of 30 /30
Past exam papers (c) Assume that the ion oscillations are so slow that the electrons remain in a Maxwell- Boltzmann distribution. If eφ/k B T e 1, show that the perturbed charge density of the electrons is given by (n 0 e 2 /k B T e )φ. (d) Use Poisson’s equation to deduce the following dispersion relation: k 2 =(n 0 e 2 /m i ε 0 ω 2 )k 2 n 0 e 2 /k B T e ε 0 (e) Recast the dispersion relation in the following form: ω 2 = ω 2 pi /(1 + 1/k 2 λ 2 D ). Discuss the low and high-k limits and compare with the Bohm-Gross dispersion relation for electron plasma waves. Question 3 (10 marks) (a) Using the steady-state force balance equation (ignore the convective derivative) show that the particle ﬂux Γ = nu for electrons and singly charged ions in a fully ionized unmagne- tized plasma is given by: Γ j = nu j = ±µ j nE D j n with mobility µ =| q | /mν where ν is the electron-ion collision frequency and diﬀusion coeﬃcient D = k B T /mν . (b) Show that the diﬀusion coeﬃcient can be expressed as D λ 2 mfp where λ mfp is the mean free path between collisions and τ is the collision time. (c) Show that the plasma resistivity is given approximately by η = m e ν/ne 2 . (d) In the presence of a magnetic ﬁeld, the mean perpedicular velocity of particles across the ﬁeld is given by u = ±µ E D n n + u E + u D 1+ ν 2 2 c with u E = E×B/B 2 , u D = −∇p×B/qnB 2 and where µ = µ/(1 + ω 2 c τ 2 ) and D = D/(1 + ω 2 c τ 2 ). Discuss the scaling with ν of each of the four terms in the expression for u . Question 4 (10 marks) (a) Show that the drift speed of a charge q in a toroidal magnetic ﬁeld can be written as v T =2k B T/qBR where R is the radius of curvature of the ﬁeld. (Hint: Consider both gradient and curvature drifts) (b) Compute the value of v T for a plasma at a temperature of 10 keV, a magnetic ﬁeld strength of 2 T and a major radius R = 1 m.
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### Transcript of Past exam papers BTe BT -... Past exam papers(c) Assume that the ion oscillations are so slow that the electrons remain in a Maxwell-Boltzmann distribution. If eφ/kBTe � 1, show that the perturbed charge density of theelectrons is given by −(n0e

2/kBTe)φ.

(d) Use Poisson’s equation to deduce the following dispersion relation:

k2 = (n0e2/miε0ω

2)k2 − n0e2/kBTeε0

(e) Recast the dispersion relation in the following form:

ω2 = ω2pi/(1 + 1/k2λ2

D).

Discuss the low and high-k limits and compare with the Bohm-Gross dispersion relationfor electron plasma waves.

Question 3 (10 marks)

(a) Using the steady-state force balance equation (ignore the convective derivative) show thatthe particle flux Γ = nu for electrons and singly charged ions in a fully ionized unmagne-tized plasma is given by:

Γj = nuj = ±µjnE − Dj∇n

with mobility µ =| q | /mν where ν is the electron-ion collision frequency and diffusioncoefficient D = kBT/mν.

(b) Show that the diffusion coefficient can be expressed as D ∼ λ2mfp/τ where λmfp is the mean

free path between collisions and τ is the collision time.

(c) Show that the plasma resistivity is given approximately by η = meν/ne2.

(d) In the presence of a magnetic field, the mean perpedicular velocity of particles across thefield is given by

u⊥ = ±µ⊥E − D⊥∇n

n+

uE + uD

1 + ν2/ω2c

with uE = E×B/B2, uD = −∇p×B/qnB2 and where µ⊥ = µ/(1 + ω2cτ

2) and D⊥ =D/(1 + ω2

cτ2). Discuss the scaling with ν of each of the four terms in the expression for

u⊥.

Question 4 (10 marks)

(a) Show that the drift speed of a charge q in a toroidal magnetic field can be written as

vT = 2kBT/qBR

where R is the radius of curvature of the field. (Hint: Consider both gradient and curvaturedrifts)

(b) Compute the value of vT for a plasma at a temperature of 10 keV, a magnetic field strengthof 2 T and a major radius R = 1 m. (c) Compute the time required by a charge to drift across a toroidal container of minor radius1 m.

(d) Suppose an electric field is applied perpendicular to the plane of the torus. Describe whathappens.

Question 5 (10 marks)

(a) Show that the MHD force balance equation ∇p = j×B requires both j and B to lie onsurfaces of constant pressure.

(b) Using Ampere’s law and MHD force balance, show that

∇(p +

B2

2µ0

)=

1

µ0

(B.∇)B

and discuss the meaning of the various terms.

(c) A straight current carrying plasma cylinder (linear pinch) is subject to a range of instabilities(sausage, kink etc.). These can be suppressed by providing an axial magnetic field Bz thatstiffens the plasma through the additional magnetic pressure B2

z/2µ0 and tension againstbending. Consider a local constriction dr in the radius r of the plasma column. Assumingthat the longitudinal magnetic flux Φ through the cross-section of the cylinder remainsconstant during the compression (dΦ = 0), show that the axial magnetic field strength isincreased by an amount dBz = −2Bzdr/r.

(d) Show that the internal magnetic pressure increases by an amount dpz = BzdBz/µ0 =−(2B2

z/µ0)dr/r. [The last step uses the result obtained in (c)].

(e) By Ampere’s law we have for the azimuthal field component rBθ(r) = constant. Show thatthe change in azimuthal field strength due the compression dr is dBθ = −Bθdr/r and thatthe associated increase in external azimuthal magnetic pressure is dpθ = −(B2

θ/µ0)dr/r.

(f) Show that the plasma column is stable against sausage distortion provided B2z > B2

θ/2.

Question 6 (10 marks)

(a) Plot the wave phase velocity as a function of frequency for plasma waves propagatingperpendicular to the magnetic field B, identifying cutoffs and resonances for both ordinaryand extraordinary modes.

(b) Using the matrix form of the wave dispersion relation

S − n2z −iD nxnz

iD S − n2x − n2

z 0nxnz 0 P − n2

x

Ex

Ey

Ez

= 0

show that the polarization state for the extraordinary wave is given by

Ex/Ey = iD/S.

Using a diagram, show the relative orientations of B, k and E for this wave. Past exam papersTHE AUSTRALIAN NATIONAL UNIVERSITY

First Semester Examination 2000

PHYSICS C17

PLASMA PHYSICS

Writing period 2 hours durationStudy period 15 minutes durationPermitted materials: Calculators

Attempt four questions. All are of equal value.Show all working and state and justify relevant assumptions.

