MEMORANDUM-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

Size: px
Start display at page:

Download "MEMORANDUM-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"

Transcription

1 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 ) [ n S)n + n S) D ] n n n + n S) = 0 where S = 1 + α ω pα ω cα ω, D = α ω pα ω cα ω ω cα ω, α = e, i Waves in plasmas B 0 = 0) Electromagnetic waves P = 1 + α. ω pα ω Dispersion relation ω = ω p + k c in plasma) ω = kc in vacuum) v g v φ = c Skin-depth δ = c ω p ω ) 1/ Waves in plasmas B 0 0) Electrostatic waves Upper hybrid frequency ω h = ω p + ω c ) 1/, ωc = eb 0 /m Lower hybrid frequency ω l = ω c Ω c ) 1/ = ω c m/m) 1/, Ω c = eb 0 /M Electrostatic ion cyclotron wave ω = Ω c + k v S, v S =ion sound speed 1

2 Electromagnetic waves k B 0 E 1 B 0 O - wave): ω = ω p + k c E 1 B 0 X - wave): ñ c v φ k c ω = 1 ω p ω ω ω p ω ω h Resonance ñ ) : ñ =refractive index ω = ω h Cut-off ñ 0 ): ω R = 1 ω L = 1 k B 0 R - wave): ñ = k c ω = 1 ω p ωω ω c ) L - wave): ñ = k c ω = 1 ω p ωω+ω c) [ ω ) ] c + 4ωp 1/ + ωc right-hand) [ ω ) ] c + 4ωp 1/ ωc left-hand)

3 Problem 4-9 A space capsule making a reentry into the earth s atmosphere suffers a communications blackout because a plasma is generated by the shock wave in front of the capsule. If the radio operates at a frequency of 300MHz, what is the minimum plasma density during the blackout? Dispersion relation: where, ω = ω p + k c 1) If Cut-off takes place when ) e 1/ n 0 ω p = ) mε 0 ω p ω ω < ωp k < 0 k = i c { Ex, t) = E 0 exp {ikx ωt)} = E 0 exp { iωt} exp ω = ω p min = ω p n 0 min ) ω p ω } x } c {{} damping ω = πf = ) e 1/ n 0 min ε 0 m n 0 min = ε 0m e πf) n 0 min = ) ) = m 3 3

4 Problem 4-13 An 8-mm microwave interferometer is used on an infinite plane-parallel plasma slab 8 cm thick see Fig P4-13 in Chen). a) If the plasma density is uniform, and a phase shift of 1/10 fringe is observed, what is the density? Note: One fringe corresponds to a 360 degrees phase shift). b) Show that if the phase shift is small, it is proportional to the density. Vacuum: ω = k 0 c, φ = π 10 k 0 = π λ 0 1) Plasma: ω = ω p + k pc ) E exp {ikx ωt)} Let us choose some fixed moment, t and direct x-axis along wave propagation with the origin point aligned with left wall of the chamber. φ 0 t=t,x=0 = φ p t=t,x=0 φ 0 t=t,x=l = k 0L ωt 3) φ p t=t,x=l = k pl ωt 4) 5) : φ = k 0 k p )L 5) k p = k 0 1 φ ) k 0 L 6) 4

5 Relation 6) into dispersion relation ): ωp = ω k0c 1 φ ) = ω 1 1 φ )) = ω φ 1 1 k 0 L k 0 L k 0 L )) φ k 0 L As far as, e n 0 ω p = mε 0 ) 1/ n 0 = ε 0m e c π) λ 0 λ 0 πl φ 1 1 φ π )) λ 0 L n 0 = 4π ε 0mc e λ 0 L φ 1 1 φ π )) λ 0 L 7) As φ << π, λ 0 << L the last term in the brackets ) φ λ 0 = << 1 π L n 0 = 4π ε 0mc φ 8) e λ 0 L For given λ 0, L and φ from 8): 5) φ n 0 n 0 = = m 3 5

