Analyzing a Cylindrically Symmetric Plasma Column

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SUMMARY

The discussion focuses on analyzing a cylindrically symmetric plasma column within a uniform magnetic field (B0 in the z direction). The electron density is defined as n(r) = n0 exp[-(r/r0)^2], and the ion and electron densities are equal, given by ni = ne = n0exp[e phi/kb Te]. The primary objective is to demonstrate that the electron velocity (Ve) and the drift velocity (VDe) are equal and opposite. The solution for VDe is derived as VDe = -2kb T r/((r0)^2 e B) in the theta direction, with an emphasis on finding an alternative method without using Poisson's equation.

PREREQUISITES
  • Understanding of plasma physics concepts, specifically cylindrical symmetry.
  • Familiarity with magnetic fields and their effects on charged particles.
  • Knowledge of electron temperature (Te) and its role in plasma behavior.
  • Basic proficiency in mathematical modeling of physical systems.
NEXT STEPS
  • Research the derivation of electron velocity in plasma using fluid dynamics principles.
  • Study the effects of magnetic fields on charged particle motion in plasma physics.
  • Explore alternative methods for solving plasma equations without Poisson's equation.
  • Learn about the implications of cylindrical symmetry in plasma confinement systems.
USEFUL FOR

Students and researchers in plasma physics, particularly those studying plasma behavior in magnetic fields and seeking alternative analytical methods for solving related equations.

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Homework Statement


A cylindrically symmetric plasma column in a uniform B field (= B0 in z direction) has
n(r) = n0 exp[-(r/r0)^2] and ni = ne = n0exp[e phi/kb Te] where phi is the potential and Te is the temp of the electrons.

(a) Show that Ve and VDe are equal and opposite


Homework Equations


We were told not to use poissons equation unless it is unavoidable (not sure if this is) but i get an answer that works using that, however i am looking for an alternate method.


The Attempt at a Solution


See attached pdf, shows my work for Ve, VDe worked out fine where VDe = -2kb T r/((r0)^2 e B) in the theta direction.

Cheers!
-Graeme
 

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