4.2.5 Non-Uniform Electric Field

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# 4.2.5 Non-Uniform Electric Field - PowerPoint PPT Presentation

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.

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Presentation Transcript
4.2.5 Non-Uniform Electric Field
• Equation for the particle motion
Non-Uniform Electric Field (II)
• Need to evaluate Ex(x) (Exat the particle position)
• Use undisturbed orbit approximation:

and

in

to obtain

Exercise 7
• Explain why the 1st order Taylor expansion for cos and sin requires krL<<1
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 (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
• Why

has a minus sign factored while the

does not?

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
• 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
• 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
• Equation for the particle motion
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)
• 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
• 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)
• 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)
• 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)
• 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
• 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
• Charge in a uniform electric field:
• Charge in an uniform magnetic field:

yields the Larmor orbit solution

where

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)
• Charge in Motion in a Gravitational Field

produces a drift of the guiding center (normally negligible)

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)
• 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)
• 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)
• Charge Motion in Non-Uniform Electric Field

the orbit-averaged solution produces

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)
• 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