Consider now a single particle trajectory with E = 0 and В constant in the space of interest. The relevant governing equation now becomes

d

m—=q(x B) (5.6)

dt

and by the definition of a vector cross product, the force on the particle is here perpendicular to both v and B. Writing Eq.(5.6) in terms of its vector components, we find

— (v* і + wy j + vz k) = ~ [(vA і + V;j + V; k) x (Bx і + By j + Bz k)] (5.7)

where i, j and к are orthogonal unit vectors along the x, у and z axes, respectively.

Orienting this coordinate system so that the z-axis is parallel to the B-field, thereby imposing Bx = By = 0, Fig. 5.3, and equating the appropriate vector components of Eq.(5.7), we obtain the following set of differential equations together with their initial conditions:

ЧВЛ —— v m у 4Bz m ,
dvx dt dvy dt
Vx(0)= Vx, o >	(5.8a) (5.8b)
vy(0)= Vyi0

and

^ = 0. vz(0)=vz, o• (5.8c)

at

The interesting feature here is that vx(t) and vy(t) are mutually coupled but that vz(t) is independent of the former. Hence, the z-component is therefore straightforward and based on Eq.(5.8c), we have

V;(f)= vz, o= V|| (5.9)

which is a constant with the parallel line subscripts referring to the direction of the В-field. Then with the reference z-coordinate as z(0) = z0, another integration gives

z(t)=Zo + V (5-Ю)

with the trajectory of the charged particle motion in the z-direction thus established.

To solve for the velocity and position components perpendicular to the B — field direction-that is vx(t), vy(t), x(t) and y(t)-we first uncouple Eqs.(5.8a) and (5.8b) by differentiation and substitution to obtain

d2 vy dt2

(5.11b)

where, upon recognizing that both Eq.(5.1 la) and Eq.(5.1 lb) describe a harmonic oscillation with the same frequency which results in a circular motion of the particle given suitable initial conditions, we have introduced the gyration frequency-sometimes also called the gyrofrequency or cyclotron frequency-as

Evidently, the following forms satisfy Eq.(5.11):

vx(t)= Ax cos(cogt) + Dx sin(0)gt),

Vy (t)= Ay cos( cog t) + Dy sinf cog t)

with Ax, Ay, Dx and Dy constants to be determined for the case of interest.

It is known that a plasma possesses diamagnetic properties. This means that the direction of the magnetic field generated by the moving charged particles is opposite to that of the externally imposed field. Therefore, not only must the equations resulting from Eq.(5.6) be satisfied but the moving charged particle representing in fact a current flow along its trajectory must generate a magnetic field in opposition to B. Some thought and analytical experimentation with Eqs.(5.13) yields velocity components for a positive ion as

vx(t)=v0cos(cOgt + ф) (5.14a)

and

y(t)= — V0 sin(cOgt + ф)

where v0 is a positive constant speed and ф is the initial phase angle. For a negatively charged particle, Eq.(5.14b) will change its sign, while Eq.(5.14a) remains unchanged. Then, with vx(t) and vy(t) thus determined, a further integration yields x(t) and y(t) as

x(t)= x0 + — sin(o)gt + $) (5.15a)

(Og

y(t)= y0 + sign(q)~ cos( (Ogt + ф) (5.15b)

(Og

describing thereby a circular orbit about the reference coordinate (x0, y0) for an arbitrary charge of magnitude q and sign designated by sign(q). This coordinate constitutes a so-called "guiding" centre which can, in view of Eq.(5.9), move at a constant speed in the z-direction. The gyration in З-dimensions would therefore appear as a helix of constant pitch.

From Eqs.(5.14) it follows that the particle speed in the plane perpendicular to the magnetic field is a conserved quantity, and hence we write

V0 = — y/v* + Vy = V_L > (5.16)

in order to denote the velocity component perpendicular to B. Evidently, if vy = 0,

the particle trajectory is purely circular, and the corresponding radius is obviously determined by

V-L _ V-L m COg |^| 3z

The label gyroradius-as a companion expression to the gyrofrequency £Og-is generally assigned to this term rg although the name Larmor radius is also often used.