Question 1 (10 marks)Attempt three of the following. Answers for each should require at most half a page.

(a) Discuss the relationship between the Boltzmann equation, the electron and ion equations ofmotion and the single fluid force balance euqation.

(b) Describe electric breakdown with reference to the parameter E/p and the role of secondaryemission.

(c) Discuss the physical meaning of the Boltzmann relation. Use diagrams to aid your expla-nation.

(d) Discuss the origin of plasma diamagnetism and its implications for magnetic plasma con-finement.

(e) Draw a Langmuir probe I-V characteristic indicating the saturation currents, plasma poten-tial and floating potential. How can the characteristic be used to estimate temperature?

(f) Describe Debye shielding and the relationship between the plasma frequency and Debyelength. Question 2 (10 marks)

(a) Consider two infinite, perfectly conducting plates A1 and A2 occupying the planes y = 0and y = d respectively. An electron enters the space between the plates through a smallhole in plate A1 with initial velocity v towards plate A2. A potential difference V betweenthe plates is such as to decelerate the electron. What is the minimum potential differenceto prevent the electron from reaching plate A2.

(b) Suppose the region between the plates is permeated by a uniform magnetic field B parallelto the plate surfaces (imagine it as pointing into the page). A proton appears at the surfaceof plate A1 with zero initial velocity. As before, the potential V between the plates is suchas to accelerate the proton towards plate A2. What is the minimum value of the magneticfield B necessary to prevent the proton from reaching plate B? Sketch what you think theproton trajectory might look like. (HINT: Energy considerations may be useful).

Question 3 (10 marks)

(a) Using the equilibrium force balance equation for electrons (assume ions are relatively im-mobile) show that the conductivity of an unmagnetized plasma is given by

σ0 =ne2

meν(10.4)

(b) What is the dependence of the conductivity on electron temperature and density in the fullyionized case?

(c) When the plasma is magnetized, the Ohm’s law for a given plasma species (electrons orions) becomes j = σ0(E + u×B) where j = nqu is the current density. Show that thefamiliar E×B drift is recovered when the collision frequency becomes very small.

(d) If E is at an angle to B, there will be current flow components both parallel and perpen-dicular to B, If ui is different from ue, there is also a nett Hall current j⊥ = en(ui⊥−ue⊥)that flows in the direction E×B. To conveniently describe all these currents, the Ohm’slaw can equivalently be expressed by the tensor relation j =

↔σ E with conductivity tensor

given by

↔σ=

σ⊥ −σH 0σH σ⊥ 00 0 σ‖

(10.5)

where

σ⊥ = σ0ν2

ν2 + ω2c

σH = σ0∓νωc

ν2 + ω2c

σ‖ = σ0 =ne2

Explain the collision frequency dependence of the perpendicular and Hall conductivities. Past exam papersQuestion 4 (10 marks) Answer either part (a) or part (b)

(a) Consider the two sets of long and straight current carrying conductors shown in confurationsA and B of Figure 1.

(i) Sketch the magnetic field line configuration for each case.

(ii) Describe the particle guiding centre drifts in each case, with particular emphasis on theconservation of the first adiabatic moment.

(iii) Charge separation will occur due to the magnetic field inhomogeneity. This in turn estab-lishes an electric field. Comment on the confining properties (or otherwise) of this electricfield.

Configuration A Configuration B

Figure 10.1: Conductors marked with a cross carry current into the page (z direction), while thedots indicate current out of the page.

(b) In a small experimental plasma device, a toroidal B-field is produced by uniformly winding120 turns of conductor around a toroidal vacuum vessel and passing a current of 250A throughit. The major radius of the torus is 0.6m.A plasma is produced in hydrogen by a radiofrequency heating field. The electrons and ionshave Maxwellian velocity distribution functions at temperatures 80eV and 10eV respectively.The plasma density at the centre of the vessel is 1016 m−3.

(i) Use Ampere’s law around a toroidal loop linking the winding to calculate the vacuum fieldon the axis of the torus.

(ii) What is the field on axis in the presence of the plasma?

(iii) Calculate the total drift for both ions and electrons at the centre of the vessel and showthe drifts on a sketch.

(iv) Explain how these drifts are compensated when a toroidal current is induced to flow. (v) The toroidal current produces a poloidal field. The combined fields result in helical magneticfield lines that encircle the torus axis. For particles not on the torus axis and which have ahigh parallel to perpendicular velocity ratio the projected guiding centre motion executesa rotation in the poloidal plane (a vertical cross-section of the torus) as it moves helicallyalong a field line. What happens to particles that have a high perpendicular to parallelvelocity ratio?

Question 5 (10 marks) There is a standard way to check the relative importance of terms inthe single fluid MHD equations. For space derivatives we choose a scale length L such thatwe can write ∂u/∂x ∼ u/L. Similarly we choose a time scale τ such that ∂u/∂t ∼ u/τ . So∇×E = ∂B/∂t becomes E/L ∼ B/τ . Introduce velocity V = L/τ so that E ∼ BV .

(a) Examine the single fluid momentum equation.

ρ∂u

∂t= j×B −∇p (10.6)

Show that the terms are in the ratio

nmiV

τ: jB :

nmev2the

Lor 1 :

jBτ

nmiV:

me

mi

v2the

V 2(10.7)

When the plasma is cold, show that this suggests V ∼ jBτ/nmi

(b) Examine the generalized Ohm’s law:

me

ne2

∂j

∂t= E + u×B − 1

nej×B +

1

ne∇pe − ηj (10.8)

Show that the terms are in the ratio

1

ωceωciτ 2: 1 : 1 :

1

ωciτ:

1

ωceτ

v2the

V 2:

νei

ωceωciτ(10.9)

(c) Which terms of the Ohm’s law can be neglected if

(i) τ � 1/ωci

(ii) τ ≈ 1/ωci

(iii) τ ≈ 1/ωce

(iv) τ � 1/ωce

When can the resistive term ηj be dropped?

