Electrostatics and reference frame

[Heimdall]

Hi,

I'm stuck with a question concerning electric fields : Can an electrostatic potential drop exist in (what I would call) a 0 dimensional system ?

Let's imagine we are in a region of space where there is nothing but a uniform magnetic field. So the problem is anisotropic but does not depend on the position. We then decide to move in a certain direction (not aligned to the magnetic field).

When we have reached a constant velocity, say V, we see, in our reference frame, a magnetic field B', almost identical to B, and an electric field which has a value given my the lorentz transformation E'=-VxB.

V and B being uniform and constant over time, E is uniform and also constant in our reference frame.

I wondered what was the "source" of the electric field seen in the moving frame. As it is a constant, it cannot be a induction field. It therefore has to be an electrostatic field. Ok but then were is the source ?

I was stuck for a while with this question when I realized that I forgot the current density consistent with the static magnetic field in initial frame. In the moving frame, a part of this current density *must* be seen as a charge density that would thus be consistent with the electrostatic field.

Ok but then, if there's an electrostatic field, where is the potential drop ? My problem does not depend on any variable, the magnetic field is uniform in all space(*), saying this must be somehow the same as saying that the gradient of the electrostatic potential is zero ? But I see an electric field... This electric field can be very strong (depends on B and V) but we continue to ignore variations (derivatives) ... this looks like a paradox to me.


(*) maybe the solution lies in the assumption of uniformity ? I mean assuming the magnetic field is completely uniform must be somehow wrong, but don't exactly see what's going on...


Thanks for your help !

Heim.

 

[Heimdall]

Hi,

I see that nobody seems inspired by my post. I'll try to sum it up by a few concise questions.

1/ An observer, defining a reference frame R, sees a uniform magnetic field B and a uniform electric field E. Both B and E are constant over time. Can he conclude that the electric field is electrostatic ? (meaning that there is a charge separation somewhere in the universe that would be consistent with this electric field).

I would say yes, because from this observer's point of view, there is no time variation of the magnetic field and no change of flux through any surface...


2/ Can I say that the transformation of the electric and magnetic fields when I change my reference frame is always the "consequence" (or consistent with) of the transformation of the sources (current density and charge density) ?


3/ In the observer's frame, the electric field is uniform and constant over time. Can he write $$\mathbf{E}=-\nabla\left(V\right)$$, where V would be the electric potential ? if so, how could there be a potential gradient in a world where nothing depends on the location ?


Thanks for helping me with these question :-)