Quote:
Originally Posted by tusenfem
In a discussion on [rul=http://www.bautforum.com/against-mainstream/74950-frozen-magnetic-field-lines-7.html#post1274740]frozen in magnetic field[/url], a question was raised by Dr Rocket, why we neglect the displacement current in Maxwell’s equations. To answer this, let us start at the beginning, i.e. Maxwell.
The electromagnetic field is described by Maxwell’s equations
Faraday’s Law: Del x E = - ∂B/∂t
Maxwell’s Law: Del x B = μ0 J + c-2 ∂E/∂t
Gauss’ Law (electric): Del • E = η/ε0
Gauss’ Law (magnetic): Del • B = 0
Del is here the derivative operator (∂/∂x, ∂/∂y, ∂/∂z) usually written a "nabla" (an upside-down Δ) which I could not find unfortunately. The last term in Maxwell’s law is the “displacement current” and this term is usually omitted from the equations. Now, the question is, under what circumstances can this term be neglected?
In the MHD regime one is looking at processes that are long with respect to the gyration time of the ions and on scales that are larger than the gyro radius of the ions. Basically, one is describing the plasma with the “guiding centre approach.”
Now we consider the low velocity approach, i.e. we look processes in the plasma with plasma speeds much smaller than the light speed (v << c). Assume that we are looking at a system with characteristic length scale L and a characteristic time scale T, then we can find from Faraday’s law:
E ~ L B / T
Now that we have this proportionality between E and B from Faraday, we can take a look at Maxwell, which shows that the last term, describing the displacement current is of the order:
L2 / c2 T2
Which is a very small number in MHD problems, and thus Maxwell’s law turns into:
Ampčre’s Law: Del x B = μ0 J
Often, this is called the MHD approximation; however, it is more general than that. Even for cases where the other MHD approximations are not valid, this one can be. Therefore, it is better to call this the low-velocity approximation, where the characteristic velocity L/T is much smaller than c.
Now, we started with talking about the MHD approximation, so what happens when we use the gyration radius and period as the characteristic scales? Well, simply said, this leaves us with the perpendicular velocity of the particle with respect to the magnetic field.
Gyration frequency: ω = q B / m
Gyration radius: ρ = m vperp / q B
So, you see that the resulting velocity is vperp, which is typically the thermal velocity of the plasma.
Thermal velocity: vth = (2 kB T / m)1/2
Okay, these are the first musings on the negligibility of the displacement current.
|
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.