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:

Ampère’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 v_perp / q B

So, you see that the resulting velocity is v_perp, which is typically the thermal velocity of the plasma.

Thermal velocity: v_th = (2 k_B T / m)^1/2

Okay, these are the first musings on the negligibility of the displacement current.

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Optimism does not change the laws of physics. (T'Pol)
A good scientist has freed himself of concepts and keeps his mind open to what is. (Dao De Jing 27)
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Martin ( http://www.geocities.com/DrMartinV )

fixed

I think the symbol grad, from its other name "gradient", is also used. Normally, people say that it's a differential operator, not a derivative operator.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

oops, i meant differential operator, sorry about that.

the symbol is called "Del" as an operator, which turns into grad when operating on a scalar, div when taking the inner product with a vector and finally rot/curl when taking the outer product with a vector.

__________________
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Optimism does not change the laws of physics. (T'Pol)
A good scientist has freed himself of concepts and keeps his mind open to what is. (Dao De Jing 27)
************************************************** *************************
Martin ( http://www.geocities.com/DrMartinV )

Which is exactly what you're doing, right?

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

not really, we are working here with vector field E and B, that is why they are in bold.

In Faraday and Maxwell you take the outer product Del x B/E, or if you want rot/curl(B/E)
In Gauss you take the inner product Del • B/E, or if you want div(B/E)

Just some basic math then.

The differential operater Del = (∂/∂x, ∂/∂y, ∂/∂z) operates in the following way:

1. On a scalar field f(x,y,z), taking the gradient
Del f(x, y, z) = grad(f(x,y,z)) = (∂f(x,y,z)/∂x, ∂f(x,y,z)/∂y, ∂f(x,y,z)/∂z)
this creates a vector field from a scalar field

2. On a vector field g(x,y,z) where one has (g_x(x,y,z), g_y(x,y,z), g_z(x,y,z)), taking the divergence
Del • g(x,y,z) = div(g) = ∂g_x/∂x + ∂g_y/∂y + ∂g_z/∂z
now the result is a scalar field from a vector field

3. On a vector field g(x,y,z) where one has (g_x(x,y,z), g_y(x,y,z), g_z(x,y,z)), taking the rotation or curl
Del x g(x,y,z) = (∂g_z/∂y - ∂g_y/∂z, ∂g_x/∂z - ∂g_z/∂x, ∂g_y/∂x - ∂g_x/∂y)
here the result is a vector field out of a vector field

And to be complete, there are a few special things you have to take into account when doing manipulations:

Del x DelΨ = 0 (the curl of a gradient is zero)
Del • (Del x a) = 0 (the divergence of a curl is zero)

__________________
************************************************** *************************
Optimism does not change the laws of physics. (T'Pol)
A good scientist has freed himself of concepts and keeps his mind open to what is. (Dao De Jing 27)
************************************************** *************************
Martin ( http://www.geocities.com/DrMartinV )

Tusenfem. For Del...search Charactermap. charmap. see:http://www.atm.ox.ac.uk/user/iwi/charmap.html

pete trying: ∇

here it is, worked first try, I'll edit and enlarge it...wait a minute..no can do.

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A third rate theory forbids
A second rate theory explains after the fact
A first rate theory predicts...A. Lomonosov

I see what you mean now. I prefer to think of them as three separate operators, and write Maxwell's equations as:

But I understand that the nabla notation has its practical advantages.

[/end technical hijack]

__________________
"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

Thanks pete!

I was looking through the abcTajpu plugin for firefox and did not find it there.

__________________
************************************************** *************************
Optimism does not change the laws of physics. (T'Pol)
A good scientist has freed himself of concepts and keeps his mind open to what is. (Dao De Jing 27)
************************************************** *************************
Martin ( http://www.geocities.com/DrMartinV )

Okay, above we discussed the reason why in plasma physics the displacement current ∂E/∂t (or more officially ∂D/∂t) is neglected, as estimates showed that in the low-velocity approach this term is of v²/c².

Now, let us look at another example of the displacement current density in a conductor. First of all, a “correction” to what we have been talking about. All the time we have said that the displacement current is ∂E/∂t, however, if you look in real electrodynamics books (e.g. Jackson, or Lorrain & Corson) you will find that the displacement is actually given by D = (ε E + P), where P is the polarization of the medium. Now, I will leave it to the interested reader to catch up with this from a book).

The example that we are now going to look at is interesting, we will compare the displacement current with the conduction current, that appear in Maxwell’s law, under the condition that the conductor is submitted to an AC electric field described by:

E = E₀ cos ω t

And the conductor has a conductivity of σ

Just using the common expression for electric current density and electric field for the conduction current density we find for the first term of the right-hand side in Maxwell’s law:

J = σ E = σ E₀ cos ω t

Fort he second term we find:

∂D/∂t = ε ∂E/∂t = - ε ω [b]E[b]₀ sin ω t

Comparing the two terms now we find that

|displ / cond| = ω ε / σ

Now, the relative permittivity ε_r of a conductor is not readily measurable, since polarization effects are usually overshadowed by conduction. However, for our estimation we will use ε_r = 1 and σ = 10⁷ mho/meter for a good conductor. This leads to:

|displ / cond| = 10^-17 f

where f = ω/2 π is the frequency of the electric field . This shows that the displacement current is a good conductor is negligible compared to the conduction current at any frequency lower than optical frequencies (f = 10¹⁵ Hz).

Another interesting thing to note is that the conduction current J is in phase with the electric field, whereas the displacement current leads the electric field by π/2 radians.

__________________
************************************************** *************************
Optimism does not change the laws of physics. (T'Pol)
A good scientist has freed himself of concepts and keeps his mind open to what is. (Dao De Jing 27)
************************************************** *************************
Martin ( http://www.geocities.com/DrMartinV )

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.

Your forms for E and B are in error. If they have the form exp(t) then the divergence equals zero since Del A = ∂A/∂x+∂A/∂y+∂A/∂z and you have nospatial dependence.

Did you mean that E and B are of the form exp(kx-wt)?

No. I may have a mistake somewhere. That is possible. But there is nothing wrong with having zero divergence. In fact unless there is a net charge concentrated somewhere one would expect the divergence of E to be zero. And if the divergence of B is not zero then there would be a problem.

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

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.

I guess can agree with these calculations

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.

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.

__________________
************************************************** *************************
Optimism does not change the laws of physics. (T'Pol)
A good scientist has freed himself of concepts and keeps his mind open to what is. (Dao De Jing 27)
************************************************** *************************
Martin ( http://www.geocities.com/DrMartinV )