Faculty of Science and Technology EXAMINATION QUESTION PAPER Exam in: Fys-2009 Introduction to Plasma Physics Date: 20161202 Time: 09.00-13.00 Place: Åsgårdvegen 9 Approved aids: Karl Rottmann: Matematisk formelsamling, (arbitrary language edition). Type of Squares. sheets (sqares/lines) : Number of The exam contains 5 pages including this pages incl. cover page cover page: Contact person during the exam: Björn Gustavsson, Phone 776 45668/40724899 Theresa Rexer, Phone 776 464500 NB! It is not allowed to submit rough paper along with the answer sheets. If you do submit rough paper it will not be evaluated. PO Box 6050 Langnes, NO-9037 Tromsø / +47 77 64 40 00 / postmottak@uit.no / uit.no
Problem 1 a (2 p), List three quantitative criteria for a plasma and explain each in a few lines. b (3 p), Use the momentum equation to derive the Boltzmann relation, ( n = n 0 exp qφ ) k B T for steady-state conditions in an unmagnetised iso-thermal plasma where the plasma pressure is: p = k B Tn Here n is the density, T is the temperature, q is the particle charge, φ is the electrostatic potential, and k B is Boltzmann s constant. c (2 p), List the two consequences labeled frozen in magnetic flux lines that follow when the condition of ideal MHD is fulfilled. d (3 p), For magnetised plasmas with constant temperature and a gradient of the plasma-pressure perpendicular to B there will be fluid drifts perpendicular to B explain the physical reason for this drift. The explanations and descriptions to each of b and d should not be more than approximately 1/3 of a page. Problem 2 a (7 p), Consider the motion of a charged test-particle in a uniform magnetic field, B = B z e z, derive the drift-velocity in the gyro-centre approximation for an electrical field that varies in the x-direction: E = E x cos(kx)e x b (8 p), Now look at charged test-particles moving in a uniform and static magnetic field, with a time-varying electrical field: E(t) = E x e x exp( iωt) Will this electrical field induce currents in the plasma? If so, describe the currents. 1
Problem 3 In this problem, small-amplitude high-frequency plasma waves are considered. A simple unmagnetised plasma is assumed, with only one ion species and singly charged particles. a (8 p), Use the two-fluid plasma model to derive the dispersion relation of electrostatic waves for the situation where the momentum transfer collision frequency ν e is small but not negligible. Clearly explain what equations you start with and how you stepwise simplify them to arrive at a dispersion relation. b (2 p), Explain the behaviour of the phase and group-velocities of this wave-mode. c (3 p), Explain what effect the collisions have on the propagation of the waves. d, (7 p) The dispersion relation for X-mode electromagnetic waves, that is: waves propagating perpendicularly to the background magnetic field with the electrical field in the plane perpendicular to the magnetic field, in a cold plasma can be written as n 2 = c2 k 2 ω 2 = 1 ω2 p w 2 (ω 2 ω 2 p ) ω 2 ω 2 H where n is the refractive index and ω p is the plasma frequency and ω H is the upper hybrid frequency, ω 2 H = ω2 p + ω2 c, where ω c is the electron gyrofrequency, assume that ω c < ω p. Sketch the dispersion diagram for this wave-mode. For which frequencies does this wave mode have a resonance? Show that the cut-off frequencies are: ω 1 = 1 [ wc +(ω c +4ωp 2 2 )1/2] ω 2 = 1 [ wc +(ω c +4ω 2 2 p) 1/2] What happens with the wave at the resonance and cutoff frequencies? For which frequencies can the wave propagate? 2
Problem 4 In kinetic theory a plasma is described by the velocity distribution function f s (r,v,t) for each particle species s. a (3 p), The dynamical evolution of the distribution function (phase-space density) f s is described by the Boltzmann equation f s +v f s r +a s f s v = ( fs Describe the concept of phase space, the meaning of the distribution function f s and explain what the terms in this equation describes? b (8 p), Derive the dispersion relation for small-amplitude high-frequency, longitudinal electrostatic waves from the Boltzmann equation for the case when the term on the right-hand side is zero. c (4 p), Even though there are no collisions waves will be damped, explain the physical cause of this wave-damping. ) c 3
The equation of motion of a particle with mass m and electric charge q in electric and magnetic fields E and B: m dv dt = q(e+v B) Maxwell s equations for electromagnetic fields B = 0 Possibly useful physical constants: E = ρ ǫ 0 E = B B = µ 0 J+ǫ 0 µ 0 E m p 1.67 10 27 (kg) m e 9.11 10 31 (kg) q e 1.60 10 19 (C) q e /m e 1.76 10 11 (C/kg) µ 0 4π 10 7 (Vs/Am) ǫ 0 1 10 9 /36/π (As/Vm) 4