Electromagnetic waves in free space
|
|
- Wesley Fletcher
- 7 years ago
- Views:
Transcription
1 Waveguide notes 017 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 use Ampere s law: = = = Use the first Maxwell equation (the "H" in LIH assures us that spatial derivatives of are zero 1 ), we obtain the wave equation with wave speed =1 = = A similar derivation gives the same equation for wave solution: = 0 exp = 0 exp + Now let s look at a plane where = the wave phase speed. By including the phase constant in the expression for we allow for a possible phase shift bewteen Inserting these expressions into Maxwell s equations, we have = 0 0 =0 (1) = 0 0 =0 Thus both are perpendicular to the direction of propagation. Faraday s law 0 exp = 0 exp + From 1 Here we also assume that is independent of 1
2 Since this relation must be true for all is real, we have =0( oscillate in phase) 0 = 0 = 1 ˆ 0 () Thus is also perpendicular to its magnitude is If the waves propagate in a vacuum, the derivation goes through in the same way the only difference is that the wave speed is =1 0 0 In an LIH medium, = where the refractive index = p 0 0 ' p 0 Electromagnetic fields in a wave guide A wave guide is a region with a conducting boundary inside which EM waves are caused to propagate. In this confined region the boundary conditions create constraints on the wave fields. We shall idealize, assume that the walls are perfect conductors. If they are not, currents flowing in the walls lead to energy loss. See Jackson Ch 8 for a discussion of this case, especially sections 1 5. The boundary conditions at the walls of our perfectly conducting guide are (see notes 1 eqns 5,7,8 10): where Σ is the free surface charge density on the wall, ˆ = Σ (3) ˆ =0 (4) ˆ =0 (5)
3 ˆ = (6) where isthefreesurfacecurrentdensity. (Notethatwehaveassumed =0 inside the conducting material. This is true here because all our fields are assumed to be time-dependent, then non-zero implies non-zero by Faraday s law. Non-zero is not allowed inside a perfect conductor, so must be zero too. ) Since we do not know Σ or equations (4) (5) will be most useful. Now we use cylindrical coordinates with ˆ along the guide in the direction of wave propagation. The transverse coordinates will be chosen to match the cross-sectional shape of the guide Cartesian for a rectangular guide polar for a circular guide. Next we assume that all fields may be written in the form = 0 () We are not making any special assumptions about the time variation, because we can always Fourier transform the fields to get combinations of terms of this form. Then Maxwell s equations in the guide take the form: = =0 =0 = Taking the curl of Faraday s law, inserting from Ampere s law, we get: = = = = (7) This equation is the same as we obtained for free space. Note that is usually a function of Next we look for solutions that take the form of waves propagating in the direction, that is: 0 () = ( ) Thewaveequation(7)thenbecomes: t + =0 (8) where t is the Laplacian operator in the two transverse coordinates ( or for example.) Thus equation (8) is an equation for the function of the two transverse coordinates. Since we were able to simplify the equations by separating the function into its dependence on the coordinates along transverse to the guide, we 3
4 nowtrytodothesamethingwiththecomponents. Atfirst glance, based on equation (1), you might want to jump to the conclusion that there is no component of a wave propagating in the direction, but in general there is. The waves are propagating between conducting boundaries, we have to allow for the possibility that waves travel at an angle to the guide center-line, bounce back forth off the walls as they travel. Since is perpendicular tothewavevector, in such a bouncing wave has a component. The total electromagnetic disturbance in the guide is a sum of such waves. The sum is a combination of waves that interfere constructively. Thus we take = ˆ + t similarly = ˆ + t This decomposition simplifies the boundary conditions, since the normal ˆ on the boundary has no component. Then eqn (5) becomes ˆ t =0on (9) However equation (4) has two components. The transverse component gives while the component gives =0on (10) ˆ t =0 on (11) Next we put these components into Maxwell s equations. The divergence equations are scalar equations, so let s start with them: µ =0= + t t evaluating the derivative, we get Similarly: + t t =0 (1) + t t =0 (13) We separate the curl equations into transverse components. Take the dot product of Faraday s law with ˆ: ˆ = =ˆ t t (14) also the cross product: ˆ =ˆ 4