6 Problem 4-8 Find the dispersion relation for electrostatic electron waves propagating at an arbitrary angle θ relative to B 0. a) Show that the answer is ω ω ω h) + ω c ω p cos θ = 0. b) Write out the two solutions of this quadratic for ω, and analyze the solutions in the limits θ 0 and θ π/. Show that in these limits, one of the two solutions is a spurious root with no physical meaning. Assumptions: n i = n 0, T i = T e = 0, V 0 = E 0 = 0 Linearization: n e = n 0 + n 1, V e = V 1, E = E 1. Linearized system of equations: n 1 t + n 0 V 1 = 0 V 1 t = e m E 1 + V 1 B 0 ) ε 0 E 1 = en 1 All variables exp {ikr ωt)} and: k E 1, k = k x, 0, k z ), E 1 = E 1x, 0, E 1z ), B 0 = 0, 0, B 0 ) Separate into components and drop subscripts: iωn + in 0 k x V x + k z V z ) = 0 1) iωv x = e m E x e m V yb 0 ) iωv y = e m V xb 0 3) iωv z = e m E z 4) iε 0 k x E x + k z E z ) = en 5) 6

7 3) V y = i eb 0 mω V x = i ω c ω V x ω c = eb ) 0 m 6) ) V x = i ω e m E x + i eb ) 0 ω c m ω V x ω e V x = i ω ωc m E x 7) 4) V z = i e ωm E z 8) Substitution of expressions for V x, V z into Eq.1) yields: n = i n 0e m kx ω ω c E x + k ) z ω E z 9) Using, k x = k sin θ, k z = k cos θ, E x = E sin θ, E z = E cos θ, where k = k x + k z, E = E x + E z we arrive at: n = i n 0e sin m ke θ ω ωc ) + cos θ ω 10) Substituting expression for n into Eq.5), we find, 1 ω p ω ω c sin θ ω p ω cos θ = 0 11) Let take Eq.11) ω ω ω c ) and make some algebra; then ω 4 ω ω h + ω pω c cos θ = 0 1) 7

8 where ω h = ω p + ω c Solution of Eq.1) with respect to ω has the following form: ω = 1 ) ωh ± ωh 4 4ω pωc cos θ Consider limit cases with respect to θ: i) θ 0 cos θ 1 k z arbitrary k x fixed) : + ω ω = 1 p ωp > ωc ω p + ωc ± ωp ωc + ωc ) = ωp < ωc - ωc ωp > ωc - ωp ωp < ωc ii) θ π cos θ 0 k z 0 k x fixed) : ω = 1 ω p + ω c ± ω p + ω c )) = { + ω h - 0 Roots ω = ω c and ω = 0 are unphysical ones see Eq.11)). It comes from the fact that by obtaining the dispersion relation we multiplied Eq.7) with ω ω ω c ). But when ω ω c and ω 0 solution do exists and is described by lower curve. 8

9 Problem 4-5 A microwave interferometer employing the ordinary wave cannot be used above the cutoff density n c. To measure higher densities, one can use the extraordinary wave. a) Write an expression for the cutoff density n cx for the X-wave. b) On a ñ vs. ω p diagram show the branch of the X-wave dispersion relation on which such an interferometer would work. a) Dispersion relation for the O-wave ω = ω p + k c 1) Cut-off takes place at ω = ω p, or n co = mε 0ω e ) n < n co ). Dispersion relation for the X-wave: ñ k c ω = 1 ω p ω ω ω p ω ω h 3) ω h = ω p + ω c ) Cut-off takes place when k = 0 we can write the equation with respect to plasma frequency, Solution is Thus, ω 4 p ω ω p + ω ω ω c ) = 0 ω p = ωω ± ω c ) = Ω ±. n cx± = mε 0 e ωω ± ω c). 9

10 The plus sign corresponds to better performance. n cx = mε 0ω 1 + ω c e ω ) = n co 1 + ω c ω ) n cx > n co 10

WaFu Notes Discussions around the cold plasma model

WaFu Notes Discussions around the cold plasma model WaFu Notes Discussions around the cold plasma model Lise-Marie Imbert-Gérard Summer 7 These notes correspond - more or less - to the presentation I gave at the WaFu summer school on July 6th, 7 in Paris.

More information

Waves in plasmas. S.M.Lea

Waves 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 information

Chapter 9 WAVES IN COLD MAGNETIZED PLASMA. 9.1 Introduction. 9.2 The Wave Equation

Chapter 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 information

Electromagnetic waves in magnetized plasma The dispersion relation

Electromagnetic waves in magnetized plasma The dispersion relation Electromagnetic waves in magnetized plasma The dispersion relation Bruno Després (LJLL-UPMC) Electromagnetic waves in magnetized plasma The dispersion relation p. 1 / 32 Vlasov-Maxwell Vectors are in bold.

