Questions and Answers of Fundamentals Of Plasma Physics

Analyze the heat flow problem in a weakly ionized plasma immersed in an externally applied magnetostatic field B0 and derive expressions for the heat flow vector qe and for the components of the
Deduce (5.22) and (5.23) starting from the definition of the electron flux vector and the expression for f1(r, v) given by (5.6), (5.11), (5.12), and (5.13).Equations fi(r, v) = F₁ (r, v) + F2(r,
Show that (3.30) and (3.31) yield, respectively, (3.37) and (3.38), when νr is independent of v for any f0(v).Equations. 0±=- 4πie² 3me Tº 4πie² 3me v3 dfo (v) w + ivr(v) Fce dv dv v3 w + iv,
Derive (3.34), from equations (3.29) to (3.33).Equations. J+ J_ J₁ = 0+ 0 0 0 0_ 0 0 0 3) ( σ || E+ E_ E (3.29)
In the expression deduced for νef in part (b) of the previous problem (high-frequency limit), consider that f0 is the Maxwell-Boltzmann distribution function and that νr(v) = ν0vn, where ν0 is a
If we define an effective collision frequency νef(ω) such that the longitudinal electric conductivity is given bythen, by comparison with (2.18), we find that(a) Show that in the low-frequency
Consider equation (2.25), which gives the AC electric conductivity of a weakly ionized plasma for a velocity-dependent collision frequency νr(v).(a) Show that in the high-frequency limit (ω2 ≫
Show that, when the collision frequency is independent of velocity, equation (2.25) reduces to (2.20). 4πie² 3me (w + ivr) for fo(v) So fo(v) 3v² dv= noe² vr me (w² + v²) inge² me (w +
Using the Maxwell-Boltzmann distribution function (2.22) and the definition (2.23), verify equation (2.24).Equations. fo(v) = no me 2пкт 3/2 mev2. (- exp 2kT (2.22)
In Cartesian coordinates in velocity space (refer to Fig. 1), with the components expressed in spherical coordinates ( v, θ, ϕ), we have(a) Show that the dyad (vv)/v2 can be written in matrix form
Consider the following expressions that define the Fokker-Planck coefficients of dynamic friction and of diffusion in velocity space:(a) With reference to Fig. 6, verify thatFor a general
Consider a gas mixture of two types of particles (α = 1, 2), each one characterized by a Maxwellian distribution functionwith its own mass, density, and temperature.(a) Make the following
Consider the case of Maxwell molecules, for which the interparticle force is of the formwhere K is a constant.(a) Without specifying the form of the distribution functions fα(v) and fβ(v1) for the
Use a Lagrange multiplier technique to show that for a system characterized by the following modified Maxwell-Boltzmann distribution functionwhere Tis constant, the entropy S, defined byis a maximum
Consider a plasma in which the electrons and the ions are characterized, respectively, by the following distribution functions:(a) Calculate the difference(b) Show that this plasma of electrons and
Consider a system consisting of a mixture of two types of particles having masses m and M, and subjected to an external force F. Denote the corresponding distribution functions by f and g,
Show that for the case of coulomb interactions (p = 2) we havewhere b0 = qq1/(4π∈0μg2), Aℓ(p) is as defined in problem 20.5, and A = λD/b0. For ⋀ ≫ 1 verify thatNote that, since K =
From the expression for σm, obtained in part (a) of problem 20.4, verify that for p = 2 the momentum transfer cross section is given bywhere A1(2) is given by (with ℓ = 1 and p = 2)Consequently,
Consider a general inverse-power interparticle force of the formwhere K is a constant and p is a positive integer number.(a) Determine expressions for the scattering angle χ, for the differential
Consider two particles whose interaction is governed by the following rectangular-well potential:(a) Calculate the differential scattering cross section σ(χ) and show that it is given by
Consider a collision between two particles of mass m and m1, in which the particle of mass m1 is initially at rest. Denote the scattering angle in the center of mass coordinate system by χ and in
For a differential scattering cross section with an angular dependence given bywhere σ0 is a constant, calculate the total cross section and the momentum transfer cross section. o(x) = 100(3 cos²x
Deduce the dispersion relation for small amplitude waves propagating at an arbitrary direction with respect to an externally applied magnetostatic field B0 = B0ẑ in a hot plasma. Carry through the
(a) Show that, starting from the Vlasov equation and the laws of electrostatics, we obtain the following dispersion relation for the quasistatic wave propagating at an arbitrary direction with
