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BACKGROUND: Maxwell’s Equations (mks) E B 0, B E , t 1 E c2 t B j neglect if v <<c E Since 2 c t We have used Bl 2 2 c t l 2 B t lc 2 2 v B lc H B ,D E 2 v c 2 2 B to eliminate H (the magnetic field) and D (the electric displacement) AS 4002 Star Formation & Plasma Astrophysics Definitions and Units The magnetic permeability The permittivity of free space 0 4 10 7 Hm -1 0 8.854 10 12 Fm -1 c2 1 0 0 Electric field Charge density Current density E Vm-1 Cm -2 j Am -3 Observers often measure magnetic induction in Gauss (rather than Tesla) and energy in erg (rather than Joules) 4 1G 10 T AS 4002 -7 1erg = 10 J Star Formation & Plasma Astrophysics THE MAGNETOHYDRODYNAMIC (MHD) EQUATIONS: DERIVATION • These equations describe how a magnetic field interacts with a plasma (ionized gas). • What happens when you put a magnetic field into a plasma? • Force (F) on a particle of charge (q) moving in an electric field (E) and a magnetic field (B): F qEr,t rÝ Br,t mrÝ Ý AS 4002 Star Formation & Plasma Astrophysics Simple case z • No electric field E = 0 • Uniform magnetic field B = (0,0,B) B y x Force equation mÝ rÝ qrÝ zˆ B has components AS 4002 mxÝ Ý qByÝ myÝ Ý qBxÝ Star Formation & Plasma Astrophysics Resultant particle motion The solution describes helical motion xÝ v cos t yÝ v sin t zÝ const where the gyro frequency (or Larmor frequency) qB m and the perpendicular velocity is constant 2 v 2 2 xÝ yÝ B The gyroradius AS 4002 r L v Star Formation & Plasma Astrophysics Consequences • Charged particles are tied to fieldlines (electrons and protons gyrate in opposite directions). • Even when the plasma is effectively collisionless ( mean free path >> typical lengthscales ) the magnetic field causes the plasma to behave collectively. • The gyroradius defines the lengthscale on which particle motions are organised. AS 4002 Star Formation & Plasma Astrophysics • Both conductivity and viscosity across magnetic field lines are much less than that along field lines. • Applying a uniform electric field perpendicular to B causes the centre of the helix to drift in the direction of E, while applying an electric field along B causes particles to be accelerated. AS 4002 Star Formation & Plasma Astrophysics The Equation of Motion • The equation of motion for charged particles is where dv nm nF p P dt n is the particle number density, m is the particle mass, F is a combination of the Lorentz force and gravity, F q E v B m g p is the pressure (assumed to be a scalar), and P describes the momentum transfer by collisions. AS 4002 Star Formation & Plasma Astrophysics Write this equation separately for electrons and ions: dv e ne me ne q E v e B ne me g pe P ei dt dv i ni mi Zni q E v i B ni mi g pi P ie dt Add these using Zni n e ni mi ne me ni mi j qZni v i n e v e ne me v e ni mi v i v ne m e ni mi AS 4002 Star Formation & Plasma Astrophysics dv e dv i ne me ni mi ne qv i v e B dt dt p ni mi ne me g to get dv dt j B p g Lorentz force This is just the same as the normal equation of motion for a fluid, except for the Lorentz force which describes the interaction between the magnetic field and the plasma. AS 4002 Star Formation & Plasma Astrophysics