Lecture10: Plasma Physics 1. APPH E6101x Columbia University
|
|
- Alicia Crawford
- 5 years ago
- Views:
Transcription
1 Lecture10: Plasma Physics 1 APPH E6101x Columbia University
2 Last Lecture - Conservation principles in magnetized plasma frozen-in and conservation of particles/flux tubes) - Alfvén waves without plasma pressure)
3 This Lecture Introduction to plasma waves Basic review of electromagnetic waves in various media conducting, dielectric, gyrotropic, ) Basic waves concepts especially plane waves) Electromagnetic waves in unmagnetized plasma Electrostatic waves in unmagnetized plasma
4 Wave Helmholtz) Equation ' '-I'" n l" II "" V 'J '41 N t' r" with c constant) ", 'P tj lt X '- 4Īi,. "- 'l 'ḷ l- J ") o ln " '; " '- It " ''K "", C\ \ n '- ' U +1 'l J. " \J\ -P.l C" \ rw \ t\, _l.1 u1 1 Q ': t i Q 1 J i 'U - J r" ') '1 t ei ) eta e " \l 3,. "" oq '- 'I'" \t 'l \,.
5 Plane Waves y. '- u. q: ù: Q, \A S- r l I- 't Ul,. u '" "- "" \Q \c ') Q v: t '- :r '- L.ø J -. ).. \ 'l " vi,. ' ;\ tj il \- 1 t 0,. '- T \1 tj i \..t \1 J \- t1 i \& '" \i v 'c ii ct 'X ) +i J! 1 'l 'I.J '", \) 0 /' VI r. v 1 '- \l u "- 1: \l \ '" '" Q Q " '" i4 l.) ') i' t ei é \i cy\ " ", t '4 :i j t \f ϕ = k r ωt "I '" I- J Ii i ta t J Q \I i i '4,- \l t t
6 Wave Eq in Multiple Dimensions The wave equation has solutions that are plane monochromatic waves of the form E = Ê exp[ik r ωt)] B = ˆB exp[ik r ωt)] j = ĵ exp[ik r ωt)]. 6.8) U '\, ':, i ",'-J.; S a '1 i l \Ò i,. ': -l r ñ ;t t L J "' \: lj.. 1. tv '" Vi ϕ = k r ωt \ tv -- y. "- i "- tv l " Here, k is the wave vector, which describes the direction of wave propagation. The magnitude of the wave vector is related to the wavelength by k = 2. The wave J- e: -i " \\ \,. Yo P \ il.a 1- \ t1, '- 3 r+\ )\ i: l c: \ ). r II '". 'l V 0 '0 C' :, r: l I, j '- \ - A. \. l 1 N t: Ql 'I "" '" \o '" \e '),, 'c. \)., 1 /" \ Q. \l,. '- "Q i II i': i. '- ", I "" r- '- "" C1 l '" 'e I 'Q '. it I l " i' 0 ".J J x "- a: 'ë.. ) " r
7 Wave Eq in Cylindrical Coordinates. '- '" í --,. 'i li rc\ "-. "' "' 1 + fi ro t" " l.".t \ "" tl " -i fl Q) "' +. ti 4- \ "" " " "',, N '\ '- t' t '- '- ') y -l, Q I, t"\ t I" L -l "Q,. '\J L u. '\,. "- U Cl., If 0 r: II l '- tv C" "\ ' \N \I - 't Vl ve -i 0.) -; 2: J tj -l r '-, i \I.- Ml - ṙ- t ". 0 i -" - '- ". n " II,,. oj \ " '- \\ i i \ IV.J v '+ O "- T J /' ),.. ii Vl - + N t'\j W V\ l \ 'ò I V\ ci\ v it td \U \t ) -\ L. a J \\ g 'l D tj '1 -
8 Wave Packets and Group Velocity '" l:. V \i \l "" W :; -- \. Cc '1 t tv i'" i J t 'l J" 1-, i - I-.l r. "' J :i,. '- II u J\ i " tj Vt 31n V tj '" t ) "" Jil C "". t /', 0,,:. X 'u ') oj 3," -- 0 rl :i II '" i I V \. t :f l1., t\ 1:. 'l '- -. J. I/ 0 Q W ø:" "' '- ) J l tt t. l) '- '" '" T,., 4- V i 'ii ". f '-.- of,,. /' )t '0 '" l' "" ij "-. " \j" :i il l :k - 'l I ') 0, ) 1 t ' -t 3'" 4 1 II 11 ti i -l.,. lc '" \",. \0- "j '- I- t.- ;i - t - ') ) q \L \! \. \U \L J
9 Wave Dispersion - P Q.. 'Q :i j J t4 \A s q ") :e \l \l t ') ") \;, 1; J.3 " \i ") i- o \ J "; \J. -a i- '" 'e.j \J '" ) "\ Q V' '" r l- ') a) ω \I '0 "" \I \J l: α b) ω \J α a- ;; dispersive medium k k dω/dk β ω non-dispersive medium dω/dk ω ). " \ i \L k \i C: J t o V\ 3i'" k i 1 \ i l. v '0 v 'l i i "" J ) q. \0. J \0 t l.l '\l "" vt, i. '" '. I': l. 0 1 '" "" t- l., '- n 't! i l'" '. '\ i: t -- u "" J VI \l 0 " \. t li i- ' \! ') ), \A,.