Question 6 (10 marks)Electromagnetic wave propagation in an unmagnetized plasma. Consider an electromagnetic

wave propagating in an unbounded, unmagnetized uniform plasma of equilibrium density n0. Weassume the bulk plasma velocity is zero (v0 = 0) but allow small drifts v1 to be induced by theone-dimensional harmonic electric field perturbation E = E1 exp [i(kx − ωt)] that is transverseto the wave propagation direction. Past exam papers(a) Assuming the plasma is also cold (∇p = 0) and collisionless, show that the momentumequations for electrons and ions give

n0mi(−iωvi1) = n0eE1

n0me(−iωve1) = −n0eE1

(b) The ion motions are small and can be neglected (why?). Show that the resulting currentdensity flowing in the plasma due to the imposed oscillating wave electric field is given by

j1 = en0(vi1 − ve1) ≈ in0e

2

meωE1. (10.10)

(c) Associated with the fluctuating current is a small magnetic field oscillation which is givenby Ampere’s law. Use the differential forms of Faraday’s law and Ampere’s law (Maxwell’sequations) to obtain the first order equations kE1 = ωB1 and ikB1 = µ0j1 − iωµ0ε0E1

(d) Use these relations to eliminate B1 and j1 to obtain the dispersion relation for plane elec-tromagnetic waves propagating in a plasma:

k2 =ω2 − ω2

pe

c2(10.11)

(d) Sketch the dispersion relation and comment on the physical significance of the dispersionnear the region ω = ωpe. THE AUSTRALIAN NATIONAL UNIVERSITY

First Semester Examination 2001

PHYSICS C17

PLASMA PHYSICS

Writing period 2 hours durationStudy period 15 minutes durationPermitted materials: Calculators

Attempt four questions. All are of equal value.Show all working and state and justify relevant assumptions.

Question 1Attempt three of the following. Answers for each should require at most half a page.

(a) Discuss the relationship between moments of the particle distribution function f and mo-ments of the Boltzmann equation. Plot f(v) for a one dimensional drifting Maxwelliandistribution, indicating pictorially the meaning of the three lowest order velocity moments.

(b) Describe electric breakdown with reference to the parameter E/p and the role of secondaryemission.

(c) Discuss the physical meaning of the Boltzmann relation. Use diagrams to aid your expla-nation.

(d) Discuss the origin of plasma diamagnetism and its implications for magnetic plasma con-finement.

(e) Elaborate the role of Coulomb collisions for diffusion in a magnetized plasma.

(f) Discuss magnetic mirrors with reference to the adiabatic invariance of the orbital magneticmoment µ.

(g) Describe Debye shielding and the relationship between the plasma frequency and Debyelength.

Question 2Consider an axisymmetric cylindrical plasma with E = Er, B = Bz and ∇pi = ∇pe =

r∂p/∂r. If we negelct (v.∇)v, the steady state two-fluid momentum-balance equations can bewritten in the form

en(E + ui×B) −∇pi − e2n2η(ui − ue) = 0

−en(E + ue×B) −∇pe + e2n2η(ui − ue) = 0

(a) From the θ components of these equations, show that uir = uer. Past exam papers(b) From the r components, show that ujθ = uE + uDj (j = i, e).

(c) Find an expression for uir showing that it does not depend on E.

Question 3The induced emf at the terminals of a wire loop that encircles a plasma measures the rate of

change of magnetic flux expelled by the plasma. You are given the following parameters:Vacuum magnetic field strength - 1 TeslaNumber of turns on the diamagnetic loop - N = 75Radius of the loop - aL = 0.075mPlasma radius - a = .05m.Given the observed diamagnetic flux loop signal shown below, calculate the plasma pressure asa function of time. If the temperature of the plasma is constant at 1 keV, what is the plasmadensity as a function of time? (HINT: use Faraday’s law relating the emf to the time derivativeof the magnetic flux)

2 4 6 8 10

12 14 16

Time (µs)

1.0

-1.0

Volts

Figure 10.2: Magnetic flux loop signal as a function of time. Question 4An infinite straight wire carries a constant current I in the +z direction. At t = 0 an electron

of small gyroradius is at z = 0 and r = r0 with v⊥0 = v‖0 (⊥ and ‖ refer to the direction relativeto the magnetic field.)

(a) Calculate the magnitude and direction of the resulting guiding centre drift velocity.

(b) Suppose the current increases slowly in time in such a way that a constant electric field isinduced in the ±z direction. Indicate on a diagram the relative directions of I, E, B andvE.

(c) Do v⊥ and v‖ increase, decrease or remain the same as the current increases? Explain youranswer.

Question 5Magnetic pumping is a means of heating plasmas that is based on the constancy of the

magnetic moment µ. Consider a magnetized plasma for which the magnetic field strength ismodulated in time according to

B = B0(1 + ε cos ωt) (10.12)

where ω � ωc and ε � 1. If U⊥ = mv2⊥/2 = (mv2

x + mv2y)/2 is the particle perpendicular kinetic

energy (electrons or ions) show that the kinetic energy is also modulated as

dU⊥dt

=U⊥B

dB

dt.

We now allow for a collisional relaxation between the perpendicular (U⊥) and parallel (U‖) kineticenergies modelled according to the coupled equations

dU⊥dt

=U⊥B

dB

dt− ν

(U⊥2

− U‖)

dU‖dt

= ν(

U⊥2

− U‖)

where ν is the collision frequency. By suitably combining these equations, one can calculate theincrement ∆U in total kinetic energy during a period ∆t = 2π/ω to obtain a nett heating rate

∆U

∆t=

ε2

6

ω2ν

9ν2/4 + ω2U ≡ αU. (10.13)

This heating scheme is called collisional magnetic pumping. Comment on the physical reasonsfor the ν-dependence of α in the cases ω � ν and ω � ν.

Assuming that the plasma is fully ionized (Coulomb collisions), and in the case ω � ν, showthat the heating rate ∆U/∆t decreases as the temperature increases. What would happen if themagnetic field were oscillating at frequency ω = ωc?

Question 6On a graph of wave frequency ω versus wavenumber k show the dispersion relations for the

ion and electron acoustic waves, and a transverse electromagnetic wave (ω > ωpe) propagatingin an unmagnetized plasma. (HINT: Draw the ion and electron plasma frequencies and linescorresponding to the electron sound speed, the ion sound speed and the speed of light.) Past exam papersConsider the case of electron plasma oscillations in a uniform plasma of density n0 in thepresence of a uniform steady magnetic field B0 = B0k. We take the background electric field tobe zero (E0 = 0) and assume the plasma is at rest u = 0. We shall consider longitudinal electronoscillations having k ‖ E1 where we take the oscillating electric field perturbation associated withthe electron wave E1 ≡ Ei to be parallel to the x-axis.

