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# sess14+15 - 14 14.1 The equations of In our preceding...

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14. Magnetohydrodynamics 14.1 The equations of magnetohydrodynamics In our preceding considerations we have treated gas independent of its state of ionization, yet we know that an ionized medium can carry and be influenced by electromagnetic fields, that should follow Maxwell’s equations. vector ∇ · vector E = 4 π ρ e vector ∇ × vector E = 1 c vector B ∂t (14 . 1 2) vector ∇ · vector B = 0 vector ∇ × vector B = 4 π c vector j e + 1 c vector E ∂t (14 . 3 4) where the charge density and current density are ρ e = Z e n i e n e (14 . 5) vector j e = Z e n i vector V i e n e vector V e (14 . 6) Implicit to Maxwell’s equations is the charge conservation, for ∂ρ e ∂t = 1 4 π vector ∇ · vector E ∂t = c 4 π vector ∇ · ( vector ∇ × vector B ) vector ∇ · vector j e ∂ρ e ∂t + vector ∇ · vector j e = 0 (14 . 7) The effect of the electromagnetic fields is different depending on the spatial scale on which they arise. Small-scale perturbations of the fields typically lead to electromagnetic waves which we will discuss later. On large scales astrophysical plasma are usual neutral, as expressed in the quasi-neutrality condition vector j e = σ vector E ∂ρ e ∂t = vector ∇ · vector j e = 4 π σ ρ e ρ e 0 on large scales (14 . 8) This implies the absence of sources of a large-scale electric field. The conductivity of astrophys- ical plasmas, σ , is high and consequently any large-scale electric field would be quickly shorted out. Currents can still exist, for only the bulk velocity of electrons and ions must be different, and the Maxwell’s fourth equation can be simply written for large scales vector ∇ × vector B = 4 π c vector j e 4 π c e n e ( vector V i vector V e ) ≃ − 4 π c e n e vector V e,rel (14 . 9) with the relative velocity of the electrons with respect to the ions, vector V e,rel = vector V e vector V i . A large-scale velocity difference of the positive and negative charges, i.e. not the difference in the gyration 1

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motion around a large-scale magnetic field, will likely be the result of a balance between an electric field and collisional resistance with collision frequency ν C , so in the rest frame of the ions (upper index ”i”) vector F = 0 = e vector E i m e ν C vector V e,rel vector V e,rel = e m e ν C vector E i vector j i = e n e vector V e,rel = n e e 2 m e ν C vector E i = σ vector E i (14 . 10) Transforming back from the rest frame of the ions to the laboratory frame we note that the current density involves only a velocity difference and hence is unchanged. Then using the Lorentz transformations for the fields we obtain v c E E B vector j e = vector j i vector B = vector B i vector E i = vector E + 1 c vector V i × vector B (14 . 11) which carries two messages: the notion of a vanishing electric field is a question of the frame of reference.
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sess14+15 - 14 14.1 The equations of In our preceding...

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