More information

Plasma Processes. m v = ee. (2)

Plasma 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 information

Heating and current drive: Radio Frequency

Heating 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 information

Let s consider nonrelativistic electrons. A given electron follows Newton s law. m v = ee. (2)

Let 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 information

Lecture11: Plasma Physics 1. APPH E6101x Columbia University

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 information

EXAMINATION QUESTION PAPER

EXAMINATION QUESTION PAPER 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

More information

Wave Phenomena Physics 15c. Lecture 17 EM Waves in Matter

Wave Phenomena Physics 15c. Lecture 17 EM Waves in Matter Wave Phenomena Physics 15c Lecture 17 EM Waves in Matter What We Did Last Time Reviewed reflection and refraction Total internal reflection is more subtle than it looks Imaginary waves extend a few beyond

More information

Superposition of electromagnetic waves

Superposition 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 information

Chemistry 24b Lecture 23 Spring Quarter 2004 Instructor: Richard Roberts. (1) It induces a dipole moment in the atom or molecule.

Chemistry 24b Lecture 23 Spring Quarter 2004 Instructor: Richard Roberts. (1) It induces a dipole moment in the atom or molecule. Chemistry 24b Lecture 23 Spring Quarter 2004 Instructor: Richard Roberts Absorption and Dispersion v E * of light waves has two effects on a molecule or atom. (1) It induces a dipole moment in the atom

More information

Waves in Linear Optical Media

Waves 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 information

On Electron-Cyclotron Waves in Relativistic Non-Thermal Tokamak Plasmas

On Electron-Cyclotron Waves in Relativistic Non-Thermal Tokamak Plasmas 1 On Electron-Cyclotron Waves in Relativistic Non-Thermal Tokamak Plasmas Lj. Nikolić and M.M. Škorić Vinča Institute of Nuclear Sciences, P.O.Box 522, Belgrade 11001, Serbia and Montenegro ljnikoli@tesla.rcub.bg.ac.yu

More information

2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1. Waves in plasmas. T. Johnson

2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1. Waves in plasmas. T. Johnson 2/8/16 Dispersive Media, Lecture 5 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasma physics Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas Transverse waves

More information

Dispersive Media, Lecture 7 - Thomas Johnson 1. Waves in plasmas. T. Johnson

Dispersive Media, Lecture 7 - Thomas Johnson 1. Waves in plasmas. T. Johnson 2017-02-14 Dispersive Media, Lecture 7 - Thomas Johnson 1 Waves in plasmas T. Johnson Introduction to plasmas as a coupled system Magneto-Hydro Dynamics, MHD Plasmas without magnetic fields Cold plasmas

More information

Plasma waves in the fluid picture II

Plasma waves in the fluid picture II Plasma waves in the fluid picture II Parallel electromagnetic waves Perpendicular electromagnetic waves Whistler mode waves Cut-off frequencies Resonance (gyro) frequencies Ordinary and extra-ordinary

More information

Part VIII. Interaction with Solids

Part 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 information

Study of Optical Properties of Tokamak Plasma

Study of Optical Properties of Tokamak Plasma Study of Optical Properties of Tokamak Plasma Sabri Naima Ghoutia 1, Benouaz Tayeb 2 1 University of Bechar, POB 417, Street Kenadsa, Bechar,08000, Algeria. 2 University of Tlemcen, POB 119, 13000, Algeria.

More information

Scattering of ECRF waves by edge density fluctuations and blobs

Scattering 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 information

Classical Mechanics/Electricity and Magnetism. Preliminary Exam. August 20, :00-15:00 in P-121

Classical Mechanics/Electricity and Magnetism. Preliminary Exam. August 20, :00-15:00 in P-121 Classical Mechanics/Electricity and Magnetism Preliminary Exam August 20, 2008 09:00-15:00 in P-121 Answer THREE (3) questions from each of the TWO (2) sections A and B for a total of SIX (6) solutions.

More information

Longitudinal Beam Dynamics

Longitudinal Beam Dynamics Longitudinal Beam Dynamics Shahin Sanaye Hajari School of Particles and Accelerators, Institute For Research in Fundamental Science (IPM), Tehran, Iran IPM Linac workshop, Bahman 28-30, 1396 Contents 1.