Replacing time derivatives by −iω and spatial gradients by ik, and ignoring pressure gradientsand the convective term (u.∇)u, show that for small amplitude perturbations, the electronmotion is governed by the linearized mass and momentum conservation equations and Maxwell’sequation:

−iωn1 + n0ikux = 0 (10.14)

−iωu = −e(E + u×B0) (10.15)

ε0ikE = −en1. (10.16)

Use Eq. (10.15) to show that the x component of the electron motion is given by

ux =eE/iωm

1 − ω2c/ω

2(10.17)

Substituting for ux from the continuity equation and eliminating the density perturbation us-ing Eq. (10.16), obtain the dispersion relation for the longitudinal electron plasma oscillationtransverse to B:

ω2 = ω2p + ω2

c . (10.18)

Why is the oscillation frequency greater than ωp? By expressing the ratio ux/uy in terms ofω and ωc show that the electron trajectory is an ellipse elongated in the x direction. THE AUSTRALIAN NATIONAL UNIVERSITY

First Semester Examination 2002

PHYSICS C17

PLASMA PHYSICS

Writing period 2 hours durationStudy period 15 minutes durationPermitted materials: Calculators

Attempt four questions. All are of equal value.Show all working and state and justify relevant assumptions.

Question 1

Discuss, using diagrams where appropriate, three of the following issues. Answers for eachshould require at most half a page.

(a) Ambipolar diffusion in unmagnetized plasma.

(b) Electric breakdown. Why is E/p important and what is the role of secondary emission?

(c) The physics underlying the Boltzmann relation.

(d) MHD waves that propagate perpendicular and parallel to B.

(e) The resistivity of weakly and fully ionized unmagnetized plasmas.

(f) Magnetic mirrors and the role of the invariance of the orbital magnetic moment µ.

(g) Debye shielding and the relationship between the plasma frequency and Debye length.

Question 2

(a) Explain using a diagram why the orbit of a particle gyrating in a magnetic field is diamag-netic.

(b) Given that the magnetic moment of a gyrating particle is µ=W⊥/B where W⊥ is the kineticenergy of the motion perpendicular to the magnetic field of strength B, find an expressionfor the magnetic moment per unit volume M in a plasma with particle density n andtemperature T immersed in a uniform magnetic field.

(c) Supposing the field inside the plasma to be reduced compared with that outside the plasmaB by µ0M � B, calculate the difference in magnetic pressure B2/2µ0 inside the plasmaand confirm that the total pressure is constant. Past exam papers(d) Describe in general terms what happens to maintain pressure balance as the plasma tem-perature is increased, keeping the total number of particles constant.

Question 3

(a) With the aid of diagrams, explain why magnetic plasma confinement is not possible in apurely toroidal magnetic field.

(b) The earth’s magnetic field may be approximated as a magnetic dipole out to a few earthradii (RE = 6370km). The magnetic field for a dipole can be written approximately as

Br =µ0

2M cos θ

r3(10.19)

Bθ =µ0

M sin θ

r3(10.20)

where θ is the polar angle from the direction of the dipole moment vector, Br is the magneticfield radial component and Bθ is the component orthogonal to Br. Using the fact that, atone of the magnetic poles (r = RE) , the field has a magnitude of 0.5 Gauss, calculate theearth’s dipole moment M .

(c) Assuming an electron is constrained to move in the earth’s magnetic equatorial plane (v‖ =0), calculate the guiding centre drift velocity, and determine the time it takes to drift oncearound the earth at a radial distance r0. What is the direction of drift.

(d) Let there be an isotropic population of 1 eV protons and 30 keV electrons each with densityn = 107 m−3 at r = 5RE in the equatorial plane. Compute the ring current density inA/m2 associated with the drift obtained in (c).

(e) Now assume that the perpendicular kinetic energy equals the parallel kinetic energy at themagnetic equatorial plane. Qualitatively describe the motion of the electron guiding centre.

Question 4

(a) Using the steady-state force balance equation (ignore the convective derivative) show thatthe particle flux Γ = nu for electrons and singly charged ions in an unmagnetized plasmais given by:

Γj = nuj = ±µjnE − Dj∇n

with mobility µ =| q | /mν where ν is the collision frequency and diffusion coefficientD = kBT/mν.

(b) Show that the diffusion coefficient can be expressed as D ∼ λ2mfp/τ where λmfp is the mean

free path between collisions and τ is the collision time.

(c) Show that the plasma resistivity is given approximately by η = meν/ne2. (d) In a weakly ionized magnetoplasma, the mean perpendicular velocity of particles across thefield is given by

u⊥ = ±µ⊥E − D⊥∇n

n+

uE + uD

1 + ν2/ω2c

with uE = E×B/B2, uD = −∇p×B/qnB2 and where µ⊥ = µ/(1 + ω2cτ

2) and D⊥ =D/(1+ω2

cτ2). Discuss the physical origin of each of these terms and their behaviour in the

limit of weak and strong magnetic fields.

Question 5 (10 marks)

The dispersion relation for low frequency magnetohydrodynamic waves in a magnetizedplasma was derived in lectures as

−ω2u1 + (V 2S + V 2

A)(k.u1)k + (k.V A)[(k.V A)u1 − (V A.u1)k − (k.u1)V A] = 0

where u1 is the perturbed fluid velocity, k is the propagataion wavevector and V A = B0/(µ0ρ0)1/2

is a velocity vector in the direction of the magnetic field with magnitude equal to the Alfvenspeed and VS is the sound speed.

(a) Deduce the dispersion relations for waves propagating parallel to the magnetic field andidentify the wave modes.

(b) Using

∂B1

∂t−∇×(u1×B0) = 0

E1 + u1×B = 0

and assuming plane wave propagation so that ∂∂t

→ −iω and ∇× → ik×, make a sketchshowing the relation between the perturbed quantities u1, E1, B1 and k and B0 for thetransverse wave propagating along B0.

Question 6 (10 marks)

Consider an electromagnetic wave propagating in an unbounded, em unmagnetized uniformplasma of equilibrium density n0. We assume the bulk plasma velocity is zero (v0 = 0) butallow small drifts v1 to be induced by the one-dimensional harmonic electric field perturbationE = E1 exp [i(kx − ωt)] that is transverse to the wave propagation direction.