For an unbounded homogeneous electron gas, characterized by the following velocity distribution function,where a0 is a constant, show that the dispersion relation for the right circularly polarized
In problem 19.4 suppose that in the equilibrium state the velocity distribution function of the electrons is given bywhich corresponds to an isotropic distribution but with the electrons drifting
Consider an electron gas immersed in a uniform magnetostatic field B0 and characterized by the following modified Maxwellian distribution function:Use this distribution function in the dispersion
Consider plane wave disturbances propagating along the magnetostatic field B0 in a hot electron gas, whose equilibrium distribution function is homogeneous and isotropic. In spherical coordinates in
Derive expression (3.61) for σzz starting from (3.60).Equations. Ozz = e2 ·∞ го : mence vi dvi poo afo дvz .2п •+∞ do f 00 vz dvz exp (-ig sin ) exp [91(ф')] do' (3.60)
Show that the first and second terms in the right-hand side of (2.16) represent, respectively, right and left circularly polarized wave fields. E = E+ (x + iŷ) √2 +E_ (x - iy) √2 + E2 (2.16)
Consider the two-stream instability using the macroscopic cold plasma equations for two beams of electrons having number densities given byand average velocities given byConsider that the electric
Consider a longitudinal wave propagating along the x direction in a plasma whose electric field is given by(a) Show that, for small displacements, the electrons that are moving with a velocity
Evaluate the integral G(C, 0), defined in (4.32) with s = 0, by the method of residues using the contours of integration in the complex plane shown in Fig. 2.Figure 2.Equation 4.32.
Solve the linearized Vlasov equation (3.9) by the method of integral transforms, taking its Laplace transform in the time domain and the Fourier transform with respect to the space variables. Then,
A longitudinal plasma wave is set up propagating in the x direction (k = kx̂) in a plasma whose equilibrium state is characterized by the following so-called resonance distribution of velocities in
(a) Show that the dispersion relation for the longitudinal plasma wave (with k = kx̂), for the case of an unbounded homogeneous plasma in which the motion of the electrons and the ions is taken into
Show thatby making a series expansion of the integrand. For C ≪ 1, that is, for (ω/k) ≪ (2kBTe/me)1/2 , show that the dispersion relation for the longitudinal plasma wave reduces toorThis result
Since the longitudinal plasma wave is an electrostatic oscillation, it is possible to derive its dispersion relation using Poisson equation, satisfied by the electrostatic potential ϕ( r, t),
Show that the resonances in a warm fully ionized magnetoplasma, neglecting collisions, occur approximately at the frequencies ω =Ωce cos θ and ω = Ωci cos θ.
Make plots analogous to Figs. 9, 10, and 11 for wave propagation in a fully ionized warm plasma, but in terms of ω as a function of the real part of k.Figure 9Figure 10.Figure 11. Vph4 U Vse V₂ V
Obtain a cubic equation in k2 , from (5.50), and analyze the dispersion relations for these three modes of wave propagation across the magnetic field in a fully ionized warm plasma.Equation S2(S1n²)
Starting from (5.12), (5.40), (5.41), and (5.42), provide all the necessary steps to obtain (5.43).Equations C.E=0 (5.12)
Show that the reflection points w′01 and w′02, for the LCP and RCP waves propagating along the magnetic field in a fully ionized warm plasma (see Fig. 9) are given, respectively, byCompare these
Make a plot analogous to Fig. 8 for wave propagation in a warm electron gas immersed in a magnetic field, but in terms of ω as a function of the real part of k.Figure 8 Vph 4 U Vse 0 RCP Sce X LCP
Show that one of the roots of the dispersion relation (2.33), at very low frequencies, corresponds to an evanescent wave.Equation V²V²k¹ + V²² (1 + T₁/Te) k² - w²wpe se si se pi = 0 (2.33)
Consider a plasma slab of thickness L and number density specified by n(x), where the x axis is normal to the slab. A plane polarized monochromatic electromagnetic wave is normally incident on the
Make a plot analogous to Fig. 20 for wave propagation in a cold magnetoplasma, but in terms of ω as a function of the real part of k.Figure 20. Vph C 0 RCP See Wo- X LONGITUDINAL OSCILLATION LCP 0 X
For a helicon wave, or a circularly polarized wave, show that the tip of the wave magnetic field vector traces out a helix.
