Object of Plasma Physics

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Part I
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• Object of Plasma Physics

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I. Object of Plasma Physics

• 1. Characterization of the Plasma State
• 2. Plasmas in Nature
• 3. Plasmas in the Laboratory

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1. Characterization of the Plasma State
BACK

• 1.1 Definition of the Plasma State
• 1.2 Historical Perspective
• 1.3 Transition to the Plasma State

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1.3 Transition to the Plasma State
BACK

• 1.3.1 Debye Shielding
• 1.3.2 The Debye Length physical intuition
• 1.3.3 Plasmas Quantitative Characterization
• 1.3.4 The Debye Length rigorous derivation

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1.3.4 Debye Length Rigorous Derivation
BACK

• Free charges in an ionized gas

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Debye Length Rigorous Derivation (II)

• Shielded charge or electric potential source
  transient showing only electron motion
• Objective find the electric potential f(x)

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Debye Length Rigorous Derivation (IV)

• Ingredients Poisson equation, electron Boltzmann
  distribution, Taylor expansion, solution of 1D
  linear differential equation with constant
  coefficients
• Problem definition find the electric potential
  produced by a charge distribution (including
  boundary conditions)

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Poisson Equation

• Gauss theorem
• Differential form of Gauss theorem (first
  Maxwell equation)
• Definition of electric potential
• Poisson equation
• or
• Laplacian operator

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Boltzmann Electron Distribution

• Boltzmann energy distribution function
• where f is probability for a particle to have
  energy E, A is a constant and, in three
  dimensions, kBT is related to the particle
  average kinetic energy by
• Electron density with potential energy Uqef
  where the density at f0 is set to n0

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Potential Energy

• Potential energy capacity for doing work which
  arises from position or configuration.

• Total energy kinetic minus potential
• Force

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Taylors Expansion

• Given a function f(x) its Taylor expansion or
  Taylor series is

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First order, linear differential equation

• Given the first order, homogenous, linear
  differential equation with constant coefficients
  a and b
• the solution is
• where A is a constant found by imposing a
  boundary condition (like f(0)f0)

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Second order, linear differential equation

• Given the second order, homogenous, linear
  differential equation with constant coefficients
  a and b
• the solution is
• where l1 and l2 are the solution of
• and A and B are constants found by imposing two
  boundary conditions

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Debye Length Rigorous Derivation (V)

• Given the total charge density rqeneqini the
  potential f is found by solving the Poisson
  equation that in one dimension is simply

• By considering qi e, qe -e it follows
  r-e(ne-ni)
• The ions are considered fixed (in the transient)
  with a constant density ni n0
• A Boltzmann distribution for the electrons is
  considered

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Debye Length Rigorous Derivation (VII)

• The total charge density will be then

• The Poisson equation can then be re-written as

• A Taylor expansion of the exponential function is
  considered for

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Debye Length Rigorous Derivation (VI)

• By retaining only the linear term of the Taylor
  expansion (linearization procedure) the Poisson
  equation becomes

• The potential is then

where lD is the Debye length
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Typical Values for the Debye Length

• Typical values of Debye Length under different
  conditions
• n m-3 TeV Debye Length
  m
• Interstellar 106 10-1 1
• Solar Wind 107 10 10
• Solar Corona 1012 102 10-1
• Solar atmosphere 1020 1 10-6
• Magnetosphere 107 103 102
• Ionosphere 1012 10-1 10-3

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Problem 1

• Prove that the dimensional analysis of the
  expression for the Debye length

yields a length (in meters)
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Solution Problem 1
,from the equation for a capacitor

• e0F/m

then
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Problem 2

• Prove that the solution of

is
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Solution Problem 2

• The solution of

is
where l1,2 are found from

• The equation

can be brought in the previous form with
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Solution Problem 2 (II)

• The eigenvalues l1,2 are then found from

• The solution will be then in the form

and (for a

• The boundary conditions are

physically meaningful potential solution)

• Therefore it will be B0 and