(a) Assuming the plasma is also cold (∇p = 0) and collisionless, show that the momentumequations for electrons and ions give

n0mi(−iωvi1) = n0eE1

n0me(−iωve1) = −n0eE1

(b) The ion motions are small and can be neglected. Show that the resulting current densityflowing in the plasma due to the imposed oscillating wave electric field is given by

j1 = en0(vi1 − ve1) ≈ in0e

2

meωE1. (10.21) Past exam papers(c) Associated with the fluctuating current is a small magnetic field oscillation which is givenby Ampere’s law. Use the differential forms of Faraday’s law and Ampere’s law (Maxwell’sequations) to obtain the first order equations kE1 = ωB1 and ikB1 = µ0j1 − iωµ0ε0E1

(d) Use these relations to eliminate B1 and j1 to obtain the dispersion relation for plane elec-tromagnetic waves propagating in a plasma:

k2 =ω2 − ω2

pe

c2(10.22)

(d) Sketch the dispersion relation and comment on the physical significance of the dispersionnear the region ω = ωpe. APPENDIX: A Glossary of Useful Formulae

Chapter 1: Basic plasma phenomena

ωpe =

√e2ne

ε0me

fpe � 9√

ne( Hz)

λD =

√ε0kBTe

nee2

ni

n� 2.4 ××1021 T

32

e

ni

exp−Ui

kBT

Chapter 2: Kinetic theory

∂f

∂t+v.∇r.f +

q

m(E+v×B).∇v.f =

(∂f

∂t

)coll

Γ = nv

j = qnv

p =2

3nUr

fM(v) = A exp(−mv2

2kBT) = A exp (−v2/v2

th)

Ur(Maxwellian) ≡ EAv =1

2kBT (1 − D)

1 eV � 11, 600 K

vrms =

√3kBT

m

vth =

√2kBT

m

pj = njkBTj

ne = ne0 exp

(eφ

kBTe

)

λmfp =1

τ =λmfp

vν = nσv

b0 =2qq0

4πε0mv2

ln Λ = ln〈λD

b0

σeicoulomb � Z2e4 ln Λ

2πε20m

2ev

4e

δEei ∼ 4Eeme

mi

Pei = −mene(ue − ui)

τei

Chapter 2: Fluid and Maxwell’s equations

σ = niqi + neqe

j = niqiui + neqeue

∂nj

∂t+ ∇.(njuj) = 0

mjnj

[∂uj

∂t+ (uj.∇)uj

]= qjnj(E + uj×B) −∇pj + Pcoll

pj = Cjnγj

j

∇.E =σ

ε0

∇×E = −∂B

∂t Past exam papers ∇.B = 0

∇×B = µ0j + µ0ε0∂E

∂t

Chapter 3: Gaseous Electronics

Γj = nuj = ±µjnE − Dj∇n

µ =|q |mν

D =kBT

mνE = ηj

η =νeime

nee2

η � Ze2√me ln Λ

6√

3πε20(kBTe)3/2

η‖ =5.2 × 10−5Z ln Λ

T32

e(eV)

I =I0e

αx

(1 − γeαx)

J =4

9

√2e

mi

ε0|φw|3/2

d2

uBohm =

√kBTe

mi

eφw

kBTe

≈ 1

2ln

(2πme

mi

)

Isi � 1

2n0eA

√kBTe

mi

Chapter 4: Single Particle Motions

F = mdv

dt= q(E + v×B)

ωc ≡ |q|Bm

rL =v⊥ωc

µ =mv2

⊥2B

vE =E×B

B2

vF =1

q

F×B

B2

vR =mv2

‖qB2

Rc×B

R2

v∇B = ±1

2v⊥rL

B×∇B

B2

vP =1

ωc

E

B

j =↔σ E

F‖ = −µ∇‖B

v‖ =[

2

m(K − µB)

]1/2

Bm

B0

=1

sin2 θm

q(r) =dφ

dθ=

rB0

RBθ

= εB0

↔σe=

ine2

meω

ω2

ω2 − ω2ce

−iωceω

ω2 − ω2ce

0

iωceω

ω2 − ω2ce

ω2

ω2 − ω2ce

0

0 0 1

↔ε= ε0

(↔I +

i

ε0ω

↔σ

)

Chapter 5: Magnetized Plasmas

u⊥ =E×B

B2+

−∇p×B

qnB2jD = (kBTi + kBTe)

B×∇n

B2 u⊥ = ±µ⊥E − D⊥∇n

n+

uE + uD

1 + ν2/ω2c

µ⊥ =µ

1 + ω2cτ

2

D⊥ =D

1 + ω2cτ

2

↔σ=

σ⊥ −σH 0σH σ⊥ 00 0 σ‖

σ⊥ = σ0ν2

ν2 + ω2c

σH = σ0∓νωc

ν2 + ω2c

σ‖ = σ0 =ne2

D⊥ =η⊥

∑nskBTs

B2

Chapter 5: Single Fluid Equations

ρ∂u

∂t= j×B −∇p + ρg

E + u×B = ηj∂ρ

∂t+ ∇.(ρu) = 0

∂σ

∂t+ ∇.j = 0

∇×E = −∂B

∂t∇×B = µ0j

p = Cnγ

Chapter 6: Magnetohydrodynamics

∇(p +

B2

2µ0

)=

1

µ0

(B.∇)B

∂B

∂t=

η

µ0

∇2B + ∇×(u×B)

RM =µ0vL

η

Chapter 7, 8, 9: Waves

vg =dω

dk

VA =

(B2

µ0ρ

)/2

VS =

(γekBTe + γikBTi

mi

)1/2

vφ =ω

k=

c

(1 − ω2pe/ω

2)1/2

ω2 = ω2pe +

3kBT

mk2

n×(n×E)+↔K .E = 0

n =c

ωk

n =|n |= ck/ω = c/vφ

↔K=

↔ε /ε0 =

S −iD 0iD S 00 0 P Past exam papersS = 1 − ∑

i,e

ω2p

ω2 − ω2c

D =∑i,e

± ω2pωc

ω(ω2 − ω2c )

P = 1 − ∑i,e

ω2p

ω2

R = S + D Right

L = S − D Left

S = (R + L)/2 Sum

D = (R − L)/2 Diff

P Plasma

tan2 θ =P (n2 − L)(n2 − R)

(n2 − P )(RL − n2S)

(c2

v2φ

)LR

= 1 − ω2pe

ω(ω ± ωce)− ω2

pi

ω(ω ∓ ωci)

ω0L =[−ωce + (ω2

ce + 4ω2pe)

1/2]/2

ω0R =[ωce + (ω2

ce + 4ω2pe)

1/2]/2

n2 =c2

v2φ

=(ω2 − ω2

0L)(ω2 − ω20R)

ω2(ω2 − ω2UH)

ωUH = (ω2pe + ω2

ce)1/2

ωLH ≈ (ωciωce)1/2

n2 =c2

v2φ

=(ω2 − ω2

0L)(ω2 − ω20R)

ω2(ω2 − ω2UH)

Useful Mathematical Identities

A.(B×C) = B.(C×A) = C.(A×B)

