Power Loss. dp loss = 1 = 1. Method 2, Ohmic heating, power lost per unit volume. Agrees with method 1. c = 2 loss per unit area is dp loss da
|
|
- Augusta Franklin
- 5 years ago
- Views:
Transcription
1 How much power is dissipated (per unit area?). 2 ways: 1) Flow of energy into conductor: Energy flow given by S = E H, for real fields E H. so 1 S ( ) = 1 2 Re E H, dp loss /da = ˆn S, so dp loss = 1 µc ω [(1 da 2 2σ ˆn Re i)(ˆn H ) H ] = µ cωδ 4 H 2 = 1 2σδ H 2 Method 2, Ohmic heating, power lost per unit volume 1 2J E = J 2 /2σ, J = σe c = 2 δ H e ξ/δ, the power loss per unit area is dp loss da = 1 δ 2 σ H 2 Agrees with method 1. 0 dξ e 2ξ/δ = 1 2δσ H 2. TEM, TE, 1 The 1, Re, * discussed in lectures B H. 2
2 In terms of surface current K eff = Thus 0 = ˆn H. dξ J(ξ) = 1 δ ˆn H dξ (1 i)e ξ(1 i)/δ dp loss da = 1 2σδ K eff 2. 1 is surface resistance (per unit area) σδ is the surface impediance Z. 0 E K eff = 1 i σδ TEM, TE,
3 For electromagnetic fields with a fixed geometry of linear materials, fourier transform decouples, we can work with frequency modes, E( x, t) = E(x, y, z) e iωt B( x, t) = B(x, y, z) e iωt Actually the fields are the real parts of these complex expressions. If ρ = 0, J = 0, Maxwell gives TEM, TE, Then E = B t = iω B, E = 0, B = 0, B = µ H = µ D t = µɛ E t = iωµɛ E. 2 E = ( E)+ ( E ) = (iω B) = ω 2 µɛ E.
4 similarly for B, so we get Helmholtz equations ( 2 + ω 2 µɛ ) E = 0, ( 2 + ω 2 µɛ ) B = 0. Consider a waveguide, a cylinder of arbitrary cross section but uniform in z. Fourier transform in z E(x, y, z, t) = E(x, y)e ikz iωt B(x, y, z, t) = B(x, y)e ikz iωt TEM, TE, k can take either sign ( a sting wave is a superposition of k = ± k ). The Helmholtz equations give [ 2 t + (µɛω 2 k 2 ) ] ( E(x, ) y) = 0, B(x, 2t := 2 y) x y 2.
5 Decompose longitudinal transverse Let E = E z ẑ + E t B = B z ẑ + B t with E t ẑ B t ẑ ( E) z = ( t E t ) z = iωb z, ( E) = ẑ E t z ẑ te z = iω B t. For any vector V, ẑ (ẑ V ) = V + ẑ(ẑ V ), so for a transverse vector ẑ (ẑ V t ) = V t. Taking ẑ last equation, E t z t E z = iωẑ B t. (1) Similarly decomposition of B = iωµɛe gives ( t B ) t = iωµɛe z z B t z t B z = iωµɛẑ E t. (2) TEM, TE,
6 Divergencelessness: t E t + E z z = 0, t B t + B z z = 0. Equations (1) (2), with the fourier transform in z, give ik E t + iωẑ B t = t E z (3) ik B t iωµɛẑ E t = t B z (4) TEM, TE, Solving 4 for B t plugging into 3, then the reverse for E t, give E t = i k t E z ωẑ t B z ω 2 µɛ k 2 (5) B t = i k t B z + ωµɛẑ t E z ω 2 µɛ k 2 (6) Unless k 2 = k 2 0 := µɛω2, E z B z determine the rest.
7 We have seen that E z B z largely determine the fields, these satisfy the two-dimensional Helmholtz equation ( 2 t + γ 2) ψ = 0 with γ 2 = µɛω 2 k 2 (7) If the walls of the waveguide are very good conductors, we may impose the perfect conductor conditions E 0 B 0 on the boundary S of the two-dimensional cross section. E z is parallel to the boundary so E z S = 0. Also the component of E t parallel to the boundary vanishes at the wall, so E t is in the ±ˆn direction. Then from the ˆn component of (2) (normal to the boundary) TEM, TE, ˆn B t z ˆn t B z = iωµɛˆn (ẑ E ) t = 0 B z n = 0, where / n is the derivative normal to the surface. So we have Dirichlet conditions on E z Neumann conditions for B z.