From the above it is evident that a heavy ion will have a larger radius than a lighter ion at the same Vj_, and, by definition of £Ug, Eq.(5.12), an electron will possess a much higher angular frequency than a proton. Figure 5.4 depicts the circular trajectory of an ion and an electron as a projection of motion on to the x — y plane under the action of a uniform magnetic field В = Bzk. A useful rule to remember is that with В pointing into the plane of the page, an ion circles counterclockwise while electrons circle clockwise. Note that this is consistent with the diamagnetic property of a plasma by noting that, at the (x0,y0) coordinate, the self-generated В-field points in the opposite direction to the external magnetic field vector B.

©

B (into page)

With a constant velocity component parallel to the В-field and a time- invariant circular projection in the perpendicular plane, the particle trajectory in З-dimensions represents a helical pattern along the В-line as suggested in Fig.5.5; note that vz o = V|| solely determines the axial motion. The pitch of the helix is also suggested in Fig.5.5 and is given by

(Og

which is constant for our case. Thus, an isolated charged particle in a constant, homogeneous magnetic field traces out a helical trajectory of constant radius, frequency and pitch.

_L

Az

T

5.1 Combined Electric and Magnetic Field

Consider next the effect of a combined E and В field on the trajectory of a charged particle. With both fields uniform and constant throughout our space of

interest, the trajectory is expected to be more complex since it will combine features of the two trajectories of Figs. 5.2 and 5.5.

With the В-field and E-field possessing arbitrary directions, it is useful to orient the Cartesian coordinate systems so that the z-axis is parallel to В with E in the x-z plane, Fig. 5.6. The equation of motion, Eq.(5.1), is now expanded to give

m~dt^!x‘l+ Vy^+ v*k) = + £zk) + (vj+ Vj j+ vzk)x(fizk)] (5.20)

so that the velocity components must satisfy

^L = — Ex + — Bzvy = — Ex + 0)g VySign(q), (5.21a)

at m m m

~~ = — — BzVx = -cogVx signfq), (5.21b)

at m

and

dvz _ q dt m

Here, the gyrofrequency C0g, Eq.(5.12), has again appeared and the charge sign of the particle is indicated by sign (q), thereby specifying the direction of gyration. The velocity components vx and vy may be uncoupled by differentiation and substitution of Eqs.(5.21a) and (5.21b):

—~ = 0 + cog~sign(q) = — o)2 vx (5.22a)

dt2 dt

^-jL=-0)g^jLsign(q) dt2 dt

With Ex, B2, and Cflg as constants, this last expression may therefore be written as

(5.23)

where

(5.24)

and

The associated (x, y,z) coordinates follow from a further integration to yield

y = yo + —co^(0gt + <p )sign(q)- — t (5,29b)

COg Bz

z = zo + y\t+ ■ (5.29c)

These equations, Eqs.(5.28) and (5.29), show that the charged particle still retains its cyclic motion but that it drifts in the negative у-direction for Ex > 0, Fig. 5.7, and oppositely if Ex < 0. Hence, the guiding centre’s drift is in the direction perpendicular to both the В-field and E-field and in contrast to the preceding (E = 0, В =£ 0) case, ion and electron motion across the field lines now occur. Additionally, the pitch increases with time-Eqs.(5.28c) and (5.29c)-so that the trajectory resembles a slanted helix with increasing pitch.

О Bz-Field

Ex-Field [1]

where a vector product identity has been utilized for the last step. Note from Eq.(5.31) that the motion of the guiding centre parallel to the magnetic field В can only be affected by the parallel component of F. Then, since vg is perpendicular to B, the velocity component vgCju = уц and is readily seen to be given by

JgcwW — vgc, ii(0) + — [ F\dt m J

Of greater interest here is the perpendicular component vgCi.

Eq.(5.34) as

_ _ FxB

V? C l q B2

because (у д • В ) = 0. This identifies what is called the drift velocity due to the force F, i. e.

