Analyzing a Cylindrically Symmetric Plasma Column

In summary, a cylindrically symmetric plasma column with a uniform B field has a density function of n(r) = n0exp[-(r/r0)^2] and equal electron and ion densities of ni = ne = n0exp[e phi/kb Te]. The potential of the electrons (Ve) and the potential of the Debye sheath (VDe) are equal and opposite, with VDe = -2kb T r/((r0)^2 e B) in the theta direction. The use of Poisson's equation was discouraged, but an alternate method was not provided.
  • #1
FunkyDwarf
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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
 

Attachments

  • Ve.pdf
    91.4 KB · Views: 382
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nobody? =(
 

What is a cylindrically symmetric plasma column?

A cylindrically symmetric plasma column is a type of plasma that has a symmetrical shape, resembling a cylinder. It is created when a gas is heated to extremely high temperatures, causing its atoms to ionize and form a plasma state.

What are the applications of analyzing a cylindrically symmetric plasma column?

Analyzing a cylindrically symmetric plasma column has many practical applications, including fusion research, plasma thrusters for spacecraft propulsion, and plasma processing for material synthesis and modification.

How is a cylindrically symmetric plasma column analyzed?

A cylindrically symmetric plasma column can be analyzed using various diagnostic techniques such as spectroscopy, interferometry, and Langmuir probes. These techniques measure different properties of the plasma, such as its temperature, density, and composition.

What factors affect the behavior of a cylindrically symmetric plasma column?

The behavior of a cylindrically symmetric plasma column is influenced by several factors, including the type of gas used, the temperature and pressure of the gas, the strength of the magnetic field, and the shape and size of the plasma column.

What are the challenges in analyzing a cylindrically symmetric plasma column?

One of the main challenges in analyzing a cylindrically symmetric plasma column is maintaining its stability. Plasma instabilities can arise due to various factors, such as uneven heating, magnetic field fluctuations, and interactions with the surrounding materials. These instabilities can affect the accuracy of the analysis and the overall behavior of the plasma column.

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