8 In general, nonzero solutions exist only for discrete values of γ, those values are generally different for Dirichlet for Neumann. So we need to consider, with E z (x, y) = B z (x, y) 0. That is, there are no longitudinal fields, both electric (E) magnetic (M) fields are purely transverse to the direction z of propagation. TE modes, E z (x, y) 0, the transverse fields are determined by the gradiant of B z = ψ, a solution of (7) with Neumann conditions. TM modes, B z (x, y) 0, the transverse fields are determined by E z = ψ, a solution of (7) with zero boundary conditions. TEM, TE,
9 With E z (x, y) = B z (x, y) 0, (5) (6) = everything vanishes or the denominator vanishes, k = ±k 0 with k 0 = µɛ ω Wave travels z with speed 1/ µɛ, same as for infinite medium. No dispersion. t E t = 0 t E t = iωb z = 0, so Φ E t = t Φ (though Φ might not be single valued) 2 Φ = 0. As E S = 0, Φ = constant on each boundary. If cross TEM, TE, section simply connected, Φ = constant, E = 0 No on simply connected cylinder Yes on coaxial cable, or two parallel wires.
10 TE TM modes Equations (5) (6) simplify for TM modes, B z = 0, γ 2 Et = ik t E z, γ 2 Bt = iµɛωẑ t E z, so H t = ɛωk 1 ẑ E t. TE modes, E z = 0, γ 2 Bt = ik t B z, γ 2 Et = iωẑ t B z, so E t = ωẑ B t /k = ẑ H t = kẑ E t /µω. TEM, TE, In either case, H t = 1 Z ẑ E t, with { k/ɛω = (k/k0 ) µ/ɛ TM Z = µω/k = (k 0 /k) µ/ɛ TE
11 To Summarize Solutions given by ψ(x, y), with ( 2 t + γ 2) ψ = 0, γ 2 = µɛω 2 k 2, by TM: E z = ψe ikz iωt, Et = ikγ 2 t ψe ikz iωt with ψ Γ = 0 TE: H z = ψe ikz iωt, Ht = ikγ 2 t ψe ikz iωt with ˆn t ψ Γ = 0 TEM, TE, By looking at 0 = A ψ ( 2 t + γ 2) ψ we can show γ 2 0. There are solutions for discrete values γ λ, so only certain wave numbers k λ for a given frequency can propagate: k 2 λ = µɛω2 γ 2 λ, only frequencies ω > ω λ := γ λ / µɛ can propagate, k λ < µɛ ω, the infinite medium wavenumber. Phase velocity v p = ω/k λ is greater than in the infinite medium.
12 : Circular Wave Guide Jackson does rectangle. You should too. Needed to do homework. We will consider a circular pipe of (inner) radius r. Of course we should use polar coordinates ρ, φ, with 2 t = 1 ρ ρ ρ ρ ρ 2, try ψ(ρ, φ) = R(ρ)Φ(φ), φ2 ( 2 t + γ 2) ψ = ( ) 1 ρ ρ ρ R ρ + γ2 R(ρ) Φ(φ) + 1 Φ(φ) ρ 2 R(ρ) 2 φ 2 = 0. Divide by R(ρ)Φ(φ) multiply by ρ 2 : ( 1 ρ ) R(ρ) ρ ρ R ρ + γ2 ρ 2 R(ρ) Φ(φ) Φ(φ) φ 2 = 0. TEM, TE,
13 : Circular Wave Guide Jackson does rectangle. You should too. Needed to do homework. We will consider a circular pipe of (inner) radius r. Of course we should use polar coordinates ρ, φ, with 2 t = 1 ρ ρ ρ ρ ρ 2, try ψ(ρ, φ) = R(ρ)Φ(φ), φ2 ( 2 t + γ 2) ψ = ( 1 ρ ρ ρ R ) ρ + γ2 R(ρ) Φ(φ) + 1 ρ 2 R(ρ) 2 Φ(φ) φ 2 = 0. Divide by R(ρ)Φ(φ) multiply by ρ 2 : ( 1 ρ ) R(ρ) ρ ρ R ρ + γ2 ρ 2 R(ρ) = C 1 2 Φ(φ) Φ(φ) φ 2 = C. TEM, TE,
14 Solving it Φ first: 2 Φ(φ) φ 2 + CΦ(φ) = 0 Φ(φ) = e ±i Cφ. Periodicity = C = m Z. Now R(ρ): (ρ ρ ρ ρ + γ2 ρ 2 m 2 ) R(ρ) = 0 TEM, TE, Bessel equation, solutions regular at origin are R(ρ) J m (γρ), so ψ(ρ, φ) = m,n A m,n J m (γ mn ρ)e imφ. γ mn is determined by boundary conditions...
