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Originally Posted by DrRocket
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.
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Well, let’s see what we can do.
Any plasma system that we look at has a characteristic length and time scale. Now, this might not be obvious at first. Suppose you have a homogeneous ball of plasma, nothing more, then what would you call a characteristic length? First of all there is the radius of the ball, which can be used. For a time scale one can look at what is driving such a ball of plasma and it would be sound waves. So, a characteristic time of the system would be the crossing time of such a sound wave. If you add a density gradient in the gravitationally bound ball of plasma, this gradient gives a length scale, if you add a magnetic field to the plasma then the curvature or gradient of the magnetic field can give a length scale, etc. etc. One looks at the system and basically characteristic lengths and times pop up. But naturally, you have to choose a pair that belongs together.
Now, one of the very nice things that Fourier has shown us, is that we can take (almost) any signal and describe it with harmonic functions, either in time or in space or in both. We can Fourier transform Maxwell’s equations and then come basically come to the same conclusions as in my first message.
Now, let’s get to the crux, the definition of MHD, is the definition of MHD that the displacement current is negligible? No, it is not. In MHD, the plasma is viewed as a fluid; the orbit of the particles is averaged out by integrating over it, which means that MHD is only valid at time scales larger than the gyration time of the ions and on length scales larger than the gyration radius. Furthermore, it is assumed that there is (quasi)neutrality and that the differences in the mean velocities of the individual species are small with respect to the fluid velocity. If you look at such a system, you cannot but find that the displacement current is negligibly small. This does not mean that is does not exist, but any effects that it will have are of order (v/c)
2. And from ideal MHD follows directly the frozen in condition.
Now let’s look at your example
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Originally Posted by DrRocket
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.
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))
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Okay, you set up an initial magnetic and electric field, with the field exponentially growing in time and E linearly growing along the y-direction. Now, this does not really have a time scale nor a spatial scale, so we cannot do any estimates in that way, but notice that you will get into infinity problems for longer times. Naturally, by implicitly assuming an exponentially growing electric field in time will automatically negate any assumption that the displacement current term can be neglected. Also, one has to think about, what this field actually means, but that is something different.
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Originally Posted by DrRocket
Del x E = (0,0,-c*exp(t)) = -1/c*∂B/∂t
Del x B = 0
Del • E = Del • B = 0
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I guess can agree with these calculations
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Originally Posted by DrRocket
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.
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Now, like I said above, by implicitly assuming that E and B are growing exponentially in time, it is rather logical that you cannot take out the displacement current. Note, that to be complete you need to also include an equation for the current density J, which you derived from Maxwell’s law, but naturally also has to obey that J = n
e v
e q
e + n
i v
i q
i, and drive this field just a little bit in time, and you will get into problems, with he velocity of the particles becoming c.
I do not see why there is a limit to the magnitude as you claim. Indeed, through Maxwell’s equations they are coupled, but there is no limit to B, because it is exponentially growing, and thus there is no limit to E and dE/dt, and there is definitely no spatial scale, unless you put in a Heavyside function into the definition somehow (e.g. you multiply everything with H(|y| = a), where a is the spatial extent of the fields in y.
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Originally Posted by DrRocket
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.
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Now, the question is, what do we want to calculate in a plasma, and naturally, that is always the question, and that always determines whether or not a term can be neglected in the equations or not. This is similar to the question of “frozen in field,” where it must be clear that this condition only holds as long as one keeps in mind that the time is shorter than the diffusion time of the plasma.
Now, why do we use MHD? Looking at the excellent (but expensive) book by Walker
Magnetohydrodynamic waves in Geospace you can see that one important thing is the calculation of wave modes in a plasma, for which MHD is excellent. Otherwise, there are processes that are most easily described by MHD, like the convection of the solar wind magnetic field with the plasma of said wind.
I do agree that one must be careful with how to apply certain approximations. That is why Alfven got “angry” and claimed that frozen in field does not exist, because, when it was first revealed, everybody just used it without thinking, fortunately, that stage has passed (mostly) and scientists are aware of the limits that are put to the theory that they are working on. That this is the case might not always be clear to the “layman” (I am not sure what your level of physics is), because usually it is not spelled out in the papers explicitly. It can be that the authors just say,
we use Hall MHD and then the referees and the specialists who read it later know what is meant, and to what level the results are applicable. Is this a flaw? Maybe.