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1) does an electrical charge (say, an electron) traveling with a constant velocity (say, in the x-direction) travel as an electromagnetic wave?

I'm thinking of an analogy with flowing mass. Suppose you have massive particles, evenly distributed through some volume, traveling with constant velocity, say the x-direction. The flux across the (y,z) surface of a volume is calculated as the integral of the current density, vector [tex]\vec{J}[/tex] over the (y,z) surface area.

where

[tex]\vec{J}=\rho\cdot\vec{v}[/tex]

and

[tex]\int\int_A\vec{J}\cdot dA[/tex]

The result is the flow of mass across the (y,z) surface area per unit time, with units of kg/s. There is no curl in this vector field.

Now, imagine that instead of flowing mass, we have flowing electric charge in the x-direction across the (y,z) surface of a volume. Assume the electrons are not being accelerated as they flow across the (y,z) surface.

With this substitution the flux of mass across the surface becomes the flux of electrons across the surface with units of q/s, which is electrical current.

Does each electron rotatate as it travels in a straight line, producing an "infintesimal" rotation, which, when integrated over the surface area (y,z), produces an overall rotation of the vector field (Stokes theorem).

Does any of this cause the electrons to radiate? I don't see any light radiating when current travels through a metal wire.

Maybe I'm confusing electromagnetic waves traveling along the electric field lines with electrons traveling through space.