Department of physics

Seminar - 4th year

Motion of a charged particle in an EM field

Author: Lojze Gacnik

Mentor: Assoc. Prof. Tomaz Gyergyek

Ljubljana, November 2011

Abstract

In this paper I will present various types of single particle motion in externally appliedelectro-magnetic fields, where they arise in nature and engineering, and discuss the applica-bility of such a model for the representation of plasmas.

Contents

1 Introduction 3

2 Uniform B field 3

2.1 E = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Constant E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Additional force F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Non-uniform B field 6

3.1 ∇B drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Curvature drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Magnetic mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Time-varying E field 12

4.1 Polarization drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Cyclotron resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Conclusion 14

References 15

2

1 Introduction

Roughly 99% of matter in the universe is in the plasma state - a gas composed of neutraland electrically charged particles. Stars, nebulae, and most of the interstellar hydrogen are allplasma. And yet we rarely encounter naturally-occuring plasma in our everyday lives - almostexclusively in the form of lightning bolts or the Aurora Borealis. This is explained by theSaha equation, which relates the amount of ionization in a gas at thermal equilibrium to itstemperature

ni

nn≈ 2.4× 1021

T 3/2

nie−Ui/KT

where ni and nn are the number densities of ionized and neutral atoms, Ui is the ionization energyof the gas, T is the temperature and K is the Boltzmann constant. For air at room temperature,this gives an ni/nn ration of 10−122, explaining why we so seldom encounter plasma [2].

To understand the behavior of a plasma it is necessary to first understand the motion ofa single charged particle in an externally imposed electro-magnetic (EM) field. While such anapproach does not explicitly touch upon the collective behavior exhibited by plasmas, it is auseful first approximation. It can reveal many faults in magnetic confinement devices, such asthose used for nuclear fusion, and tells us about the behavior of space plasmas, such as theVan-Allen radiation belts in the Earth’s magnetosphere, which is relevant to artificial satellites,since the highly energetic plasma trapped in the belts can cause damage to both electronics andorganic life.

Throughout this paper I will take E to be the electric field, B the magnetic field, v thevelocity, m will be mass, and q the electric charge. Vectors are written as boldface, such as B,and we take the same non-boldface variable to mean the absolute value, so that B = ‖B‖. Wewill also adopt the convention that A ·B = AB.

2 Uniform B field

2.1 E = 0

In this case a particle has a simple cyclotron gyration. The equation of motion is

v =q

mv×B

We can assume B = (0, 0, B), which yields

vx =qB

mvy (2.1)

vy = −qB

mvx

vz = 0

vx =qB

mvy = −

(qB

m

)2

vx

vy = −qB

mvx = −

(qB

m

)2

vy (2.2)

With the exception of describing velocity, not position, these equations are otherwise identicalto those of a harmonic oscillator, with a frequency of

ωc =qB

m

3

also known as the angular cyclotron frequency [3]. Typical cyclotron frequencies inside a tokamakare a few tens of GHz for electrons and a few tens of MHz for ions. The solution will be of theform

vx = a cos(ωct+ δx) (2.3)

vy = b cos(ωct+ δy) (2.4)

Inserting eq. 2.3 into eq. 2.1, we find that a = b = v⊥ and δx = δ = δy + π2. Setting t to

0, we divide eq. 2.4 by eq. 2.3, which yields vy0/vx0 = − tan δ, where vx0 = vx(t = 0) andvy0 = vy(t = 0). This gives us

δ = − arctanvy0vx0

vx = v⊥ cos(ωct+ δ)

vy = −v⊥ sin(ωct+ δ)

vz = vz0

Since v2x + v2y = v2⊥, it follows that v⊥ is the velocity in the plane perpendicular to B. To findthe position, we simply integrate the above equations

x = rL sin(ωct+ δ) + xc

y = rL cos(ωct+ δ) + yc

Where xc and yc are integrating constants derived from δ and position at t = 0, and

rL =v⊥ωc

is the Larmor radius. These equations produce a guiding center motion around the point(xc, yc, z), and a constant velocity parallel to B. Particles thus follow a helical trajectory. Thedirection of gyration is always such that the magnetic field produced by the particle is oppositethe external field. Plasmas are therefore diamagnetic [2].