5 Let s investigate the triple cross product on the left. Since ˆ is a constant, we may move it through the operator in the BAC-CAB rule: ˆ = ˆ ˆ =ˆ t The derivative times ˆ appears in both terms in the middle, so cancels, leaving: t t = ˆ t evaluating the derivative, we get t t = ˆ t (15) Similarly, from Ampere s law, we have the transverse component: t t = ˆ t (16) the component =ˆ t t (17) Equations (1), (13), (17) (14) show that the longitudinal components act as sources of the transverse fields t t. Now we can simplify by looking at the normal modes of the system. Transverse Electric (TE) (or magnetic) modes. In these modes there is no longitudinal component of : 0 everywhere Thus boundary condition (10) is automatically satisfied. The remaining boundary conditions are (9) (11), we can find the version that we need by taking the dot product of ˆ with equation (16): ˆ t ˆ t = ˆ ˆ t The second term is zero on (eqn 9), we rearrange the triple scalar product on the right, leaving: = ˆ ˆ t =0on (18) where we used the other boundary condition (11) for Transverse Magnetic (TM) (or electric) modes. In these modes there is no longitudinal component of : 0 everywhere 5
6 Thus the boundary condition (18) is trivially satisfied, we must impose the remaining condition (10) =0on S. Since the Maxwell equations are linear, we can form superpositions of these two sets of modes to obtain fields in the guide with non-zero longitudinal components of both Thesemodesaretheresultoftheconstructiveinterference mentioned above. Transverse electromagnetic (TEM) modes In these modes both are zero everywhere. Then from (1) (14), t t t t are zero everywhere. This means we can express t as the gradient of a scalar function Φ that satisfies Laplace s equation in two dimensions. The boundary condition (11) becomes µ ˆ Φ Φ =ˆ ˆ =0 on where is a coordinate parallel to the surface Since ˆ ˆ are perpendicular, ˆ ˆ is not zero, so Φ = constant on therefore is constant everywhere inside the volume making t =0 Thus these modes cannot exist inside a hollow guide. They may exist, in fact become the dominant modes, inside a guide with a separate inner boundary, like a coaxial cable. We will not consider them further here. Now let s see how the equations simplify for the TE TM modes.. TM modes We start by findinganequationfor Since 0 equation (16) simplifies to: t = ˆ t t = ˆ t (19) we substitute this result back into equation (15). t + t = ˆ ˆ t t = t t = µ1 t (0) finally we substitute this result for back into equation (1): t + t 1 = 0 t + = 0 6
7 or t + =0 (1) with () Equation (1) is the defining differential equation for Once we have solved for we can find t from equation (0) then t from equation (19). TE modes The argument proceeds similarly. We start with equation (15) with =0, to get: t = ˆ t (3) substitute into equation (16) t + t = ˆ ˆ t µ t = t (4) Then finally from equation (13) we have: + t t = 0 t + = 0 which is the same differential equation that we found for in the TM modes. The solutions are different because the boundary conditions are different. Thus thesolutionforthetwomodesproceedsasfollows: TM modes TE modes assumed 0 0 differential equation t + =0 t + =0 This is an eigenvalue/ boundary condition =0on S =0on S eigenfunction problem. next find = t = t then find t = ˆ t t = ˆ t The differential equation plus boundary condition is an eigenvalue problem that produces a set of eigenfunctions (or ) a set of eigenvalues The wave number is then determined from equation (): This equation is the Helmholtz equation. = (5) 7
8 Clearly if is greater than = where is the wave phase speed in unbounded space, becomes imaginary the wave does not propagate. There is a cut-off frequency for each mode, given by = If is the lowest eigenvalue for any mode, the corresponding frequency is the cutoff frequency for the guide, waves at lower frequencies cannot propagate in the guide. A few things to note: the wave number is always less than the freespace value thus the wavelength is always greater than the free-space wavelength. The phase speed in the waveguide is = = 1 1 p 1 1 = we can differentiate eqn (5) to get = Thenthegroupspeedintheguideis = = = 1 = 1 r 1 1 = Thus information travels more slowly than if the wave were to propagate in free space. TM modes in a rectangular wave guide We use Cartesian coordinates with origin at one corner of the guide. Let the guide have dimensions in the -direction by in the direction. Let the interior be full of air so 0 = 0 ' 1 Then the differential equation for is (1) µ + + =0 As usual we look for a separated solution, choosing = () () to obtain: =0 Each term must separately be constant, so we have: 00 = 00 = + =0 8