15 Boundary conditions: For TM, ψ(r, φ) = 0 = J m (γr) = 0, so γmn TM = x mn /r where x mn is the n th value of x > 0 for which J m (x) = 0, given on page 114. For TE, ˆn t ψ(r, φ) = 0 = djm dr (γr) = 0, so γmn TM = x mn/r where x mn is the n th value of x > 0 for which dj m (x)/dx = 0, given on page 370. J 0 J 1 J2 J 3 TEM, TE, Thus the lowest cutoff frequency is the m = 1 TE mode, with x 11 = while the lowest TM mode or circularly symmetric mode has x 01 =
16 For a waveguide 5 cm in diameter, with air or vacuum inside, the cutoff frequencies are f = ω 2π = 3.5 GHz for the lowest TE 4.6 GHz for the lowest TM modes. TEM, TE,
Joel A. Shapiro January 21, 2010
Joel A. shapiro@physics.rutgers.edu January 21, 20 rmation Instructor: Joel Serin 325 5-5500 X 3886, shapiro@physics Book: Jackson: Classical Electrodynamics (3rd Ed.) Web home page: www.physics.rutgers.edu/grad/504
More informationPhysics 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 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 informationWaveguides and Cavities
Waveguides and Cavities John William Strutt also known as Lord Rayleigh (1842-1919) September 17, 2001 Contents 1 Reflection and Transmission at a Conducting Wall 2 1.1 Boundary Conditions...........................
More informationElectromagnetic waves in free space
Waveguide notes 018 Electromagnetic waves in free space We start with Maxwell s equations for an LIH medum in the case that the source terms are both zero. = =0 =0 = = Take the curl of Faraday s law, then
More informationFiber Optics. Equivalently θ < θ max = cos 1 (n 0 /n 1 ). This is geometrical optics. Needs λ a. Two kinds of fibers:
Waves can be guided not only by conductors, but by dielectrics. Fiber optics cable of silica has nr varying with radius. Simplest: core radius a with n = n 1, surrounded radius b with n = n 0 < n 1. Total
More informationLecture 5 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lecture 5 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell 1. Waveguides Continued - In the previous lecture we made the assumption that
More informationHomework Assignment #1 Solutions
Physics 56 Winter 8 Textook prolems: h. 8: 8., 8.4 Homework Assignment # Solutions 8. A trnsmission line consisting of two concentric circulr cylinders of metl with conductivity σ nd skin depth δ, s shown,
More informationRF 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 informationChapter 5 Cylindrical Cavities and Waveguides
Chapter 5 Cylindrical Cavities and Waveguides We shall consider an electromagnetic field propagating inside a hollow (in the present case cylindrical) conductor. There are no sources inside the conductor,
More information1 Lectures 10 and 11: resonance cavities
1 1 Lectures 10 and 11: resonance cavities We now analyze cavities that consist of a waveguide of length h, terminated by perfectly conducting plates at both ends. The coordinate system is oriented such
More informationAdvanced Electrodynamics Exercise 11 Guides
Advanced Electrodynamics Exercise 11 Guides Here we will calculate in a very general manner the modes of light in a waveguide with perfect conductor boundary-conditions. Our derivations are widely independent
More informationPhysics 504, Lecture 7 Feb 14, Energy Flow, Density and Attenuation. 1.1 Energy flow and density. Last Latexed: February 11, 2011 at 10:54 1
Last Latexe: February, at :54 Physics 54, Lecture 7 Feb 4, Energy Flow, Density an ttenuation We have seen that there are iscrete moes for electromagnetic waves with E, B e ikz it corresponing, for real
More informationChapter 5 Cylindrical Cavities and Waveguides
Chapter 5 Cylindrical Cavities and Waveguides We shall consider an electromagnetic field propagating inside a hollow (in the present case cylindrical) conductor. There are no sources inside the conductor,
