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fundamentals of plasma physics

Fundamentals Of Plasma Physics 3rd Edition J. A. Bittencourt - Solutions

For the plasma sheath region formed in the vicinity of a plane wall immersed in a plasma, assume that the ions at the plasma edge of the sheath can be described by a shifted Maxwellian distribution functionwith drift velocity u0 = u0x̂. Prove that the ion flux Гix, at the edge of the sheath, is
From an experimental current-potential curve of a Langmuir probe of area A immersed in a plasma, such as shown in Fig. 5, where the electric potentials are measured with respect to a fixed reference potential, explain how you can determine Ji, Jeo, the space potential ϕs , the floating potential
The Langmuir plasma probe has been widely used in satellites to measure space plasma properties. In one valuable technique, circuits are arranged that measure directlywhere Ip = JpA and A is the probe's area. Use (5.3) to show thatwhich gives directly the electron temperature Te. Next show that Je0
An electron gas (Lorentz gas), in a background of stationary ions, is acted upon by a weak, externally applied electric field E, under steady state conditions. Using the Boltzmann equation for the electrons, with the relaxation model for the collision term (considering a constant collision
Using the distribution function of the previous problem, evaluate the electric current density J to show that the presence of a temperature gradient gives rise to an electric current associated with thermoelectric effects.Data from problem 10An electron gas (Lorentz gas), in a background of
Consider problem 11.10, but taking E = 0 and, instead of the adiabatic case, consider a constant kinetic pressure(a) Show that the electron distribution function is given by(b) Evaluate the heat flux vector q and show that it can be written aswhere the thermal conductivity K is equal to
In the previous problem, consider that n = constant and that f0 is the following modified Maxwell-Boltzmann distribution function:Calculate the electron distribution function f(r, v) and show that the heat flux vector is given byDetermine the expression for the thermal conductivity K.Data from
In problem 11.12, include the presence of an external magnetostatic field B in the z direction and deduce the following expression for the nonequilibrium distribution function:Show that the heat flux vector q can be expressed aswhere K is the dyadic thermal conductivity, which in matrix form can be
The coefficient of viscosity η is defined as the shear stress produced by unit velocity gradient. For the pxz component of the kinetic pressure dyad, for example, we haveAssume the following form for the equilibrium velocity distribution function of the electronswhich indicates the presence of an
Consider the following heat flow equation, derived in problem 8.11 (in Chapter 8), for a stationary electron gas immersed in a magnetic field,Show that this equation can be written in the formwhere K denotes the dyadic thermal conductivity coefficient, given bywhereData from Problem 8.11.Using the
Consider a weakly ionized plasma immersed in a uniform magnetostatic field B0 oriented along the z axis of a Cartesian coordinate system.(a) Show that the diffusion equation for the electrons (with Due/Dt = 0) in the presence of the space charge electric field is given bywith De⊥, DeH, and De∥
Show that the solution of the diffusion equation in the case of cylindrical geometry (see Fig. 5),can be written in terms of Bessel's functions Jm(kr) . Explain how k must be determined so that n(r, t) satisfies the boundary condition n = 0 at r = R0.Figure 5. d² S(r) dr² + 1dS(r) r dr +k² S(r)
From the momentum conservation equation with the MHD approximation [see (6.6)], and the generalized Ohm's law in the simplified form (6.10), but without considering the Hall effect term, derive the following equation:Solve this equation, considering that E = 0 and p = constant, to show that the
Using the relaxation model (or Krook collision model) for the collision term,and the ideal gas law Pe = nekTe, show that the heat flow equation of problem 8.10 becomeswhereis the thermal conductivity.Data from problem 8.10.From the heat flow equation, obtain the following simplified equation for