More information

20. Alfven waves. ([3], p ; [1], p ; Chen, Sec.4.18, p ) We have considered two types of waves in plasma:

20. Alfven waves. ([3], p ; [1], p ; Chen, Sec.4.18, p ) We have considered two types of waves in plasma: Phys780: Plasma Physics Lecture 20. Alfven Waves. 1 20. Alfven waves ([3], p.233-239; [1], p.202-237; Chen, Sec.4.18, p.136-144) We have considered two types of waves in plasma: 1. electrostatic Langmuir

More information

Propagation of EM Waves in material media

Propagation of EM Waves in material media Propagation of EM Waves in material media S.M.Lea 09 Wave propagation As usual, we start with Maxwell s equations with no free charges: D =0 B =0 E = B t H = D t + j If we now assume that each field has

More information

Derivation of Appleton s equation Aug 2010 W. Gekelman

Derivation 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 information

in Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD

in Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD 2141418 Numerical Method in Electromagnetics Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD ISE, Chulalongkorn University, 2 nd /2018 Email: charusluk.v@chula.ac.th Website: Light

More information

Plasma Effects. Massimo Ricotti. University of Maryland. Plasma Effects p.1/17

Plasma Effects. Massimo Ricotti. University of Maryland. Plasma Effects p.1/17 Plasma Effects p.1/17 Plasma Effects Massimo Ricotti ricotti@astro.umd.edu University of Maryland Plasma Effects p.2/17 Wave propagation in plasma E = 4πρ e E = 1 c B t B = 0 B = 4πJ e c (Faraday law of

More information

ELECTROMAGNETISM SUMMARY

ELECTROMAGNETISM SUMMARY Review of E and B ELECTROMAGNETISM SUMMARY (Rees Chapters 2 and 3) The electric field E is a vector function. E q o q If we place a second test charged q o in the electric field of the charge q, the two

More information

Plasma waves in the fluid picture I

Plasma waves in the fluid picture I Plasma waves in the fluid picture I Langmuir oscillations and waves Ion-acoustic waves Debye length Ordinary electromagnetic waves General wave equation General dispersion equation Dielectric response

More information

Problem Set 10 Solutions

Problem 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 information

Theoretische Physik 2: Elektrodynamik (Prof. A-S. Smith) Home assignment 9

Theoretische 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 information

Vlasov-Maxwell Equations and Cold Plasma Waves

Vlasov-Maxwell Equations and Cold Plasma Waves Astronomy 53 Spring 016) uening Bai Mar.,, 016 Vlasov-Maxwell Equations and Cold Plasma Waves The Vlasov-Maxwell equations Consider a plasma as a collection of N charged particles, each particle i has

More information

Space 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 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 information

Set 5: Classical E&M and Plasma Processes

Set 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 information

(a) Show that the amplitudes of the reflected and transmitted waves, corrrect to first order

(a) Show that the amplitudes of the reflected and transmitted waves, corrrect to first order Problem 1. A conducting slab A plane polarized electromagnetic wave E = E I e ikz ωt is incident normally on a flat uniform sheet of an excellent conductor (σ ω) having thickness D. Assume that in space

More information

Relativistic description of electron Bernstein waves

Relativistic description of electron Bernstein waves PSFC/JA-6-7 Relativistic description of electron Bernstein waves J. Decker and A.K. Ram October 6 Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge MA 39 USA This work was

More information

The Interaction of Light and Matter: α and n

The Interaction of Light and Matter: α and n The Interaction of Light and Matter: α and n The interaction of light and matter is what makes life interesting. Everything we see is the result of this interaction. Why is light absorbed or transmitted

More information

Solutions: Homework 7

Solutions: Homework 7 Solutions: Homework 7 Ex. 7.1: Frustrated Total Internal Reflection a) Consider light propagating from a prism, with refraction index n, into air, with refraction index 1. We fix the angle of incidence

More information

Solution for Problem Set 19-20

Solution 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 information

Linear second-order differential equations with constant coefficients and nonzero right-hand side

Linear second-order differential equations with constant coefficients and nonzero right-hand side Linear second-order differential equations with constant coefficients and nonzero right-hand side We return to the damped, driven simple harmonic oscillator d 2 y dy + 2b dt2 dt + ω2 0y = F sin ωt We note

More information

Full-Wave Maxwell Simulations for ECRH

Full-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 information

A theory for localized low-frequency ideal MHD modes in axisymmetric toroidal systems is generalized to take into account both toroidal and poloidal

A theory for localized low-frequency ideal MHD modes in axisymmetric toroidal systems is generalized to take into account both toroidal and poloidal MHD spectra pre-history (selected results I MHD spectra pre-history (selected results II Abstract A theory for localized low-frequency ideal MHD modes in axisymmetric toroidal systems is generalized to

