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2018년 4월 10일 화요일

[플라즈마물리][Plasma Physics]CH2 - Single Particle Motion - Nonuniform B-field Grad-B drift

Case Five: Non Uniform Magnetic Field

Let us assume gradient B-field exist in the z-direction with gradient along the y-direction.

B(r)=B(y)ˆz

Assume B is slightly in-homogeneous.

rL|BB|<<1

B can be expressed by the Taylor series.

B=B0+(r)B+...

so that we have

B=B0+(r)B=B0+yBy

From the Lorentz force, we will compute the average force and determine the guiding center drift.

dvxdt=qmvyB(y)ˆxdvydt=qmvxB(y)ˆy

vx=vcosωctvy=vsinωct

x=rLsin(ωct)y=±rLcosωct

Hence, using above equation, we can approximate the Lorentz force in inhomogeneous B-field.

Fx=qvyB=qvsin(ωct)(B0±rLcos(ωct)By)Fy=qvxB=qvcosωct(B0±rLcos(ωct)By)

Since

<sin(ωt)>=1TT0sinωtdt=0<cos(ωt)>=1TT0cosωtdt=0<sin2(ωt)>=1TT0sin2ωtdt=1TT01cos(2ωt)2dt=12<cos2(ωt)>=1TT0cos2ωtdt=1TT01+cos(2ωt)2dt=12<sinωtcosωt>=1TT0cos(ωt)sin(ωt)dt=1TT012sin(2ωt)dt=0

we get

<Fx>=1TT0Fxdt=0<Fy>=1TT0Fydt=1TT0Fydt=qvrLBy12=μBy

Notice that μ=±12qvrL. By plugging this into the guiding center drift equation for general case,

vB=F×BqB2=Fyˆy×BˆzqB2=FyqBˆx=μqBByˆx

The expression can be generalized as

vB=μqB×BB2

or

vB=12mv2qBB×BB2

The grad-B drift depends of the charge q, it can cause the plasma currents and charge separation.

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