Problem 12

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Problem 12

• Explain under what conditions a diffusion
  equation can be derived from the continuity
  equation plus a Ficks law

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Solution Problem 12

• The Ficks law states (for a generic species)

• The continuity equation is

and by using Ficks law becomes
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Solution Problem 12 (II)

• If the diffusion coefficient in the Ficks law is
  uniform in space it can be taken outside the
  divergence operator and a diffusion equation can
  be written as

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Problem 13

• Determine the dimension (in MKSA) of the electron
  mobility
• Find a relationship between the electron mobility
  and the plasma conductivity (reciprocal of the
  resistivity) for the case in which the plasma
  current is carried only by the electrons and the
  density is uniform

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Solution Problem 13

• The electron mobility, for a given collision
  frequency, is defined as

• The plasma conductivity is defined as s1/h while
  the resistivity appears in the Ohms law as

• The electron mobility was introduced from the
  fluid equation of motion as

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Solution Problem 13 (II)

• For a uniform density the gradient is zero and
  the mobility for the electrons will satisfy

• While for the conductivity, considering the
  definition of current density, it can be written

• Then

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Problem 14

• Write the MHD momentum equation for steady state
  and no gravitational effect.
• A cylindrical plasma with uniform axial magnetic
  field is confined in these conditions
• Intense heating is applied in the plasma core
  region (the inner, centered section of the plasma
  along the axis)
• Is it possible to maintain the initial steady
  state conditions while the heating takes place?
  Explain why yes or why not (assume plasmaideal
  gas)
• What happens to the current density j in the MHD
  momentum equation when the plasma is heated?

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Solution Problem 14

• The MHD momentum equation for steady state and no
  gravitational effect is

• If the plasma temperature is being raised around
  the axis the pressure gradient will increase (the
  temperature is non uniform and will have a
  gradient)
• To maintain the steady state the force balance
  will require to increase the magnetic field
  (improving the confinement)

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Solution Problem 14 (II)

• The current density (diamagnetic current) is the
  result of the BXgrad p force and will increase
  self-consistently to maintain the force balance

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Problem 15

• For the case of a diffusion across a magnetic
  field what happens to the perpendicular diffusion
  coefficient when the magnetic field is increased?
• Explain this trend in physical terms (from a
  particle or fluid point of view)

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Solution Problem 15

• The perpendicular diffusion coefficient for a
  magnetized plasma is

• The Larmor frequency wc is defined as

then an increase in the magnetic field reduces
the diffusion coefficient
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Solution Problem 15 (II)

• From the Ficks law, in general,

• it is clear that a smaller diffusion coefficient
  reduces the flux. Therefore an increased magnetic
  field also reduces the flux across the field
  itself
• This is equivalent to say that the increase in
  the magnetic field improves the plasma confinement

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Problem 16

• An electrostatic perturbation in a non-magnetized
  plasma is detected propagating at a frequency of
  10GHz. The plasma has a density of 1018 m-3 and a
  temperature of 3 eV
• Find the wavelength associated to the perturbation

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Solution Problem 16

• The frequency of the oscillations is given by the
  dispersion relation

where

• The detected frequency corresponds to an angular
  frequency

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Solution Problem 16 (II)

• The plasma frequency is given by

where

• The wavenumber is found inverting the dispersion
  relation

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Solution Problem 16 (III)

• The wavelength is finally found from