Chapter 3 Dynamics of the Electromagnetic Fields
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Lecture 5: Displacement Current and Ampère's Law
Whites, EE 382 Lecture 5 Page 1 of 8 Lecture 5: Displacement Current and Ampère’s Law. One more addition needs to be made to the governing equations of electromagnetics before we are finished. Specifically, we need to clean up a glaring inconsistency. From Ampère’s law in magnetostatics, we learned that H J (1) Taking the divergence of this equation gives 0 H J That is, J 0 (2) However, as is shown in Section 5.8 of the text (“Continuity Equation and Relaxation Time), the continuity equation (conservation of charge) requires that J (3) t We can see that (2) and (3) agree only when there is no time variation or no free charge density. This makes sense since (2) was derived only for magnetostatic fields in Ch. 7. Ampère’s law in (1) is only valid for static fields and, consequently, it violates the conservation of charge principle if we try to directly use it for time varying fields. © 2017 Keith W. Whites Whites, EE 382 Lecture 5 Page 2 of 8 Ampère’s Law for Dynamic Fields Well, what is the correct form of Ampère’s law for dynamic (time varying) fields? Enter James Clerk Maxwell (ca. 1865) – The Father of Classical Electromagnetism. He combined the results of Coulomb’s, Ampère’s, and Faraday’s laws and added a new term to Ampère’s law to form the set of fundamental equations of classical EM called Maxwell’s equations. It is this addition to Ampère’s law that brings it into congruence with the conservation of charge law. -
This Chapter Deals with Conservation of Energy, Momentum and Angular Momentum in Electromagnetic Systems
This chapter deals with conservation of energy, momentum and angular momentum in electromagnetic systems. The basic idea is to use Maxwell’s Eqn. to write the charge and currents entirely in terms of the E and B-fields. For example, the current density can be written in terms of the curl of B and the Maxwell Displacement current or the rate of change of the E-field. We could then write the power density which is E dot J entirely in terms of fields and their time derivatives. We begin with a discussion of Poynting’s Theorem which describes the flow of power out of an electromagnetic system using this approach. We turn next to a discussion of the Maxwell stress tensor which is an elegant way of computing electromagnetic forces. For example, we write the charge density which is part of the electrostatic force density (rho times E) in terms of the divergence of the E-field. The magnetic forces involve current densities which can be written as the fields as just described to complete the electromagnetic force description. Since the force is the rate of change of momentum, the Maxwell stress tensor naturally leads to a discussion of electromagnetic momentum density which is similar in spirit to our previous discussion of electromagnetic energy density. In particular, we find that electromagnetic fields contain an angular momentum which accounts for the angular momentum achieved by charge distributions due to the EMF from collapsing magnetic fields according to Faraday’s law. This clears up a mystery from Physics 435. We will frequently re-visit this chapter since it develops many of our crucial tools we need in electrodynamics. -
Magnetohydrodynamics 1 19.1Overview
Contents 19 Magnetohydrodynamics 1 19.1Overview...................................... 1 19.2 BasicEquationsofMHD . 2 19.2.1 Maxwell’s Equations in the MHD Approximation . ..... 4 19.2.2 Momentum and Energy Conservation . .. 8 19.2.3 BoundaryConditions. 10 19.2.4 Magneticfieldandvorticity . .. 12 19.3 MagnetostaticEquilibria . ..... 13 19.3.1 Controlled thermonuclear fusion . ..... 13 19.3.2 Z-Pinch .................................. 15 19.3.3 Θ-Pinch .................................. 17 19.3.4 Tokamak.................................. 17 19.4 HydromagneticFlows. .. 18 19.5 Stability of Hydromagnetic Equilibria . ......... 22 19.5.1 LinearPerturbationTheory . .. 22 19.5.2 Z-Pinch: Sausage and Kink Instabilities . ...... 25 19.5.3 EnergyPrinciple ............................. 28 19.6 Dynamos and Reconnection of Magnetic Field Lines . ......... 29 19.6.1 Cowling’stheorem ............................ 30 19.6.2 Kinematicdynamos............................ 30 19.6.3 MagneticReconnection. 31 19.7 Magnetosonic Waves and the Scattering of Cosmic Rays . ......... 33 19.7.1 CosmicRays ............................... 33 19.7.2 Magnetosonic Dispersion Relation . ..... 34 19.7.3 ScatteringofCosmicRays . 36 0 Chapter 19 Magnetohydrodynamics Version 1219.1.K.pdf, 7 September 2012 Please send comments, suggestions, and errata via email to [email protected] or on paper to Kip Thorne, 350-17 Caltech, Pasadena CA 91125 Box 19.1 Reader’s Guide This chapter relies heavily on Chap. 13 and somewhat on the treatment of vorticity • transport in Sec. 14.2 Part VI, Plasma Physics (Chaps. 20-23) relies heavily on this chapter. • 19.1 Overview In preceding chapters, we have described the consequences of incorporating viscosity and thermal conductivity into the description of a fluid. We now turn to our final embellishment of fluid mechanics, in which the fluid is electrically conducting and moves in a magnetic field. -
