Dale said:
One of the basic assumptions of circuit theory is that none of the components have a net charge. So this analysis would seem to indicate that circuit theory is inherently non relativistic.
I basically agree. In particular, I think Kirchoff's current law has some issues - basically, it does not factor in the relativity of simultaneity. In case our OP is still with us, I'm going to go over some of the fundamentals as I see them. The key point as I see it is understanding how charge densities and currents densities transform between frames. Charge densities are just charge per unit volume, i.e. charge/m^3, current densities are charges/second across some area A, so they have units of charge / (area * second) or (charge / m^2 s).
Consider two frames, one of which is moving in the x direction with some velocity ##\beta = v/c## relative to the other. We will call the charge density ##\rho## or ##\rho'## depending on whether we are in the primed or unprimed frames, and we will write the current density components as ##j_x, j_y, j_z##. In the non-relativistic Newtonian analysis would have for the relation between primed and unprimed frames:
$$\rho' = \rho \quad j_x' =j_x + \beta c \rho \quad j_y' = j_y \quad j_z' = j_z$$
You may see other versions with a minus sign in front of ##\beta##, this is just a matter of sign convention, it indicates that we need to replace ##\beta## with ##-\beta## depending on how exactly we've defined the velocity between the primed and unprimed frames.
Basically what this says is that the charge density ##\rho## stays unchanged, and moving charge creates a current density of ##\rho## * velocity.
The relativistic analysis tells us, however, that
$$\rho' = \gamma (\rho + \beta j_x/c) \quad j_x' = \gamma(j_x + \beta c \rho) \quad j_y' = j_y \quad j_z' = j_z$$
where ##\gamma## is the relativistic factor ##1/\sqrt{1-\beta^2}##.
Note that ##\rho c## has the same dimensions as j, as it is (charge / ##m^3##) * (m/s) = charge / (##m^2 s)##
The short-hand version of explaining this is to say that charge and current transforms "as a 4-vector". Here I've written out the 4-vector transformations explicitly.
For the benefit of Jartsa, I would draw attention to the equation ##\rho' = \gamma (\rho + \beta j_x/c)##. We can divide this into two terms: ##\gamma \rho## and ##\gamma \beta j_x / c## which add together to give the total charge density in the primed frame.
The origin of the first term can be explained by "length contraction". The origin of the second term is more mysterious, but it's absolutely necessary to include it to get a correct analysis. I would describe the ultimate origin of this term as being due to the "relativity of simultaneity".
So - to summarize the issue as I see it. The challenging part of the relativistic analysis is how the relativity of simultaneity affects the expression of the physics. A second challenge in communicating this issue - getting people to pay attention to the relativity of simultaneity. My working hypothesis is that the term "relativity of simultaneity" is very abstract, an tends not to be understood. The specific example of how charge density transforms between a moving and non-moving frame is less abstract, but as far as communication goes, it seems to have the opposite problem. It's very detail oriented, so the big picture gets lost. However, without a shared understanding of what the abstract term "the relativity of simultaneity" means, going over the details is the only way I can think of to try to explain the fundamental underlying issue.