From the heat flow equation, derived in problem 8.7, obtain the following simplified equation for heat flow in a stationary (u = 0) electron gas:State all the assumptions necessary to obtain this result.Data from problem 8.7.Consider the general transport equation of the previous problem and let
Verify that the energy conservation equation, for the random kinetic energy 1/2mαc2α can be obtained from the viscous stress equation (see problem 8.8) by letting j = k, and summing over k.Data from Problem 8.8In the general transport equation, consider that the property χ(r, v, t) is the random
In the general transport equation of problem 8.6, consider that the property χ(r, v, t) is the random momentum flux, that is, mαCαjCαk· Show that, in this case,where the summation convention on repeated indices is being used. Plug these results in the general transport equation to derive the
Explain the reason why there is no term containing the magnetic flux density B in the energy equation (5.17). De (3) + Dt 2 3pa.ua + (Pa. V) .ua + V· qa= 2 Maua Aa + u Sa . (5.17)
Verify that plane wave solutions to the diffusion equationyields the following dispersion relation between k and ω,Then show that for free electron diffusion we obtainwhereis the isothermal speed of sound in the electron gas and kB is Boltzmann's constant. Next show that for ambipolar diffusion we
Consider the solution of the diffusion equation by separation of variables in the linear geometry of the plasma slab indicated in Fig. 4. Show that the solutions of the equationthat satisfy the boundary condition S = 0 at x = ±L, areandExplain why the solution as a sine series is not a physically
(a) In order to solve the diffusion equationby the method of separation of variables, letand show thatwhere k2 is the separation constant and T0 is a constant.(b) Assuming that S depends only on the x coordinate, show thatwhere k can be either positive or negative, and thatwhere n0(x) = n(x, 0) is
Imagine a horizontally stratified ionosphere in the absence of a magnetic field, constituted only of electrons (density n, temperature T, charge –e, mass me) and one type of ions (density n, temperature T, charge +e, mass mi), subjected to the gravitational field (g), vertical pressure gradient
Same as problem 10.6, but including also a constant and uniform magnetic field B0 .Data from Problem 10.6.Consider the electrons in a plasma acted upon by a small, constant, and uniform external electric field E. Under steady-state conditions with no spatial gradients, obtain an expression for the
Consider the electrons in a plasma acted upon by a small, constant, and uniform external electric field E. Under steady-state conditions with no spatial gradients, obtain an expression for the nonequilibrium distribution function f for the electrons, by applying a perturbation technique to the
Write expressions for the components of the dielectric dyad ε of a multiconstituent magnetized plasma.
What is the physical meaning of a complex conductivity, as given in (5.7) and (5.8)? Consider, for example, that E(r, t) = E(r) exp (–iωt), and calculate the real parts of E(r, t) and of J(r, t) = S · E(r, t). Interpret physically the results considering the phase differences between J and E.
Consider the equation J = S · E, with S as given in (4.23). If we choose a Cartesian coordinate system such that(refer to Fig. 2), verify that in this coordinate system we haveInterpret physically this result with reference to Fig. 2.Equation.Figure 2 Ex = E₁, Ey = 0, Ez = E, and Bo = Boz
Assume that the average velocities of the electrons and ions in a completely ionized plasma, in the presence of constant and uniform electric (E) and magnetic (B0) fields, satisfy, respectively, the following equations of motion:(a) Determine expressions for the steady-state DC conductivities σH,
Consider a solid-state plasma with the same number of electrons (e) and holes (h). Using the linearized Langevin equation (with α = e, h)taking me= mh, νce = νch, assuming a time dependence for both E and Uα of the form exp (–iωt), and choosing a Cartesian coordinate system with the z axis
For a perfectly conducting fluid characterized by a scalar pressure, under steady-state conditions, use the equation of motion (6.6) and the generalized Ohm's law (6.10) to derive the following equation for the fluid velocity component perpendicular to B:Equations Tn B² (E -1 Vp) x ne E- - x B
Derive an energy equation, of higher order than (4.14), involving the total time rate of change of the total pressure dyad, that is, DP/Dt.Equation 4.14 D/3p Dt 2 + 3Pv.u+V.q+(P. V) . u = J'. E' 2 (4.14)