More information

CERN Accelerator School. RF Cavities. Erk Jensen CERN BE-RF

CERN Accelerator School. RF Cavities. Erk Jensen CERN BE-RF CERN Accelerator School RF Cavities Erk Jensen CERN BE-RF CERN Accelerator School, Varna 010 - "Introduction to Accelerator Physics" What is a cavity? 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities Lorentz

More information

Propagation of Radio Frequency Waves Through Density Filaments

Propagation of Radio Frequency Waves Through Density Filaments PSFC/JA-15-13 Propagation of Radio Frequency Waves Through Density Filaments A. K. Ram and K. Hizanidis a May 015 a National Technical University of Athens (part of HELLAS) School of Electrical and Computer

More information

Lecture10: Plasma Physics 1. APPH E6101x Columbia University

Lecture10: Plasma Physics 1. APPH E6101x Columbia University Lecture10: Plasma Physics 1 APPH E6101x Columbia University Last Lecture - Conservation principles in magnetized plasma frozen-in and conservation of particles/flux tubes) - Alfvén waves without plasma

More information

13.1 Ion Acoustic Soliton and Shock Wave

13.1 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 information

PHYS 110B - HW #5 Fall 2005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased

PHYS 110B - HW #5 Fall 2005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased PHYS 0B - HW #5 Fall 005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased [.] Imagine a prism made of lucite (n.5) whose cross-section is a

More information

Problem set 3. Electromagnetic waves

Problem set 3. Electromagnetic waves Second Year Electromagnetism Michaelmas Term 2017 Caroline Terquem Problem set 3 Electromagnetic waves Problem 1: Poynting vector and resistance heating This problem is not about waves but is useful to

More information

Using a Microwave Interferometer to Measure Plasma Density Mentor: Prof. W. Gekelman. P. Pribyl (UCLA)

Using 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 information

Lecture 2 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell

Lecture 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 information

Chapter 2 Basic Optics

Chapter 2 Basic Optics Chapter Basic Optics.1 Introduction In this chapter we will discuss the basic concepts associated with polarization, diffraction, and interference of a light wave. The concepts developed in this chapter

More information

Electrodynamics 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 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 information

Geophysical Applications GPR Ground Penetrating Radar

Geophysical Applications GPR Ground Penetrating Radar Overview: Basics of GPR Radar-wave velocity, attenuation and skin depth Modes of acquisition The Radar-range equation Dielectric properties of materials and relation to porosity Case studies [Archeology,

More information

Characterization of Left-Handed Materials

Characterization of Left-Handed Materials Characterization of Left-Handed Materials Massachusetts Institute of Technology 6.635 lecture notes 1 Introduction 1. How are they realized? 2. Why the denomination Left-Handed? 3. What are their properties?

More information

Macroscopic dielectric theory

Macroscopic dielectric theory Macroscopic dielectric theory Maxwellʼs equations E = 1 c E =4πρ B t B = 4π c J + 1 c B = E t In a medium it is convenient to explicitly introduce induced charges and currents E = 1 B c t D =4πρ H = 4π

More information

Jackson 7.6 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 7.6 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jackson 7.6 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell PROBLEM: A plane wave of frequency ω is incident normally from vacuum on a semi-infinite slab of material

More information

What does the Sun tell us about circular polarization on stars? Stephen White

What does the Sun tell us about circular polarization on stars? Stephen White What does the Sun tell us about circular polarization on stars? Stephen White The Radio Sun at 4.6 GHz Combination of: optically thick upper chromosphere, optically thick coronal gyroresonance where B>500

More information

Physics 506 Winter 2004

Physics 506 Winter 2004 Physics 506 Winter 004 G. Raithel January 6, 004 Disclaimer: The purpose of these notes is to provide you with a general list of topics that were covered in class. The notes are not a substitute for reading

More information

Electromagnetic (EM) Waves

Electromagnetic (EM) Waves Electromagnetic (EM) Waves Short review on calculus vector Outline A. Various formulations of the Maxwell equation: 1. In a vacuum 2. In a vacuum without source charge 3. In a medium 4. In a dielectric

More information

INSTABILITY GROWTH RATE DEPENDENCE ON INPUT PARAMETERS DURING THE BEAM TARGET PLASMA INTERACTION. Miroslav Horký

INSTABILITY GROWTH RATE DEPENDENCE ON INPUT PARAMETERS DURING THE BEAM TARGET PLASMA INTERACTION. Miroslav Horký Acta Polytechnica 53):74 78 3 Czech Technical University in Prague 3 available online at http://ctn.cvut.cz/ap/ INSTABILITY GROWTH RATE DEPENDENCE ON INPUT PARAMETERS DURING THE BEAM TARGET PLASMA INTERACTION

