We have a simple expression for magnetic pressure P= B(square)/2μ0. It is not having any charge density of plasma. So, the magnetic pressure or magnetic force is independent of the degree of ionization of the plasma? Can anybody give some good references?
Hi Srinivasa,
As has been rightly pointed out, the magnetic pressure is equivalent to the field energy density (it’s a quick exercise to check the units of energy density and pressure) and always exists as a concept even without the presence of a magnetic field. The key to this question and I believe the source of the confusion is that the pressure is only observable through a force and pressure forces only come about with gradients (i.e. changes) of pressures.
below I put links to the relevant equations because there doesn't seem to be Latex support...
The formula you give is derived as a part of MHD http://en.wikipedia.org/wiki/Magnetohydrodynamics, and comes from examining the lorentz force. Here it is F = J x B. This says that the magnetic field exerts a force on any current flowing in the plasma perpendicular to both the current and the field.
We expand the J x B term by invoking Amperes law (http://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_law#Differential_form) which basically states that a curling (read changing) magnetic field has to be due to the presence of some current.
When we substitute Ampere’s law into J x B, we are keeping the fact that there is a current in the term for the FORCE exerted by the field by instead counting the change in the field instead of the current directly.
If you now look at the expansion of the J x B term in the first link, you may notice two things.
1) the magnetic pressure term only exerts a force when there is a changing field
2) the pressure force is accompanied by a second force which is the magnetic tension force and is broadly interpreted as the tension force along field lines
Thus we can only feel forces due to the magnetic pressure when two things happen together:
1) there exists a changing magnetic field
2) the pressure force is not exactly balanced by the tension force.
This ONLY occurs in the presence of currents, which mean there must be some charge carriers.
In summary,
1) You are correct, magnetic pressure is independent of the ionisation state of the plasma. It can be thought of as the energy density of the field.
2) We only notice the effect of magnetic pressure when it exerts a force.
3) It can only exert a force on currents, which imply some charge carriers (in your case some ionisation)
If you care to link this to the degree of ionisation you must find a formula to describe the currents in the plasma and this will involve the degree of ionisation, # of collisions etc...
I hope this is helpful to you.
Simon
It's a simplified equation. If you say the magnetic pressure is independent of the degree of ionization, ask yourself how large the magnetic pressure is in case the ionization degree is zero.
This equation holds only for steady state in which the time derivative d/dt = 0 and without any external forces applied to the plasma.
The steady state approximation causes the mass density-term rho (in which the electron and ion densities are embedded) to vanish:
rho = n_i M + n_e m (where n_i is the ion density with the ion mass M and n_e the elctron density with electron mass m).
this goes into the MHD equations with: rho * dv/dt
as dv/dt vanishes in steady state, this term drops out of the equation, regardless of the charge densities involved.
As you calculate your simple approximation for magnetic pressure from these equations, you will loose this part.
but if the time derivative does not vanish, the magnetic pressure will be indeed dependent on n_i and n_e.
Actually, this expression tells the energy density of the magnetic field. And this is general ! This energy density interacts with an ideal plasma to pushing with this pressure onto the plasma.
With simplified free energy density of the plasma of neKTe+nikTi you can calculate equilibrium situation of plasma compressed by magnetic field. This is frequently applied to calculate typical densities and temperatures of a plasma.
However, energy density only leads to pressure (of same formula) when the two manifestations (particles, fields) realy strogly interact in a "thin" layer e.g. with collisions like gas molecules on pistons.or chamber walls.
Hence, magnetic pressure onto magnetically and electrical neutral gas is zero !
In a partial ionized gas, electrons and ions are confirmed by a strong magnetic field while neutrals can escape by penetrating the field.
Similar situation my be with dark metter or neutrons in the universe. If existing, both certainly have energy density. But due to the very low interaction, they dont apply pressure onto the baryon matter nor the photons. If there would be strong interaction "shock fronts" between the two types of matter be generated.
"Collissions" of gravitational interaction are of such large free path length .that no shock fronts and no "pressure" is visible.
In steady state (as Johannes Gruenwald told you), yes, absolutely. This is the reason why in fusion devices you want the magnitude of B to be as high as possible.
Johannes Gruenwald and
Rainer Lebert Thank you , very much, can you please provide some good references ( books or papers )
Fundamentally the magnetic pressure term in the MHD equations emerges from the jxB term in the momentum equation and so clearly there is a hidden dependence on rho through j. It comes from jxB approx (curl B/mu0) \cross B, i.e it neglects the displacement current. The MHD model contains many approximations and are simply not valid under many conditions and should applied (and interpreted) with caution. However, within the framework of the MHD model, there is no explicit charge density dependence in the magnetic pressure term.