More informationGUIDED 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 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 informationExam in TFY4240 Electromagnetic Theory Wednesday Dec 9, :00 13:00
NTNU Page 1 of 5 Institutt for fysikk Contact during the exam: Paul Anton Letnes Telephone: Office: 735 93 648, Mobile: 98 62 08 26 Exam in TFY4240 Electromagnetic Theory Wednesday Dec 9, 2009 09:00 13:00
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 informationMathematical Tripos, Part IB : Electromagnetism
Mathematical Tripos, Part IB : Electromagnetism Proof of the result G = m B Refer to Sec. 3.7, Force and couples, and supply the proof that the couple exerted by a uniform magnetic field B on a plane current
More informationPHYS4210 Electromagnetic Theory Spring Final Exam Wednesday, 6 May 2009
Name: PHYS4210 Electromagnetic Theory Spring 2009 Final Exam Wednesday, 6 May 2009 This exam has two parts. Part I has 20 multiple choice questions, worth two points each. Part II consists of six relatively
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 information8.03 Lecture 12. Systems we have learned: Wave equation: (1) String with constant tension and mass per unit length ρ L T v p = ρ L
8.03 Lecture 1 Systems we have learned: Wave equation: ψ = ψ v p x There are three different kinds of systems discussed in the lecture: (1) String with constant tension and mass per unit length ρ L T v
More informationECE357H1S ELECTROMAGNETIC FIELDS TERM TEST March 2016, 18:00 19:00. Examiner: Prof. Sean V. Hum
UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING The Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE357H1S ELECTROMAGNETIC FIELDS TERM TEST 2 21 March 2016, 18:00
More informationUSPAS Accelerator Physics 2017 University of California, Davis
USPAS Accelerator Physics 2017 University of California, Davis Chapter 9: RF Cavities and RF Linear Accelerators Todd Satogata (Jefferson Lab) / satogata@jlab.org Randika Gamage (ODU) / bgama002@odu.edu
More informationEECS 117. Lecture 23: Oblique Incidence and Reflection. Prof. Niknejad. University of California, Berkeley
University of California, Berkeley EECS 117 Lecture 23 p. 1/2 EECS 117 Lecture 23: Oblique Incidence and Reflection Prof. Niknejad University of California, Berkeley University of California, Berkeley
More informationPart IB. Electromagnetism. Year
Part IB Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2017 10 Paper 2, Section I 6C State Gauss s Law in the context of electrostatics. A spherically symmetric capacitor consists
More informationWaves. Daniel S. Weile. ELEG 648 Waves. Department of Electrical and Computer Engineering University of Delaware. Plane Waves Reflection of Waves
Waves Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Waves Outline Outline Introduction Let s start by introducing simple solutions to Maxwell s equations
More information5 Electromagnetic Waves
5 Electromagnetic Waves 5.1 General Form for Electromagnetic Waves. In free space, Maxwell s equations are: E ρ ɛ 0 (5.1.1) E + B 0 (5.1.) B 0 (5.1.3) B µ 0 ɛ 0 E µ 0 J (5.1.4) In section 4.3 we derived
More informationModule I: Electromagnetic waves
Module I: Electromagnetic waves Lecture 3: Time dependent EM fields: relaxation, propagation Amol Dighe TIFR, Mumbai Outline Relaxation to a stationary state Electromagnetic waves Propagating plane wave
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 informationEECS 117 Lecture 20: Plane Waves
University of California, Berkeley EECS 117 Lecture 20 p. 1/2 EECS 117 Lecture 20: Plane Waves Prof. Niknejad University of California, Berkeley University of California, Berkeley EECS 117 Lecture 20 p.