Retarded Potential, Radiation Field and Power for a Short Dipole Antenna
Module 1- Antenna: Retarded potential, radiation field and power for a short dipole antenna ELL 212 Instructor: Debanjan Bhowmik Department of Electrical Engineering Indian Institute of Technology Delhi Abstract An antenna is a device that acts as interface between electromagnetic waves prop- agating in free space and electric current flowing in metal conductor. It is one of the most beautiful devices that we study in electrical engineering since it combines the concepts of flow of electricity in circuits and propagation of waves in free space. The governing physics behind antenna, e.g. how and why antenna radiates power, can be confusing to learn. It is only after a careful study of the Maxwell's equations that we can start understanding the physics of antenna. In this module we shall discuss the physics of radiation of an antenna in details. We will first learn Green's functions because that will help us in understanding the concept of retarded vector potential, without which we will not be able to derive the radiation field for time varying charge and current and show its "1/r" dependence. We will then derive the expressions for radiated field and power for time varying current flowing through a short dipole antenna. (Reference: a) Classical electrodynamics- J.D. Jackson b) Electromagnetics for Engineers- T. Ulaby) 1 We need the Maxwell's equations throughout the module. So let's list them here first (for vacuum): ρ r~ :E~ = (1) 0 r~ :B~ = 0 (2) @B~ r~ × E~ = − (3) @t 1 @E~ r~ × B~ = µ J~ + (4) 0 c2 @t Also scalar potential φ and vector potential A~ are defined as follows: @A~ E~ = −r~ φ − (5) @t B~ = r~ × A~ (6) Note that equation (1)-(4) are independent equations but equation (5) is dependent on equation (3) and equation (6) is dependent on equation (4). -
Meaning of the Maxwell Tensor
Meaning of the Maxwell Tensor We define the Maxwell Stress Tensor as 1 1 1 T = ε E E − δ E E + B B − δ B B ij 0 i j ij k k µ i j ij k k 2 0 2 While it looks scary, the Maxwell tensor is just a different way of writing the fact that electric fields exert forces on charges, and that magnetic fields exert forces on cur- rents. Dimensionally, Momentum T = (Area)(Time) This is the dimensionality of a pressure: Force = dP dt = Momentum Area Area (Area)(Time) Meaning of the Maxwell Tensor (2) If we have a uniform electric field in the x direction, what are the components of the tensor? E = (E,0,0) ; B = (0,0,0) 2 1 0 0 1 0 0 ε 1 0 0 2 1 1 0 E = ε 0 0 0 − 0 1 0 + ( ) = 0 −1 0 Tij 0 E 0 µ − 0 0 0 2 0 0 1 0 2 0 0 1 This says that there is negative x momentum flowing in the x direction, and positive y momentum flowing in the y direction, and positive z momentum flowing in the z di- rection. But it’s a constant, with no divergence, so if we integrate the flux of any component out of (not through!) a closed volume, it’s zero Meaning of the Maxwell Tensor (3) But what if the Maxwell tensor is not uniform? Consider a parallel-plate capacitor, and do the surface integral with part of the surface between the plates, and part out- side of the plates Now the x-flux integral will be finite on one end face, and zero on the other. -
Mutual Inductance
Chapter 11 Inductance and Magnetic Energy 11.1 Mutual Inductance ............................................................................................ 11-3 Example 11.1 Mutual Inductance of Two Concentric Coplanar Loops ............... 11-5 11.2 Self-Inductance ................................................................................................. 11-5 Example 11.2 Self-Inductance of a Solenoid........................................................ 11-6 Example 11.3 Self-Inductance of a Toroid........................................................... 11-7 Example 11.4 Mutual Inductance of a Coil Wrapped Around a Solenoid ........... 11-8 11.3 Energy Stored in Magnetic Fields .................................................................. 11-10 Example 11.5 Energy Stored in a Solenoid ........................................................ 11-11 Animation 11.1: Creating and Destroying Magnetic Energy............................ 11-12 Animation 11.2: Magnets and Conducting Rings ............................................. 11-13 11.4 RL Circuits ...................................................................................................... 11-15 11.4.1 Self-Inductance and the Modified Kirchhoff's Loop Rule....................... 11-15 11.4.2 Rising Current.......................................................................................... 11-18 11.4.3 Decaying Current..................................................................................... 11-20 11.5 LC Oscillations .............................................................................................. -