From (5.25) show that the polarization of the waves propagating at an angle θ with respect to B0 (considering the perpendicular electric field vector component) is determined byFrom this result
From the dispersion relations obtained in problem 16.8 show that, in the limit of ω ≪ Ωci, we obtain the dispersion relation for the (shear) Alfven wave (cold plasma limit of the magnetosonic
Using the results of the previous problem, analyze the various modes of propagation for the particular cases when θ = 0 and θ = π/2. Compare the results with those for a cold electron gas. Make a
Consider the problem of wave propagation at an arbitrary direction in a cold magnetoplasma, but including the motion of the ions (one type only).(a) Show that the dispersion relation is obtained from
Use (6.17) and (6.18) for the group velocities of the left and right circularly polarized waves, respectively, to show that the group velocity vanishes at the resonances and reflection
Consider the propagation of high-frequency waves in a solid-state plasma with equal number of electrons and holes (considering me = mh and νe = νh), immersed in a magnetostatic field B0. Let k =
Use the dispersion relation (4.10), for the transverse mode of propagation in a cold isotropic electron gas (with B0 = 0), to calculate the damping factor α = I{k}. Show that, when ω ≫ ωpe, the
Derive expressions for the phase velocity and group velocity from the dispersion relation (5.26), for wave propagation at arbitrary angles in a cold magnetoplasma.Equation (S sin²0 + P cos²0) n -
Consider a plane electromagnetic wave incident normally on an infinite plane plasma slab occupying the space 0 ≤ x ≤ L, with vacuum for x < 0 and x > L, as indicated in Fig. 30. Use the
Consider a plane electromagnetic wave incident normally on a semi infinite plasma occupying the semi-space x > 0, with vacuum for x < 0, as illustrated in Fig. 29. Denote the incident,
Consider the following closed set of MHD equations in the so-called Chew, Goldberger, and Low approximation,In the equations of this set, involving the pressure tensor P, it is considered that(a)
For the fast and slow MHD waves, let uℓ and ut denote the components of the mass flow velocity that are longitudinal and transverse, respectively, to the direction of propagation. Show that uℓ
A plane electromagnetic wave is incident normally on the surface of a conducting fluid of large but finite conductivity (σ), immersed in a uniform magnetic field B0 such that k ⊥ B0 . Assume that
Include the effect of finite conductivity in the derivation of the equations for the plane Alfven wave propagating along the magnetic field. Show that the linearized equations are satisfied by
For the pure Alfven wave propagating at an angle θ with respect to the magnetostatic field B0, with phase velocity given by (5.10), determine relations between the associated field components B1y,
Show that the Alfven wave propagating along the magnetic field is circularly polarized.
Calculate the speed of an Alfven wave for the following cases:(a) In the Earth's ionosphere, considering that ne = 105 cm–3 , B = 0.5 gauss, and that the positive charge carriers are atomic oxygen
Consider an evanescent plane electromagnetic wave (for which k = iαx̂, with α real), with the wave field vectors E and H proportional to exp [i(k · r – ωt)]. Show that the average value of the
Calculate ψ(x, 0) for a one-dimensional wave packet, when the amplitude function A(k) is given byand when it is given byFor both cases, verify the validity of the uncertainty principle for wave
Consider a one-dimensional wave packet at the instant t = 0, whose amplitude function A(k) is given by the Gaussian functionwhere a and k0 are constants.(a) Show that ψ(x, 0) is also a Gaussian
Consider a one-dimensional wave packet at the instant t = 0, whose amplitude function A(k) is given byShow thatMake a plot of both A(k) and ψ(x, 0) and verify that the uncertainties in x and k
Calculate the Green function of problem 14.8 for the case of free space, for which ω = ck, where cis the speed of light in free space. Show that, in this case, the initial wave packet ψ(r, t0)
Analyze the meaning of the Green function of problem 14.8 for the case when ψ(r, t0) is given by the Dirac delta functionData from Problem 14.8.Show that the time evolution of a wave packet ψ(r, t)
Show that the time evolution of a wave packet ψ(r, t) can be expressed in terms of the initial form of the wave packet ψ(r, t0) aswhere G(r, t; r′, t0) denotes the Green function or kernel of the
Generalize equations (6.1) through (6.5) for the three-dimensional case in Cartesian coordinates. r+∞ 4 (5₁ t) = 1500 A(k) exp [i(kCwt)] dk (6.1)
Consider the superposition (E = E1 + E2) of the following waves:Analyze the resultant polarization for the following cases: E₁₁E exp (ia₁) exp [i(kr - wt)] E2 = 2E2 exp (ia2) exp [i(kr - wt)]
Show that in a general medium (not free space), the time average of the energy density (over one cycle) for harmonic plane waves is given byConsequently, show that the average Poynting vector (energy