Obtain an expression for the heat flux triad Q for the plasma as a whole, defined aswhere Cα0 = Cα + wα, in terms of a summation over the heat flux triad for each species Qα and of terms involving the diffusion velocity wα. Then, simplify this expression for the isotropic case. Ω=Σpma < Cao
In equations (1.5) and (1.6), explain the reason why the mass flux Jm is given by ρmU, whereas the electric charge flux J is not given by ρu.Equations Jm = Σnamaua = Pmu a (1.5)
Show that when there is no heat flow (q = 0), no joule heating (J′ · E′ = 0), and when the pressure tensor is isotropic given by P = p1, the energy equation (4.14) reduces to the following adiabatic equation:Equation 4.14 -5/3 PPm = constant
Show that the total kinetic energy density of all species in a fluid can be written as the sum of the thermal energy density of the whole fluid plus the kinetic energy of the mass motion, that iswhere Σ' έPma Σέρπα α 3р 2 + Σipmana a
Consider the general transport equation of the previous problem and let the property χ(r, v, t) be the random flux of thermal kinetic energy, that is,Show that (considering the Lorentz force for F)Using these results in the general transport equation, derive the following equation, known as the
Derive the following general transport equation, similar to (2.13), for the case when the quantity χ depends on r, v, and t,Equation 2.13 Ə əx :(na a) - na < Ət Ət +V. (na a) - na < (v⋅V)x >a −na < (a · √₂)X >a = 5(no a) | coll [8(na [8(na St
Consider a uniform mixture of different fluids (all spatial derivatives vanish), with no external forces, such that the equation of motion for the a species becomes(a) Show that the time rate of change of the total fluid kinetic energy density, wk, is given bywhere(b) Consider now the total fluid
In order to investigate the effect of the collision term (4.11) in the macroscopic fluid motion, consider a uniform mixture of different fluids (all spatial derivatives vanish), with no external forces, so that the equation of motion for the a species reduces toSolve this equation to determine
(a) From Maxwell equations,where E and H denote the electric and magnetic fields in a plasma, p denotes the electric charge density nq, and J the electric current density nqu, show thatwhere ∈0μ0 (E x H) is the electromagnetic momentum density, and T is the electromagnetic stress dyad whose
Consider the following simplified steady-state equation of motion, for each species in a fluid plasma,where the electric (E) and magnetic (B) fields are uniform, but the number density (n) and the kinetic pressure (p) have a spatial gradient. Taking the cross-product of this equation with B show
Show that the average thermal energy per particle, for a gas in thermodynamic equilibrium, is equal to 1.292 × w–4 eV/K.
Use the laws of conservation of momentum and of energy in a collision to show that the Maxwell-Boltzmann distribution functionsatisfies the following equation of detailed balance 3/2 f(v) = n(27KT) exp m(v - u)²- 2kT
Consider two large chambers that communicate with each other only through a small aperture of area A in a very thin wall, as indicated in Fig. 11. The chambers contain an ideal gas at a very low pressure, such that the particle mean free path is much larger than the dimensions of A. The
The temperature of a plasma, in thermal equilibrium with a neutral gas, can be determined experimentally by measuring the electron density ne with a microwave transmission experiment, for example, and the neutrals number density in a particular excited state through the rate of transitions to a
Consider the particles in the Earth's atmosphere under equilibrium conditions in the presence of the Earth's gravitationl field. Assume a horizontally stratified (x, y plane) atmosphere with constant temperature T and consider a constant valuefor the acceleration due to gravity. Derive an
A plasma is in equilibrium under the action of an external electrostatic field E and a gravitational field g. Consider that the plasma as a whole is moving with constant velocity u, with respect to the observer's frame of reference. Write down the distribution function for the species of type a for
A gas of O2 molecules is in the equilibrium state with number density n and absolute temperature T. Calculate the average value of the reciprocal of the particle velocity, that is, < 1/v >.
