1 Chapter 10 Past exam papers2 184 Plasma Physics C : Final Examination Attempt four questions. All six are of equal value. The best four marks will b...

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Plasma Physics C17 1993: Final Examination Attempt four questions. All six are of equal value. The best four marks will be considered, but candidates are discouraged from answering all six questions because it is unlikely that there will be suﬃcient time. Show all working and state and justify relevant assumptions brieﬂy. Question 1 (10 marks): tions)

(answer both parts, illustrate with appropriate equa-

a List three quantitative criteria for a plasma and explain each in a few lines. b Describe three out of four of the following phenomena, and their relation to adiabatic invariants. i adiabatic compression ii Fermi acceleration iii ion cyclotron heating iv transit time magnetic pumping Question 2 (10 marks): Discuss one of the following: answers are not restricted to material from the specialist lectures a Plasma fusion and magnetic conﬁnement devices b Extraterrestrial plasma and plasma phenomena c Plasma diagnostics using laser radiation d Describe the process of electrical breakdown between electrodes in gas at pressures near 1 Torr, including relevant equations. Explain why secondary emission is important, and at which electrode. Question 3 (10 marks): Derive an expression for the Debye length in planar (1-D slab) geometry taking into account both Te and Ti . Assume time scales long enough so that both species have equilibrium (Maxwellian) distributions. Discuss the validity of your treatment of the ions. Question 4 (10 marks): (5/10) aUsing the single ﬂuid MHD equations and Fick’s law (Γ = −D∇n), obtain the coeﬃcient of diﬀusion perpendicular to magnetic ﬁeld lines. (2/10) b Explain how and why this diﬀusion depends on plasma resistivity.

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(3/10) c In a few sentences, explain neoclassical diﬀusion qualitatively with the aid of a few sketches. Question 5 (10 marks): Consider a high frequency plane transverse electromagnetic wave in an unmagnetized plasma. (B0 = 0) a From the two ﬂuid electron equation, show that j1 =

ie2 n0 E 1 mω

b and continue, by considering Maxwell’s equations, to derive the dispersion relation. c Calculate the group velocity and sketch both the group and phase velocities on graphs with labels and numerical scales for ne = 1 × 1018 ± 3. Question 6 (10 marks): planar geometry.

Consider the plasma sheath region near a wall in

a Write down Poisson’s equation including both electron and ion terms, explaining and justifying your assumptions. b Justify under what conditions the electron contribution in (a) can be ignored, and solve the equation for those conditions to obtain a relation between V (or Φ) the sheath width d, and J.

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Plasma Physics C17 1994: Final Examination Attempt four questions. All are of equal value. Candidates are discouraged from answering all six questions because it is unlikely that there will be suﬃcient time. Show all working and state and justify relevant assumptions brieﬂy. Question 1 (10 marks): tions)

(answer both parts, illustrate with appropriate equa-

a List three quantitative criteria for a plasma and explain each in a few lines. b Describe three out of ﬁve of the following phenomena. i Debye shielding. ii Boltzmann’s relation for electrons. iii Energy transfer from a plasma to a conducting wall. iv Mechanisms for plasma generation, conﬁnement, and loss. v Discuss an example of a plasma heating scheme that relies on conservation of an adiabatic invariant, and one that relies on the breaking of an adiabatic invariant. Question 2 (10 marks): Discuss one of the following: answers are not restricted to material from the specialist lectures a Plasma fusion and magnetic conﬁnement devices b Low temperature plasma, and its use in materials processing. c Plasma diagnostics - measurements of density, temperature etc. d Discuss Coulomb collisions, explaining the basic properties of the collisions, the range of the interaction, the eﬀect on plasma resistivity, runaway electrons, indicating scaling (e.g. with n, T etc.) where appropriate. Question 3 (10 marks): a Show that the electrical resistivity of a fully ionized plasma can be expressed in the form νei me η= ne e2 where νei is the electron-ion collision frequency. Do not attempt to ﬁnd an expression for νei or derive Coulomb scattering!

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b Explain why η is almost independent of ne even though ne appears in the equation for η. Why ”almost independent”? c Explain why η decreases as the electron temperature Te increases. Question 4 (10 marks): a Sketch the motion of ions and electrons in a magnetic ﬁeld B, directed out of the page, when B increases in a direction vertically up the page as shown below. Label the sketches with appropriate dimensions. b An electron moves from point P to point Q in a magnetic mirror. At point P, 2 the magnetic ﬁeld is 0.5 Tesla, the perpendicular energy ( 12 mv⊥ = 200 eV), 1 2 and the parallel energy ( 2 mv = 600 eV). What is the magnetic ﬁeld at point Q if the electron is reﬂected at this point? c Obtain and expression for the curvature drift of an electron travelling at velocity v along a circular magnetic ﬁeld line of radius Rc . How are curvature drift and grad-B drift related? Question 5 (10 marks): For the electromagnetic mode with E 1 ⊥ B 0 and k B 0 it can be shown that Ex (ω 2 − c2 k 2 − α) + Ey iαωc /ω = 0 Ey (ω 2 − c2 k 2 − α) − Ex iαωc /ω = 0 (Equn 6.24 in notes), where α=

ωp2 1 − ωc2 /ω 2

(3/10) a Continue to obtain the dispersion relation for this wave (in the form given in the formula handout for the exam) (4/10) b show (brieﬂy) that the modes are right and left hand circularly polarized, and identify which is which. (3/10) c Deﬁne and obtain the cutoﬀ frequencies. Question 6 (10 marks): aUsing the single ﬂuid MHD equations and Fick’s law (Γ = −D∇n), obtain the coeﬃcient of diﬀusion perpendicular to magnetic ﬁeld lines.