10 E ik Ê, E ik Ê, PDE s become algebraic! "- \. \ " \ - 'o C r: 0 \. o o ) '0 "" 'J f' \f t' Phasors, Sine, Cosine 'i "Î c: 'i \ 0-./ Ii 'i 'l J j IJ '" ; "" fv 0 J 0 ) 0 \ L \ "- '- I j I ii I '" i "- v- li c: \) \. ". \. -v \\ /' /' '+ \) -J '\ j J t E iωê J \ \D "- -L. Î \\ \ "- t- \ ;: \: :) " \) ' L Q l \J 'J,,,.. ) U 'V '- ' o. '" 'i J l x +- i' '\ x If,J "- Q l" '- "- \/ 1I. ' \) r- l./ /" -- Y. l./ y. ') N 't lj /' i. -\ ' II l :r 1 ':r Cl - '- J J _- 0 r. Q v' r 't ) -irj 1I t; r- " c- l/ '- c.- - VI ct r" -- 0 Il 2$ II \1 "" I -;rj c: Il) \\ "v \\ ". tj j \. \l V\ c;. CD 1 't F QJ "" cr i \l C C: \)
11 Review of EM Waves E = B t ) E B = µ 0 j + ε 0 t E) = B t = t B) = µ 0 ε 0 2 E t 2 µ 0 j t
12 Review of EM Waves E = B t ) E B = µ 0 j + ε 0 t ε ω = I + i ωε 0 σ ω. ε 0 E t + j = ε 0 εω) E t jω) = σ ω) Eω) εω) = 1 + i ωε 0 σ ω)
13 Review of EM Waves E) + 1 c 2 2 E t 2 = µ 0 ik Ê = iω ˆB j t ik ˆB = iωε 0 µ 0 Ê + µ 0 ĵ n, we discuss the wave equat 0 k k Ê) = kk k ] 2 I)Ê udes 6.7) can be transforme {kk k 2 I + ω2 c 2 I + iωµ 0σ ω) } } {kk k 2 I + ω2 c 2 εω) All of the plasma physics here Ê = 0 Ê = 0
14 k x k x k 2 + ω2 c 2 ε xx k y k x + ω2 c 2 ε yx k z k x + ω2 c 2 ε zx Normal Modes Dispersion Relation) k x k y + ω2 c 2 ε xy k y k y k 2 + ω2 c 2 ε yy k z k y + ω2 c 2 ε zy = 0 = Dω, k) = det k x k z + ω2 c 2 ε xz k y k z + ω2 c 2 ε yz k z k z k 2 + ω2 c 2 ε zz ] [kk k 2 I + ω2 c 2 εω) Ê x Ê y Ê z = ). 6.29)
15 Waves in Unmagnetized Plasma Collisionless reactive) m dv dt = qêeik r ωt) ĵ = nqˆv = i ne2 ωm Ê σ xx = σ yy = σ zz = i ne2 ωm 90 out of phase ˆv = m iω + ν m ) ˆv = q Ê [ ν m ω 2 + ν 2 m In phase low-frequency) Collisional resistive) + iω ω 2 + ν 2 m ] q m Ê ε xx = ε yy = ε zz = 1 + i ωε 0 σ yy = 1 ω2 pe ω 2 ω pe = ne 2 ε 0 m e ) 1/2
16 Waves in Unmagnetized Plasma ] 0 = Dω, k) = det [kk k = = = 2 I + ω2 c 2 εω). ω 2 c 2 1 ω2 pe ω 2 ) 0 k 2 + ω2 c 2 1 ω2 pe ω 2 ) k 2 + ω2 c 2 1 ω2 pe ω 2 ) 0 ) ) Ê x Ê y Ê z = 0. ω 2 = ω 2 pe + k2 c ) v ϕ = ω 2 pe k 2 + c2 ) 1/2 v gr = kc 2 ω 2 pe + k2 c 2 ) 1/2
17 Waves in Unmagnetized Plasma ) ˆv = m iω + ν m ) ˆv = q Ê [ ] ν m iω q ω 2 + νm 2 + ω 2 + νm 2 m Ê ω 2 = ω 2 pe + k2 c 2 k = 1 c ω 2 ωpe 2 ) 1/2 1 + iν m /ω) Collisional resistive)
18 Interferometry a) laser Plasma detector microwave source µw Plasma ϕ = 2π λ horn antenna [N x) 1] dx b) directional coupler phase shifter attenuator detector Fig. 6.5 a) Laser interferometer in Mach-Zehnder arrangement, b) microwave interferometer. The optical arrangement uses partially-reflecting and fully-reflecting mirrors. The analog to a partially reflecting mirror is the directional coupler for microwaves
19 N = Interferometry ϕ = 2π λ 1 ω 2 pe /ω [N x) 1] dx ω 2 pe ω 2 = Table 6.1 Cut-off densities for microwave and laser interferometers Wavelength Frequency Cut-off-density Source λ f n co m 3 ) Microwave 3 cm 10 GHz mm 37 GHz mm 75 GHz HCN-laser 337 µm 890GHz CO 2 laser 10.6 µm 28THz He-Ne laser 3.39 µm 88THz µm 474THz n n co
20 Interferometry N = ϕ = 2π λ [N x) 1] dx 1 ω 2 pe /ω ω 2 pe ω 2 = n n co interferometer signal 0 current pulse cut-off a) 4096 necm 3 ) b) ncut-off t µs) t ms) Fig. 6.6 a) Interferogram in a pulsed gas discharge. b) Reconstruction of the decaying electron density by counting interferometer fringes
21 Electrostatic Waves n t + x nv) = 0 m v = q dφ dx γ n dnk B T ) dx Electron Pressure Force
22 Electrostatic Plasma Waves n t + x nv) = 0 iω ˆn + ikn 0 ˆv = 0 m v = q dφ dx γ n dnk B T ) dx iωm ˆv = ikq ˆφ ikγ k B T ˆn Electron Pressure Force
23 Electrostatic Plasma Waves ω = ω 2 pe k2 v 2 Te) 1/2 = ω pe 1 + 3k 2 λ 2 ) 1/2 De Electron Pressure Force Electron Pressure Force
24 Electrostatic Ion Sound Waves Ion Pressure Force iωm i ˆv i = eê ik n i0 γ i k B T i ) ˆn i 0 = eê ik n e0 k B T e ) ˆn e ˆn i = ˆn e = ek iω 2 m i + ik 2 Ê γ i k B T i e Ê, ikk B T e Look! No electron acceleration Electron Pressure Force ik Ê = ni0 e 2 ) ε 0 m i k iω 2 + ik 2 γ i k B T i /m i Ê + ne0 e 2 ε 0 k B T e ) 1 ik Ê
25 Electrostatic Ion Sound Waves εk, ω) = 1 ω 2 pi ω 2 k 2 γ i k B T i /m i + 1 k 2 λ 2 De ) ω 2 = k 2 γ i k B T i + ω2 pi λ2 De m i 1 + k 2 λ 2 De ω kc s 1 + k 2 λ 2 De T e T i. d C s = ω pi λ De stic wave.
26 Important Wave Concepts Linear vs. nonlinear Dispersion Phase and group velocity W = Polarization and wave structure Energy & intensity Poynting s Theorem) Inhomogeneity W t dv H B + E D) V N ds = dv J E, S V
27 Next Lecture Chapter 6: Plasma Waves Waves in magnetized plasma
