perpendicular to L? Consider a cylindrical shape of the conductor. In such a case the magnetic field will be uniform just like the electric field arising from a long charged plane plate.