4 2 5 non uniform electric field
Download
Skip this Video
Download Presentation
4.2.5 Non-Uniform Electric Field

Loading in 2 Seconds...

play fullscreen
1 / 26

4.2.5 Non-Uniform Electric Field - PowerPoint PPT Presentation


  • 151 Views
  • Uploaded on

4.2.5 Non-Uniform Electric Field. Equation for the particle motion. Non-Uniform Electric Field (II). Need to evaluate E x (x) ( E x at the particle position ) Use undisturbed orbit approximation:. and. in. to obtain. Exercise 7.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about '4.2.5 Non-Uniform Electric Field' - gina


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
4 2 5 non uniform electric field
4.2.5 Non-Uniform Electric Field
  • Equation for the particle motion
non uniform electric field ii
Non-Uniform Electric Field (II)
  • Need to evaluate Ex(x) (Exat the particle position)
  • Use undisturbed orbit approximation:

and

in

to obtain

exercise 7
Exercise 7
  • Explain why the 1st order Taylor expansion for cos and sin requires krL<<1
non uniform electric field iv
Non-Uniform Electric Field (IV)
  • Use orbitaveraging: expecting a drift perpendicular to both E and B
  • Velocity along x averages to zero
  • Oscillating term of velocity along y averages to zero:
  • 1st order Taylor expansion for cos and sin for krL<<1 yields
non uniform electric field iv5
Non-Uniform Electric Field (IV)
  • The orbitaveraging makes the sin term to vanish
  • The average of the cos term yields:
  • This expression was obtained as special case with E non-uniformity perpendicular to y and z
  • The general expression for the ExB modified by the inomogeneity is:
exercise 8
Exercise 8
  • Why

has a minus sign factored while the

does not?

non uniform electric field v physics understanding
Non-Uniform Electric Field (V): Physics Understanding
  • The modification of the ExB due to the inomogeneity is decreasing the ExB drift itself for a cos(kx) field
  • If an ion spends more time in regions of weaker |E|, then its average drift will be less than the pure ExBamount computed at the guiding center
  • If the field has a linear dependence on x, that is depends on the first derivative dE/dx, itwill cause contributions of weaker and larger E to be averaged out and the drift correction (in this case depending on E and dE/dx) will be zero
  • Then drift correction must have a dependence on the second derivative for this reduced drift to take place
non uniform electric field vi physics understanding
Non-Uniform Electric Field (VI): Physics Understanding
  • The 2nd derivative of a cos(kx) field is always negative w.r.t. the field itself, as required in
  • An arbitrary field variation (instead of cos shaped) can be always expressed as a harmonic (Fourier) series of cos and sin functions (or exp(ikx) functions)
  • For such a series

or in a vector form

non uniform electric field vii physics understanding
Non-Uniform Electric Field (VII): Physics Understanding
  • Finally, the expression

can be then rewritten for an arbitrary field variation as

where the finite Larmor radius effect is put in evidence

  • This drift correction is much larger for ions (in general)
  • It is more relevant at large k, that is at smaller length scales
4 2 6 time varying electric field
4.2.6 Time-Varying Electric Field
  • Equation for the particle motion
time varying electric field ii
Time-Varying Electric Field (II)
  • Define an oscillating drift
  • The equation for vy has been previously found as
  • It can be verified that solutions of the form

apply in the assumption of slow E variation:

time varying electric field iii
Time-Varying Electric Field (III)
  • The polarization drift is different for ion and electrons: in general
  • It causes a plasma polarization current:
  • The polarization effect is similar to what happens in a solid dielectric: in a plasma, however, quasineutrality prevents any polarization to occur for a fixed E
4 2 7 time varying magnetic field
4.2.7 Time-Varying Magnetic Field
  • A time-varying magnetic field generates an electric field according to Faraday’s law:
  • To study the motion perpendicular to the magnetic field:

or, considering a vector l along the perpendicular trajectory,

time varying magnetic field ii
Time-Varying Magnetic Field (II)
  • By integrating over one gyration period the increment in perpendicular kinetic energy is:
  • Approximation: slow-varying magnetic field
  • For slow-varying B the time integral can be approximated by an integral over the unperturbed orbit
  • Apply Stoke’s theorem
time varying magnetic field iii
Time-Varying Magnetic Field (III)
  • The surface S is the area of a Larmor orbit
  • Because the plasma diamagnetismB·dS<0 for ions and vice-versa for electrons. Then:
  • Define the change of B during the period of one orbit as:
  • Recalling the definition of the magnetic moment m:
time varying magnetic field iv
Time-Varying Magnetic Field (IV)
  • The slowly varying magnetic field implies the invariance of the magnetic moment
  • Slowly-varying B cause the Larmor radius to expand or contract loss or gain of perpendicular particle kinetic energy
  • The magnetic flux through a Larmor orbit is

is then constant when the magnetic moment m is constant

time varying magnetic field v adiabatic compression
Time-Varying Magnetic Field (V): Adiabatic Compression
  • The adiabatic compression is a plasma heating mechanism based on the invariance of m
  • If a plasma is confined in a mirror field by increasing B through a coil pulse the plasma perpendicular energy is raised (=heating)
4 3 particle motion summary
4.3 Particle Motion Summary
  • Charge in a uniform electric field:
  • Charge in an uniform magnetic field:

yields the Larmor orbit solution

where

particle motion summary ii
Particle Motion Summary (II)
  • Charge in Uniform Electric and Magnetic Fields

produces the ExB drift of the guiding center

  • Charge Uniform Force Field and Magnetic Field

produces the (1/q)FxB drift of the guiding center

particle motion summary iii
Particle Motion Summary (III)
  • Charge in Motion in a Gravitational Field

produces a drift of the guiding center (normally negligible)

particle motion summary iv
Particle Motion Summary (IV)
  • Charge Motion in Non Uniform Magnetic Field: Grad-B Perpendicular to the Magnetic Field

the orbit-averaged solution gives a grad B drift of the guiding center

particle motion summary v
Particle Motion Summary (V)
  • Charge Motion in Non Uniform Magnetic Field: Curvature Drift due to Curved Magnetic Field
  • The particles in a curved magnetic field will be then always subjected to a gradB drift
  • An additional drift is due to the centrifugal force
particle motion summary vi
Particle Motion Summary (VI)
  • Charge Motion in Non Uniform Magnetic Field: Grad-B Parallel to the Magnetic Field:in a mirror geometry, defining the magnetic moment

the orbit-averaged solution of

provides a force directed against the gradB

particle motion summary vii
Particle Motion Summary (VII)
  • Charge Motion in Non-Uniform Electric Field

the orbit-averaged solution produces

particle motion summary viii
Particle Motion Summary (VIII)
  • Charge Motion in a Time-Varying Electric Field:

the solution in the assumption of slow E variation

yields a polarization drift that is different for ions and electrons

particle motion summary ix
Particle Motion Summary (IX)
  • Charge Motion in a Time-Varying Magnetic Field: solution of

in the perpendicular (w.r.t. B) plane and under the assumption of slow B variation shows a motion constrained by the invariance of the magnetic moment

ad