Conversely, we can solve the above equations using vector analysis. We can write v = v⊥+v‖,where v⊥ is perpendicular, and v‖ is parallel, to B. The equation of motion is thus split in two:

v‖ =q

m(v‖ ×B) = 0

v⊥ =q

m(v⊥ ×B)

The first equation tells us that v‖ is constant. The second equation tells us that v⊥ is alwaysperpendicular to v⊥. Thus v⊥, and consequently |v⊥ × B|, are constant. This means thatacceleration is constant in magnitude and always perpendicular to B and v⊥, and thereforedescribes circular motion. [4]

2.2 Constant E

We introduce a constant external field E = (Ex, 0, Ez). The equation of motion is now

mv = q(v×B+E)

Or separately for each component

vx =q

mEx + ωcvy (2.5)

vy = −ωcvx (2.6)

vz =q

mEz

4

Figure 1: Guiding-center motion of a positive ion [3].

The solution for the z-component is simple acceleration

vz =qEz

mt+ vz0

Taking the time derivative of 2.5 and 2.6 we get

vx = ωcvy = −ω2cvx

vy = −ωcvx = −ωc

( q

mEx + ωcvy

)

= −ω2c

(Ex

B+ vy

)

The second equation is equivalent to

d2

dt2

(

vy +Ex

B

)

= −ω2c

(

vy +Ex

B

)

If we replace vy + Ex/B with vy, the above expression becomes identical to eq. 2.2. Thus thesolution is

vx = v⊥ cos(ωct+ δ)

vy = −v⊥ sin(ωct+ δ)−Ex

B

In addition to the previous motion, we now have an additional drift parallel to y. To find thedirection of this drift in general, let us recall that B = (0, 0, B) and E = (Ex, 0, Ez), meaningthat this drift is perpendicular to both B and E, meaning it is parallel to E×B = (0,−ExB, 0).If we merely divide this by B2, we obtain

vE×B =E×B

B2= (0,−

Ex

B, 0) (2.7)

which we shall call the E ×B drift [2], and is exactly the additional velocity introduced by E

in the direction perpendicular to B. Since we can transform any reference frame into one where

5

Figure 2: E×B of a positive ion, with additional acceleration parallel to B [2].

B‖z and E ⊥ y, it holds that the vector form of the above equation is always applicable (foruniform and constant B and E).

Thus the motion of a charged particle in this case is that of a slanted spiral - the particlecircles a guiding center, which is accelerated in the z direction by E, and moves at a velocity ofvE in the y direction. Note that vE does not depend on particle mass, or even charge, meaningboth positively and negatively charged particles will drift in the same direction.

2.3 Additional force F

If we introduce an additional force F of arbitrary origin, the equation of motion now reads

mv = q(v×B) + F = q(v×B+F

q)

Thus the additional drift caused by F can be obtained by replacing E with F/q in eq. 2.7, whichyields

vF =F×B

qB2(2.8)

Note that unlike vE , vF depends upon the charge q, and thus points in opposite directions forpositively and negatively charged particles, which results in a net current and subsequent chargeseparation. F can, among other things, represent a systemic force in a non-inertial referenceframe.

If it represents gravity it is called the gravitational drift [1], in which case its magnitudeis on the order of 10−11 m/s in typical fusion setups, and 10−5 m/s in the Earth’s Van-Allenradiation belts, and is in both cases negligible compared to the curvature and gradient drifts.

3 Non-uniform B field

3.1 ∇B drift

To calculate the motion in a non-uniform magnetic field, it is necessary to make some approx-imations. We shall assume that the inhomogeneity of our field is small on scales of rL, so that

6

‖B(r0 + r1)−B(r0)‖ � B(r0) for any r1 ≤ rL. We write

r = r0 + r1

v = v0 + v1

v0 = r0,v1 = r1

where r0 and v0 describe simple guiding center motion in a homogenous magnetic field, and r1and v1 are small perturbations due to inhomogeneity. Next we split B into two parts

B = B0 +B1

Where B0 = (0, 0, B0z) is the main part, and B1 is a small disturbance - the source of ourinhomogeneity. The equation of motion becomes

mv = q(v ×B)

= q((v0 + v1)× (B0 +B1))

= q(v0 ×B0) + q(v0 ×B1) + q(v1 ×B0) + q(v1 ×B1)