Consider a gas mixture containing ne electrons and ni oxygen ions per unit volume, all in thermal equilibrium at a temperature T and having no drift velocity.(a) Resolve the motion of the particle species into the motion in space of the center of mass plus the relative motion of one species with
Derive an expression for the Doppler intensity profile (thermal broadening) of a spectral line emitted near the central frequency ν0, assuming that the emitting atoms have a Maxwellian velocity distribution. Ignore all other factors that contribute to the shape of the line.(1) The change in
The entropy of a system can be expressed in terms of the distribution function asProve that, for a Maxwellian distribution function, the entropy satisfies the following thermodynamic relations:where N is the total number of particles in the system, V is the total volume, and E = 3NkT/2 is the total
The distribution of thermal kinetic energies E, for a gas in the Maxwellian state, is given byCalculate the most probable energy and show that the velocity of the particles, which have this energy, is equal to (kT/m )1/2. G(E) = 2n E1/2 п1/2(kT)3/2 exp E kT
Consider a gas of particles consisting of only one species and characterized by the Maxwell-Boltzmann equilibrium distribution function (with u = 0)(a) Show that the total number of particles crossing a unit area per unit time, lying within an element dΩ of solid angle, is given bywhere θ denotes
A two-dimensional gas, consisting of only one species and whose particles are restricted to move in a plane (the z = 0 plane), is characterized by a homogeneous, isotropic, two-dimensional Maxwell-Boltzmann distribution function (with u = 0),where n0 represents the number of particles per unit
Consider (5.6.4), which is the solution of the Boltzmann equation with the relaxation model for the collision term, in the absence of external forces and spatial gradients, and when fα0 and the relaxation time τ are time-independent. Show that, according to this simplified equation, we
(a) Show that the time rate of increase of momentum in an infinitesimal volume element d3r = dx dy dz inside a gas of number density n, as a result of particles of mass m entering d3r with average velocity u, is given by –∇ · (nmuu) d3r.(b) If the infinitesimal volume element d3r moves with
A plasma is made up of a mixture of various particle species, the type α species having mass mα, number density nα, average macroscopic velocity uα, random velocity Cα = v – uα, scalar pressure Pα = nαkTα, temperatureand heat flow vector qα = (nαmα/2) < cα2cα >. Similar
Verify, by symmetry arguments, that there are only ten independent elements in the thermal energy flux triad Q. Note that, according to its definition, Qijk = nm < ci cj ck > is symmetric under the interchange of any two of its three indices.
For the loss-cone distribution function of problem 5.3 (in Chapter 5), show thatCompare these results with those of problem 6.2(d) and provide physical arguments to justify the difference in the perpendicular part of the average thermal energy.Data from Problem 5.3.The electrons inside a system of
Suppose that the peculiar (random) velocities of the electrons in a given plasma satisfy the following modified Maxwell-Boltzmann distribution function (considering u = 0),(a) Verify that the electron number density is given by n0.(b) Considering a Cartesian coordinate system with the z axis
Consider a system of particles characterized by the distribution function given in problem 5.1 (in Chapter 5).(a) Show that the absolute temperature of the system is given bywhere m is the mass of each particle and k is Boltzmann's constant.(b) Obtain the following expression for the pressure
Consider a one-dimensional harmonic oscillator whose total energy can be expressed bywhere c is a constant and x its displacement coordinate. Show that the trajectory described by the representative point of the oscillator, in phase space, is an ellipse. = // (mv² + cx²) E =
The entropy of a system can be expressed, in terms of the distribution function, asShow that, for a system that obeys the collisionless Boltzmann equation, the total time derivative of the entropy vanishes. k [ [ ƒ ln (f) V S = -k f ln(f) d³r d³v
Show that the Vlasov equation for a homogeneous plasma under the influence of a uniform external magnetostatic field B0, in the equilibrium state, is satisfied by any homogeneous distribution function, f (v∥ , v⊥), which is cylindrically symmetric with respect to the magnetostatic field.