More information

Light and Matter. Thursday, 8/31/2006 Physics 158 Peter Beyersdorf. Document info

Light and Matter. Thursday, 8/31/2006 Physics 158 Peter Beyersdorf. Document info Light and Matter Thursday, 8/31/2006 Physics 158 Peter Beyersdorf Document info 3. 1 1 Class Outline Common materials used in optics Index of refraction absorption Classical model of light absorption Light

More information

Waves in Cold Plasmas: Two-Fluid Formalism

Waves in Cold Plasmas: Two-Fluid Formalism Contents 1 Waves in Cold Plasmas: Two-Fluid Formalism 1 1.1 Overview...................................... 1 1. Dielectric Tensor, Wave Equation, and General Dispersion Relation..... 3 1.3 Two-Fluid Formalism...............................

More information

Cold plasma waves. Waves in non-magnetized plasma Cold plasma dispersion equation Cold plasma wave modes

Cold plasma waves. Waves in non-magnetized plasma Cold plasma dispersion equation Cold plasma wave modes Cold plasma waves Waves in non-magnetized plasma Cold plasma dispersion equation Cold plasma wave modes EM wave propagation through and interaction with plasmas belong to central issues of plasma physics.

More information

2 u 1-D: 3-D: x + 2 u

2 u 1-D: 3-D: x + 2 u c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience 2013-14 Onde 1 1 Waves 1.1 wave propagation 1.1.1 field Field: a physical quantity (measurable, at least in principle) function

More information

Waves in plasma. Denis Gialis

Waves 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 information

Fundamentals of wave kinetic theory

Fundamentals 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 information

Electron Cyclotron Emission Simulation from TCABR Plasmas

Electron Cyclotron Emission Simulation from TCABR Plasmas 1602 Brazilian Journal of Physics, vol. 34, no. 4B, December, 2004 Electron Cyclotron Emission Simulation from TCABR Plasmas Eduardo H. Lyvio and P. R. da S. Rosa Departamento de Física, UFMS, Caixa Postal

More information

Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces

Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces Plasmonics Plasmon: Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces Femius Koenderink Center for Nanophotonics AMOLF, Amsterdam

More information

Simulation Study of High-Frequency Magnetosonic Waves Excited by Energetic Ions in Association with Ion Cyclotron Emission )

Simulation Study of High-Frequency Magnetosonic Waves Excited by Energetic Ions in Association with Ion Cyclotron Emission ) Simulation Study of High-Frequency Magnetosonic Waves Excited by Energetic Ions in Association with Ion Cyclotron Emission ) Mieko TOIDA 1),KenjiSAITO 1), Hiroe IGAMI 1), Tsuyoshi AKIYAMA 1,2), Shuji KAMIO

More information

Magnetically Induced Transparency and Its Application as an Accelerator

Magnetically Induced Transparency and Its Application as an Accelerator Magnetically Induced Transparency and Its Application as an Accelerator M.S. Hur, J.S. Wurtele and G. Shvets University of California Berkeley University of California Berkeley and Lawrence Berkeley National

More information

Po-Han Chen, and Bing-Hung Chen. Institute of Electronic Engineering,

Po-Han Chen, and Bing-Hung Chen. Institute of Electronic Engineering, Simulation of EM wave propagating p g in a nanocylinder-base localized surface plasma resonance senor Po-Han Chen, and Bing-Hung Chen Institute of Electronic Engineering, National Dong Hwa University,

More information

Report submitted to Prof. P. Shipman for Math 540, Fall 2009

Report submitted to Prof. P. Shipman for Math 540, Fall 2009 Dynamics at the Horsetooth Volume 1, 009. Three-Wave Interactions of Spin Waves Aaron Hagerstrom Department of Physics Colorado State University aaronhag@rams.colostate.edu Report submitted to Prof. P.