9 The boundary condition is =0on so: =0at =0 = =0at =0 = Thus the appropriate solutions are =sin =sin with eigenvalues chosen to fit the second boundary condition in each coordinate: = = Thus, putting back the dependence on we have = sin sin (6) = + (7) Notice that the lowest possible values of are 1 in each case, since taking or =0would render identically zero. Thus the lowest eigenvalue is r 1 11 = + 1 the cutoff frequency for the TM modes is: r 1 c,tm = 11 = + 1 = r 1+ Jackson solves for the TE modes (pg 361). The eigenvalues are the same, but in this case it is possible for one (but not both) of to be zero, leadingtoalowercutoff frequency for the TE modes: TE = assuming This would be the cutoff frequency for the guide. IntheTMmode,theremainingfields are (eqns 0 6): t = t sin sin = ˆ cos sin + ˆ sin cos (eqn 19) t = ˆ ˆ cos sin + ˆ sin cos = ˆ cos sin ˆ sin cos 9 (8) (9)
10 where (eqns 5 7) r = As usual, the physical fields are given by the real part of each mathematical expression, so that, for example, t sin ( ). You should verify that these fields satisfy the boundary conditions at =0 at =0 Power The power transmitted by the waves in the guide is: () = 1 0 where here we must take the real,physical fields. Usually we are interested in the time-averaged Poynting flux, which is given by =Re 1 0 (30) where the fields on the right are the complex functions we have just found. Proof of this result: If = 0 =ˆ 0 similarly for where 0 0 are real, then = 1 Re Re 0 the time average is = ˆ ˆ 0 0 cos ( )cos( ) 0 = ˆ ˆ 0 0 (cos cos +sin sin )(cos cos +sin sin ) 0 = ˆ ˆ 0 0 cos cos cos + 0 (cos sin +cos sin )cos sin +sin sin sin = ˆ ˆ (cos cos +sin sin )=ˆ ˆ cos ( ) = Re 1 0 so the two results are the same. (See also Notes page 11.) 10
11 Using the solution (8, 9), the components of are: =Re 1 0 = Re 1 0 ( ) cos sin + sin cos = 0 4 cos sin + sin cos p = 0 4 cos sin + sin cos q = 0 i cos sin + sin cos h + Check the dimensions! is positive for all values of showing that power is propagating continuously along the guide in the positive -direction. The transverse component is: =Re 1 µ 0 0 sin sin ( ) = Re 1 0 = 0 1 =Re 0 0 cos sin Because there is no real part, the time averaged power flowing across the guide is zero. Power sloshes back forth, but there is no net energy transfer. Fields in a parallel plate wave guide By simplifying the shape of the guide even more, we can demonstrate how thewavemodesareformedbyreflection of waves at the guide walls. Let this guide exist in the region 0 with infinite extent in For the TM modes, theequationtobesatisfied is: t + =0 with =0at =0 = Because the region is infinite in the direction, the appropriate solution has no dependence: = sin (31) with = Thus the cutoff frequency for these modes is TM = 1 = (3) 11
12 Then the other components of the fields are: = = cos ˆ = cos ˆ (33) = ˆ = cos ˆ = cos ˆ (34) Let s look at the electric field first. We write the sine cosine as combinations of complex exponentials. From (31), µ = = µ + from (33) = µ + Thus we can write the electric field as a superposition total = where the two superposed fields are Similarly: with t = ˆ Now define the four vectors 1 = (ˆ ˆ) = (ˆ + ˆ) µ + exp ( + ) exp ( ) = = ˆ exp [ ( ± )] 1 = ˆ ˆ; 1 = ˆ + ˆ; = ˆ + ˆ = ˆ + ˆ 1
13 Then for =1 thevectorsareperpendicular: =0 1 1 =(ˆ + ˆ) (ˆ ˆ) = ˆ + = ˆ = where we used equation (5). The vectors are shown in the diagram below. The two electric field components are then: while 1 = 1 exp ( 1 ) = exp ( ) 1 1 = ˆ exp [ ( + )] = 1 consistent with () (34). Each of these sets of fields ( 1 1 ) has the form of a free-space wave propagating in the direction given by the vectors 1 respectively with wave number 1 = = p + =. These waves are moving across the guide at an angle given by tan = = ± = ± q that is, the waves are reflecting off the plates at =0asshowninthe diagram. When the angle becomes the wave ceases to propagate along the guide, but just bounces back forth. This happens when tan or = This gives the cut-off frequency (3) we found before. See also Lea Burke pages
6 J - vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J - vector electric current density (A/m2 ) M - vector magnetic current density (V/m 2 ) Some problems
More informationMode Patterns of Parallel plates &Rectangular wave guides Mr.K.Chandrashekhar, Dr.Girish V Attimarad
International Journal of Scientific & Engineering Research Volume 3, Issue 8, August-2012 1 Mode Patterns of Parallel plates &Rectangular wave guides Mr.K.Chandrashekhar, Dr.Girish V Attimarad Abstract-Parallel
More informationLecture 10: TEM, TE, and TM Modes for Waveguides. Rectangular Waveguide.
