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Dr. Peter T. Gallagher
Astrophysics Research Group
Trinity College Dublin
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o  Motion in a uniform B field
o  E x B drift
o  Motion in non-uniform B fields
o  Gradient drift
o  Curvature drift
o  Other gradients of B
o  Time-varying E field
o  Time-varying B field
o  Adiabatic Invariants
Today’s lecture
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o  In last lecture, solved equations of motion for particles in uniform B and E fields.
o  Showed that drift of guiding-centre is:
vE =
E"B
B2
which can be extended to a form for a general force F:
!
vf =
o  In a gravitation field for example,
1 F"B
q B2
vg =
m g"B
q B2
!
o  Similar to the drift vE, in that drift is perpendicular to both forces, but in this case
particles of opposite charge drift in opposite direction.
!
o  Uniform fields provide poor descriptions for many phenomena, such as planetary
fields, coronal loops, tokamaks, which have spatially and temporally varying fields.
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o  Particle drifts in inhomogeneous fields are classified in several ways.
o  In this lecture, consider two drifts associated with spatially non-uniform B:
gradient drift and curvature drift. There are many others!
o  As soon as introduce inhomogeneity, too complicated to obtain exact
solutions for guiding centre drifts.
o  Therefore use orbit theory approximation:
o  Within one Larmor orbit, B is approximately uniform, i.e., the typical
length-scale, L, over which B varies is such that L >> rL.=> gyro-orbit
is nearly a circle.
E$-&=F*?$2@**
o  Assume lines of forces are straight, but their density increases in y direction (see
figure below from Chen, Page 27).
y -direction
o  Gradient in |B| causes the Larmor radius (rL = mv/qB) to be larger at the bottom of
the orbit than the top, which leads to a drift.
o  Drift should be perpendicular to grad B and B and ions and electrons drift in
opposite directions.
E$-&=F*?$2@**
o  Consider spatially-varying magnetic field, B = ( 0, 0, Bz(y) ). i.e., B only has zcomponent, the strength of which varies with y.
o  Assume that E = 0, so equation of motion is F = q(v " B)
o  Separating into components,
Fx = q(vy Bz )
F !
= "q(v B )
y
x
z
(4.1a)
(4.1b)
Fz = 0
o  Now the gradient of Bz is
!
dBz Bz
B
"
<< z
dy
L
rL
dB
=> rL z << Bz
dy
o  This means that the magnetic field strength can be expanded in a Taylor expansion
for distances y < rL:
dB
!
Bz (y) = B0 + y z + ...
dy
!
E$-&=F*?$2@**
o  Expanding Bz to first order in Eqns. 4.1a and 4.1b:
"
dB %
Fx = qvy $ B0 + y z '
dy &
#
"
dB %
Fy = (qvx $ B0 + y z '
dy &
#
(4.2)
o  Particles in B-field travelling about a guide centre (0,0) have a helical trajectory:
x = rL sin("c t)
y = ±rL cos("c t)
!
o  The velocities can be written in a similar form: vx = v"cos(#c t)
vy = ±v"sin(#c t)
!
%
dB (
o  Substituting these into Eqn. 4.2 gives: Fx = "qv#sin($c t)' B0 ± rL cos($c t) z *
dy )
&
!
%
dB (
Fy = "qv# cos($c t)' B0 ± rL cos($c t) z *
dy )
&
!
E$-&=F*?$2@**
o  Since we are only interested in the guiding centre motion, we average force over a
gyroperiod. Therefore, in the x-direction:
%
dBz (
Fx = "qv#'B0 sin($c t) ± rL sin($c t)cos($c t)
*
dy )
&
o  But < sin(!ct) > = 0 and < sin(!ct) cos(!ct) > = 0 => <Fx> = 0.
!
o  In the y-direction:
%
dBz (
Fy = "qv#'B0 cos($c t) ± rL cos2 ($c t)
*
dy )
&
qv r dB
= ! #L z
(4.3)!
2 dy
where < cos(!ct) > = 0 but < cos2 (!ct) > =1/2.
!
E$-&=F*?$2@**
o  In general, drift of guiding-centre is
vf =
o  Subing in using Eqn 4.3:
v "B =
!
1 F"B
q B2
1 Fy yˆ # B z zˆ
q
Bz2
=!
v$ rL dBz
xˆ
2Bz dy
o  So +ve particles drift in –x direction and –ve particles drift in +x.
!
o  In 3D, the result can be generalised
to:
1
B $ "B
v "B = ± v# rL
2
B2
Grad-B Drift
o  The ± stands for sign of charge. The grad-B drift in in opposite directions for
electrons and ions and causes a current transverse to B.
!
E$-&=F*?$2@**
o  Below are shown particle drifts due to a magnetic field gradient, where B(y) = z Bz
(y). Page 31 of Inan & Golkowski.
o  Consider rL = mv/qB. Local gyroradius is large where B is small and visa versa
which gives rise to a drift. Note also that charge determines direction of drift.
E$-&=F*?$2@G*+,-"<#-$1*H2";*I'$$<"#*
o  Gradient drift is responsible for current in inner parts of planetary magnetospheres.
o  Approximate field is a dipole:
µ 0 2m
cos(# )
4" r 3
µ 2m
B# = 0 3 sin(# )
4" r
Br =
!
o  In equatorial plane, Br = 0 and B" = B0 / r3. There is therefore a positive gradient in
B" directed radialy outwards.
o  There is therefore a grad-B drift perpendicular to B and grad-B, which produces a
ring current circulating about a planet.
E$-&=F*?$2@G*+,-"<#-$1*H2";*I'$$<"#*
o  Right: Ring current viewed from
north pole with NASA’s Image satellite.
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o  When charged particles move along curved magnetic field lines, experience
centrifugal force perpendicular to magnetic field (Fig Inan & Golkowski, Page 34).
o  Assume radius of curvature (Rc) is >> rL
o  The outward centrifugal force is
mv||2
Fcf =
rˆ
Rc
o  This can be directly inserted into general form
for guiding-centre drift
!
vf =
1 F"B
q B2
mv||2 Rc " B
=> v R = 2
qRc B2
Curvature Drift
!
o  Drift is therefore into or out of page depending on sign of q.
!
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o  Type of field configuration studied above – having curved but parallel field lines –
will never occur in reality, since ∇䏙B # 0 for this field.
o  In practice, curved field lines will always be converging/diverging. Thus a
curvature drift will always be accompanied by a grad-B drift.
o  Where ∇B is in the opposite direction to Rc, the grad-B and curvature drifts act in
the same direction.
o  Right is a cross-section of a tokamak magnetic field.
o  Dashed lines are field lines. In the highlighed
section, the field is lower at edge than in
the centre. So curvature and gradient drift
act in same direction - outwards for ions –
so ions hit the vessel walls and are lost.