More informationELECTROMAGNETIC THEORY
ELECTROMAGNETIC THEORY A. B. Lahanas University of Athens, Physics Department, Nuclear and Particle Physics Section, Athens 157 71, Greece Abstract An introduction to Electromagnetic Theory is given with
More informationCHAPTER 4 ELECTROMAGNETIC WAVES IN CYLINDRICAL SYSTEMS
CHAPTER 4 ELECTROMAGNETIC WAVES IN CYLINDRICAL SYSTEMS The vector Helmholtz equations satisfied by the phasor) electric and magnetic fields are where. In low-loss media and for a high frequency, i.e.,
More informationPhysics 504, Lecture 9 Feb. 21, 2011
Last Latexed: February 17, 011 at 15:8 1 1 Ionosphere, Redux Physics 504, Lecture 9 Feb. 1, 011 Let us return to the resonant cavity formed by the surface of the Earth a spherical shell of radius r = R
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 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 informationCylindrical Dielectric Waveguides
03/02/2017 Cylindrical Dielectric Waveguides Integrated Optics Prof. Elias N. Glytsis School of Electrical & Computer Engineering National Technical University of Athens Geometry of a Single Core Layer
More informationÜbungen zur Elektrodynamik (T3)
Arnold Sommerfeld Center Ludwig Maximilians Universität München Prof. Dr. vo Sachs SoSe 8 Übungen zur Elektrodynamik T3 Übungsblatt Bearbeitung: Juni - Juli 3, 8 Conservation of Angular Momentum Consider
More informationCartesian Coordinates
Cartesian Coordinates Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Cartesian Coordinates Outline Outline Separation of Variables Away from sources,
More informationPHYS4210 Electromagnetic Theory Quiz 1 Feb 2010
PHYS4210 Electromagnetic Theory Quiz 1 Feb 2010 1. An electric dipole is formed from two charges ±q separated by a distance b. For large distances r b from the dipole, the electric potential falls like
More informationPhysics 442. Electro-Magneto-Dynamics. M. Berrondo. Physics BYU
Physics 44 Electro-Magneto-Dynamics M. Berrondo Physics BYU 1 Paravectors Φ= V + cα Φ= V cα 1 = t c 1 = + t c J = c + ρ J J ρ = c J S = cu + em S S = cu em S Physics BYU EM Wave Equation Apply to Maxwell
More informationGraduate Diploma in Engineering Circuits and waves
9210-112 Graduate Diploma in Engineering Circuits and waves You should have the following for this examination one answer book non-programmable calculator pen, pencil, ruler No additional data is attached
More informationElectrical and optical properties of materials
Electrical and optical properties of materials John JL Morton Part 4: Mawell s Equations We have already used Mawell s equations for electromagnetism, and in many ways they are simply a reformulation (or
More informationLongitudinal 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 informationPhysics 3323, Fall 2014 Problem Set 13 due Friday, Dec 5, 2014
Physics 333, Fall 014 Problem Set 13 due Friday, Dec 5, 014 Reading: Finish Griffiths Ch. 9, and 10..1, 10.3, and 11.1.1-1. Reflecting on polarizations Griffiths 9.15 (3rd ed.: 9.14). In writing (9.76)
More informationTypical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields
Lecture 6: Polarimetry 1 Outline 1 Polarized Light in the Universe 2 Fundamentals of Polarized Light 3 Descriptions of Polarized Light Polarized Light in the Universe Polarization indicates anisotropy
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More information1 Chapter 8 Maxwell s Equations
Electromagnetic Waves ECEN 3410 Prof. Wagner Final Review Questions 1 Chapter 8 Maxwell s Equations 1. Describe the integral form of charge conservation within a volume V through a surface S, and give
More informationGUIDED WAVES IN A RECTANGULAR WAVE GUIDE
GUIDED WAVES IN A RECTANGULAR WAVE GUIDE Consider waves propagating along Oz but with restrictions in the and/or directions. The wave is now no longer necessaril transverse. The wave equation can be written
More informationEECS 117 Lecture 26: TE and TM Waves