Whites, EE 481/581 Lecture 10 Page 1 of 10 Lecture 10: TEM, TE, and TM Modes for Waveguides. Rectangular Waveguide. We will now generalie our discussion of transmission lines by considering EM waveguides.
More informationThe waveguide adapter consists of a rectangular part smoothly transcending into an elliptical part as seen in Figure 1.
Waveguide Adapter Introduction This is a model of an adapter for microwave propagation in the transition between a rectangular and an elliptical waveguide. Such waveguide adapters are designed to keep
More informationarxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014
Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationFundamentals of Electromagnetic Fields and Waves: I
Fundamentals of Electromagnetic Fields and Waves: I Fall 2007, EE 30348, Electrical Engineering, University of Notre Dame Mid Term II: Solutions Please show your steps clearly and sketch figures wherever
More informationWAVES AND FIELDS IN INHOMOGENEOUS MEDIA
WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society,
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationElectromagnetism Laws and Equations
Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E- and D-fields............................................. Electrostatic Force............................................2
More informationA wave lab inside a coaxial cable
INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 25 (2004) 581 591 EUROPEAN JOURNAL OF PHYSICS PII: S0143-0807(04)76273-X A wave lab inside a coaxial cable JoãoMSerra,MiguelCBrito,JMaiaAlves and A M Vallera
More informationEdmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H).
INDUCTANCE MUTUAL INDUCTANCE If we consider two neighbouring closed loops and with bounding surfaces respectively then a current through will create a magnetic field which will link with as the flux passes
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationON THE COMPUTATION OF TM AND TE MODE ELECTROMAGNETIC FIELDS
PERIODICA POLYTECHNICA SER. EL. ENG. VOL. 98, NO. 2, PP. 71-77 (1994) ON THE COMPUTATION OF TM AND TE MODE ELECTROMAGNETIC FIELDS Istvan VAGO Attila u. 23 1013 Budapest, Hungary Received: June 3, 1994
More informationPhysics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings
1 of 11 9/7/2012 1:06 PM Logged in as Julie Alexander, Instructor Help Log Out Physics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings Course Home Assignments Roster Gradebook Item Library
More informationElectromagnetism - Lecture 2. Electric Fields
Electromagnetism - Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric
More information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More informationCHAPTER 24 GAUSS S LAW
CHAPTER 4 GAUSS S LAW 4. The net charge shown in Fig. 4-40 is Q. Identify each of the charges A, B, C shown. A B C FIGURE 4-40 4. From the direction of the lines of force (away from positive and toward
More information4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES HW/Study Packet
4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES HW/Study Packet Required: READ Hamper pp 115-134 SL/HL Supplemental: Cutnell and Johnson, pp 473-477, 507-513 Tsokos, pp 216-242 REMEMBER TO. Work through all
More informationA METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS
A METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS Joseph J. Stupak Jr, Oersted Technology Tualatin, Oregon (reprinted from IMCSD 24th Annual Proceedings 1995) ABSTRACT The
More informationWaves - Transverse and Longitudinal Waves
Waves - Transverse and Longitudinal Waves wave may be defined as a periodic disturbance in a medium that carries energy from one point to another. ll waves require a source and a medium of propagation.