188 b Use the single ﬂuid MHD Equation of motion, and the mass continuity equation to calculate the phase velocity of an ion-acoustic wave in an unmagnetized plasma with Te >> Ti . spare,10) The plasma potential is usually a few KTe above the potential of its (conducting) container. Explain, and, justifying your assumptions, obtain an approximate relation for the diﬀerence in potential Φp − Φw . (DON’T derive the ¯ Langmuir-Child law.)

Past exam papers

Plasma Physics C17 1996: Final Examination Attempt four questions. All are of equal value. Candidates are discouraged from answering all six questions as it is unlikely there will be suﬃcient time. Show all working and state and justify relevant assumptions. Question 1 (10 marks) a Describe oscillation at the plasma frequency and the Debye length and the relation between them. b Discuss distribution functions, the Boltzmann equation and the relationship between their respective zeroth, ﬁrst and second order velocity moments. c Using the equilibrium ﬂuid equation of motion (set the convective derivative to zero) and in the absence of collisions, show that the particle number density distribution for Maxwellian electrons at temperature Te is described by the Boltzmann relation ne = n0 exp (eφ/kTe ) where E = −∇φ is the plasma electric ﬁeld. What is the physical interpretation of this formula. Use pictures to illustrate. Question 2 (10 marks) a Show that the resistivity of a very weakly ionized plasma can be expressed in the form νen me η = 1/σ = ne e2 where νen is the electron-neutral collision frequency. b Assuming that the electron-neutral collision cross section is independent of particle velocity, calculate the scaling of the resistivity with Te and ne for constant neutral density and compare and contrast with those scalings for a fully ionized plasma. Question 3 (10 marks) a By considering the circular motion of an ion in a magnetic ﬁeld B as a current loop of magnetic moment µ = IA where I is the circulating ion current and A is the area of the orbit, show that the diamagnetic ﬂux associated with the particle motion is proportional to the particle perpendicular kinetic energy. Discuss how can this eﬀect be used to estimate the plasma internal perpendicular thermal energy. b Draw orbits for electrons and ions in orthogonal electric and magnetic ﬁelds for both weak and strong electric ﬁelds. Explain why there is no net current associated with the particle drifts.

189

190 c The grad B drift is in opposite directions for electrons and ions. Show, with the aid of diagrams, how this drift renders impossible plasma conﬁnement in a purely toroidal magnetic ﬁeld. Question 4 (10 marks) a Consider the following simpliﬁed steady state equation of motion for each species in a ﬂuid plasma 0 = qn(E + u×B) − ∇p where the electric and magentic ﬁelds are uniform but the number density and pressure have a gradient. Taking the cross product of this equation with B show that, besides the E×B drift, there is also a diamagnetic drift given by uD = (1/n)∇p×B/(qB 2 ). b Provide physical arguments to justify the reason for this drift. Explain if there is any motion of the particle guiding centres associated with this ﬂuid drift and why it does not appear in the particle orbit theory. Question 5 (10 marks) a Using the single ﬂuid equilibrium MHD equations and Fick’s law (Γ = −D∇n where Γ = nu is the particle ﬂux and D the diﬀusion coeﬃcient), obtain the coeﬃcient of diﬀusion perpendicular to a magnetic ﬁeld. b Explain how and why this diﬀusion depends on plasma resistivity. c Consider the non-equilibrium case. Ignoring gravity and Hall currents, combine the single ﬂuid equation of motion and the Ohm’s law to obtain ρ

∂u = σ(E×B) + σ(u×B)×B − ∇p ∂t

By considering E = 0 and p = constant, solve this equation to show that the ﬂuid velocity perpendicular to B is given by u⊥ = u⊥ (0) exp (−t/τ ) where τ , the characteristic time for damping of the ﬂuid ﬂow across the ﬁeld lines is given by τ = ρ/(σB 2 ). Comment on the scaling of τ with B and σ.

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Question 6 (10 marks) a Plot the wave phase velocity as a function of frequency for plasma waves propagating along the direction of the magnetic ﬁeld B, identifying cutoﬀs and resonances for both electromagnetic and electrostatic wave modes. b The dispersion relation for an em wave propagating in an unmagnetized plasma is 2 vφ2 = c2 /n2 = ω 2 /k 2 = c2 /(1 − ωpe /ω 2). 2 show that the phase shift suﬀered by such a wave (compared For ω 2 ωpe with vacuum) on propagation through a plasma of length L is given by

ω φ=− 2ncr c

L 0

ne d

where ncr is the cutoﬀ plasma density (at which ω = ωpe ).