(A×B)×C = B(C.A) − A(C.B)

∇.(φA) = A.∇φ + φ∇.A

∇×(φA) = ∇φ×A + φ∇×A

A×(∇×B) = ∇(A.B) − (A.∇)B

− (B.∇)A − B×(∇×A)

(A.∇)A = ∇(1

2A2) − A×(∇×A)

∇.(A×B) = B.(∇×A) − A.(∇×B)

∇×(A×B) = A(∇.B) − B(∇.A)

+ (B.∇)A − (A.∇)B

∇×(∇×A) = ∇(∇.A) − (∇.∇)A

∇×∇φ = 0

∇.(∇×A) = 0∫ ∞

−∞v2 exp (−av2)dv =

1

2

√π

a3

Cylindrical coordinates

∇φ =∂φ

∂rr +

1

r

∂φ

∂θθ +

∂φ

∂zz

∇2φ =1

r

∂r

(r∂φ

∂r

)+

1

r2

∂2φ

∂θ2+

∂2φ

∂z2

∇.A =1

r

∂r(rAr) +

1

r

∂Aθ

∂θ+

∂Az

∂z

∇×A =

(1

r

∂Az

∂θ− ∂Aθ

∂z

)r +

(∂Ar

∂z− ∂Az

∂r

)

+

[1

r

∂r(rAθ) − 1

r

∂Ar

∂θ

]z  Bibliography

 R. D. Hazeltine and F. L. Waelbroeck, The Framework of Plasma Physics (Perseus Books,Reading, Massachusetts, 1998).

 D. J. ROSE and M. CLARK, Plasmas and Controlled Fusion (John Wiley and Sons, NewYork, 1961).

 J. A. ELLIOT, in Plasma Physics - An Introductory Course, edited by R. O. DENDY (PressSyndicate of the University of Cambridge, Cambridge, 1993), pp. 29–53.

 F. F. CHEN, Introduction to Plasma Physics and Controlled Fusion (Plenum Press, NewYork, 1984), Vol. 1.

 C. L. HEMENWAY, R. W. HENRY, and M. CAULTON, Physical Electronics (Wiley Inter-national, New York, 1967).

 G. BEKEFI, Radiation Processes in Plasmas (John Wiley and Sons, New York, 1966).

 J. A. BITTENCOURT, Fundamentals of Plasma Physics (Pergamon Press, New York, 1986).  List of Figures

1.1 The electric field applied between electrodes inserted into a plasma is screened byfree charge carriers in the plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Electric field generated by a line of charge . . . . . . . . . . . . . . . . . . . . . . 10

1.3 The figure shows regions of unbalanced charge and the resulting electric field profilethat results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Principle of the Mach Zehnder interferometer . . . . . . . . . . . . . . . . . . . . 12

1.5 Curve showing the dependence of the fractional ionization of a hydrogen (Ui=13.6eV) as a function of temperature and number density. . . . . . . . . . . . . . . . . 14

1.6 Unless alternatives can be found, a serious shortfall in expendable energy reserves will beapparent by the middle of the next century. (reproduced from http://FusEdWeb.pppl.gov/)16

1.7 The terrestrial fusion reaction is based on the fusion of deuterium and tritium with therelease of a fast neutron and an alpha particle. (reproduced from http://FusEdWeb.pppl.gov/)17

1.8 Comaprison of fuel needs and waste by-products for 1GW coal fired and fusion powerplants. (reproduced from http://FusEdWeb.pppl.gov/) . . . . . . . . . . . . . . . . 18

1.9 The principal means for confining hot plasma are gravity (the sun), inertia (laserfusion) and using electric and magnetic fields (magnetic fusion). (reproduced fromhttp://FusEdWeb.pppl.gov/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.10 H-1NF. (a) Artist’s impression (the plasma is shown in red), (b) during construc-tion and (c) coil support structure. . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.11 Computed magnetic surfaces (left) and surfaces measured using electron gun and fluo-rescent screen (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.12 Joint European Torus (JET) tokamak. (a) CAD view and (b) inside the vacuumvessel. From JET promotional material. . . . . . . . . . . . . . . . . . . . . . . . 25

1.13 Large Helical Device (LHD) heliotron. . . . . . . . . . . . . . . . . . . . . . . . . 26

1.14 Coil system and vacuum vessel for the Wendelstein 7-X (W 7-X) modular helias. . 26

1.15 Coil system and plasma of the H-1NF flexible heliac. Key: PFC: poloidal fieldcoil, HCW: helical control winding, TFC: toroidal field coil, VFC: vertical field coil. 27

1.16 Plasma and coil system for the Heliotron-E torsatron. . . . . . . . . . . . . . . . . 27

2.1 Left: A configuration space volume element dr = dxdydz at spatial positionr. Right: The equivalent velocity space element. Together these two elementsconstitute a volume element dV = drdv at position (r,v) in phase space. . . . . . 30

2.2 Evolution of phase space volume element under collisions . . . . . . . . . . . . . . 33

2.3 Examples of velocity distribution functions . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Imaginary box containing plasma at temperature T . . . . . . . . . . . . . . . . . 39 2.5 Electrostatic Coulomb potential and Debye potential as a function of distancefrom a test charge Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.6 The contributions (i) and (ii) noted in the text  . . . . . . . . . . . . . . . . . . 462.7 Diagram showing the electron potential energy −eφ in an electron plasma wave.

Regions labelled A accelerate the electron while in B, the electron is decelerated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.8 Particles moving from the left and impinging on a gas undergo collisions. . . . . . 502.9 The trajectory taken by an electron as it makes a glancing impact with a massive

test charge Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.10 Collison between ion and electron in centre of mass frame. . . . . . . . . . . . . . 532.11 The electron impact ionization cross-section for hydrogen as a function of electron

energy. Note the turn-on at 13.6 eV.  . . . . . . . . . . . . . . . . . . . . . . . . 562.12 Electron secondary emission for normal incidence on a typical metal surface  . 57

3.1 Illustrating the Boltzmann relation. Because of the pressure gradient, fast mobileelectrons move away, leaving ions behind. The nett positive charge generates anelectric field. The force F e opposes the pressure gardient force F p. . . . . . . . 66

3.2 Schematic diagram showing plasma in a container of length 2L with particle den-sity vanishing at the wall.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.3 High spatial frequency features are quickly washed out by diffusion as the plasmadensity relaxes towards its lowest order profile. . . . . . . . . . . . . . . . . . . 73

3.4 Elastic collision cross-section of electrons in Ne, A, Kr and Xe. . . . . . . . . . 743.5 Drift velocity of electrons in hydrogen and deuterium. . . . . . . . . . . . . . . 753.6 The ionization coefficient α/p for hydrogen. Note the exponential behaviour. . . 763.7 An electron avalanche as a function of time. . . . . . . . . . . . . . . . . . . . . 773.8 (a) Paschen’s curve showing breakdown voltage as a function of the product pd.