EECS 117 Lecture 26: TE and TM Waves Prof. Niknejad University of California, Berkeley University of California, Berkeley EECS 117 Lecture 26 p. 1/2 TE Waves TE means that e z = 0 but h z 0. If k c 0,
More informationTC412 Microwave Communications. Lecture 8 Rectangular waveguides and cavity resonator
TC412 Microwave Communications Lecture 8 Rectangular waveguides and cavity resonator 1 TM waves in rectangular waveguides Finding E and H components in terms of, WG geometry, and modes. From 2 2 2 xye
More informationPlasma 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 informationA GENERAL APPROACH FOR CALCULATING COUPLING IMPEDANCES OF SMALL DISCONTINUITIES
Presented at 16th IEEE Particle Accelerator Conference (PAC 95) and Int l Conference on on High Energy Accelerators, 5/1/1995-5/5/1995, Dallas, TX, USA SLAC-PUB-9963 acc-phys/9504001 A GENERAL APPROACH
More informationPractice Exam in Electrodynamics (T3) June 26, 2018
Practice Exam in Electrodynamics T3) June 6, 018 Please fill in: Name and Surname: Matriculation number: Number of additional sheets: Please read carefully: Write your name and matriculation number on
More informationChap. 1 Fundamental Concepts
NE 2 Chap. 1 Fundamental Concepts Important Laws in Electromagnetics Coulomb s Law (1785) Gauss s Law (1839) Ampere s Law (1827) Ohm s Law (1827) Kirchhoff s Law (1845) Biot-Savart Law (1820) Faradays
More informationDepartment of Physics Preliminary Exam January 2 5, 2013
Department of Physics Preliminary Exam January 2 5, 2013 Day 2: Electricity, Magnetism and Optics Thursday, January 3, 2013 9:00 a.m. 12:00 p.m. Instructions: 1. Write the answer to each question on a
More informationShort Introduction to (Classical) Electromagnetic Theory
Short Introduction to (Classical) Electromagnetic Theory (.. and applications to accelerators) Werner Herr, CERN (http://cern.ch/werner.herr/cas/cas2013 Chavannes/lectures/em.pdf) Why electrodynamics?
More informationCERN Accelerator School Wakefields. Prof. Dr. Ursula van Rienen, Franziska Reimann University of Rostock
CERN Accelerator School Wakefields Prof. Dr. Ursula van Rienen, Franziska Reimann University of Rostock Contents The Term Wakefield and Some First Examples Basic Concept of Wakefields Basic Definitions
More informationTorque on a wedge and an annular piston. II. Electromagnetic Case
Torque on a wedge and an annular piston. II. Electromagnetic Case Kim Milton Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA Collaborators: E. K. Abalo,
More informationGuided Waves. Daniel S. Weile. Department of Electrical and Computer Engineering University of Delaware. ELEG 648 Guided Waves
Guided Waves Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Guided Waves Outline Outline The Circuit Model of Transmission Lines R + jωl I(z + z) I(z)
More informationCERN 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 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 informationLecture 3 Fiber Optical Communication Lecture 3, Slide 1
Lecture 3 Optical fibers as waveguides Maxwell s equations The wave equation Fiber modes Phase velocity, group velocity Dispersion Fiber Optical Communication Lecture 3, Slide 1 Maxwell s equations in
More informationProblem 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 informationSpherical Waves. Daniel S. Weile. Department of Electrical and Computer Engineering University of Delaware. ELEG 648 Spherical Coordinates
Spherical Waves Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648 Spherical Coordinates Outline Wave Functions 1 Wave Functions Outline Wave Functions 1
More informationHelmholtz Wave Equation TE, TM, and TEM Modes Rect Rectangular Waveguide TE, TM, and TEM Modes Cyl Cylindrical Waveguide.
Waveguides S. R. Zinka zinka@vit.ac.in School of Electronics Engineering Vellore Institute of Technology April 26, 2013 Outline 1 Helmholtz Wave Equation 2 TE, TM, and TEM Modes Rect 3 Rectangular Waveguide
More informationPHYS 408, Optics. Problem Set 1 - Spring Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015.