Section 2: Maxwell Equations
Section 2: Maxwell’s equations Electromotive force We start the discussion of time-dependent magnetic and electric fields by introducing the concept of the electromotive force . Consider a typical electric circuit. There are two forces involved in driving current around a circuit: the source, fs , which is ordinarily confined to one portion of the loop (a battery, say), and the electrostatic force, E, which serves to smooth out the flow and communicate the influence of the source to distant parts of the circuit. Therefore, the total force per unit charge is a circuit is = + f fs E . (2.1) The physical agency responsible for fs , can be any one of many different things: in a battery it’s a chemical force; in a piezoelectric crystal mechanical pressure is converted into an electrical impulse; in a thermocouple it’s a temperature gradient that does the job; in a photoelectric cell it’s light. Whatever the mechanism, its net effect is determined by the line integral of f around the circuit: E =fl ⋅=d f ⋅ d l ∫ ∫ s . (2.2) The latter equality is because ∫ E⋅d l = 0 for electrostatic fields, and it doesn’t matter whether you use f E or fs . The quantity is called the electromotive force , or emf , of the circuit. It’s a lousy term, since this is not a force at all – it’s the integral of a force per unit charge. Within an ideal source of emf (a resistanceless battery, for instance), the net force on the charges is zero, so E = – fs. The potential difference between the terminals ( a and b) is therefore b b ∆Φ=− ⋅ = ⋅ = ⋅ = E ∫Eld ∫ fs d l ∫ f s d l . -
Lecture 4: Magnetohydrodynamics (MHD), MHD Equilibrium, MHD Waves
HSE | Valery Nakariakov | Solar Physics 1 Lecture 4: Magnetohydrodynamics (MHD), MHD Equilibrium, MHD Waves MHD describes large scale, slow dynamics of plasmas. More specifically, we can apply MHD when 1. Characteristic time ion gyroperiod and mean free path time, 2. Characteristic scale ion gyroradius and mean free path length, 3. Plasma velocities are not relativistic. In MHD, the plasma is considered as an electrically conducting fluid. Gov- erning equations are equations of fluid dynamics and Maxwell's equations. A self-consistent set of MHD equations connects the plasma mass density ρ, the plasma velocity V, the thermodynamic (also called gas or kinetic) pressure P and the magnetic field B. In strict derivation of MHD, one should neglect the motion of electrons and consider only heavy ions. The 1-st equation is mass continuity @ρ + r(ρV) = 0; (1) @t and it states that matter is neither created or destroyed. The 2-nd is the equation of motion of an element of the fluid, "@V # ρ + (Vr)V = −∇P + j × B; (2) @t also called the Euler equation. The vector j is the electric current density which can be expressed through the magnetic field B. Mind that on the lefthand side it is the total derivative, d=dt. The 3-rd equation is the energy equation, which in the simplest adiabatic case has the form d P ! = 0; (3) dt ργ where γ is the ratio of specific heats Cp=CV , and is normally taken as 5/3. The temperature T of the plasma can be determined from the density ρ and the thermodynamic pressure P , using the state equation (e.g. -
Application Limits of the Airgap Maxwell Tensor
Application Limits of the Airgap Maxwell Tensor Raphaël Pile, Guillaume Parent, Emile Devillers, Thomas Henneron, Yvonnick Le Menach, Jean Le Besnerais, Jean-Philippe Lecointe To cite this version: Raphaël Pile, Guillaume Parent, Emile Devillers, Thomas Henneron, Yvonnick Le Menach, et al.. Application Limits of the Airgap Maxwell Tensor. 2019. hal-02169268 HAL Id: hal-02169268 https://hal.archives-ouvertes.fr/hal-02169268 Preprint submitted on 1 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Application Limits of the Airgap Maxwell Tensor Raphael¨ Pile1,2, Guillaume Parent3, Emile Devillers1,2, Thomas Henneron2, Yvonnick Le Menach2, Jean Le Besnerais1, and Jean-Philippe Lecointe3 1EOMYS ENGINEERING, Lille-Hellemmes 59260, France 2Univ. Lille, Arts et Metiers ParisTech, Centrale Lille, HEI, EA 2697 - L2EP - Laboratoire d’Electrotechnique et d’Electronique de Puissance, F-59000 Lille, France 3Univ. Artois, EA 4025, Laboratoire Systemes` Electrotechniques´ et Environnement (LSEE), F-62400 Bethune,´ France Airgap Maxwell tensor is widely used in numerical simulations to accurately compute global magnetic forces and torque but also to estimate magnetic surface force waves, for instance when evaluating magnetic stress harmonics responsible for electromagnetic vibrations and acoustic noise in electrical machines. -
Chapter 6. Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws
Chapter 6. Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws 6.1 Displacement Current The magnetic field due to a current distribution satisfies Ampere’s law, (5.45) Using Stokes’s theorem we can transform this equation into an integral form: ∮ ∫ (5.47) We shall examine this law, show that it sometimes fails, and find a generalization that is always valid. Magnetic field near a charging capacitor loop C Fig 6.1 parallel plate capacitor Consider the circuit shown in Fig. 6.1, which consists of a parallel plate capacitor being charged by a constant current . Ampere’s law applied to the loop C and the surface leads to ∫ ∫ (6.1) If, on the other hand, Ampere’s law is applied to the loop C and the surface , we find (6.2) ∫ ∫ Equations 6.1 and 6.2 contradict each other and thus cannot both be correct. The origin of this contradiction is that Ampere’s law is a faulty equation, not consistent with charge conservation: (6.3) ∫ ∫ ∫ ∫ 1 Generalization of Ampere’s law by Maxwell Ampere’s law was derived for steady-state current phenomena with . Using the continuity equation for charge and current (6.4) and Coulombs law (6.5) we obtain ( ) (6.6) Ampere’s law is generalized by the replacement (6.7) i.e., (6.8) Maxwell equations Maxwell equations consist of Eq. 6.8 plus three equations with which we are already familiar: (6.9) 6.2 Vector and Scalar Potentials Definition of and Since , we can still define in terms of a vector potential: (6.10) Then Faradays law can be written ( ) The vanishing curl means that we can define a scalar potential satisfying (6.11) or (6.12) 2 Maxwell equations in terms of vector and scalar potentials and defined as Eqs. -
31295005665780.Pdf
DESIGN AND CONSTRUCTION OF A THREE HUNDRED kA BREECH SIMULATION RAILGUN by BRETT D. SMITH, B.S. in M.E. A THESIS IN -ELECTRICAL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEERING Jtpproved Accepted December, 1989 ACKNOWLEDGMENTS I would like to thank Dr. Magne Kristiansen for serving as the advisor for my grad uate work as well as serving as chairman of my thesis committee. I am also appreciative of the excellent working environment provided by Dr. Kristiansen at the laboratory. I would also like to thank my other two thesis committee members, Dr. Lynn Hatfield and Dr. Edgar O'Hair, for their advice on this thesis. I would like to thank Greg Engel for his help and advice in the design and construc tion of the experiment. I am grateful to Lonnie Stephenson for his advice and hard work in the construction of the experiment. I would also like to thank Danny Garcia, Mark Crawford, Ellis Loree, Diana Loree, and Dan Reynolds for their help in various aspects of this work. I owe my deepest appreciation to my parents for their constant support and encour agement. The engineering advice obtained from my father proved to be invaluable. 11 CONTENTS .. ACKNOWLEDGMENTS 11 ABSTRACT lV LIST OF FIGURES v CHAPTER I. INTRODUCTION 1 II. THEORY OF RAILGUN OPERATION 4 III. DESIGN AND CONSTRUCTION OF MAX II 27 Railgun System Design 28 Electrical Design and Construction 42 Mechanical Design and Construction 59 Diagnostics Design and Construction 75 IV. -
Comparison of Main Magnetic Force Computation Methods for Noise and Vibration Assessment in Electrical Machines
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2017 1 Comparison of main magnetic force computation methods for noise and vibration assessment in electrical machines Raphael¨ PILE13, Emile DEVILLERS23, Jean LE BESNERAIS3 1 Universite´ de Toulouse, UPS, INSA, INP, ISAE, UT1, UTM, LAAS, ITAV, F-31077 Toulouse Cedex 4, France 2 L2EP, Ecole Centrale de Lille, Villeneuve d’Ascq 59651, France 3 EOMYS ENGINEERING, Lille-Hellemmes 59260, France (www.eomys.com) In the vibro-acoustic analysis of electrical machines, the Maxwell Tensor in the air-gap is widely used to compute the magnetic forces applying on the stator. In this paper, the Maxwell magnetic forces experienced by each tooth are compared with different calculation methods such as the Virtual Work Principle based nodal forces (VWP) or the Maxwell Tensor magnetic pressure (MT) following the stator surface. Moreover, the paper focuses on a Surface Permanent Magnet Synchronous Machine (SPMSM). Firstly, the magnetic saturation in iron cores is neglected (linear B-H curve). The saturation effect will be considered in a second part. Homogeneous media are considered and all simulations are performed in 2D. The technique of equivalent force per tooth is justified by finding similar resultant force harmonics between VWP and MT in the linear case for the particular topology of this paper. The link between slot’s magnetic flux and tangential force harmonics is also highlighted. The results of the saturated case are provided at the end of the paper. Index Terms—Electromagnetic forces, Maxwell Tensor, Virtual Work Principle, Electrical Machines. I. INTRODUCTION TABLE I MAGNETO-MECHANICAL COUPLING AMONG SOFTWARE FOR N electrical machines, the study of noise and vibrations due VIBRO-ACOUSTIC I to magnetic forces first requires the accurate calculation of EM Soft Struct.