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Plasma Physics C17 1997: Final Examination Attempt four questions. All are of equal value. Candidates are discouraged from answering all six questions as it is unlikely there will be suﬃcient time. Show all working and state and justify relevant assumptions. Question 1 (10 marks) a Assume a perturbation to charge neutrality in an unmagnetized plasma such that, under the action of the restoring Coulomb force, an oscillating electric ﬁeld E = E0 exp (−iωt) is established. By considering the force felt by an electron in such a ﬁeld, show that an oscillating electron current j = ne e2 E0 /(iωme ) results. Using Maxwell’s equation

∇×b = µ0

∂E j + ε0 ∂t

where b is the associated magnetic perturbation, and assuming the perturbation not to propagate (set the left side to zero), obtain an alternative expression for j. By equating these expressions, obtain a formula for the oscillation frequency ω. What is this frequency? b Describe Debye shielding and the relationship between the plasma frequency and Debye length. c Brieﬂy discuss distribution functions and the Boltzmann equation. What is the relationship between their respective zeroth and ﬁrst order velocity moments. Question 2 (10 marks) Discuss two of the following a Electric breakdown. Discuss to the signiﬁcance of the parameter E/p and the role of secondary emission. b Boltzmann’s relation for electrons c Ambipolar diﬀusion in an unmagnetized plasma d Faraday rotation of an electromagnetic wave traversing a magnetized plasma. e “Frozen-in” magnetic ﬁelds and resistive diﬀusion. Question 3 (10 marks)

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a The Ohm’s law for an unmagnetized plasma in steady state is given by E = ηj where η is the resistivity. Show that the resistivity of a fully ionized plasma can be expressed in the form νei me η = 1/σ = ne e2 where νei is the electron-ion collision frequency. Do not derive the expression for the Coulomb collision frequency. With reference to the formula sheet, explain why η is “almost” independent of density and why it decreases with increasing temperature. b In the time-varying case, and including magnetic eﬀects, the equations of motion for the ions and electrons can be combined to give the single ﬂuid force balance and Ohm’s Laws: ρ

∂u = j×B ∂t

E − ηj = −u×B +

me ∂j 1 + j×B 2 ne e ∂t ne e

(10.1)

In these expressions, we have ignored the plasma kinetic pressure gradient terms. Assuming a time variation of the form ∂/∂t = −iω compare the magnitudes of the various terms on the right side of the Ohm’s law as a function of frequency. Which terms dominate for ω ωci . What about for ω ωce ? (HINT: you will need to use the force equation for u). c Combine the steady state Ohm’s law (neglecting Hall current) E + u×B = ηj and the force balance equation to obtain the the ﬂuid ﬂow velocity perpendicular to the magnetic ﬁeld u⊥ =

E×B η⊥ − ∇p. B2 B2

Use Fick’s diﬀusion law Γ⊥ = D⊥ ∇n to obtain an expression for the classical perpendicular diﬀusion coeﬃcient for a fully ionized plasma and discuss its scaling with temperature. If the Hall current were retained in the Ohm’s law, which additional component of the ﬂuid ﬂow would have been obtained? Question 4 (10 marks) a Draw orbits for electrons and ions in orthogonal electric and magnetic ﬁelds for both weak and strong electric ﬁelds. Explain why there is no net current associated with the particle drifts.

194 b The grad B drift is in opposite directions for electrons and ions. Show, with the aid of diagrams, how this drift renders impossible plasma conﬁnement in a purely toroidal magnetic ﬁeld. c For slow time variations ω ωci , the polarization drift velocity for ions and electrons respectively is given by vp = ±

1 dE ωc B dt

where ωc is the associated cycoltron frequency. (i) Explain the origin of this eﬀect (ii) Calculate the current which ﬂows as a result of a time varying electric ﬁeld. (iii) Identifying this polarization current as equivalent to the electric displacement current density for solid dielectrics j D = ε0 εr

∂E ∂t

where εr is the relative permittivity, express the relative permittivity for the plasma in terms of the Alfven wave speed. Question 5 (10 marks) Consider a particle that is gyrating in a circular orbit in a substantially uniform magnetic ﬁeld. a Obtain an expression for the radius of the Larmor orbit of the particle in terms 2 of the orbital magnetic moment µ = mv⊥ /(2B). b Calculate the magnetic ﬂux linked by this orbit as B is slowly changed? Comment on its dependence on B. c Evaluate the volume of magnetic ﬁeld that has the same energy as the kinetic energy of the particle. Consider the cylinder that has this volume and has the same radius as the orbit of the particle. What is the height of this cylinder? Do you recognize this expression? d Suppose that a mirror ﬁeld increases slowly with time. What will happen to a particle that is conﬁned between the magnetic mirrors? e Why is the adiabatic invariant broken for time variations comparable with or faster than the cyclotron frequency?

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Question 6 (10 marks) a Plot the wave phase velocity as a function of frequency for electromagnetic plasma waves propagating along the direction of the magnetic ﬁeld B, identifying cutoﬀs and resonances. b Radio signals from pulsars pass through the interstellar medium that contains free electrons. Assume that the dispersion relation for an em wave propagating in the interstellar plasma is 2 /ω 2 ). vφ2 = ω 2/k 2 = c2 /(1 − ωpe

(i) What is the plasma frequency if the mean interstellar electron density is ne = 104.5 m−3 ? (ii) Show that the wave group velocity is given by vg = c2 /vφ . 2 ω 2 show that the arrival time t(ν) of a signal will be (iii) Assuming ωpe a function of frequency of the form

t(ν) = Dν −2 + constant where ν is the frequency in Hz and the “dispersion coeﬃcient” D is expressible as D = C ne d

where the integral represents the integral of the electron density along the propagation path of the radio signal. (iv) Find the coeﬃcient C (v) For a particular pulsar it is found that the signal at 100 MHz arrives 2 seconds later than the signal at 200 MHz. What is the value of D for that pulsar? Given ne as in part (i), what is the distance to the pulsar? (vi) What complicating factors are neglected in deriving the above simple expression for the time delay as a function of frequency?