(b) A plot of plasma current versus applied voltage. Note the dramatic increaseat the onset of breakdown.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.9 (a) The plasma potential distribution for a plasma confined electrostatically and(b) the corresponding density distribution of ions and electrons. The ion densityis higher than electrons near the wall due to the negative electric field establishedthere by the escaping electron flux. . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.10 The structure of the potential distribution near the plasma boundary. . . . . . . 803.11 Potential variation in the wall edge of the sheath compared with that for uniform

and point like charge distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . 823.12 Schematic diagram showing the potential drops around the plasma circuit. . . . . 843.13 The Langmuir probe I − V characteristic showing the electron and ion contribu-

tions and the plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.14 The circuit shows the typical measurement arrangement using Langmuir probes.

The probe bias is adjusted using the variable resistor. . . . . . . . . . . . . . . . . 86

4.1 Electrons and ions spiral about the lines of force. The ions are left-handed andelectrons right. The magnetic field is taken out of the page . . . . . . . . . . . . . 90

4.2 When immersed in orthogonal electric and magnetic fields, electrons and ions driftin the same direction and at the same velocity. . . . . . . . . . . . . . . . . . . . . 91

4.3 The orbit in 3-D for a charged particle in uniform electric and magnetic fields. . . 934.4 The cylindrical plasma rotates azimuthally as a result of the radial electric and

axial magnetic fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 LIST OF FIGURES4.5 When the electric field is changed at time t = 0, ions and electrons suffer anadditional displacement as shown. The effect is opposite for each species. . . . . . 94

4.6 The decomposition of E⊥ into left and right handed components. . . . . . . . . . 974.7 When the magnetic field changes in time, the induced electric field does work on

the cyclotron orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.8 The grad B drift is caused by the spatial inhomogeneity of B. It is in opposite

directions for electrons and ions but of same magnitude. . . . . . . . . . . . . . . 1034.9 The curvature drift arises due to the bending of lines of force. Again this force

depends on the sign of the charge. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.10 The grad B drift for a cylindrical field. . . . . . . . . . . . . . . . . . . . . . . . . 1064.11 Schematic diagram showing lines of force in a magnetic mirror device. . . . . . . . 1064.12 Top:The flux linked by the particle orbit remains constant as the particle moves

into regions of higher field. The particle is reflected at the point where v‖ = 0.Bottom: Showing plasma confined by magnetic mirror . . . . . . . . . . . . . . . 109

4.13 Particles having velocities in the loss cone are preferentially lost . . . . . . . . . . 1104.14 The grad B drift separates vertically the electrons and ions. The resulting electric

field and E/B drift pushes the plasma outwards. . . . . . . . . . . . . . . . . . . . 1114.15 A helical twist (rotational transform) of the toroidal lines of force is introduced

with the induction of toroidal current in the tokamak. Electrons follow the mag-netic lines toroidally and short out the charge separation caused by the grad Bdrift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.16 Diagram showing toridal magnetic geometry . . . . . . . . . . . . . . . . . . . . . 1124.17 Top: Schematic diagram of trajectory of “banana orbit” in a tokamak field. Bot-

tom: The projection of passing and banana-trapped orbits onto the poloidal plane. 116

5.1 Left: Diamagnetic current flow in a plasma cylinder. Right: more ions movingdownwards than upwards gives rise to a fluid drift perpendicular to both thedensity gradient and B. However, the guiding centres remain stationary. . . . . . 121

5.2 Schematic showing the parallel and perpendicular electron and ion fluxes for amagnetized plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Top: The directions of current flow and their associated conductivities in a weakly-ionized magnetoplasma. Bottom: The collision frequency dependence of the per-pendicular conductivty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.4 Left: Schematic showing particle displacements in direct Coulomb collisions be-tween like species in a magnetized plasma. Right: Collisons between unlike parti-cles effectively displace guiding centres. . . . . . . . . . . . . . . . . . . . . . . . . 131

5.5 The theoretical perpendicular diffusion coefficient versus collision frequency fora tokamak. The region of enhanced diffusion occurs in the so called ”plateau”regime centered about the particle bounce frequency in the magnetic mirrors. . . . 132

6.1 In a cylindrical magnetized plasma column, the pressure gradient is supported bythe diamagnetic current j. In time, however, the gradient is dissipated throughradial diffusion u⊥ = ur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2 In a tokamak, the equilibrium current density and magnetic field lie in nestedsurfaces of constant pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.3 The plasma thermal pressure gradient is exactly balanced by a radial variation inthe magnetic pressure. This variation is generated by diamagnetic currents thatflow azimuthally. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.4 The magnetic lines of force for wires carrying parallel currents. . . . . . . . . . . . 1386.5 The geometric interpretation of the magnetic tension due to curvature of lines of

force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.6 An unmagnetized linear pinch showing sausage instability. . . . . . . . . . . . . . 1406.7 The kinking of a plasma column under magnetic forces. . . . . . . . . . . . . . . . 1406.8 Left:The flux through surface S as it is convected with plasma velocity u remains

constant in time. Right: showing the area element dA swept out by the plasmamotion. Note that this area vanishes when u is parallel to the circumferentialelement d�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.9 Showing the process of magnetic reconnection. . . . . . . . . . . . . . . . . . . . . 145

7.1 The phase and group velocities of a wave can be determined from its dispersionrelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2 Torisonal Alfven waves in a compressible conducting MHD fluid propagating alongthe lines of force. The fluid motion and magnetic perturbations are normal to thefield lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.3 Longitudinal sound waves propagate along the magnetic field lines in a compress-ible conducting magnetofluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.4 The magnetoacoustic wave propagates perpendicularly to B compressing and re-leasing both the lines of force and the conducting fluid which is tied to the field. . 153

7.5 The perturbed components associated with the compressional magnetoacousticwave propagating perpendicular to B0 . . . . . . . . . . . . . . . . . . . . . . . . 156

7.6 The perturbed components associated with the torsional or shear wave propagat-ing along B0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.7 The perturbed components associated with the torsional wave in a cylindricalplasma column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.8 Wave normal diagrams for the fast, slow and pure Alfven waves for (a) VA > VS

and (b) VA < VS. The length of the radius from the origin to a point on theassociated closed curve is proportional to the wave phase velocity . . . . . . . . . 159