Since r1 � r0 and v1 � v0, the right-most term in the above equation is a second-order correctionand can be discarded. Joining the 1st and 3rd term then yields

mv ≈ q(v0 ×B0) + q(v0 ×B1) + q(v1 ×B0)

≈ q(v ×B0) + q(v0 ×B1)

Moving into a stationary reference frame where the guiding center is at ~0 at time 0, and settingB0 to be the total field at ~0 means B(~0) = B0 and B1(~0) = ~0, thus we may write

B1(r) ≈ (r∇)B

We introduced the notation B(~0) = B. This gives the equation

mv ≈ q(v ×B0) + q(v0 × (r∇)B)

≈ q(v ×B0) + F(r,v0)

Which is very similar to 2.8, except that now F is a function of position and velocity. It ispossible to further simplify the expression for F:

F = q(v0 × (r∇)B)

= q(v0 × ((r0 + r1)∇)B)

= q(v0 × ((r0∇)B) + q(v0 × ((r1∇)B)

≈ q(v0 × ((r0∇)B)

In the last step we dropped the final term, since it is of second-order. Thus F becomes a functionof only r0 and v0. To use equation 2.8, we will average F over a cyclotron gyration. First wedefine r0 and v0:

r0 = (x0, y0, z0) v0 = r0

x0 = rL sin (ωct+ ϕ)

y0 = rL cos (ωct+ ϕ)

z0 = vz0t

7

Figure 3: ∇B-drift of a positive and negative particle [2].

We used the total field at ~0, B, to calculate ωc. Next we average over a cyclotron gyration

Fpavg =1

t0

∫t0

2

−t0

2

F dt =mv⊥2B

−∂xBzv⊥ − 2∂zBzvz0 cosϕ∂yBzv⊥ − 2∂zBzvz0 sinϕ

−∂zBzv⊥ + 2∂zBxvz0 cosϕ− 2∂zByvz0 sinϕ

Where t0 = 2π/ωc, and we used ∇B = 0 to replace ∂xBx + ∂yBy with −∂zBz. This has givenus a force that is dependent upon the arbitrary angle ϕ, which is unexpected. Since a particle’sinitial angle ϕ is equally likely to have any value between 0 and 2π we again average over ϕ:

Favg =1

2π

∫ π

−πFavg1 dϕ = −

W⊥

B

∂xBz

∂yBz

∂zBz

Where W⊥ = mv2⊥/2. Immediately we find it produces a constant acceleration parallel to B.Let’s try to write this expression using general vector notation:

(∂xBz, ∂yBz, ∂zBz) = ∇Bz = ∇(Bb)

= ∇(BB

B) = ∇B

Where b is a unit vector parallel to B. Thus

Favg = −W⊥∇B

B

Inserting Favg into 2.8 gives the drift velocity

v∇B =W⊥

q

B×∇B

B3

This is called the gradient B drift [3]. We notice that it depends on q, giving opposite valuesfor positively and negatively charged particles and produces a net current and charge separationin a plasma. In addition to its dependence on W⊥, the magnitude of the gradient drift falls as1/B, meaning it is smaller in stronger magnetic fields. Typical values inside a tokamak at fusiontemperatures are on the order of 400 m/s, while electrons with energies of 0.1 MeV in the outerVan-Allen radiation belt have a drift speed on the order of 3 km/s.

3.2 Curvature drift

However, this result is incomplete. We assumed the guiding center velocity is parallel to B, eventhough the direction of B changes as the particle moves. We will deal with this by viewing the

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particle in an accelerated frame that keeps the direction of B at the particle’s position constant.This introduces a systemic centrifugal force Fcf , while keeping our current equations of motionaccurate. We can write

vdrift =Fcf ×B

qB2+

W⊥

q

B×∇B

B3

All that is left now is to express Fcf as a function of v and B. Since we are following thedirection of B with speed v‖, the particle is affected by a centrifugal force

Fcf = mv2‖

Rc

Rc

Rc

where Rc is the local curvature radius vector of B.

Figure 4: Illustration of the curvature vector Rc [2]

We express Rc with B using db/ds = −Rc/R2c , where s measures length along the field line,

so that b = B/B. Since d/ds is the derivative along the b direction, we can write

−Rc

R2c

= (b∇)b

GivingFcf = −mv2‖(b∇)b

Thus the drift equals

vcurv =2W‖

q

B× [(b∇)b]

B2(3.1)

This is known as the curvature drift [2]. Note that since Fcf‖Rc ⊥ B, the only effect of this forceis a drift, with no acceleration parallel to B. Also note that once again the drift velocity dependsupon q, and will thus drive a current and produce charge separation. Typical magnitudes for afusion plasma in a tokamak are on the order of 400 m/s, and 800 m/s for electrons in the outerVan-Allen radiation belt.