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Plasma Physics C17 1998: Final Examination Attempt four questions. All are of equal value. Only the ﬁrst four answers will be marked. Nominal time allowed is 2 hours. 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) Show that the pressure for a Maxwellian electron gas is given by p = nkB Te . (b) Describe electric breakdown with reference to the parameter E/p and the role of secondary emission. (c) Discuss the physics of Landau damping of electron plasma waves. (d) Discuss ambipolar diﬀusion in an unmagnetized plasma (e) In terms of the magnetic Reynold’s number, explain “Frozen-in” magnetic ﬁelds and resistive diﬀusion. (f) Describe the Boltzmann equation and its relation to distribution functions and their moments. (g) Describe Debye shielding and the relationship between the plasma frequency and Debye length. Question 2 (10 marks) (a) The distance between electrons in a plasma is of order n−1/3 . Show that e the potential energy of electrons that are this close is much less than their kinetic energy provided ne λ3D >> 1. What is the signiﬁcance of this condition? (b) Use the parallel component (parallel to B) of the equilibrium equation of motion for electrons in the absence of collisions to show that the number density for Maxwellian electrons at temperature Te in an electric potential φ is given by the Boltzmann relation: ne = n0 exp(eφ/kB Te ). (c) Suppose a small varying electric potential φ = φ1 sin kx is created in an initially uniform neutral plasma (eφ1 kB Te ). Show that the electrons will come to equilibrium with ne (x) = n0 + ne1 sin kx where ne1 /n0 = eφ1 /kB Te 1. Using Poisson’s equation, show that the ion density will be given by ni (x) = n0 + ni1 sin kx with (ni1 − ne1 )/ne1 = k 2 λ2D . Explain this result.

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Question 3 (10 marks) (a) Using the collisionless equation of motion for a speces of charge q in a plasma with uniform electric and magnetic ﬁelds, show that the particles have drift velocity E×B ∇p×B vD = − . B2 qnB 2 (b) With the aid of diagrams, show the origin of the E/B drift and explain why it is independent of species charge and mass. (c) Show that for a neutral plasma consisting of electrons and singly charged ions, the diamagnetic drift results in a current ﬂow jD = (B×∇p)/B 2 . Explain how this current, which is not due to a guiding centre drift can arise. Question 4 (10 marks) Consider a plasma cylinder of radius a with uniform axial vacuum magnetic ﬁeld B0 . Assume the plasma has a parabolic radial pressure proﬁle p = p0 (1 − r 2 /a2 ). (a) What is maximum value of p0 ? (b) Using this value of p0 and Ampere’s law, obtain an expression for the magnetic ﬁeld B(r) and plot it on a graph for r < a and r > a. (c) What is the diamagnetic current density jD (r) ? (d) Obtain an expression for the associated ∇B and curvature drifts. (e) Show that | v∇B (r) | / | vD (r) | is in the ratio of the kinetic and magnetic pressures. Question 5 (10 marks) The dispersion relation for low frequency magnetohydrodynamic waves in a magnetized plasma was derived in lectures as −ω 2 u1 + (VS2 + VA2 )(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 = B 0 /(µ0 ρ0 )1/2 is a velocity vector in the direction of the magnetic ﬁeld with magnitude equal to the Alfv´en speed and VS is the sound speed. (a) Derive the following dispersion relations: vφ =

ω = (VS2 + VA2 )1/2 k

for k.V A = 0

(10.2)

198 for k V A and u1 V A for k V A and u1 .V A = 0

vφ = VS vφ = VA

(10.3) and identify the wave modes. (b) Using ∂B 1 − ∇×(u1 ×B 0 ) = 0 ∂t E 1 + u1 ×B = 0 ∂ → −iω and ∇× → ik×, and assuming plane wave propagation so that ∂t make a sketch showing the relation between the perturbed quantities u1 , E 1 , B 1 and k and B 0 for wave propagation perpendicular to B 0 . What is the nature of this wave?

Question 6 (10 marks) (a) Plot the wave phase velocity as a function of frequency for plasma waves propagating along the direction of the magnetic ﬁeld B, identifying cutoﬀs and resonances for both electromagnetic and electrostatic wave modes. (b) Starting with the dispersion relation for L and R waves in the form n2 = S ± D, show that the phase velocity for both waves is given by vφ2

VA2 = 1 + VA2 /c2

in the low frequency limit. HINT: You must consider both ions and electrons.

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THE AUSTRALIAN NATIONAL UNIVERSITY First Semester Examination 1999 PHYSICS C17 PLASMA PHYSICS Writing period 2 hours duration Study period 15 minutes duration Permitted 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 moments of the particle distribution function f and moments of the Boltzmann equation. Draw a contour plot of f (vx , vy ) for an anisotropic electron velocity distribution and for a beam of electrons propagating in the x direction. (b) Describe electric breakdown with reference to the parameter E/p and the role of secondary emission. (c) Describe the physics of Landau damping of electron plasma waves. (d) Discuss the role of Coulomb collisions for diﬀusion in a magnetized plasma. (e) Discuss magnetic mirrors with reference to the adiabatic invariance of the orbital magnetic moment µ. (g) Describe Debye shielding and the relationship between the plasma frequency and Debye length. Question 2 (10 marks) Low frequency ion oscillations: Let n0 be the equilibrium number density of singly charged ions and electrons and assume a one-dimensional harmonic perturbation of the form φ˜ = φ exp [i(kx − ωt)]. We assume the plasma is collisionless and that the ion temperature is small and can be neglected.