(a) Show that the Boltzmann equation, in cylindrical coordinates, can be written aswhere the dot over the symbols stands for the time derivative operator d/dt and where(b) Show, by direct substitution, that in the presence of an azimuthally symmetric magnetic field (in the z direction) a function
Consider the motion of charged particles, in one dimension only, in the presence of an electric potential V(x). Show, by direct substitution, that a function of the formis a solution of the Boltzmann equation under steady-state conditions. f = f(mv² +qV)
The electrons inside a system of two coaxial magnetic mirrors can be described by the so-called loss-cone distribution functionwhere v∥ and v⊥ denote the magnitudes of the electron velocities in the directions parallel and perpendicular to the magnetic bottle axis, respectively, and where
Consider the following two-dimensional Maxwellian distribution function:(a) Verify that n0 represents correctly the particle number density, that is, the number of particles per unit area.(b) Sketch, in a three-dimensional perspective view, the surface for this distribution function, plotting f(vx,
Consider a system of particles uniformly distributed in space, with a constant particle number density n0, and characterized by a velocity distribution function f(v) such thatwhere K0 is a nonzero positive constant. Determine the value of K0 in terms of n0 and v0. f(v) = Ko f(v) = 0 otherwise, for
Consider a rocket once it is beyond the Earth's gravitational field.Let:v = constant velocity of the exhaust gas relative to the rocket.u(t) = instantaneous velocity of the rocket.M(t) = instantaneous total mass of the rocket.–dM/dt =constant time rate of decrease of M(t), that is, the mass
Solve the equation of motion to determine the velocity and the trajectory of an electron in the presence of a uniform magnetostatic field B = B0ẑ, and an oscillating electric field given byConsider the same assumptions and initial conditions as in the previous problem.Data from Problem
Using the Maxwell equation (1.5.3) (Eq. 5.3 in Chapter 1) and the equation (3.34) which defines the plasma conductivity dyad S, and considering the time variation indicated in (3.1), show thatwhere ε is the plasma electric permittivity dyad given bywhere 1 denotes the unit dyad, which in Cartesian
For an electron with initial velocity v0x̂ and initial position x0x̂, acted upon by an electric field E = x̂E cos( kx – ωt), show that its velocity is given byUsing a perturbation approach, in which to lowest order E = 0, show thatNotice that the velocity perturbation will be large only when
(a) Assume that f(t), in problem 4.9, is given by exp (–αt). Show that, in this case, ξ(t) satisfies the Bessel equation of zero order,where τ = (Ωc/2α) exp (–αt). Determine the two solutions of this equation which satisfy the initial conditions stated in problem 4.9 and interpret them
Consider the motion of a charged particle in a spatially uniform magnetic field that varies slowly in time as compared to the particle cyclotron period.(a) Show that the equation of motion can be written in vector form aswhere Ωc(t) = –qB(t)/m.(b) Considering that B(t) = ẑB0f(t), where B0 is
Consider the motion of an electron in a spatially uniform magnetic field B = Bzẑ, such that Bz has a slow time variation given bywhere B0 and α are positive constants, and |αt| ≪ 1. Assume the followinginitial conditions: r(0) = (rc, 0, 0) and v(0) = (0, v⊥0, 0), where rc is the Larmor
Consider the motion of an electron in the presence of a uniform magnetostatic field B = B0ẑ, and an electric field that oscillates in time at the electron cyclotron frequency Ωc, according to(a) What type of polarization has this electric field?(b) Obtain the following uncoupled differential
Integrate (3.49) and (3.50) to determine the particle trajectory in the plane normal to Band sketch the path of the particle for q > 0 and q V₁ = Vm + ¹/E₁L t e-inet m (3.49)
Consider an electron acted upon by a constant and uniform magnetic field B = B0ẑ, and a uniform but time-varying electric field E = ŷEy0 sin(ωt). Assume that the initial conditions are such that the motion takes place in the (x, y) plane and that at t = 0 the electron is at rest (v0 = 0) at
Solve the equation of motion to determine the transient response of a charged particle in the presence of a spatially uniform AC electric field E(t) = x̂E sin(ωt), which is switched on at t = 0. Assume that initially, at t = 0, the particle is at rest at the origin. Make a plot of the particle
Describe, in a semiquantitative way, the motion of an electron in the presence of a constant magnetic field B = B0ẑ and a time-varying electric field given bywhere E0 and B0 are positive constants and Ωc = eB0/me. What type of polarization has this electric field? = Eo(x+iỹ) e-inct