More informationFrequency-dependent underground cable model for electromagnetic transient simulation
Frequency-dependent underground cable model for electromagnetic transient simulation Vasco José Realista Medeiros Soeiro Abstract A computation method for transients in a three-phase underground power-transmission
More informationElectrostatic Fields: Coulomb s Law & the Electric Field Intensity
Electrostatic Fields: Coulomb s Law & the Electric Field Intensity EE 141 Lecture Notes Topic 1 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University
More information2 Session Two - Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationChapter 4. Electrostatic Fields in Matter
Chapter 4. Electrostatic Fields in Matter 4.1. Polarization A neutral atom, placed in an external electric field, will experience no net force. However, even though the atom as a whole is neutral, the
More informationVector surface area Differentials in an OCS
Calculus and Coordinate systems EE 311 - Lecture 17 1. Calculus and coordinate systems 2. Cartesian system 3. Cylindrical system 4. Spherical system In electromagnetics, we will often need to perform integrals
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationElectromagnetic waves
Chapter 8 lectromagnetic waves David Morin, morin@physics.harvard.edu The waves we ve dealt with so far in this book have been fairly easy to visualize. Waves involving springs/masses, strings, and air
More informationpotential in the centre of the sphere with respect to infinity.
Umeå Universitet, Fysik 1 Vitaly Bychkov Prov i fysik, Electricity and Waves, 2006-09-27, kl 16.00-22.00 Hjälpmedel: Students can use any book. Define the notations you are using properly. Present your
More informationPolarization of Light
Polarization of Light References Halliday/Resnick/Walker Fundamentals of Physics, Chapter 33, 7 th ed. Wiley 005 PASCO EX997A and EX999 guide sheets (written by Ann Hanks) weight Exercises and weights
More informationChapter 7: Polarization
Chapter 7: Polarization Joaquín Bernal Méndez Group 4 1 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationMODULE VII LARGE BODY WAVE DIFFRACTION
MODULE VII LARGE BODY WAVE DIFFRACTION 1.0 INTRODUCTION In the wave-structure interaction problems, it is classical to divide into two major classification: slender body interaction and large body interaction.
More informationWhen the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.
Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs
More informationMagnetostatics (Free Space With Currents & Conductors)
Magnetostatics (Free Space With Currents & Conductors) Suggested Reading - Shen and Kong Ch. 13 Outline Review of Last Time: Gauss s Law Ampere s Law Applications of Ampere s Law Magnetostatic Boundary
More informationQuestion based on Refraction and Refractive index. Glass Slab, Lateral Shift.
Question based on Refraction and Refractive index. Glass Slab, Lateral Shift. Q.What is refraction of light? What are the laws of refraction? Ans: Deviation of ray of light from its original path when
More informationTeaching Electromagnetic Field Theory Using Differential Forms
IEEE TRANSACTIONS ON EDUCATION, VOL. 40, NO. 1, FEBRUARY 1997 53 Teaching Electromagnetic Field Theory Using Differential Forms Karl F. Warnick, Richard H. Selfridge, Member, IEEE, and David V. Arnold
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
More informationDynamics at the Horsetooth, Volume 2A, Focussed Issue: Asymptotics and Perturbations
Dynamics at the Horsetooth, Volume A, Focussed Issue: Asymptotics and Perturbations Asymptotic Expansion of Bessel Functions; Applications to Electromagnetics Department of Electrical Engineering Colorado
More informationPHYSICAL QUANTITIES AND UNITS
1 PHYSICAL QUANTITIES AND UNITS Introduction Physics is the study of matter, its motion and the interaction between matter. Physics involves analysis of physical quantities, the interaction between them