Lecture11: Plasma Physics 1. APPH E6101x Columbia University
Lecture11: Plasma Physics 1 APPH E6101x Columbia University 1 Last Lecture Introduction to plasma waves Basic review of electromagnetic waves in various media (conducting, dielectric, gyrotropic, ) Basic
More informationAPPH 4200 Physics of Fluids
APPH 42 Physics of Fluids Problem Solving and Vorticity (Ch. 5) 1.!! Quick Review 2.! Vorticity 3.! Kelvin s Theorem 4.! Examples 1 How to solve fluid problems? (Like those in textbook) Ç"Tt=l I $T1P#(
More informationChapter 9 WAVES IN COLD MAGNETIZED PLASMA. 9.1 Introduction. 9.2 The Wave Equation
Chapter 9 WAVES IN COLD MAGNETIZED PLASMA 9.1 Introduction For this treatment, we will regard the plasma as a cold magnetofluid with an associated dielectric constant. We then derive a wave equation using
More informationHeating and current drive: Radio Frequency
Heating and current drive: Radio Frequency Dr Ben Dudson Department of Physics, University of York Heslington, York YO10 5DD, UK 13 th February 2012 Dr Ben Dudson Magnetic Confinement Fusion (1 of 26)
More informationFuture Self-Guides. E,.?, :0-..-.,0 Q., 5...q ',D5', 4,] 1-}., d-'.4.., _. ZoltAn Dbrnyei Introduction. u u rt 5,4) ,-,4, a. a aci,, u 4.
te SelfGi ZltAn Dbnyei Intdtin ; ) Q) 4 t? ) t _ 4 73 y S _ E _ p p 4 t t 4) 1_ ::_ J 1 `i () L VI O I4 " " 1 D 4 L e Q) 1 k) QJ 7 j ZS _Le t 1 ej!2 i1 L 77 7 G (4) 4 6 t (1 ;7 bb F) t f; n (i M Q) 7S
More information( ). One set of terms has a ω in
Laptag Class Notes W. Gekelan Cold Plasa Dispersion relation Suer Let us go back to a single particle and see how it behaves in a high frequency electric field. We will use the force equation and Maxwell
More informationElectrodynamics II: Lecture 9
Electrodynamics II: Lecture 9 Multipole radiation Amol Dighe Sep 14, 2011 Outline 1 Multipole expansion 2 Electric dipole radiation 3 Magnetic dipole and electric quadrupole radiation Outline 1 Multipole
More informationMicroscopic-Macroscopic connection. Silvana Botti
relating experiment and theory European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Temporary Address: Centre for Computational
More informationWaves in Linear Optical Media
1/53 Waves in Linear Optical Media Sergey A. Ponomarenko Dalhousie University c 2009 S. A. Ponomarenko Outline Plane waves in free space. Polarization. Plane waves in linear lossy media. Dispersion relations
More informationElectromagnetically Induced Flows in Water
Electromagnetically Induced Flows in Water Michiel de Reus 8 maart 213 () Electromagnetically Induced Flows 1 / 56 Outline 1 Introduction 2 Maxwell equations Complex Maxwell equations 3 Gaussian sources
More informationModule I: Electromagnetic waves
Module I: Electromagnetic waves Lectures 10-11: Multipole radiation Amol Dighe TIFR, Mumbai Outline 1 Multipole expansion 2 Electric dipole radiation 3 Magnetic dipole and electric quadrupole radiation
More informationIntroduction to Plasma Physics
Mitglied der Helmholtz-Gemeinschaft Introduction to Plasma Physics CERN School on Plasma Wave Acceleration 24-29 November 2014 Paul Gibbon Outline Lecture 1: Introduction Definitions and Concepts Lecture
More informationSimple medium: D = ɛe Dispersive medium: D = ɛ(ω)e Anisotropic medium: Permittivity as a tensor
Plane Waves 1 Review dielectrics 2 Plane waves in the time domain 3 Plane waves in the frequency domain 4 Plane waves in lossy and dispersive media 5 Phase and group velocity 6 Wave polarization Levis,
More information9 The conservation theorems: Lecture 23
9 The conservation theorems: Lecture 23 9.1 Energy Conservation (a) For energy to be conserved we expect that the total energy density (energy per volume ) u tot to obey a conservation law t u tot + i
More informationA Brief Revision of Vector Calculus and Maxwell s Equations
A Brief Revision of Vector Calculus and Maxwell s Equations Debapratim Ghosh Electronic Systems Group Department of Electrical Engineering Indian Institute of Technology Bombay e-mail: dghosh@ee.iitb.ac.in
More information6.1. Linearized Wave Equations in a Uniform Isotropic MHD Plasma. = 0 into Ohm s law yields E 0
Chapter 6. Linear Waves in the MHD Plasma 85 Chapter 6. Linear Waves in the MHD Plasma Topics or concepts to learn in Chapter 6:. Linearize the MHD equations. The eigen-mode solutions of the MHD waves
More informationModule I: Electromagnetic waves
Module I: Electromagnetic waves Lecture 9: EM radiation Amol Dighe Outline 1 Electric and magnetic fields: radiation components 2 Energy carried by radiation 3 Radiation from antennas Coming up... 1 Electric