With reference to a magnetic field pointing along the z axis (B = B0ẑ), describe the type of polarization of the following electric field:Make a drawing that shows the orientation of E for the instants t = 0, t = π/(2ω), and t = π/ω. How can you represent this electric field in complex
For the magnetic mirror system of problem 3.3 suppose that the axial magnetic field changes in time, that is Baxial = B(z, t)ẑ. Considering that the magnetic momentis an adiabatic invariant (note that its value is the same at z = 0 and atshow that the longitudinal adiabatic invariant can be
Consider a system of two coaxial magnetic mirrors whose axis coincides with the z axis, being symmetrical about the plane z = 0, as shown schematically in Fig. 20. Describe semiquantitatively the motion of a charged particle in this magnetic mirror system considering that at z = 0 the particle has
Describe semiquantitatively the motion of an electron under the presence of a constant electric field in the x direction,and a space varying magnetic field given bywhere E0, B0, and α are positive constants, |αx|≪ 1 and |αz| ≪ 1. Assume that initially the electron moves with constant
Verify if there is any drift velocity for a charged particle in a magnetic field given bywhere By(x) and ∂By/∂x are very small quantities. Does this field satisfy the Maxwell equation ∇ x B = 0? B = By(x)ŷ + Boz
Consider the magnetic mirror system shown in Fig. 20. Suppose that the axial magnetic field is given bywhere B0 and a0 are positive constants, and that the mirroring planes are given by z = –zm and z = zm·(a) For a charged particle trapped in this mirror system, show that the z component of the
Consider a toroidal magnetic field, as shown in Fig. 21.(a) Show that the magnetic flux density along the axis of the torus is given by where Ba denotes the magnitude of B at the radial distance r = a.(b) In what direction is the gradient drift associated with the radial variation of Bϕ? Examine
Consider a spatially nonuniform magnetostatic field expressed in terms of a Cartesian coordinate system bywhere B0 and α are positive constants, |αx| ≪ 1 and |αz| ≪ 1.(a) Show that this magnetic field is consistent with Maxwell equations, so that both gradient and curvature terms are
The Earth's magnetic field can be represented, in a first approximation, by a magnetic dipole placed in the Earth's center, at least up to distances of a few Earth radii (RE)·(a) Using the fact that, at one of the magnetic poles, the field has a magnitude of approximately 0.5 gauss near the
Imagine an infinite straight wire carrying a current I and uniformly charged to a negative electrostatic potential ϕ. Analyze the motion of an electron in the vicinity of this wire using first-order orbit theory. Sketch the path described by the electron, indicating the relative directions of the
The field of a magnetic monopole can be represented bywhere λ is a constant. Solve the equation of motion to determine the trajectory of a charged particle in this field. B(r) = A r 73
Analyze the motion of a charged particle in the field of a magnetic dipole. Determine the two constants of the motion and analyze their physical meaning.
In general the trajectory of a charged particle in crossed electric and magnetic fields is a cycloid. Show that, if v = vox̂, B = Boẑ, and E = Eoŷ, then for vo = Eo/Bo the path is a straight line. Explain how this situation can be exploited to design a mass spectrometer.
Write down, in vector form, the relativistic equation of motion for a charged particle in the presence of a uniform magnetostatic field B = Boẑ, and show that its Cartesian components are given bywhereand where β = ν/c. Show that the velocity and trajectory of the charged particle are given by
Analyze the motion of a relativistic charged particle in the presence of crossed electric (E) and magnetic (B) fields that are constant in time and uniform in space. What coordinate transformation must be made in order to transform away the transversal electric field? Derive equations for the
Derive the relativistic equation of motion in the form (1.4), starting from (1.1) and the relation (1.2). dp dt = F =q(E+vx B) (1.1)
Consider a particle of mass m and charge q moving in the presence of constant and uniform electromagnetic fields given by E = Eoŷ and B = Boẑ. Assuming that initially (t = 0) the particle is at rest at the origin of a Cartesian coordinate system, show that it moves on the cycloidPlot the
For an electron and an oxygen ion O+ in the Earth's ionosphere, at 300 km altitude in the equatorial plane, where B ≃ 0.5 * 10–4 tesla, calculate:(a) The gravitational drift velocity vg .(b) The gravitational current density Jg , considering ne = ni = 1012 m–3 .Assume that g is perpendicular
Calculate the cyclotron frequency and the cyclotron radius for:(a) An electron in the Earth's ionosphere at 300 km altitude, where the magnetic flux density B ≃ 0.5 * 10–4 tesla, considering that the electron moves at the thermal velocity (kT/m), with T = 1000 K, where k is Boltzmann's

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