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationInterferometers. OBJECTIVES To examine the operation of several kinds of interferometers. d sin = n (1)
Interferometers The true worth of an experimenter consists in his pursuing not only what he seeks in his experiment, but also what he did not seek. Claude Bernard (1813-1878) OBJECTIVES To examine the
More informationChapter 22: Electric Flux and Gauss s Law
22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationPHY3128 / PHYM203 (Electronics / Instrumentation) Transmission Lines. Repeated n times I L
Transmission Lines Introduction A transmission line guides energy from one place to another. Optical fibres, waveguides, telephone lines and power cables are all electromagnetic transmission lines. are
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More informationINTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS
INTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS I. Dolezel Czech Technical University, Praha, Czech Republic P. Karban University of West Bohemia, Plzeft, Czech Republic P. Solin University of Nevada,
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics
More informationFiber Optics: Fiber Basics
Photonics Technical Note # 21 Fiber Optics Fiber Optics: Fiber Basics Optical fibers are circular dielectric wave-guides that can transport optical energy and information. They have a central core surrounded
More informationSound. References: L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol. 2, Gas Dynamics, Chapter 8
References: Sound L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol., Gas Dynamics, Chapter 8 1 Speed of sound The phenomenon of sound waves is one that
More informationCBE 6333, R. Levicky 1 Differential Balance Equations
CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,
More informationWaves: Recording Sound Waves and Sound Wave Interference (Teacher s Guide)
Waves: Recording Sound Waves and Sound Wave Interference (Teacher s Guide) OVERVIEW Students will measure a sound wave by placing the Ward s DataHub microphone near one tuning fork A440 (f=440hz). Then
More informationChapter 18. Electric Forces and Electric Fields
My lecture slides may be found on my website at http://www.physics.ohio-state.edu/~humanic/ ------------------------------------------------------------------- Chapter 18 Electric Forces and Electric Fields
More informationAntennas. Antennas are transducers that transfer electromagnetic energy between a transmission line and free space. Electromagnetic Wave
Antennas Transmitter I Transmitting Antenna Transmission Line Electromagnetic Wave I Receiver I Receiving Antenna Transmission Line Electromagnetic Wave I Antennas are transducers that transfer electromagnetic
More informationTHE ELECTROMAGNETIC FIELD DUE TO THE ELECTRONS.
THE ELECTROMAGNETIC FIELD DUE TO THE ELECTRONS 367 Proceedings of the London Mathematical Society Vol 1 1904 p 367-37 (Retyped for readability with same page breaks) ON AN EXPRESSION OF THE ELECTROMAGNETIC
More informationThe Dot and Cross Products
The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and
More informationElasticity Theory Basics
G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationAP1 Waves. (A) frequency (B) wavelength (C) speed (D) intensity. Answer: (A) and (D) frequency and intensity.
1. A fire truck is moving at a fairly high speed, with its siren emitting sound at a specific pitch. As the fire truck recedes from you which of the following characteristics of the sound wave from the
More informationThe Vector or Cross Product
The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero
More informationPhysics 202 Problems - Week 8 Worked Problems Chapter 25: 7, 23, 36, 62, 72
Physics 202 Problems - Week 8 Worked Problems Chapter 25: 7, 23, 36, 62, 72 Problem 25.7) A light beam traveling in the negative z direction has a magnetic field B = (2.32 10 9 T )ˆx + ( 4.02 10 9 T )ŷ
More informationThe purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.