200 (a) Show that the perturbed ion velocity is given by vi = (ek/mi ω)φ where φ is the electric potential perturbation. (b) From the equation of continuity, show that the perturbation charge density of the ions is obtained as (n0 e2 k 2 /mi ω 2)φ . (c) Assume that the ion oscillations are so slow that the electrons remain in a Maxwell-Boltzmann distribution. If eφ/kB Te 1, show that the perturbed charge density of the electrons is given by −(n0 e2 /kB Te )φ. (d) Use Poisson’s equation to deduce the following dispersion relation: k 2 = (n0 e2 /mi ε0 ω 2)k 2 − n0 e2 /kB Te ε0 (e) Recast the dispersion relation in the following form: 2 ω 2 = ωpi /(1 + 1/k 2 λ2D ).

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 unmagnetized plasma is given by: Γj = nuj = ±µj nE − Dj ∇n with mobility µ =| q | /mν where ν is the electron-ion collision frequency and diﬀusion coeﬃcient D = kB T /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 = me ν/ne2 . (d) In the presence of a magnetic ﬁeld, the mean perpedicular velocity of particles across the ﬁeld is given by u⊥ = ±µ⊥ E − D⊥

uE + uD ∇n + n 1 + ν 2 /ωc2

with uE = E×B/B 2 , uD = −∇p×B/qnB 2 and where µ⊥ = µ/(1 + ωc2τ 2 ) and D⊥ = D/(1 + ωc2 τ 2 ). Discuss the scaling with ν of each of the four terms in the expression for u⊥ .

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Question 4 (10 marks) (a) Show that the drift speed of a charge q in a toroidal magnetic ﬁeld can be written as vT = 2kB T /qBR where R is the radius of curvature of the ﬁeld. (Hint: Consider both gradient and curvature drifts) (b) Compute the value of vT for a plasma at a temperature of 10 keV, a magnetic ﬁeld strength of 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 radius 1 m. (d) Suppose an electric ﬁeld is applied perpendicular to the plane of the torus. Describe what happens. Question 5 (10 marks) (a) Show that the MHD force balance equation ∇p = j×B requires both j and B to lie on surfaces of constant pressure. (b) Using Ampere’s law and MHD force balance, show that

B2 ∇ p+ 2µ0

=

1 (B.∇)B µ0

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 that stiﬀens the plasma through the additional magnetic pressure Bz2 /2µ0 and tension against bending. Consider a local constriction dr in the radius r of the plasma column. Assuming that the longitudinal magnetic ﬂux Φ through the cross-section of the cylinder remains constant during the compression (dΦ = 0), show that the axial magnetic ﬁeld strength is increased by an amount dBz = −2Bz dr/r. (d) Show that the internal magnetic pressure increases by an amount dpz = Bz dBz /µ0 = −(2Bz2 /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 that the change in azimuthal ﬁeld strength due the compression dr is dBθ = −Bθ dr/r and that the associated increase in external azimuthal magnetic pressure is dpθ = −(Bθ2 /µ0 )dr/r.

202 (f) Show that the plasma column is stable against sausage distortion provided Bz2 > Bθ2 /2. Question 6 (10 marks) (a) Plot the wave phase velocity as a function of frequency for plasma waves propagating perpendicular to the magnetic ﬁeld B, identifying cutoﬀs and resonances for both ordinary and extraordinary modes. (b) Using the matrix form of the wave dispersion relation

S − n2z −iD nx nz Ex 2 2 iD S − nx − nz 0 Ey = 0 0 P − n2x Ez nx nz 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.

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THE AUSTRALIAN NATIONAL UNIVERSITY First Semester Examination 2000 PHYSICS C17 PLASMA PHYSICS Writing period 2 hours duration Study period 15 minutes duration Permitted 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 of motion and the single ﬂuid force balance euqation. (b) Describe electric breakdown with reference to the parameter E/p and the role of secondary emission. (c) Discuss the physical meaning of the Boltzmann relation. Use diagrams to aid your explanation. (d) Discuss the origin of plasma diamagnetism and its implications for magnetic plasma conﬁnement. (e) Draw a Langmuir probe I-V characteristic indicating the saturation currents, plasma potential and ﬂoating potential. How can the characteristic be used to estimate temperature? (f) Describe Debye shielding and the relationship between the plasma frequency and Debye length.

204 Question 2 (10 marks) (a) Consider two inﬁnite, perfectly conducting plates A1 and A2 occupying the planes y = 0 and y = d respectively. An electron enters the space between the plates through a small hole in plate A1 with initial velocity v towards plate A2 . A potential diﬀerence V between the plates is such as to decelerate the electron. What is the minimum potential diﬀerence to prevent the electron from reaching plate A2 . (b) Suppose the region between the plates is permeated by a uniform magnetic ﬁeld B parallel to the plate surfaces (imagine it as pointing into the page). A proton appears at the surface of plate A1 with zero initial velocity. As before, the potential V between the plates is such as to accelerate the proton towards plate A2 . What is the minimum value of the magnetic ﬁeld B necessary to prevent the proton from reaching plate B? Sketch what you think the proton 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 immobile) show that the conductivity of an unmagnetized plasma is given by ne2 σ0 = (10.4) me ν (b) What is the dependence of the conductivity on electron temperature and density in the fully ionized case? (c) When the plasma is magnetized, the Ohm’s law for a given plasma species (electrons or ions) becomes j = σ0 (E +u×B) where j = nqu is the current density. Show that the familiar 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 perpendicular 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’s law can equivalently be ↔ expressed by the tensor relation j = σ E with conductivity tensor given by