More informationNONLINEAR CONTINUUM FORMULATIONS CONTENTS
NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell
More informationFull-Wave Maxwell Simulations for ECRH
Full-Wave Maxwell Simulations for ECRH H. Hojo Plasma Research Center, University of Tsukuba in collaboration with A. Fukuchi, N. Uchida, A. Shimamura, T. Saito and Y. Tatematsu JIFT Workshop in Kyoto,
More informationUniform Plane Waves Page 1. Uniform Plane Waves. 1 The Helmholtz Wave Equation
Uniform Plane Waves Page 1 Uniform Plane Waves 1 The Helmholtz Wave Equation Let s rewrite Maxwell s equations in terms of E and H exclusively. Let s assume the medium is lossless (σ = 0). Let s also assume
More informationAPPH 4200 Physics of Fluids
APPH 4200 Physics of Fluids Rotating Fluid Flow October 6, 2011 1.!! Hydrostatics of a Rotating Water Bucket (again) 2.! Bath Tub Vortex 3.! Ch. 5: Problem Solving 1 Key Definitions & Concepts Ω U Cylindrical
More informationPart VIII. Interaction with Solids
I with Part VIII I with Solids 214 / 273 vs. long pulse is I with Traditional i physics (ICF ns lasers): heating and creation of long scale-length plasmas Laser reflected at critical density surface Fast
More informationA DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any
Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»
More informationLecture 2 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lecture Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell 1. Dispersion Introduction - An electromagnetic wave with an arbitrary wave-shape
More informationWaves in plasmas. S.M.Lea
Waves in plasmas S.M.Lea 17 1 Plasma as an example of a dispersive medium We shall now discuss the propagation of electromagnetic waves through a hydrogen plasm an electrically neutral fluid of protons
More informationWorked Examples Set 2
Worked Examples Set 2 Q.1. Application of Maxwell s eqns. [Griffiths Problem 7.42] In a perfect conductor the conductivity σ is infinite, so from Ohm s law J = σe, E = 0. Any net charge must be on the
More information2nd Year Electromagnetism 2012:.Exam Practice
2nd Year Electromagnetism 2012:.Exam Practice These are sample questions of the type of question that will be set in the exam. They haven t been checked the way exam questions are checked so there may
More informationEM waves: energy, resonators. Scalar wave equation Maxwell equations to the EM wave equation A simple linear resonator Energy in EM waves 3D waves
EM waves: energy, resonators Scalar wave equation Maxwell equations to the EM wave equation A simple linear resonator Energy in EM waves 3D waves Simple scalar wave equation 2 nd order PDE 2 z 2 ψ (z,t)
More informationMODELING THE ELECTROMAGNETIC FIELD
MODELING THE ELECTROMAGNETIC FIELD IN ANISOTROPIC INHOMOGENEOUS MAGNETIZED PLASMA OF ECR ION SOURCES G. Torrisi (1), D. Mascali (1), A. Galatà (2), G. Castro (1), L. Celona (1), L. Neri (1), G. Sorbello
More informationTheoretische Physik 2: Elektrodynamik (Prof. A-S. Smith) Home assignment 9
WiSe 202 20.2.202 Prof. Dr. A-S. Smith Dipl.-Phys. Ellen Fischermeier Dipl.-Phys. Matthias Saba am Lehrstuhl für Theoretische Physik I Department für Physik Friedrich-Alexander-Universität Erlangen-Nürnberg
More informationOverview in Images. 5 nm
Overview in Images 5 nm K.S. Min et al. PhD Thesis K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (000) J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997) S. Lin et al, Nature, vol. 394, p. 51-3,
More informationMEMORANDUM-4. n id (n 2 + n2 S) 0 n n 0. Det[ɛ(ω, k)]=0 gives the Dispersion relation for waves in a cold magnetized plasma: ω 2 pα ω 2 cα ω2, ω 2
Fundamental dispersion relation MEMORANDUM-4 n S) id n n id n + n S) 0 } n n 0 {{ n P } ɛω,k) E x E y E z = 0 Det[ɛω, k)]=0 gives the Dispersion relation for waves in a cold magnetized plasma: n P ) [
More information3 Constitutive Relations: Macroscopic Properties of Matter
EECS 53 Lecture 3 c Kamal Sarabandi Fall 21 All rights reserved 3 Constitutive Relations: Macroscopic Properties of Matter As shown previously, out of the four Maxwell s equations only the Faraday s and
More informationPHYS463 Electricity& Magnetism III ( ) Problems Solutions (assignment #3) r n+1
. (Problem 3.38, p.6) Solution: Use equation (3.95) PHYS463 Electricity& Magnetism (3-4) Problems Solutions (assignment #3) Φ 4π² X n ³ r n Pn ³cos ³ ϑ ρ r dτ r n+ Now λ Q/a a
More informationECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 7
ECE 634 Intermediate EM Waves Fall 16 Prof. David R. Jackson Dept. of ECE Notes 7 1 TEM Transmission Line conductors 4 parameters C capacitance/length [F/m] L inductance/length [H/m] R resistance/length
More informationĞ ğ ğ Ğ ğ Öğ ç ğ ö öğ ğ ŞÇ ğ ğ
Ğ Ü Ü Ü ğ ğ ğ Öğ ş öğ ş ğ öğ ö ö ş ğ ğ ö ğ Ğ ğ ğ Ğ ğ Öğ ç ğ ö öğ ğ ŞÇ ğ ğ l _.j l L., c :, c Ll Ll, c :r. l., }, l : ö,, Lc L.. c l Ll Lr. 0 c (} >,! l LA l l r r l rl c c.r; (Y ; c cy c r! r! \. L : Ll.,
More informationi.ea IE !e e sv?f 'il i+x3p \r= v * 5,?: S i- co i, ==:= SOrq) Xgs'iY # oo .9 9 PE * v E=S s->'d =ar4lq 5,n =.9 '{nl a':1 t F #l *r C\ t-e
fl ) 2 ;;:i c.l l) ( # =S >' 5 ^'R 1? l.y i.i.9 9 P * v ,>f { e e v? 'il v * 5,?: S 'V i: :i (g Y 1,Y iv cg G J :< >,c Z^ /^ c..l Cl i l 1 3 11 5 (' \ h 9 J :'i g > _ ^ j, \= f{ '{l #l * C\? 0l = 5,
More informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
More information4: birefringence and phase matching
/3/7 4: birefringence and phase matching Polarization states in EM Linear anisotropic response χ () tensor and its symmetry properties Working with the index ellipsoid: angle tuning Phase matching in crystals
More informationPhysics 322 Midterm 2
Physics 3 Midterm Nov 30, 015 name: Box your final answer. 1 (15 pt) (50 pt) 3 (0 pt) 4 (15 pt) total (100 pt) 1 1. (15 pt) An infinitely long cylinder of radius R whose axis is parallel to the ẑ axis
More informationFORMULA SHEET FOR QUIZ 2 Exam Date: November 8, 2017
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II November 5, 207 Prof. Alan Guth FORMULA SHEET FOR QUIZ 2 Exam Date: November 8, 207 A few items below are marked
More informationLecture 21 Reminder/Introduction to Wave Optics
Lecture 1 Reminder/Introduction to Wave Optics Program: 1. Maxwell s Equations.. Magnetic induction and electric displacement. 3. Origins of the electric permittivity and magnetic permeability. 4. Wave
More informationLight matter interaction. Ground state spherical electron cloud. Excited state : 4 quantum numbers n principal (energy)
Light matter interaction Hydrogen atom Ground state spherical electron cloud Excited state : 4 quantum numbers n principal (energy) L angular momentum, 2,3... L L z projection of angular momentum S z projection
More informationSet 5: Classical E&M and Plasma Processes
Set 5: Classical E&M and Plasma Processes Maxwell Equations Classical E&M defined by the Maxwell Equations (fields sourced by matter) and the Lorentz force (matter moved by fields) In cgs (gaussian) units
More informationSolution for Problem Set 19-20
Solution for Problem Set 19-0 compiled by Dan Grin and Nate Bode) April 16, 009 A 19.4 Ion Acoustic Waves [by Xinkai Wu 00] a) The derivation of these equations is trivial, so we omit it here. b) Write
More informationFinite Element Method (FEM)
Finite Element Method (FEM) The finite element method (FEM) is the oldest numerical technique applied to engineering problems. FEM itself is not rigorous, but when combined with integral equation techniques
More informationFORMULATION OF ANISOTROPIC MEDIUM IN SPATIAL NETWORK METHOD. N. Yoshida, S. Koike, N. Kukutsu, and T. Kashiwa
Progress In Electromagnetics Research, PIER 13, 293 362, 1996 FORMULATION OF ANISOTROPIC MEDIUM IN SPATIAL NETWORK METHOD N. Yoshida, S. Koike, N. Kukutsu, and T. Kashiwa 1. Introduction 2. Spatial Network
More informationLASER GENERATED THERMOELASTIC WAVES IN AN ANISOTROPIC INFINITE PLATE
LASER GENERATED THERMOELASTIC WAVES IN AN ANISOTROPIC INFINITE PLATE H. M. Al-Qahtani and S. K. Datta University of Colorado Boulder CO 839-7 ABSTRACT. An analysis of the propagation of thermoelastic waves
More informationDielectric wave guides, resonance, and cavities
Dielectric wave guides, resonance, and cavities 1 Dielectric wave guides Instead of a cavity constructed of conducting walls, a guide can be constructed of dielectric material. In analogy to a conducting
More informationETIKA V PROFESII PSYCHOLÓGA
P r a ž s k á v y s o k á š k o l a p s y c h o s o c i á l n í c h s t u d i í ETIKA V PROFESII PSYCHOLÓGA N a t á l i a S l o b o d n í k o v á v e d ú c i p r á c e : P h D r. M a r t i n S t r o u
More informationUniversity of Illinois at Chicago Department of Physics
University of Illinois at Chicago Department of Physics Electromagnetism Qualifying Examination January 4, 2017 9.00 am - 12.00 pm Full credit can be achieved from completely correct answers to 4 questions.
More informationRigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient
Rigid body dynamics Rigid body simulation Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body simulation Unconstrained system no contact Constrained
More informationDerivation of Appleton s equation Aug 2010 W. Gekelman
Derivation of Appleton s equation Aug 010 W. Gekelman This is a derivation of Appleton s equation, which is the equation for the index of refraction of a cold plasma for whistler waves. The index of refraction
More information1 Fundamentals of laser energy absorption
1 Fundamentals of laser energy absorption 1.1 Classical electromagnetic-theory concepts 1.1.1 Electric and magnetic properties of materials Electric and magnetic fields can exert forces directly on atoms
More informationUsing a Microwave Interferometer to Measure Plasma Density Mentor: Prof. W. Gekelman. P. Pribyl (UCLA)
Using a Microwave Interferometer to Measure Plasma Density Avital Levi Mentor: Prof. W. Gekelman. P. Pribyl (UCLA) Introduction: Plasma is the fourth state of matter. It is composed of fully or partially
More information13.1 Ion Acoustic Soliton and Shock Wave
13 Nonlinear Waves In linear theory, the wave amplitude is assumed to be sufficiently small to ignore contributions of terms of second order and higher (ie, nonlinear terms) in wave amplitude In such a
More informationElectromagnetism II Lecture 7
Electromagnetism II Lecture 7 Instructor: Andrei Sirenko sirenko@njit.edu Spring 13 Thursdays 1 pm 4 pm Spring 13, NJIT 1 Previous Lecture: Conservation Laws Previous Lecture: EM waves Normal incidence
More informationElectrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Electro Dynamic
Electrodynamics I Final Exam - Part A - Closed Book KSU 2005/12/12 Name Electro Dynamic Instructions: Use SI units. Short answers! No derivations here, just state your responses clearly. 1. (2) Write an
More informationGuided waves - Lecture 11
Guided waves - Lecture 11 1 Wave equations in a rectangular wave guide Suppose EM waves are contained within the cavity of a long conducting pipe. To simplify the geometry, consider a pipe of rectangular
More informationBasics of Radiation Fields
Basics of Radiation Fields Initial questions: How could you estimate the distance to a radio source in our galaxy if you don t have a parallax? We are now going to shift gears a bit. In order to understand
More informationSub: Submission of the copy of Investor presentation under regulation 30 of SEBI (Listing Obligations & Disclosure Reguirements) Regulations
matrimny.cm vember 1, 218 tinal Stck xchange f India Ltd xchan laza, 5th Flr Plt : C/1, Sand Krla Cmplex, Sa Mmbai - 4 51 Crpte Relatinship Department SS Ltd., Phirze Jeejheebhy Twers Dalal Street, Mmbai
More informationAPPENDIX Z. USEFUL FORMULAS 1. Appendix Z. Useful Formulas. DRAFT 13:41 June 30, 2006 c J.D Callen, Fundamentals of Plasma Physics
APPENDIX Z. USEFUL FORMULAS 1 Appendix Z Useful Formulas APPENDIX Z. USEFUL FORMULAS 2 Key Vector Relations A B = B A, A B = B A, A A = 0, A B C) = A B) C A B C) = B A C) C A B), bac-cab rule A B) C D)
More informationWaves in plasma. Denis Gialis
Waves in plasma Denis Gialis This is a short introduction on waves in a non-relativistic plasma. We will consider a plasma of electrons and protons which is fully ionized, nonrelativistic and homogeneous.