More information

AST 553. Plasma Waves and Instabilities. Course Outline. (Dated: December 4, 2018)

AST 553. Plasma Waves and Instabilities. Course Outline. (Dated: December 4, 2018) AST 553. Plasma Waves and Instabilities Course Outline (Dated: December 4, 2018) I. INTRODUCTION Basic concepts Waves in plasmas as EM field oscillations Maxwell s equations, Gauss s laws as initial conditions

More information

Full-wave simulation of tokamak plasma heating

Full-wave simulation of tokamak plasma heating Full-wave simulation of tokamak plasma heating Institut Elie Cartan de Nancy, UMR 7502, Université de Lorraine, CNRS & INRIA (project-team CALVI). B.P. 70239, 54506 Vandœuvre les Nancy Cedex, France. Université

More information

RF cavities (Lecture 25)

RF cavities (Lecture 25) RF cavities (Lecture 25 February 2, 2016 319/441 Lecture outline A good conductor has a property to guide and trap electromagnetic field in a confined region. In this lecture we will consider an example

More information

Electron Acceleration by Microwave Radiation Inside a Rectangular Waveguide

Electron Acceleration by Microwave Radiation Inside a Rectangular Waveguide Plasma Science and Technology, Vol.13, No.3, Jun. 2011 Electron Acceleration by Microwave Radiation Inside a Rectangular Waveguide B. F. MOHAMED, A. M. GOUDA Plasma Physics Department, N.R.C., Atomic Energy

More information

UNIVERSITY OF SOUTHAMPTON. Answer all questions in Section A and two and only two questions in. Section B.

UNIVERSITY OF SOUTHAMPTON. Answer all questions in Section A and two and only two questions in. Section B. UNIVERSITY OF SOUTHAMPTON PHYS2023W1 SEMESTER 1 EXAMINATION 2009/10 WAVE PHYSICS Duration: 120 MINS Answer all questions in Section A and two and only two questions in Section B. Section A carries 1/3

More information

Wave Phenomena Physics 15c

Wave Phenomena Physics 15c Wave Phenomena Physics 15c Lecture 15 lectromagnetic Waves (H&L Sections 9.5 7) What We Did Last Time! Studied spherical waves! Wave equation of isotropic waves! Solution e! Intensity decreases with! Doppler

More information

Ph304 Final Examination Electrodynamics

Ph304 Final Examination Electrodynamics Princeton University Ph304 Final Examination Electrodynamics Prof: Kirk T. McDonald (1:30-4:30 pm, May, 00) Do all work you wish graded in the exam booklets provided. mcdonald@puphep.princeton.edu http://puhep1.princeton.edu/

More information

MCQs E M WAVES. Physics Without Fear.

MCQs E M WAVES. Physics Without Fear. MCQs E M WAVES Physics Without Fear Electromagnetic Waves At A Glance Ampere s law B. dl = μ 0 I relates magnetic fields due to current sources. Maxwell argued that this law is incomplete as it does not

More information

arxiv:cond-mat/ v1 22 Jul 2002

arxiv:cond-mat/ v1 22 Jul 2002 Propagation of waves in metallic photonic crystals at low frequencies and some theoretical aspects of left-handed materials arxiv:cond-mat/0207535v1 22 Jul 2002 Abstract A. L. Pokrovsky, A. L. Efros, Department

More information

INTERACTION OF LIGHT WITH MATTER

INTERACTION OF LIGHT WITH MATTER INTERACTION OF LIGHT WITH MATTER Already.the speed of light can be related to the permittivity, ε and the magnetic permeability, µ of the material by Rememberε = ε r ε 0 and µ = µ r µ 0 where ε 0 = 8.85

More information

The Theory of the Photon Planck, Einstein the origins of quantum theory Dirac the first quantum field theory The equations of electromagnetism in

The Theory of the Photon Planck, Einstein the origins of quantum theory Dirac the first quantum field theory The equations of electromagnetism in The Theory of the Photon Planck, Einstein the origins of quantum theory Dirac the first quantum field theory The equations of electromagnetism in empty space (i.e., no charged particles are present) using

More information

Light as a Transverse Wave.

Light as a Transverse Wave. Waves and Superposition (Keating Chapter 21) The ray model for light (i.e. light travels in straight lines) can be used to explain a lot of phenomena (like basic object and image formation and even aberrations)

More information

NYU Physics Preliminary Examination in Electricity & Magnetism Fall 2011

NYU Physics Preliminary Examination in Electricity & Magnetism Fall 2011 This is a closed-book exam. No reference materials of any sort are permitted. Full credit will be given for complete solutions to the following five questions. 1. An impenetrable sphere of radius a carries

More information

Electromagnetic Waves

Electromagnetic Waves Electromagnetic Waves Our discussion on dynamic electromagnetic field is incomplete. I H E An AC current induces a magnetic field, which is also AC and thus induces an AC electric field. H dl Edl J ds

More information

Mode conversion of waves in the ion cyclotron frequency range in magnetospheric plasmas