Vgc,± = V£>F — (5.37)

This generalization allows us to specify an expression if the additional force acting in a static magnetic field domain is an electric force F = qE, so that the drift velocity due to a stationary electric field is

vfl, r = ^. (5.38)

B2

The label "E X В drift" is frequently used in this context. Note that the drift is evidently independent of the mass and charge of the particle of interest, and consistent with the drift velocity term in Eq.(5.28b), i. e. (- Ex / Bz).

5.2 Spatially Varying Magnetic Field

Each of the preceding cases involved a В-field taken to be uniform throughout the space of interest. Such an idealization is, in practice, impossible although it
can be approached to some degree by using sufficiently large magnet or coil arrangements. In general, however, it often becomes necessary to incorporate a space dependent В-field which governs the particles’ motion. The associated equation of motion is hence written as

m— = q[vx B(r)l. (5.39)

dt

A comprehensive analysis of this equation for a most general case may obscure some important instructional aspects so that a more intuitive approach has merit; this approach also provides for a convenient association with the preceding analysis while still leading to some useful generalizations.

We consider therefore a dominant magnetic field in the z-direction characterized by an increasing strength in the у-direction; that is, a constant non­zero gradient of the magnetic field strength, VB, exists everywhere given by

VB = ^j. (5.40)

dy

Figure 5.8 provides a graphical depiction of this В-field showing also a test ion with specified initial velocity components.

From the previous analysis of the motion of a charged particle in a uniform В-field, it is known that the radius of gyration in the plane perpendicular to the local В-field is given by

»■« = — ==ГЬ (5-41)

ft), I qB

where В is the local magnetic field magnitude at the point where the charged particle’s guiding centre is located; thus, the gyroradius rg varies inversely with

B. Any motion in the z-direction will be additive and depends upon the initial vy component; for reasons of algebraic simplification and pictorial clarity, we may therefore take vy = 0.

Referring to Eq.(5.41) and Fig. 5.8, it is evident therefore that an isolated
charged test particle will trace out a trajectory whose radius of curvature becomes larger in the weaker magnetic field and smaller where the В-field is stronger. We can translate this effect into a process which tends to transform a circular path into a shifting cycloid-like path with the consequence that the particle trajectory now tends to cross magnetic field lines as suggested in Fig. 5.9.

Note that the radius of gyration, Eq.(5.41), is directly proportional to the mass of the particle and that the sign of the charge enters into the direction of the trajectory. This is also indicated in Fig. 5.9 for a positively and negatively charged particle, although not to scale. The feature that the gradient causes a drift of the positive ions in one direction and for electrons in the opposite direction has important consequences because the resulting charge separation introduces an electric field which, in a plasma, can contribute to plasma oscillations and instability, which we will further discuss in subsequent chapters.

With an intuitive approach of the gradient-B drift phenomenon thus suggested, we consider next an approximate though useful analytical representation. The first approximation to be introduced is that for the case of a weak magnetic field inhomogeneity over distances corresponding to a gyroradius, that is, IVBI/B « l/rg, only the first two terms of a Taylor expansion of Bz(y) are retained, i. e.

where y0 corresponds to the у-coordinate of the guiding centre. Extending to the vector form for Eq.(5.42), we write

Ъ(У)~В0 + (У-У0Щ — Bo

where now B0 = B(y0). We next expand the equation of motion, Eq.(5.39), and write

This latter expression is a general formulation since the у-axis was arbitrarily chosen due to the geometry of Sec. 5.3.

Finally, recalling Eqs.(5.36) and (5.37), we find the grad-B drift velocity as

vl В x VB

vD, vB=SI?’1^Ti——————- T~ (5-51)

2(og B2

and thus the positive ions and negative electrons drift across В-field lines in opposite directions in a non-uniform magnetic field as displayed in Fig. 5.9.

In subsequent chapters, we will discover that fusion devices which utilize magnetic fields for confinement of the reacting ions involve very complex magnetic field topologies. Not only are there electric fields and spatially varying magnetic fields present-thus giving rise to the drift velocities vD, E and vd, Vb just discussed-but there are additional complexities due to curvature in the magnetic field lines and gradients along these field lines. We thus now formulate expressions for the effective forces and resulting drift velocities due to these inhomogeneities in the В-fields, analogous to Eqs.(5.38) and (5.51).