More informationGoal: The theory behind the electromagnetic radiation in remote sensing. 2.1 Maxwell Equations and Electromagnetic Waves
Chapter 2 Electromagnetic Radiation Goal: The theory behind the electromagnetic radiation in remote sensing. 2.1 Maxwell Equations and Electromagnetic Waves Electromagnetic waves do not need a medium to
More informationI. Rayleigh Scattering. EE Lecture 4. II. Dipole interpretation
I. Rayleigh Scattering 1. Rayleigh scattering 2. Dipole interpretation 3. Cross sections 4. Other approximations EE 816 - Lecture 4 Rayleigh scattering is an approximation used to predict scattering from
More informationElectromagnetism. Christopher R Prior. ASTeC Intense Beams Group Rutherford Appleton Laboratory
lectromagnetism Christopher R Prior Fellow and Tutor in Mathematics Trinity College, Oxford ASTeC Intense Beams Group Rutherford Appleton Laboratory Contents Review of Maxwell s equations and Lorentz Force
More informationSolar Physics & Space Plasma Research Center (SP 2 RC) MHD Waves
MHD Waves Robertus vfs Robertus@sheffield.ac.uk SP RC, School of Mathematics & Statistics, The (UK) What are MHD waves? How do we communicate in MHD? MHD is kind! MHD waves are propagating perturbations
More informationFundamentals of wave kinetic theory
Fundamentals of wave kinetic theory Introduction to the subject Perturbation theory of electrostatic fluctuations Landau damping - mathematics Physics of Landau damping Unmagnetized plasma waves The plasma
More informationYell if you have any questions
Class 36: Outline Hour 1: Concept Review / Overview PRS Questions Possible Exam Questions Hour : Sample Exam Yell if you have any questions P36-1 Before Starting All of your grades should now be posted
More information~,. :'lr. H ~ j. l' ", ...,~l. 0 '" ~ bl '!; 1'1. :<! f'~.., I,," r: t,... r':l G. t r,. 1'1 [<, ."" f'" 1n. t.1 ~- n I'>' 1:1 , I. <1 ~'..
,, 'l t (.) :;,/.I I n ri' ' r l ' rt ( n :' (I : d! n t, :?rj I),.. fl.),. f!..,,., til, ID f-i... j I. 't' r' t II!:t () (l r El,, (fl lj J4 ([) f., () :. -,,.,.I :i l:'!, :I J.A.. t,.. p, - ' I I I
More informationNDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.
CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo
More informationSTUDY OF LOSS EFFECT OF TRANSMISSION LINES AND VALIDITY OF A SPICE MODEL IN ELECTROMAG- NETIC TOPOLOGY
Progress In Electromagnetics Research, PIER 90, 89 103, 2009 STUDY OF LOSS EFFECT OF TRANSMISSION LINES AND VALIDITY OF A SPICE MODEL IN ELECTROMAG- NETIC TOPOLOGY H. Xie, J. Wang, R. Fan, andy. Liu Department
More informationNonlinear wave-wave interactions involving gravitational waves
Nonlinear wave-wave interactions involving gravitational waves ANDREAS KÄLLBERG Department of Physics, Umeå University, Umeå, Sweden Thessaloniki, 30/8-5/9 2004 p. 1/38 Outline Orthonormal frames. Thessaloniki,
More informationELECTROMAGNETIC WAVES
Physics 4D ELECTROMAGNETIC WAVE Hans P. Paar 26 January 2006 i Chapter 1 Vector Calculus 1.1 Introduction Vector calculus is a branch of mathematics that allows differentiation and integration of (scalar)
More informationElectromagnetic optics!
1 EM theory Electromagnetic optics! EM waves Monochromatic light 2 Electromagnetic optics! Electromagnetic theory of light Electromagnetic waves in dielectric media Monochromatic light References: Fundamentals
More information(tnaiaun uaejna) o il?smitfl?^ni7wwuiinuvitgviisyiititvi2a-a a imaviitjivi5a^ qw^ww^i fiaa!i-j?s'u'uil?g'ijimqwuwiijami.wti. a nmj 1,965,333.
0 fltu77jjiimviu«7mi^ gi^"ijhm?'ijjw?flfi^ V m 1 /14 il?mitfl?^i7wwuiinuvitgviiyiititvi2- imviitvi^ qw^ww^i fi!i-j?'u'uil?g'iqwuwiijmi.wti twwrlf^ imii2^
More informationFourier transforms, Generalised functions and Greens functions
Fourier transforms, Generalised functions and Greens functions T. Johnson 2015-01-23 Electromagnetic Processes In Dispersive Media, Lecture 2 - T. Johnson 1 Motivation A big part of this course concerns
More informationElectron-Acoustic Wave in a Plasma
Electron-Acoustic Wave in a Plasma 0 (uniform ion distribution) For small fluctuations, n ~ e /n 0
More informationT h e C S E T I P r o j e c t
T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T
More informationSensors Plasma Diagnostics
Sensors Plasma Diagnostics Ken Gentle Physics Department Kenneth Gentle RLM 12.330 k.gentle@mail.utexas.edu NRL Formulary MIT Formulary www.psfc.mit.edu/library1/catalog/ reports/2010/11rr/11rr013/11rr013_full.pdf
More informationSuperposition of electromagnetic waves
Superposition of electromagnetic waves February 9, So far we have looked at properties of monochromatic plane waves. A more complete picture is found by looking at superpositions of many frequencies. Many
More informationPhysics 7A Lecture 2 Fall 2014 Final Solutions. December 22, 2014
Physics 7A Lecture Fall 04 Final Solutions December, 04 PROBLEM The string is oscillating in a transverse manner. The wave velocity of the string is thus T s v = µ, where T s is tension and µ is the linear
