A laboratory deuterium plasma column ( ,

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Fluid and magnetohydrodynamic description · Fluid and magnetohydrodynamic description

Plasma Physics, Ch Magnetohydrodynamics: Ohm's law for a moving plasma, frozen flux and magnetic pressure

Problem

A laboratory deuterium plasma column (

m_i = 2m_p

n_i = n_e = 1.0\times10^{20}\,\mathrm{m^{-3}}

, resistivity

\eta = 5.0\times10^{-6}\,\Omega\,\mathrm{m}

) sits in a straight field along

z

whose magnitude varies only across it, in

x

. At

x=0

the pressure is

p(0) = 2000\,\mathrm{Pa}

and

B(0) = 0.30\,\mathrm{T}

; at

x = L = 0.050\,\mathrm{m}

the plasma pressure has fallen to

p(L) = 0

. The plasma is in static perpendicular force balance, so the total pressure

p + B^2/2\mu_0

is the same at both points. ((a)) Find

B(L)

. ((b)) Because the field is straight,

\mathbf{j}\times\mathbf{B}

here is the perpendicular-gradient (magnetic pressure) term alone. Find its magnitude (take it uniform over

0\le x\le L

) and check that it equals

|dp/dx|

, as force balance requires. ((c)) Using the two-fluid diamagnetic current

j_{\rm dia} = |\nabla p|/B

(chapter 4), estimate the current density needed to produce this force, evaluating

B

at the mid-point

x=L/2

. ((d)) Using

B(0)

and

L

, find the Alfven speed

v_A

and the Lundquist number

S = \mu_0Lv_A/\eta

for this column.

Browse the problem library