Finally we can write the total guiding center drift as

vdrift = v∇B + vcurv + vF

=W⊥

q

B×∇B

B3+

2W‖

q

B× [(b∇)b]

B2+

F×B

qB2

(3.2)

And the guiding center acceleration as

vgc = −W⊥

∇‖B

B+

1

mF‖ = v‖ (3.3)

9

Figure 5: ∇B and curvature drifts inside a tokamak. [3]

Where ∇‖B denotes the rate of change of B along field lines, that is, ∇‖B = b((b∇)B), andany force caused by E is included in F. Note that the acceleration is parallel to B.

We encounter such drifts when working with tokamaks - toroidal fusion devices. A tokamakis, in its most basic form, a simple magnetic coil shaped into a torus, creating a toroidal magneticfield. This is intended to trap charged particles without the losses that occur in a magnetic mirror(covered later). In a toroidal coil B is larger closer to the inner wall, which means both ∇Band the curvature vector Rc point towards the inner wall, creating a vertical current due to the∇B and curvature drifts, until the E field caused by charge separation balances the two drifts.However, this field gives rise to an E × B drift that drives the plasma towards the outer wall.[3]

Figure 6: Combination of poloidal and toroidal B field in a tokamak. [3]

We can avoid this by adding a poloidal B field, either by driving a current through theplasma, as is the case with tokamaks, or with additional or more complex coils, as is the casewith stellarators. This causes particles to follow a helical path, mixing those on the top andthose on the bottom, and thus undoing any charge separation that might occur [3].

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3.3 Magnetic mirror

We will show that particles can be reflected when traveling into a region with higher B. Firstwe find the magnetic moment caused by our particle, since its gyration forms a current loop

µ =mv2⊥2B

Let us assume that F = 0,E = 0, so that the only field acting upon our particle is B. Writingthe guiding center acceleration, we get

md

dtv‖ = −W⊥

∇‖B

B= −

mv2⊥2B

∇‖B

Multiplying both sides by b yields

mdv‖

dt= −µ

∂B

∂s

Where B(s) runs parallel to B. Since dsdt = v‖ we may multiply the left side by v‖ and the right

by dsdt .

mv‖dv‖

dt=

d

dt(1

2mv2‖) = −µ

∂B

∂s

ds

dt= −µ

dB

dt(3.4)

dBdt is the variation of B as seen by the particle, while B itself is not time-dependent. Since theonly field is B, the energy of the particle must be conserved

d

dt(1

2mv2‖ +

1

2mv2⊥) = −

d

dt(1

2mv2‖ + µB)

Where we used the relation 12mv2⊥ = µB. Combining this with eq. 3.4 yields

−µdB

dt+

d

dt(µB) = 0

Which finally givesdµ

dt= 0 (3.5)

Note that this only holds if B, as seen by the particle, changes slowly compared to ωc. Thiswill serve as the basis for constructing a magnetic mirror. We can arrange two magnetic coilssymetrically, so that they form a field B0 in their center, and a stronger field Bmax nearer eachcoil. Suppose a particle is in the central B0 region with v‖ = v‖0 and v⊥ = v⊥0. It will bereflected when v‖ = 0, v⊥ = v′⊥ and B = B′. The invariance of µ yields

µ0 =mv2⊥0

2B0

= µ′ =mv′2⊥2B′

And the conservation of energy yields

v′2⊥ = v2⊥0 + v2‖0 = v20

Combining the above equations gives

B0

B′=

v2⊥0

v′2⊥=

v2⊥0

v20= sin2 θ

B′ =B0

sin2 θWhere θ is the angle between B and v in the weak-field region. This means that when the

field reaches B′, a particle will be reflected, and if B′ > Bmax, a particle will not be reflected atall. This is the source of particle losses in a magnetic bottle made from two magnetic mirrors[2].

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Figure 7: A simple magnetic mirror trap [1].

Figure 8: A banana orbit between two points on the high-field side of a tokamak [3].