More informationSpace Physics. ELEC-E4520 (5 cr) Teacher: Esa Kallio Assistant: Markku Alho and Riku Järvinen. Aalto University School of Electrical Engineering
Space Physics ELEC-E4520 (5 cr) Teacher: Esa Kallio Assistant: Markku Alho and Riku Järvinen Aalto University School of Electrical Engineering The 6 th week: topics Last week: Examples of waves MHD: Examples
More informationContribution of the Hanbury Brown Twiss experiment to the development of quantum optics
Contribution of the Hanbury Brown Twiss experiment to the development of quantum optics Kis Zsolt Kvantumoptikai és Kvantuminformatikai Osztály MTA Wigner Fizikai Kutatóközpont H-1121 Budapest, Konkoly-Thege
More informationPHYS General Physics for Engineering II FIRST MIDTERM
Çankaya University Department of Mathematics and Computer Sciences 2010-2011 Spring Semester PHYS 112 - General Physics for Engineering II FIRST MIDTERM 1) Two fixed particles of charges q 1 = 1.0µC and
More informationnecessita d'interrogare il cielo
gigi nei necessia d'inegae i cie cic pe sax span s inuie a dispiegaa fma dea uce < affeandi ves i cen dea uce isnane " sienzi dei padi sie veic dei' anima 5 J i f H 5 f AL J) i ) L '3 J J "' U J J ö'
More informationECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 6
ECE 6340 Intermediate EM Waves Fall 016 Prof. David R. Jackson Dept. of ECE Notes 6 1 Power Dissipated by Current Work given to a collection of electric charges movg an electric field: ( qe ) ( ρ S E )
More informationLet s consider nonrelativistic electrons. A given electron follows Newton s law. m v = ee. (2)
Plasma Processes Initial questions: We see all objects through a medium, which could be interplanetary, interstellar, or intergalactic. How does this medium affect photons? What information can we obtain?
More informationElectromagnetic Properties of Materials Part 2
ECE 5322 21 st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) 747 6958 rcrumpf@utep.edu Lecture #3 Electromagnetic Properties of Materials Part 2 Nonlinear
More informationScattering of ECRF waves by edge density fluctuations and blobs
PSFC/JA-14-7 Scattering of ECRF waves by edge density fluctuations and blobs A. K. Ram and K. Hizanidis a June 2014 Plasma Science and Fusion Center, Massachusetts Institute of Technology Cambridge, MA
More informationLecture 3 : Electrooptic effect, optical activity and basics of interference colors with wave plates
Lecture 3 : Electrooptic effect, optical activity and basics of interference colors with wave plates NW optique physique II 1 Electrooptic effect Electrooptic effect: example of a KDP Pockels cell Liquid
More informationSummary of Beam Optics
Summary of Beam Optics Gaussian beams, waves with limited spatial extension perpendicular to propagation direction, Gaussian beam is solution of paraxial Helmholtz equation, Gaussian beam has parabolic
More informationRadio Propagation Channels Exercise 2 with solutions. Polarization / Wave Vector
/8 Polarization / Wave Vector Assume the following three magnetic fields of homogeneous, plane waves H (t) H A cos (ωt kz) e x H A sin (ωt kz) e y () H 2 (t) H A cos (ωt kz) e x + H A sin (ωt kz) e y (2)
More informationEnergy mode distribution: an analysis of the ratio of anti-stokes to Stokes amplitudes generated by a pair of counterpropagating Langmuir waves.
Energy mode distribution: an analysis of the ratio of anti-stokes to Stokes amplitudes generated by a pair of counterpropagating Langmuir waves. Fernando J. R. Simões Júnior a, M. Virgínia Alves a, Felipe
More informationELECTROMAGNETIC FIELDS AND WAVES
ELECTROMAGNETIC FIELDS AND WAVES MAGDY F. ISKANDER Professor of Electrical Engineering University of Utah Englewood Cliffs, New Jersey 07632 CONTENTS PREFACE VECTOR ANALYSIS AND MAXWELL'S EQUATIONS IN
More informationPoynting Vector and Energy Flow W14D1
Poynting Vector and Energy Flow W14D1 1 Announcements Week 14 Prepset due online Friday 8:30 am PS 11 due Week 14 Friday at 9 pm in boxes outside 26-152 Sunday Tutoring 1-5 pm in 26-152 2 Outline Poynting
More informationScattering of Electromagnetic Radiation. References:
Scattering of Electromagnetic Radiation References: Plasma Diagnostics: Chapter by Kunze Methods of experimental physics, 9a, chapter by Alan Desilva and George Goldenbaum, Edited by Loveberg and Griem.
More informationProblem Set 10 Solutions
Massachusetts Institute of Technology Department of Physics Physics 87 Fall 25 Problem Set 1 Solutions Problem 1: EM Waves in a Plasma a Transverse electromagnetic waves have, by definition, E = Taking
More informationPlasma Processes. m v = ee. (2)
Plasma Processes In the preceding few lectures, we ve focused on specific microphysical processes. In doing so, we have ignored the effect of other matter. In fact, we ve implicitly or explicitly assumed
More informationTechnische Universität Graz. Institute of Solid State Physics. 22. Crystal Physics
Technische Universität Graz Institute of Solid State Physics 22. Crystal Physics Jan. 7, 2018 Hall effect / Nerst effect Technische Universität Graz Institute of Solid State Physics Crystal Physics Crystal
More information