Mode conversion of waves in the ion cyclotron frequency range in magnetospheric plasmas Mode conversion of waves in the ion cyclotron frequency range in magnetospheric plasmas Ye.O. Kazakov and T. Fülöp Department of Applied Physics, Chalmers University of Technology and Euratom-VR Association,

More information

Magnetohydrodynamic waves in a plasma

Magnetohydrodynamic waves in a plasma Department of Physics Seminar 1b Magnetohydrodynamic waves in a plasma Author: Janez Kokalj Advisor: prof. dr. Tomaž Gyergyek Petelinje, April 2016 Abstract Plasma can sustain different wave phenomena.

More information

Electromagnetism. Christopher R Prior. ASTeC Intense Beams Group Rutherford Appleton Laboratory

Electromagnetism. 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 information

12. Nonlinear optics I

12. Nonlinear optics I 1. Nonlinear optics I What are nonlinear-optical effects and why do they occur? Maxwell's equations in a medium Nonlinear-optical media Second-harmonic generation Conservation laws for photons ("Phasematching")

More information

kg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides.

kg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides. II. Generalizing the 1-dimensional wave equation First generalize the notation. i) "q" has meant transverse deflection of the string. Replace q Ψ, where Ψ may indicate other properties of the medium that

More information

PHYS 1444 Section 003 Lecture #23

PHYS 1444 Section 003 Lecture #23 PHYS 1444 Section 3 Lecture #3 Monday, Nov. 8, 5 EM Waves from Maxwell s Equations Speed of EM Waves Light as EM Wave Electromagnetic Spectrum Energy in EM Waves Energy Transport The epilogue Today s homework

More information

H ( E) E ( H) = H B t

H ( E) E ( H) = H B t Chapter 5 Energy and Momentum The equations established so far describe the behavior of electric and magnetic fields. They are a direct consequence of Maxwell s equations and the properties of matter.

More information

M01M.1 Massive Spring

M01M.1 Massive Spring Part I Mechanics M01M.1 Massive Spring M01M.1 Massive Spring A spring has spring constant K, unstretched length L, and mass per unit length ρ. The spring is suspended vertically from one end in a constant

More information

MCQs of Plasma Physics. by Prof. V.K. Tripathi, IIT Delhi, New Delhi. Lecture 1

MCQs of Plasma Physics. by Prof. V.K. Tripathi, IIT Delhi, New Delhi. Lecture 1 MCQs of Plasma Physics by Prof. V.K. Tripathi, IIT Delhi, New Delhi. Lecture 1 Problem 1: Consider a singly ionized sphere of electron density n o, radius R and electron temperature T. Due to thermal motions

More information

Waves Part 3B: Interference

Waves Part 3B: Interference Waves Part 3B: Interference Last modified: 31/01/2018 Contents Links Interference Path Difference & Interference Light Young s Double Slit Experiment What Sort of Wave is Light? Michelson-Morley Experiment

More information

1. Types of Waves. There are three main types of waves:

1. Types of Waves. There are three main types of waves: Chapter 16 WAVES I 1. Types of Waves There are three main types of waves: https://youtu.be/kvc7obkzq9u?t=3m49s 1. Mechanical waves: These are the most familiar waves. Examples include water waves, sound

More information

GUIDED MICROWAVES AND OPTICAL WAVES

GUIDED MICROWAVES AND OPTICAL WAVES Chapter 1 GUIDED MICROWAVES AND OPTICAL WAVES 1.1 Introduction In communication engineering, the carrier frequency has been steadily increasing for the obvious reason that a carrier wave with a higher

More information

Mandl and Shaw reading assignments

Mandl and Shaw reading assignments Mandl and Shaw reading assignments Chapter 2 Lagrangian Field Theory 2.1 Relativistic notation 2.2 Classical Lagrangian field theory 2.3 Quantized Lagrangian field theory 2.4 Symmetries and conservation

More information

Derivation of Eigen value Equation by Using Equivalent Transmission Line method for the Case of Symmetric/ Asymmetric Planar Slab Waveguide Structure

Derivation of Eigen value Equation by Using Equivalent Transmission Line method for the Case of Symmetric/ Asymmetric Planar Slab Waveguide Structure ISSN 0974-9373 Vol. 15 No.1 (011) Journal of International Academy of Physical Sciences pp. 113-1 Derivation of Eigen value Equation by Using Equivalent Transmission Line method for the Case of Symmetric/

More information