Alec, good contribution. I was about to write pretty much the same now! Equations are very often too quickly taken as an exact representation without considering the assumptions made during derivation.
The dependency vanishes in the MHD framework (under steady state cond.), but does not imply general validity.
Hi Srinivasa,
As has been rightly pointed out, the magnetic pressure is equivalent to the field energy density (it’s a quick exercise to check the units of energy density and pressure) and always exists as a concept even without the presence of a magnetic field. The key to this question and I believe the source of the confusion is that the pressure is only observable through a force and pressure forces only come about with gradients (i.e. changes) of pressures.
below I put links to the relevant equations because there doesn't seem to be Latex support...
The formula you give is derived as a part of MHD http://en.wikipedia.org/wiki/Magnetohydrodynamics, and comes from examining the lorentz force. Here it is F = J x B. This says that the magnetic field exerts a force on any current flowing in the plasma perpendicular to both the current and the field.
We expand the J x B term by invoking Amperes law (http://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_law#Differential_form) which basically states that a curling (read changing) magnetic field has to be due to the presence of some current.
When we substitute Ampere’s law into J x B, we are keeping the fact that there is a current in the term for the FORCE exerted by the field by instead counting the change in the field instead of the current directly.
If you now look at the expansion of the J x B term in the first link, you may notice two things.
1) the magnetic pressure term only exerts a force when there is a changing field
2) the pressure force is accompanied by a second force which is the magnetic tension force and is broadly interpreted as the tension force along field lines
Thus we can only feel forces due to the magnetic pressure when two things happen together:
1) there exists a changing magnetic field
2) the pressure force is not exactly balanced by the tension force.
This ONLY occurs in the presence of currents, which mean there must be some charge carriers.
In summary,
1) You are correct, magnetic pressure is independent of the ionisation state of the plasma. It can be thought of as the energy density of the field.
2) We only notice the effect of magnetic pressure when it exerts a force.
3) It can only exert a force on currents, which imply some charge carriers (in your case some ionisation)
If you care to link this to the degree of ionisation you must find a formula to describe the currents in the plasma and this will involve the degree of ionisation, # of collisions etc...
I hope this is helpful to you.
Simon
Ah, the last three replies didn't come up until just now. My reply is a long-winded way of saying what Alec said.
Only I would like to add that this is not dependent on the assumptions of MHD (you may need quasi-neutrality). The only necessary equations are the Lorentz force and Amperes law. It all stems from those two. Any magnetic field has an intrinsic energy density and this may as well be called a pressure as they are unitarily equivalent.
Yes it depends, but its dependence is implicit in magnetic inbuction intensity B since it is a self consistent field which derives from the Amper's law ∇ ×B =μ0J+μ0ε0 ∂E/∂t. Here plasma density is included in defenition of current density J.
Dear Srinivasa Raju,
you can find a detailed description of the derivation for example in Francis F. Chen's book: Introduction to Plasma Physics and controlled Fusion
But this is quite the same as Alec, Simon and Afzal have written here - also my comment was just a short version of their descriptions.
FF Chen's book is best source for understanding concepts of plasma physics phenomena s...
If the plasma is full ionized it acts as a superconductor, then when changing the magnetic field, it appear Eddy currents in plasma surface that oppose to the field, so the magnetic pressure affects to the plasma surface, then it compress the inner plasma, but the currents are generated in the plasma surface
Sorry Javier, but a fully ionised plasma is far from being a superconductor. The electrical conductivity of such a plasma has an order of magnitude of 10^4 / (Ohm m) (see for example: Electrical conductivity of nondegenerate, fully ionized plasmas
G Röpke, R Redmer - Physical Review A, 1989).
Most metals have conductivities in the order of 10^6 - 10^7 / (Ohm m). Thus actually plasma, even if it's fully ionised is a worse conductor than a common metal.
I am sorry, perhaps my mistake cames from I usually work at Z-pinches and obtained currents of about a million amps but after reaching breakdown.
In this paper there are resistivity of less than 5e-6, but it is high enough to avoid Eddy currents at low frequency:
"Electrical Conductivity of Fully-Ionized Multicomponent Real and Model Plasmas"
Wow, that's really a low resistivity - I didn't expect such high currents after the breakdown.
Again I learned something - thanks Javier.
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