σ⊥ −σH 0 ↔ σ= σH σ⊥ 0 0 0 σ

(10.5)

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where σ⊥ σH σ

ν2 = σ0 2 ν + ωc2 ∓νωc = σ0 2 ν + ωc2 ne2 = σ0 = mν

Explain the collision frequency dependence of the perpendicular and Hall conductivities. Question 4 (10 marks) Answer either part (a) or part (b) (a) Consider the two sets of long and straight current carrying conductors shown in confurations A 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 the conservation of the ﬁrst adiabatic moment. (iii) Charge separation will occur due to the magnetic ﬁeld inhomogeneity. This in turn establishes an electric ﬁeld. Comment on the conﬁning properties (or otherwise) of this electric ﬁeld.

Configuration A

Configuration B

Figure 10.1: Conductors marked with a cross carry current into the page (z direction), while the dots indicate current out of the page. (b) In a small experimental plasma device, a toroidal B-ﬁeld is produced by uniformly winding 120 turns of conductor around a toroidal vacuum vessel and

206 passing a current of 250A through it. The major radius of the torus is 0.6m. A plasma is produced in hydrogen by a radiofrequency heating ﬁeld. The electrons and ions have 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 ﬁeld on 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 show the 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 magnetic ﬁeld lines that encircle the torus axis. For particles not on the torus axis and which have a high parallel to perpendicular velocity ratio the projected guiding centre motion executes a rotation in the poloidal plane (a vertical cross-section of the torus) as it moves helically along a ﬁeld line. What happens to particles that have a high perpendicular to parallel velocity ratio? Question 5 (10 marks) There is a standard way to check the relative importance of terms in the single ﬂuid MHD equations. For space derivatives we choose a scale length L such that we 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. ∂u = j×B − ∇p ∂t Show that the terms are in the ratio 2 2 nme vthe jBτ me vthe V or 1 : : nmi : jB : τ L nmi V mi V 2 ρ

(10.6)

(10.7)

When the plasma is cold, show that this suggests V ∼ jBτ /nmi (b) Examine the generalized Ohm’s law: me ∂j 1 1 = E + u×B − j×B + ∇pe − ηj ne2 ∂t ne ne Show that the terms are in the ratio 2 1 vthe 1 νei 1 : : 1 : 1 : : 2 2 ωce ωci τ ωci τ ωce τ V ωce ωci τ

(10.8)

(10.9)

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(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 . We assume the bulk plasma velocity is zero (v 0 = 0) but allow small drifts v1 to be induced by the one-dimensional harmonic electric ﬁeld perturbation E = 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 momentum equations for electrons and ions give n0 mi (−iωvi1 ) = n0 eE1 n0 me (−iωve1 ) = −n0 eE1 (b) The ion motions are small and can be neglected (why?). Show that the resulting current density ﬂowing in the plasma due to the imposed oscillating wave electric ﬁeld is given by j1 = en0 (vi1 − ve1 ) ≈ i

n0 e2 E1 . me ω

(10.10)

(c) Associated with the ﬂuctuating current is a small magnetic ﬁeld oscillation which is given by Ampere’s law. Use the diﬀerential forms of Faraday’s law and Ampere’s law (Maxwell’s equations) to obtain the ﬁrst order equations kE1 = ωB1 and ikB1 = µ0 j1 − iωµ0ε0 E1 linking B1 , E1 and j1 . (d) Use these relations to eliminate B1 and j1 to obtain the dispersion relation for plane electromagnetic waves propagating in a plasma: k2 =

2 ω 2 − ωpe c2

(10.11)

(d) Sketch the dispersion relation and comment on the physical signiﬁcance of the dispersion near the region ω = ωpe .

208

THE AUSTRALIAN NATIONAL UNIVERSITY First Semester Examination 2001 PHYSICS C17 PLASMA PHYSICS Writing period 2 hours duration Study period 15 minutes duration Permitted materials: Calculators Attempt four questions. All are of equal value. Show all working and state and justify relevant assumptions.

Question 1 Attempt 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 moments of the Boltzmann equation. Plot f (v) for a one dimensional drifting Maxwellian distribution, 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 secondary emission. (c) Discuss the physical meaning of the Boltzmann relation. Use diagrams to aid your explanation. (d) Discuss the origin of plasma diamagnetism and its implications for magnetic plasma conﬁnement. (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 magnetic moment µ. (g) Describe Debye shielding and the relationship between the plasma frequency and Debye length.