260 17-1 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this
More informationReflection and Refraction
Equipment Reflection and Refraction Acrylic block set, plane-concave-convex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,
More information7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )
34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using
More informationStack Contents. Pressure Vessels: 1. A Vertical Cut Plane. Pressure Filled Cylinder
Pressure Vessels: 1 Stack Contents Longitudinal Stress in Cylinders Hoop Stress in Cylinders Hoop Stress in Spheres Vanishingly Small Element Radial Stress End Conditions 1 2 Pressure Filled Cylinder A
More informationDivergence and Curl of the Magnetic Field
Divergence and Curl of the Magnetic Field The static electric field E(x,y,z such as the field of static charges obeys equations E = 1 ǫ ρ, (1 E =. (2 The static magnetic field B(x,y,z such as the field
More informationAmplification of the Radiation from Two Collocated Cellular System Antennas by the Ground Wave of an AM Broadcast Station
Amplification of the Radiation from Two Collocated Cellular System Antennas by the Ground Wave of an AM Broadcast Station Dr. Bill P. Curry EMSciTek Consulting Co., W101 McCarron Road Glen Ellyn, IL 60137,
More informationVibrations of a Free-Free Beam
Vibrations of a Free-Free Beam he bending vibrations of a beam are described by the following equation: y EI x y t 4 2 + ρ A 4 2 (1) y x L E, I, ρ, A are respectively the Young Modulus, second moment of
More informationModern Classical Optics
Modern Classical Optics GEOFFREY BROOKER Department of Physics University of Oxford OXPORD UNIVERSITY PRESS Contents 1 Electromagnetism and basic optics 1 1.1 Introduction 1 1.2 The Maxwell equations 1
More informationDispersion diagrams of a water-loaded cylindrical shell obtained from the structural and acoustic responses of the sensor array along the shell
Dispersion diagrams of a water-loaded cylindrical shell obtained from the structural and acoustic responses of the sensor array along the shell B.K. Jung ; J. Ryue ; C.S. Hong 3 ; W.B. Jeong ; K.K. Shin
More informationBASIC ELECTRONICS AC CIRCUIT ANALYSIS. December 2011
AM 5-202 BASIC ELECTRONICS AC CIRCUIT ANALYSIS December 2011 DISTRIBUTION RESTRICTION: Approved for Pubic Release. Distribution is unlimited. DEPARTMENT OF THE ARMY MILITARY AUXILIARY RADIO SYSTEM FORT
More informationCHAPTER 2. Eigenvalue Problems (EVP s) for ODE s
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More informationSection 1.7 22 Continued
Section 1.5 23 A homogeneous equation is always consistent. TRUE - The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE - The equation
More informationAlternating Current and Direct Current
K Hinds 2012 1 Alternating Current and Direct Current Direct Current This is a Current or Voltage which has a constant polarity. That is, either a positive or negative value. K Hinds 2012 2 Alternating
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More informationLecture - 4 Diode Rectifier Circuits
Basic Electronics (Module 1 Semiconductor Diodes) Dr. Chitralekha Mahanta Department of Electronics and Communication Engineering Indian Institute of Technology, Guwahati Lecture - 4 Diode Rectifier Circuits
More information2.2 Magic with complex exponentials
2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More informationv = fλ PROGRESSIVE WAVES 1 Candidates should be able to :
PROGRESSIVE WAVES 1 Candidates should be able to : Describe and distinguish between progressive longitudinal and transverse waves. With the exception of electromagnetic waves, which do not need a material
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More informationTwo-Dimensional Conduction: Shape Factors and Dimensionless Conduction Heat Rates
Two-Dimensional Conduction: Shape Factors and Dimensionless Conduction Heat Rates Chapter 4 Sections 4.1 and 4.3 make use of commercial FEA program to look at this. D Conduction- General Considerations
More informationA RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM
Progress In Electromagnetics Research, PIER 69, 287 304, 2007 A RIGOROU AND COMPLETED TATEMENT ON HELMHOLTZ THEOREM Y. F. Gui and W. B. Dou tate Key Lab of Millimeter Waves outheast University Nanjing,
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationHW6 Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case.
HW6 Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case. Tipler 22.P.053 The figure below shows a portion of an infinitely
More informationExamples of Uniform EM Plane Waves
Examples of Uniform EM Plane Waves Outline Reminder of Wave Equation Reminder of Relation Between E & H Energy Transported by EM Waves (Poynting Vector) Examples of Energy Transport by EM Waves 1 Coupling
More informationSecond Order Linear Partial Differential Equations. Part I
Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationCoupling Impedance of SIS18 and SIS100 beampipe CERN-GSI-Webmeeting
Coupling Impedance of SIS18 and SIS100 beampipe CERN-GSI-Webmeeting 23 October 2011 TU Darmstadt Fachbereich 18 Institut Theorie Elektromagnetischer Felder Uwe Niedermayer 1 Contents Motivation / Overview
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationLecture L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationScientific Programming
1 The wave equation Scientific Programming Wave Equation The wave equation describes how waves propagate: light waves, sound waves, oscillating strings, wave in a pond,... Suppose that the function h(x,t)
More information