PHYS 408, Optics Problem Set 1 - Spring 2016 Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015. 1. An electric field in vacuum has the wave equation, Let us consider the solution, 2 E 1 c 2 2 E =
More informationBack to basics : Maxwell equations & propagation equations
The step index planar waveguide Back to basics : Maxwell equations & propagation equations Maxwell equations Propagation medium : Notations : linear Real fields : isotropic Real inductions : non conducting
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 information(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 informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATH-GA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More informationELECTROMAGNETISM SUMMARY. Maxwell s equations Transmission lines Transmission line transformers Skin depth
ELECTROMAGNETISM SUMMARY Maxwell s equations Transmission lines Transmission line transformers Skin depth 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#4 Magnetostatics: The static magnetic field Gauss
More informationCHAPTER 9 ELECTROMAGNETIC WAVES
CHAPTER 9 ELECTROMAGNETIC WAVES Outlines 1. Waves in one dimension 2. Electromagnetic Waves in Vacuum 3. Electromagnetic waves in Matter 4. Absorption and Dispersion 5. Guided Waves 2 Skip 9.1.1 and 9.1.2
More informationModule 6 : Wave Guides. Lecture 40 : Introduction of Parallel Waveguide. Objectives. In this course you will learn the following
Objectives In this course you will learn the following Introduction of Parallel Plane Waveguide. Introduction of Parallel Plane Waveguide Wave Guide is a structure which can guide Electro Magnetic Energy.
More informationTransmission Lines, Waveguides, and Resonators
Chapter 7 Transmission Lines, Waveguides, and Resonators 1 7.1. General Properties of Guided Waves 7.. TM, TE, and TEM Modes 7.3. Coaxial Lines 7.4. Two-Wire Lines 7.5. Parallel-Plate Waveguides 7.6. Rectangular
More informationD B. An Introduction to the Physics and Technology of e+e- Linear Colliders. Lecture 2: Main Linac. Peter (PT) Tenenbaum (SLAC)
An Introduction to the Physics and Technology of e+e- Linear Colliders Lecture : Main Linac Peter (PT) Tenenbaum (SLAC) Nick Walker DESY DESY Summer Student Lecture USPAS Santa Barbara, CA, 16-7 31 st
More information1. Consider the biconvex thick lens shown in the figure below, made from transparent material with index n and thickness L.
Optical Science and Engineering 2013 Advanced Optics Exam Answer all questions. Begin each question on a new blank page. Put your banner ID at the top of each page. Please staple all pages for each individual
More informationFor the magnetic field B called magnetic induction (unfortunately) M called magnetization is the induced field H called magnetic field H =
To review, in our original presentation of Maxwell s equations, ρ all J all represented all charges, both free bound. Upon separating them, free from bound, we have (dropping quadripole terms): For the
More informationLecture 1 Introduction to RF for Accelerators. Dr G Burt Lancaster University Engineering
Lecture 1 Introduction to RF for Accelerators Dr G Burt Lancaster University Engineering Electrostatic Acceleration + - - - - - - - + + + + + + Van-de Graaff - 1930s A standard electrostatic accelerator
More informationEECS 117. Lecture 25: Field Theory of T-Lines and Waveguides. Prof. Niknejad. University of California, Berkeley
EECS 117 Lecture 25: Field Theory of T-Lines and Waveguides Prof. Niknejad University of California, Berkeley University of California, Berkeley EECS 117 Lecture 25 p. 1/2 Waveguides and Transmission Lines
More informationElectromagnetic Waves Retarded potentials 2. Energy and the Poynting vector 3. Wave equations for E and B 4. Plane EM waves in free space
Electromagnetic Waves 1 1. Retarded potentials 2. Energy and the Poynting vector 3. Wave equations for E and B 4. Plane EM waves in free space 1 Retarded Potentials For volume charge & current = 1 4πε
More informationBASIC WAVE CONCEPTS. Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, Giancoli?
1 BASIC WAVE CONCEPTS Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, 9.1.2 Giancoli? REVIEW SINGLE OSCILLATOR: The oscillation functions you re used to describe how one quantity (position, charge, electric field,
More informationNondiffracting Waves in 2D and 3D
Nondiffracting Waves in 2D and 3D A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics from the College of William and Mary by Matthew Stephen
More informationDO NOT WRITE YOUR NAME OR STUDENT NUMBER ON ANY SHEET!
