Originally Posted by DrRocket

I think that perhaps there is still more to be revealed with regard to the MHD approximation that neglects the displacement current, c-2 ∂E/∂t.

It is not clear to me how you are using the terms “characteristic length” and “characteristic time”, particularly in regard to and application in Faraday’s Law. If you start with Faraday’s law and apply Stokes’ Theorem, you can get a very rough estimate by integrating over a square with side L, to get something like ll Emax ll >/= Const * L* ll ∂B/∂t ll if you assume that ∂B/∂t is constant over the time period of interest. But I don’t quite see how that either helps much or why such an assumption is valid. I suspect that the assumptions being made are rather subtle substitutes for ∂E/∂t = 0. Jackson or Landau and Lifshi tz make use of characteristic dimensions and times only after making the assumption that ∂E/∂t = 0, and reducing the problem to a solution of the heat/diffusion equation. Alfven and Falthammer are even a bit more circumspect, and simply call physics with ∂E/∂t=0 magnetohydrodyamics and otherwise call it plasma physics.

Just as an exercise to look at the implications of Maxwell’s equations one might consider the following case. I will use Maxwell’s equations in Gaussian units, largely to avoid confusing myself.

Faraday’s Law: Del x E = -1/c*∂B/∂t

Ampere’s Law: Del x B = 4pi/c*J + 1/c * ∂E/∂t

Gauss’ Law (electric): Del • E =4pi* η

Gauss’ Law (magnetic): Del • E = 0

Let E =(c*exp(t)*y,0,0) , then ∂E/∂t = (c*exp(t)*y,0,0)

Let B = (0,0,c^2*exp(t), then ∂B/∂t = (0,0, c^2*exp(t))

Then we can verify that

Del x E = (0,0,-c*exp(t)) = -1/c*∂B/∂t

Del x B = 0

Del • E = Del • B = 0

From this we can calculate that J = (-c/(4pi)*exp(t)y,0,0) and the displacement current is equal in magnitude to the ordinary current density, which probably should not be neglected.

Note that here the magnitudes of E and ∂E/∂t are limited by the magnitude of B over a spatial extent described by the y coordinate, a limitation consistent with some notion of characteristic length.

Now this example does not show that it is wrong to neglect the displacement in magnetohydrodynamics nor does it show that it is wrong to neglect it in plasma physics. But I think it does show that that greater care is needed to justify that approximation.

I think perhaps the notion of characteristic length and characterisic time need to be more closely attached to specific physical phenomena, but I am not quite sure what those phenomena are.