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Question 2 Consider an axisymmetric cylindrical plasma with E = Eˆ r , B = Bˆ z and ∇pi = ∇pe = rˆ ∂p/∂r. If we negelct (v.∇)v, the steady state two-ﬂuid momentumbalance equations can be written in the form en(E + ui ×B) − ∇pi − e2 n2 η(ui − ue ) = 0 −en(E + ue ×B) − ∇pe + e2 n2 η(ui − ue ) = 0 ˆ components of these equations, show that uir = uer . (a) From the θ (b) From the rˆ components, show that ujθ = uE + uDj (j = i, e). (c) Find an expression for uir showing that it does not depend on E. Question 3 The 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 Tesla Number of turns on the diamagnetic loop - N = 75 Radius of the loop - aL = 0.075m Plasma radius - a = .05m. Given the observed diamagnetic ﬂux loop signal shown below, calculate the plasma pressure as a function of time. If the temperature of the plasma is constant at 1 keV, what is the plasma density as a function of time? (HINT: use Faraday’s law relating the emf to the time derivative of the magnetic ﬂux)

Volts 1.0

12 2

4

6

8

14

16

10 Time (µs)

-1.0

Figure 10.2: Magnetic ﬂux loop signal as a function of time.

210 Question 4 An 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 = v0 (⊥ and refer to the direction relative to 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 is induced in the ±z direction. Indicate on a diagram the relative directions of I, E, B and v E . (c) Do v⊥ and v increase, decrease or remain the same as the current increases? Explain your answer. Question 5 Magnetic 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 is modulated in time according to B = B0 (1 + cos ωt)

(10.12)

2 where ω ωc and 1. If U⊥ = mv⊥ /2 = (mvx2 + mvy2 )/2 is the particle perpendicular kinetic energy (electrons or ions) show that the kinetic energy is also modulated as dU⊥ U⊥ dB = . dt B dt We now allow for a collisional relaxation between the perpendicular (U⊥ ) and parallel (U ) kinetic energies modelled according to the coupled equations

U⊥ dB U⊥ dU⊥ = −ν − U dt B dt 2

dU U⊥ = ν − U dt 2

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

2 ω2ν U ≡ αU. = ∆t 6 9ν 2 /4 + ω 2

(10.13)

This heating scheme is called collisional magnetic pumping. Comment on the physical reasons for 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, show that the heating rate ∆U/∆t decreases as the temperature

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increases. What would happen if the magnetic ﬁeld were oscillating at frequency ω = ωc ? Question 6 On 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 ) propagating in an unmagnetized plasma. (HINT: Draw the ion and electron plasma frequencies and lines corresponding to the electron sound speed, the ion sound speed and the speed of light.) Consider the case of electron plasma oscillations in a uniform plasma of density ˆ We take the n0 in the presence of a uniform steady magnetic ﬁeld B 0 = B0 k. background electric ﬁeld to be zero (E 0 = 0) and assume the plasma is at rest u = 0. We shall consider longitudinal electron oscillations having k E 1 where we take the oscillating electric ﬁeld perturbation associated with the electron wave E 1 ≡ Eˆi to be parallel to the x-axis. Replacing time derivatives by −iω and spatial gradients by ik, and ignoring pressure gradients and the convective term (u.∇)u, show that for small amplitude perturbations, the electron motion is governed by the linearized mass and momentum conservation equations and Maxwell’s equation: −iωn1 + n0 ikux = 0 −iωu = −e(E + u×B 0 ) ε0 ikE = −en1 .

(10.14) (10.15) (10.16)

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

eE/iωm 1 − ωc2 /ω 2

(10.17)

Substituting for ux from the continuity equation and eliminating the density perturbation using Eq. (10.16), obtain the dispersion relation for the longitudinal electron plasma oscillation transverse to B: ω 2 = ωp2 + ωc2 .

(10.18)

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

212

APPENDIX: A Glossary of Useful Formulae Chapter 1: Basic plasma phenomena

ωpe =

e2 ne ε0 me

λD =

ε0 kB Te ne e2 3

ni Te2 −Ui 2.4 × ×1021 exp n ni kB T

√

fpe 9 ne ( Hz)

Chapter 2: Kinetic theory ∂f q +v.∇r .f + (E+v×B).∇v .f = ∂t m

∂f ∂t

coll

eφ ne = ne0 exp kB Te

Γ = n¯ v j = qn¯ v 2 p = nU¯r 3 −mv 2 2 fM (v) = A exp( ) ) = A exp (−v 2 /vth 2kB T 1 U¯r (Maxwellian) ≡ EAv = kB T (1 − D) 2 1 eV 11, 600 K

vrms =

3kB T m

2kB T vth = m pj = nj kB Tj

λmfp =

1 nσ

λmfp v ν = nσv 2qq0 b0 = 4πε0mv 2 λD ln Λ = ln b0 τ=

ei σcoulomb

δEei ∼ Pei = −

Z 2 e4 ln Λ 2πε20m2e ve4 4Ee me mi

me ne (ue − ui ) τei

Chapter 2: Fluid and Maxwell’s equations σ = ni qi + ne qe j = ni qi ui + ne qe ue ∂nj + ∇.(nj uj ) = 0

∂t ∂uj + (uj .∇)uj = qj nj (E + uj ×B) − ∇pj + Pcoll mj nj ∂t

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213 γ

pj = Cj nj j σ ∇.E = ε0 ∂B ∇×E = − ∂t ∇.B = 0 ∇×B = µ0 j + µ0 ε0

∂E ∂t

Chapter 3: Gaseous Electronics Γj = nuj = ±µj nE − Dj ∇n |q | µ= mν kB T D= mν E = ηj νei me η= ne e2 √ Ze2 me ln Λ η √ 6 3πε20 (kB Te )3/2 5.2 × 10−5 Z ln Λ η = 3 2 Te(eV)

I= 4 J= 9

I0 eαx (1 − γeαx )

