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1 "#$()"*#*+,-./-*+01.2(.* * "#$"()*+,-."/012#."3424, Dr. Peter T. Gallagher Astrphysics Research Grup Trinity Cllege Dublin :2";,<=+-$)(,<*>)"* Mtin in a unifrm B field E x B drift Mtin in nn-unifrm B fields Gradient drift Curvature drift Other gradients f B Tday s lecture Time-varying E field Time-varying B field Adiabatic Invariants

2 In last lecture, slved equatins f mtin fr particles in unifrm B and E fields. Shwed that drift f guiding-centre is: which can be extended t a frm fr a general frce F: In a gravitatin field fr example, v E = E " B Similar t the drift v E, in that drift is perpendicular t bth frces, but in this case particles f ppsite charge drift in ppsite directin. v f = 1 q F " B v g = m q g " B Unifrm fields prvide pr descriptins fr many phenmena, such as planetary fields, crnal lps, tkamaks, which have spatially and temprally varying fields.

3 Particle drifts in inhmgeneus fields are classified in several ways. In this lecture, cnsider tw drifts assciated with spatially nn-unifrm B: gradient drift and curvature drift. There are many thers As sn as intrduce inhmgeneity, t cmplicated t btain exact slutins fr guiding centre drifts. Therefre use rbit thery apprximatin: Within ne Larmr rbit, B is apprximately unifrm, i.e., the typical length-scale, L, ver which B varies is such that L >> r L. => gyr-rbit is nearly a circle.

4 Assume lines f frces are straight, but their density increases in y directin (see figure belw frm Chen, Page 27). y -directin Gradient in B causes the Larmr radius (r L = mv/qb) t be larger at the bttm f the rbit than the tp, which leads t a drift. Drift shuld be perpendicular t grad B and B and ins and electrns drift in ppsite directins. Cnsider spatially-varying magnetic field, B = ( 0, 0, B z (y) ). i.e., B nly has z- cmpnent, the strength f which varies with y. Assume that E = 0, s equatin f mtin is F = q(v " B) Separating int cmpnents, F x = q(v y B z ) (4.1a) F y = "q(v x B z ) (4.1b) F z = 0 Nw the gradient f B z is db z dy " B z L << B z r L => r L db z dy << B z This means that the magnetic field strength can be expanded in a Taylr expansin fr distances y < r L : B z (y) = B 0 + y db z dy +...

5 Expanding B z t first rder in Eqns. 4.1a and 4.1b: " F x = qv y B 0 + y db z $ # dy " F y = (qv x B 0 + y db z $ # dy (4.2) Particles in B-field travelling abut a guide centre (0,0) have a helical trajectry: x = r L sin(" c t) y = ±r L cs(" c t) The velcities can be written in a similar frm: v x = v " cs(# c t) v y = ±v " sin(# c t) Substituting these int Eqn. 4.2 gives: F x = "qv # sin($ c t) B 0 ± r L cs($ c t) db ( z * dy ) F y = "qv # cs($ c t) B 0 ± r L cs($ c t) db ( z * dy ) Since we are nly interested in the guiding centre mtin, we average frce ver a gyrperid. Therefre, in the x-directin: F x = "qv # B 0 sin($ c t) ± r L sin($ c t)cs($ c t) db ( z * dy ) But < sin( c t) > = 0 and < sin( c t) cs( c t) > = 0 => <F x > = 0. In the y-directin: F y = "qv # B 0 cs($ c t) ± r L cs 2 ($ c t) db ( z * dy ) = qv #r L 2 db z dy (4.3) where < cs( c t) > = 0 but < cs 2 ( c t) > =1/2.

6 In general, drift f guiding-centre is Subing in using Eqn 4.3: v f = 1 q v "B = 1 q F " B F y y ˆ # B z z ˆ B z 2 = v $ r L db z 2B z dy x ˆ S +ve particles drift in x directin and ve particles drift in +x. In 3D, the result can be generalised t: Grad-B Drift The ± stands fr sign f charge. The grad-b drift in in ppsite directins fr electrns and ins and causes a current transverse t B. v "B = ± 1 2 v #r L B $ "B Belw are shwn particle drifts due t a magnetic field gradient, where B(y) = z B z (y). Page 31 f Inan Glkwski. Cnsider r L = mv/qb. Lcal gyrradius is large where B is small and visa versa which gives rise t a drift. Nte als that charge determines directin f drift.

7 Gradient drift is respnsible fr current in inner parts f planetary magnetspheres. Apprximate field is a diple: B r = µ 0 2m 4" r cs(#) 3 B # = µ 0 2m 4" r sin(#) 3 In equatrial plane, B r = 0 and B " = B 0 / r 3. There is therefre a psitive gradient in B " directed radialy utwards. There is therefre a grad-b drift perpendicular t B and grad-b, which prduces a ring current circulating abut a planet. Right: Ring current viewed frm nrth ple with NASA s Image satellite.

8 When charged particles mve alng curved magnetic field lines, experience centrifugal frce perpendicular t magnetic field (Fig Inan Glkwski, Page 34). Assume radius f curvature (R c ) is >> r L The utward centrifugal frce is This can be directly inserted int general frm fr guiding-centre drift F cf = mv 2 r ˆ R c v f = 1 q F " B => v R = mv 2 R c " B 2 qr c Curvature Drift Drift is therefre int r ut f page depending n sign f q. Type f field cnfiguratin studied abve having curved but parallel field lines will never ccur in reality, since B # 0 fr this field. In practice, curved field lines will always be cnverging/diverging. Thus a curvature drift will always be accmpanied by a grad-b drift. Where B is in the ppsite directin t R c, the grad-b and curvature drifts act in the same directin. Right is a crss-sectin f a tkamak magnetic field. Dashed lines are field lines. In the highlighed sectin, the field is lwer at edge than in the centre. S curvature and gradient drift act in same directin - utwards fr ins s ins hit the vessel walls and are lst.

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