While not obvious at first glance, magnetic mirrors can appear in tokamaks as well. Sincethe magnetic field lines, augmented by the poloidal field, now follow a helical trajectory in- andout-of the inner, high-field side, a particle may bounce when it encounters the stronger B field.The path such a particle follows is called a banana orbit [3].

The Van-Allen radiation belts, containing charged particles trapped in the Earth’s magne-tosphere, are also a consequence of the mirror effect. As particles follow the field lines from theequatorial region towards the polar regions, they encounter a stronger magnetic field. Thosethat are reflected form the radiation belts, while those that are not produce the Aurora Borealiswhen they encounter the atmosphere [2].

4 Time-varying E field

4.1 Polarization drift

We take E and B to be spatially uniform, where B is constant, while E varies in time with afrequency much slower than ωc, and observe the effects such a setup gives. We recall the formula

12

Figure 9: Charged particle motion in the magnetosphere [2].

for E×B drift

vE×B =E×B

B2

Since E now varies with time, so does vE×B

vE×B =d

dt

E×B

B2

In the guiding center frame this is equivalent to a force F = −mvE×B, which produces yetanother drift

vP =F×B

qB2= −

m

qB4

d

dt(E×B)×B

= −m

qB4

d

dt((EB)B−B2E)

=m

qB2

d

dt(E− (Eb)b) =

m

qB2E⊥

This is called the polarization drift, and is dependent upon charge, and thus in opposite directionsfor positively and negatively charged particles, which produces a polarization current [4].

4.2 Cyclotron resonance

We shall now look at the effects of an E field that varies with the cyclotron frequency. We setB = (0, 0, B) and E = (E cos(ωct), 0, 0). The equations of motion become

vx = ωcvy + qE cos(ωct)

vy = −ωcvx (4.1)

vz = 0

vx = ωcvy − ωcqE sin(ωct) = −ω2cvx − ωcqE sin(ωct) (4.2)

vy = −ωcvx = −ω2cvy − ωcqE cos(ωct) (4.3)

Lets guess that the solution will be of the form

vx = At cos(ωct) (4.4)

vy = At cos(ωct+ δ) (4.5)

Inserting 4.4 and 4.5 into 4.1 gives δ = π/2, while inserting 4.4 into 4.2 yields

−2Aωc sin(ωct)−Atω2c cos(ωct) = −Atω2

c cos(ωct)− Eωcq sin(ωct)

13

From which we can derive A to be

A =1

2Eq

Thus the solution is

vx =1

2Eqt cos(ωct)

vy = −1

2Eqt sin(ωct)

We can express v⊥ and W⊥ as

v⊥ =√

v2x + v2y =1

2Eqt

W⊥ =1

2mv2⊥ =

1

8mE2q2t2

Figure 10: Spiral trajectory of a particle undergoing cyclotron heating.

We see that W⊥ increases as t2! This is the basis for cyclotron resonance heating - plasmais irradiated with radio waves of frequency ωc, thus heating it. Since ions and electrons havedifferent ωc, we can choose which of them to heat directly. Furthermore, since B decreases withdistance from the inner wall of a tokamak, so does ωc, meaning we can control where the energyof cyclotron heating is deposited by choosing an appropriate frequency. [3]

5 Conclusion

We have shown that charged particles follow complex paths in an EM-field, and some of theeffects that make containing charged particles difficult. However, missing from our analysisare multi-particle effects, such as collisions with other particles, and collective behavior - theinteraction of a plasma with its own EM-fields, which further complicates the containment ofa hot plasma. Nonetheless such simple theory can often produce workable results, especially ifthe particle density is low enough to be able to neglect collision effects, and the EM-field causedby the plasma is small compared to the external field, as is the case in some space plasmas.

14

References

[1] R. J. Goldston, P. H. Rutherford, Introduction to Plasma Physics. Institute of Physics Pub-lishing, Bristol and Philadelphia, 1995.

[2] F. Chen, Introduction to Plasma Physics and Controlled Fusion. 1974.

[3] A. A. Haarms, K. F. Schoepf, G. H. Miley, D. R. Kingdon, Principles of Fusion Energy:

An Introduction to Fusion Energy for Students of Science and Engineering. World ScientificPublishing, Singapore, 2000.

[4] J. A. Bittencourt Fundamentals of Plasma Physics. Springer-Verlag, New York, 2004

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