2e ε0 |φw |3/2 mi d2

uBohm =

kB Te mi

eφw 2πme 1 ≈ ln kB Te 2 mi

kB Te 1 Isi n0 eA 2 mi

Chapter 4: Single Particle Motions F =m

dv = q(E + v×B) dt |q|B ωc ≡ m v⊥ rL = ωc

2 mv⊥ 2B E×B vE = B2 1 F ×B vF = q B2

µ=

vR =

mv2 Rc ×B qB 2 R2

B×∇B 1 v ∇B = ± v⊥ rL 2 B2 vP =

˙ 1 E ωc B ↔

j =σ E F = −µ∇ B

2 v = (K − µB) m

1/2

1 Bm = B0 sin2 θm q(r) =

rB0 dφ B0 = = dθ RBθ Bθ

214

ine2 σ e= me ω

↔

ω2 2 ω 2 − ωce iωce ω 2 2 ω − ωce 0

−iωce ω 0 2 ω 2 − ωce 2 ω 0 2 ω 2 − ωce 0 1

↔

↔

ε = ε0 I +

i ↔ σ ε0 ω

Chapter 5: Magnetized Plasmas

E×B −∇p×B u⊥ = + B2 qnB 2 j D = (kB Ti + kB Te ) u⊥ = ±µ⊥ E − D⊥

B×∇n B2

µ 1 + ωc2 τ 2

D⊥ =

D 1 + ωc2 τ 2

ν2 ν 2 + ωc2 ∓νωc = σ0 2 ν + ωc2 ne2 = σ0 = mν

σ⊥ = σ0

uE + uD ∇n + n 1 + ν 2 /ωc2

µ⊥ =

σ⊥ −σH 0 ↔ 0 σ= σH σ⊥ 0 0 σ

σH σ

D⊥ =

η⊥

ns kB Ts B2

Chapter 5: Single Fluid Equations ∂u ∂t E + u×B ∂ρ + ∇.(ρu) ∂t ∂σ + ∇.j ∂t ρ

= j×B − ∇p + ρg = ηj = 0 = 0

∂B ∂t ∇×B = µ0 j p = Cnγ ∇×E = −

Chapter 6: Magnetohydrodynamics

B2 ∇ p+ 2µ0

=

1 (B.∇)B µ0

η ∂B = ∇2 B + ∇×(u×B) ∂t µ0

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215

µ0 vL η

RM =

Chapter 7, 8, 9: Waves dω dk /2 B2 VA = µ0 ρ

R L S D P

vg =

1/2

γe kB Te + γi kB Ti VS = mi ω c vφ = = 2 k (1 − ωpe /ω 2)1/2 2 ω 2 = ωpe +

tan2 θ =

3kB T 2 k m

↔

n×(n×E)+ K .E = 0 c n= k ω n =| n |= ck/ω = c/vφ

S = 1−

i,e

D =

i,e

c2 vφ2

L

=1−

R

i,e

P (n2 − L)(n2 − R) (n2 − P )(RL − n2 S) 2 2 ωpe ωpi − ω(ω ± ωce ) ω(ω ∓ ωci )

n2 =

ωp2 ω 2 − ωc2

ωp2 ω2

2 2 1/2 ω0R = ωce + (ωce + 4ωpe ) /2 2 2 )(ω 2 − ω0R ) c2 (ω 2 − ω0L = 2 2 2 2 vφ ω (ω − ωU H ) 2 2 1/2 ωU H = (ωpe + ωce )

ωp2 ωc ± ω(ω 2 − ωc2 )

P = 1−

S+D Right S −D Left (R + L)/2 Sum (R − L)/2 Diﬀ Plasma

2 2 1/2 ω0L = −ωce + (ωce + 4ωpe ) /2

S −iD 0 ↔ ↔ 0 K= /ε0 = iD S 0 0 P

= = = =

ωLH ≈ (ωci ωce )1/2 n2 =

2 2 c2 (ω 2 − ω0L )(ω 2 − ω0R ) = 2 2 vφ ω 2 (ω 2 − ωU H )

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

1 (A.∇)A = ∇( A2 ) − A×(∇×A) 2 ∇.(A×B) = B.(∇×A) − A.(∇×B)

∇×(φA) = ∇φ×A + φ∇×A ∇×(A×B) = A(∇.B) − B(∇.A) A×(∇×B) = ∇(A.B) − (A.∇)B + (B.∇)A − (A.∇)B − (B.∇)A − B×(∇×A)

216 ∇×(∇×A) = ∇(∇.A) − (∇.∇)A ∇×∇φ = 0 ∞ −∞

∇.(∇×A) = 0 v 2 exp (−av 2 )dv =

1 2

π a3

Cylindrical coordinates ∇φ =

1 ∂φ ˆ ∂φ ∂φ ˆ rˆ + z θ+ ∂r r ∂θ ∂z

1 ∂ ∂φ 1 ∂2φ ∂2φ ∇ φ= r + 2 2+ 2 r ∂r ∂r r ∂θ ∂z 2

∇.A =

1 ∂ 1 ∂Aθ ∂Az (rAr )+ + r ∂r r ∂θ ∂z

∂Aθ 1 ∂Az ∂Ar ∂Az rˆ + ∇×A = − − r ∂θ ∂z ∂z ∂r

1 ∂ 1 ∂Ar ˆ (rAθ ) − z + r ∂r r ∂θ

ˆ θ

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