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+y∂→B∂y
From the Lorentz force, we will compute the average force and determine the guiding center drift.
dvxdt=qmvyB(y)ˆxdvydt=−qmvxB(y)ˆy
vx=v⊥cosωctvy=∓v⊥sinωct
x=rLsin(ωct)y=±rLcosωct
Hence, using above equation, we can approximate the Lorentz force in inhomogeneous B-field.
Fx=qvyB=∓qv⊥sin(ωct)(B0±rLcos(ωct)∂B∂y)Fy=−qvxB=−qv⊥cosωct(B0±rLcos(ωct)∂B∂y)
Since
<sin(ωt)>=1T∫T0sinωtdt=0<cos(ωt)>=1T∫T0cosωtdt=0<sin2(ωt)>=1T∫T0sin2ωtdt=1T∫T01−cos(2ωt)2dt=12<cos2(ωt)>=1T∫T0cos2ωtdt=1T∫T01+cos(2ωt)2dt=12<sinωtcosωt>=1T∫T0cos(ωt)sin(ωt)dt=1T∫T012sin(2ωt)dt=0
we get
<Fx>=1T∫T0Fxdt=0<Fy>=1T∫T0Fydt=1T∫T0Fydt=∓qv⊥rL∂B∂y12=−μ∂B∂y
Notice that μ=±12qv⊥rL. By plugging this into the guiding center drift equation for general case,
→v∇B=→F×→BqB2=Fyˆy×BˆzqB2=FyqBˆx=−μqB∂B∂yˆx
The expression can be generalized as
→v∇B=μq→B×∇BB2
or
→v∇B=12mv2⊥qB→B×∇BB2
The grad-B drift depends of the charge q, it can cause the plasma currents and charge separation.
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