The electric field vector of an elliptically polarized plane wave, propagating in free space, can be expressed as(a) Show that the associated magnetic induction vector is(b) Show that the time
Consider a plane electromagnetic harmonic wave propagating along the positive x direction in free space, which can be decomposed into the sum of two waves,Show that E = ŷE₁ exp (ikx - iwt) +2E₂
Consider a plane electromagnetic harmonic wave traveling towards the positive x direction in free space, having the frequency v = 5 x 1015 hertz.(a) What is the associated wavelength?(b) Calculate
Derive the vector wave equations, analogous to (1.8) and (1.9) for the electric and magnetic fields, considering a medium in which there is a space charge density distribution ρ(r, t) and a charge
Consider the following basic equation for the equilibrium of a plasma column with cylindrical geometry (see problem 12.9 in Chapter 12)(a) Verify that, for the theta pinch, this equation reduces
(a) Show that a force-free magnetic field satisfies the relation(b) Let ∇ x B = α(r) Band show that(c) Verify that the surfaces α = constant are made up of magnetic field lines.(d) Show that
In the longitudinal equilibrium pinch shown schematically in Fig. 1, assume that the radial dependence of the current density Jz(r) is such thatCalculate p(r) and Bθ(r) and make a plot showing their
Use the equation for the fluid velocity component (u⊥) normal to B, derived in problem 9. 7 in Chapter 9, to determine the relative orientations of u, B, E, J, and ∇p in a theta-pinch device.Data
For the equilibrium theta pinch produced by an azimuthal current in the theta direction (Jθ), as illustrated in Fig. 6 of Chapter 12, determine expressions for the radial distributions of Jθ(r) and
For the equilibrium Bennett pinch with cylindrical geometry, calculate Bθ(r) using (2.4) and the expression for n(r) given in (3.8). Make a plot showing the radial distributions of p(r), Jz(r), and
Consider a cylindrically symmetric plasma column (∂/∂z = 0, ∂/∂θ = 0) under equilibrium conditions, confined by a magnetic field. Verify that in cylindrical coordinates the radial component
The boundary of the Earth's magnetosphere, in the direction of the Earth-sun line, occurs at a distance where the kinetic pressure of the solar wind particles is equal to the (modified) Earth's
Use (4.1), for a perfectly conducting fluid, and the nonlinear equation of continuity (1.1), to show that the change of B with time in a fluid element is related to changes of density according toUse
Consider a plasma in the form of a straight circular cylinder with a helical magnetic field given byShow that the force per unit volume, associated with the inward magnetic pressure for this
Calculate the diffusion time (τD) and the magnetic Reynolds number (Rm) for a typical MHD generator, with L = 0.1 m, u = 103 m/s, and σ0 = 100 mho/m. Verify that in this case τD is very short, so
A plasma is confined by a unidirectional magnetic induction B of magnitude 5 weber/m2 . Considering that the plasma temperature is 10 keV and β = 0.4, calculate the particle number density. If the
Calculate the minimum intensity of the magnetic induction (B0) necessary to confine a plasma at(a) An internal pressure of 100 atm;(b) A temperature of 10 keV and density of 8 x 1021m–3 .
Derive an energy conservation equation, similar to (1.64), but considering the Parker modified momentum equation and the CGL energy equations, instead of (1.2) and (1.3).Equations Du Pm Dt = J x B -
Consider the energy equation involving the time rate of change of the total pressure dyad P, derived in problem 9.6 in Chapter 9. Show that when this equation is contracted with t he unit dyad 1
Consider a stationary plasma (electrons and one type of ions) under steady-state conditions at a uniform temperature T0 , when perturbed by a point charge +Q placed at the origin of a coordinate
Analyze the Debye potential problem considering only the motion of the electrons (ions stay immobile) and show that in this case the differential equation for the electric potential ϕ(r) is
When the macroscopic neutrality of a plasma is instantaneously perturbed by external means, the electrons react in a such a way as to give rise to oscillations at the electron plasma
Evaluate the negative electrostatic potential ϕw that appears on an infinite plane wall immersed in a plasma consisting of electrons of charge –e and ions of charge Ze, under steady-state
Deduce an expression for the Debye potential for a test particle of charge +Q immersed in a plasma consisting of electrons (charge –e) and ions of charge Ze, the temperature of the electrons and
Using the following expressions for the electron and ion number densitiesin the plasma sheath region formed between an infinite plane and a semi infinite plasma, deduce the differential equation

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