tatistical Mechanics ubject Exam, Wednesday, May 3, 2017 DO NOT WRITE YOUR NAME OR TUDENT NUMBER ON ANY HEET! a ( b c) = b( a c) c( a b), a ( b c) = b ( c a) = c ( a b), ( a b) ( c d) = ( a c)( b d) (
More informationElectromagnetic wave propagation. ELEC 041-Modeling and design of electromagnetic systems
Electromagnetic wave propagation ELEC 041-Modeling and design of electromagnetic systems EM wave propagation In general, open problems with a computation domain extending (in theory) to infinity not bounded
More informationPhys. 506 Electricity and Magnetism Winter 2004 Prof. G. Raithel Problem Set 1 Total 30 Points. 1. Jackson Points
Phys. 56 Electricity nd Mgnetism Winter 4 Prof. G. Rithel Prolem Set Totl 3 Points. Jckson 8. Points : The electric field is the sme s in the -dimensionl electrosttic prolem of two concentric cylinders,
More informationTransmission Lines. Plane wave propagating in air Y unguided wave propagation. Transmission lines / waveguides Y. guided wave propagation
Transmission Lines Transmission lines and waveguides may be defined as devices used to guide energy from one point to another (from a source to a load). Transmission lines can consist of a set of conductors,
More informationElectromagnetic Waves Across Interfaces
Lecture 1: Foundations of Optics Outline 1 Electromagnetic Waves 2 Material Properties 3 Electromagnetic Waves Across Interfaces 4 Fresnel Equations 5 Brewster Angle 6 Total Internal Reflection Christoph
More informationUNIT I ELECTROSTATIC FIELDS
UNIT I ELECTROSTATIC FIELDS 1) Define electric potential and potential difference. 2) Name few applications of gauss law in electrostatics. 3) State point form of Ohm s Law. 4) State Divergence Theorem.
More informationWaveguide systems. S. Kazakov 19/10/2017, JAS 2017
Waveguide systems S. Kazakov 19/10/017, JAS 017 What is waveguide systems? Let s define a waveguide system as everything between a source of electromagnetic power and power consumer. For example: the Sun
More informationPhysics 3323, Fall 2014 Problem Set 12 due Nov 21, 2014
Physics 333, Fall 014 Problem Set 1 due Nov 1, 014 Reading: Griffiths Ch. 9.1 9.3.3 1. Square loops Griffiths 7.3 (formerly 7.1). A square loop of wire, of side a lies midway between two long wires, 3a
More informationWave Phenomena Physics 15c. Lecture 8 LC Transmission Line Wave Reflection
Wave Phenomena Physics 15c Lecture 8 LC Transmission Line Wave Reflection Midterm Exam #1 Midterm #1 has been graded Class average = 80.4 Standard deviation = 14.6 Your exam will be returned in the section
More informationGreen s functions for planarly layered media
Green s functions for planarly layered media Massachusetts Institute of Technology 6.635 lecture notes Introduction: Green s functions The Green s functions is the solution of the wave equation for a point
More information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
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 informationPlateau-Rayleigh Instability of a Cylinder of Viscous Liquid (Rayleigh vs. Chandrasekhar) L. Pekker FujiFilm Dimatix Inc., Lebanon NH USA
Plateau-Rayleigh Instability of a Cylinder of Viscous Liquid (Rayleigh vs. Chandrasekhar) L. Pekker FujiFilm Dimatix Inc., Lebanon NH 03766 USA Abstract In 1892, in his classical work, L. Rayleigh considered
More informationCavity basics. 1 Introduction. 2 From plane waves to cavities. E. Jensen CERN, Geneva, Switzerland
Cavity basics E. Jensen CERN, Geneva, Switerland Abstract The fields in rectangular and circular waveguides are derived from Maxwell s equations by superposition of plane waves. Subsequently the results
More informationCausality. but that does not mean it is local in time, for = 1. Let us write ɛ(ω) = ɛ 0 [1 + χ e (ω)] in terms of the electric susceptibility.
We have seen that the issue of how ɛ, µ n depend on ω raises questions about causality: Can signals travel faster than c, or even backwards in time? It is very often useful to assume that polarization
More informationElectromagnetic Theory: PHAS3201, Winter Maxwell s Equations and EM Waves
Electromagnetic Theory: PHA3201, Winter 2008 5. Maxwell s Equations and EM Waves 1 Displacement Current We already have most of the pieces that we require for a full statement of Maxwell s Equations; however,
More informationFIBER OPTICS. Prof. R.K. Shevgaonkar. Department of Electrical Engineering. Indian Institute of Technology, Bombay. Lecture: 07
FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 07 Analysis of Wave-Model of Light Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of
More information