8.1 The longitudinal and transverse electric field perturbations for waves in a coldelectron plasma are decoupled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.2 Phase velocity versus oscillation frequency for the transverse electron plasma wave.Note reciprocal behaviour of vg and vφ and the region of nonpropagation. . . . . . 164

8.3 The form of the complex wavenumber for transverse electron plasma waves. . . . . 1658.4 Dispersion relations for the three wave modes supported in an isotropic (unmag-

netized) warm plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

9.1 The geometry for analysis of plane waves in cold magnetized plasma. . . . . . . . 1729.2 (a) Near a cutoff, the wave field swells, the wavelength increases and the wave is ul-

timately reflected. (b) near a resonance, the wavefield diminishes, the wavelengthdecreases and the wave enrgy is absorbed. . . . . . . . . . . . . . . . . . . . . . . 175

9.3 A plot of wave phase velocity versus frequency for waves propagating parallel tothe magnetic field for a cold plasma. . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.4 The principle of Faraday rotation for an initially plane polarized wave propagatingparallel to the magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9.5 The ordinary wave is a transverse electromagnetic wave having its electric vectorparallel to B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 LIST OF FIGURES9.6 The relationship between the propagation vector, magnetic field and wave com-ponents for the extraordinary wave. The wave exhibits an electric field in thedirection of motion and so is partly electrostatic in character. . . . . . . . . . . . . 179

9.7 A plot of wave phase velocity versus frequency for waves propagating perpendicularto the magnetic field for a cold plasma. . . . . . . . . . . . . . . . . . . . . . . . . 181

10.1 Conductors marked with a cross carry current into the page (z direction), whilethe dots indicate current out of the page. . . . . . . . . . . . . . . . . . . . . . . . 205

10.2 Magnetic flux loop signal as a function of time. . . . . . . . . . . . . . . . . . . . 209 Index

adiabatic changes, 58adiabatic invariants, 85ambipolar diffusion coefficient, 69ambipolar electric field, 68anomalous diffusion, 129average velocity, 32

banana orbit, 113Bohm diffusion coefficient, 129Bohm sheath criterion, 79Bohm speed, 79Bohm-Gross dispersion relation, 44Boltzmann

factor, 77relation, 64

Boltzmann factor, 40Boltzmann relation, 39breakdown, 71–75

Child-Langmuir Law, 80collision frequency, 48collision time, 48collisions, 47–55

Coulomb, 48, 129cross section, 48electron-ion, 58electron-neutral, 72fusion, 52mean free path, 47

conductivitytensor, 123

conductivity tensor, 97configuration space, 28convective derivative, 30Coulomb

collisions, 48–52force, 48

currentdensity, 66diamagnetic, 102, 118, 133

Hall, 128ion saturation, 82polarization, 93

current density, 33cyclotron frequency, 15, 86

Debye length, 12Debye shielding, 40Debye shielding, 11diamagnetic current, 117diamagnetic drift, 118diamagnetism, 87, 135dielectric

low frequency susceptibility, 93susceptibility, 168tensor, 124

dielectric tensor, 98diffusion, 67–70

ambipolar, 121, 128Fick’s law, 65, 128fully ionized, magnetized, 128neoclassical, 130perpendicular, 120resistive, 140–143

diffusion coefficient, 65dispersion

electron plasma waves, 44extraordinary wave, 175ion acoustic wave, 154left hand em wave, 172magnetoacoustic wave, 152ordinary wave, 175right hand em wave, 172torsional Alfven wave, 154

dispersion relationAlfven waves, 152cold magnetized plasma, 171

distribution average, 32distribution function, 28

electric sheath, 76, 80 INDEXelectromagnetic waves, 145electron

plasma waves, 42sound speed, 165

electron plasma frequency, 10electron saturation current, 83electrostatic waves, 145equation of continuity, 56equation of motion, 57equation of state, 58

generalized Ohm’s law, 127group velocity, 147guiding centre, 85

H-1 heliac, 16Hall current, 122heating

ohmic, 67, 142

ideal MHD equations, 133inelastic collisions, 47interferometry, 11ion saturation current, 82ionization

electron impact, 54photo, 53

isothermal, 59

kink instability, 138

Landau damping, 42Langmuir probes, 81–83Larmor radius, 87light scattering, 13, 52Lorentz equation, 85loss cone, 106lower hybrid frequency, 178

magneticdiffusion, 140–143dipole moment, 100flux, frozen-in, 140islands, 143mirror ratio, 106pressure, 135reconnection, 143

tension, 135magnetic flux surface, 18magnetic mirrors, 104magnetic Reynold’s number, 141magnetohydrodynamics, 125Maxwell-Boltzmann distribution, 35mean free path, 48mean speed, 38mobility

perpendicular, 120mobility, 65mobility tensor, 96

neoclassical diffusion, 113nuclear fusion, 52

particle flux, 33particle number density, 32Paschen’s law, 75passing particles, 110phase space, 28, 29phase velocity, 146phasor, 146photo-ionization, 70plasma

approximation, 60confinement, 15convection, 139heating, 14oscillations, 161potential, 81stability, 137, 143

plasma parameter, 9plasma sound speed, 165polarization

left handed, 96right handed, 96

polarization current, 91Poynting flux, 161pre-sheath, 79pressure, 37, 57pressure tensor, 33

radiative recombination, 70random particle flux, 38ratio of specific heats, 59, 164recombination, 70refractive index, 168 resistivity, 66rms thermal speed, 34rotational transform, 108, 111

safety factor, 111Saha equation, 13sausage instability, 138secondary emission, 54sheath, 76single fluid equations, 125, 127speed

mean 〈v〉, 38rms vrms, 34, 36thermal vth, 35

stellarator, 111Stellarators, 16superposition, 146

thermal equilibrium, 29, 35tokamak, 108tokamak, 16

upper-hybrid frequency, 177

velocitydrift

curvature vR, 102E/B vE, 90Grad B v∇B, 102polarization vP , 93toroidal, 107

group, 147phase, 146

velocity moments, 32velocity space, 28Vlasov, 30Vlasov equation, 42

waveAlfven, 153cutoff, 162, 172electron plasma, 42extraordinary, 176ion acoustic, 148, 153left hand, 171magnetoacoustic, 150, 152ordinary, 176resonance, 172right hand, 171

sound, 148torsional Alfven, 148Whistler, 174

wave normal, 156wave:ion acoustic, 149winding number, 111

zeroth order velocity moment, 32