Notes on Magnetohydrodynamics Dana Longcope 12/02/02
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Chapter 3 Dynamics of the Electromagnetic Fields
Chapter 3 Dynamics of the Electromagnetic Fields 3.1 Maxwell Displacement Current In the early 1860s (during the American civil war!) electricity including induction was well established experimentally. A big row was going on about theory. The warring camps were divided into the • Action-at-a-distance advocates and the • Field-theory advocates. James Clerk Maxwell was firmly in the field-theory camp. He invented mechanical analogies for the behavior of the fields locally in space and how the electric and magnetic influences were carried through space by invisible circulating cogs. Being a consumate mathematician he also formulated differential equations to describe the fields. In modern notation, they would (in 1860) have read: ρ �.E = Coulomb’s Law �0 ∂B � ∧ E = − Faraday’s Law (3.1) ∂t �.B = 0 � ∧ B = µ0j Ampere’s Law. (Quasi-static) Maxwell’s stroke of genius was to realize that this set of equations is inconsistent with charge conservation. In particular it is the quasi-static form of Ampere’s law that has a problem. Taking its divergence µ0�.j = �. (� ∧ B) = 0 (3.2) (because divergence of a curl is zero). This is fine for a static situation, but can’t work for a time-varying one. Conservation of charge in time-dependent case is ∂ρ �.j = − not zero. (3.3) ∂t 55 The problem can be fixed by adding an extra term to Ampere’s law because � � ∂ρ ∂ ∂E �.j + = �.j + �0�.E = �. j + �0 (3.4) ∂t ∂t ∂t Therefore Ampere’s law is consistent with charge conservation only if it is really to be written with the quantity (j + �0∂E/∂t) replacing j. -
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
Magnetohydrodynamics J.W.Haverkort April 2009 Abstract This summary of the basics of magnetohydrodynamics (MHD) as- sumes the reader is acquainted with both fluid mechanics and electro- dynamics. The content is largely based on \An introduction to Magneto- hydrodynamics" by P.A. Davidson. The basic equations of electrodynam- ics are summarized, the assumptions underlying MHD are exposed and a selection of aspects of the theory are discussed. Contents 1 Electrodynamics 2 1.1 The governing equations . 2 1.2 Faraday's law . 2 2 Magnetohydrodynamical Equations 3 2.1 The assumptions . 3 2.2 The equations . 4 2.3 Some more equations . 4 3 Concepts of Magnetohydrodynamics 5 3.1 Analogies between fluid mechanics and MHD . 5 3.2 Dimensionless numbers . 6 3.3 Lorentz force . 7 1 1 Electrodynamics 1.1 The governing equations The Maxwell equations for the electric and magnetic fields E and B in vacuum are written in terms of the sources ρe (electric charge density) and J (electric current density) ρ r · E = e Gauss's law (1) "0 @B r × E = − Faraday's law (2) @t r · B = 0 No monopoles (3) @E r × B = µ J + µ " Ampere's law (4) 0 0 0 @t with "0 the electric permittivity and µ0 the magnetic permeability of free space. The electric field can be thought to consist of a static curl-free part Es = @A −∇V and an induced or in-stationary part Ei = − @t which is divergence-free, with V the electric potential and A the magnetic vector potential. The force on a charge q moving with velocity u is given by the Lorentz force f = q(E+u×B). -
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. -
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 .............................................................................................. -
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. -
Astrophysical Fluid Dynamics: II. Magnetohydrodynamics
Winter School on Computational Astrophysics, Shanghai, 2018/01/30 Astrophysical Fluid Dynamics: II. Magnetohydrodynamics Xuening Bai (白雪宁) Institute for Advanced Study (IASTU) & Tsinghua Center for Astrophysics (THCA) source: J. Stone Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 2 Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 3 What is a plasma? Plasma is a state of matter comprising of fully/partially ionized gas. Lightening The restless Sun Crab nebula A plasma is generally quasi-neutral and exhibits collective behavior. Net charge density averages particles interact with each other to zero on relevant scales at long-range through electro- (i.e., Debye length). magnetic fields (plasma waves). 4 Why plasma astrophysics? n More than 99.9% of observable matter in the universe is plasma. n Magnetic fields play vital roles in many astrophysical processes. n Plasma astrophysics allows the study of plasma phenomena at extreme regions of parameter space that are in general inaccessible in the laboratory. 5 Heliophysics and space weather l Solar physics (including flares, coronal mass ejection) l Interaction between the solar wind and Earth’s magnetosphere l Heliospheric -
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. -
Magnetohydrodynamics
MAGNETOHYDRODYNAMICS International Max-Planck Research School. Lindau, 9{13 October 2006 {4{ Bulletin of exercises n◦4: The magnetic induction equation. 1. The induction equation: Starting from Faraday's law and Ampere's law (neglecting the displacement current) and making use of Ohm's constitutive relation, eliminate the electric field and the current density to obtain the inducion equation for a plasma in the MHD approximation: @B c2 = rot (v B) rot rot B : (1) @t ^ − 4πσe ! Show that if the electrical conductivity is uniform, the induction equation can be cast into the form @B = rot (v B) + η 2 B ; (2) @t ^ r def 2 where η = c =(4πσe) is the \magnetic diffusivity." 2. Combined form of the induction and the continuity equations. Show that the induction equation can be combined with the equation of continuity into one single differential equation which gives the time evolution of B/ρ following a fluid element, viz: D B B 1 = v rot (η rot B) : (3) D t ρ ! ρ · r! − ρ In the case of an ideal MHD-plasma, Eq. (3) simplifies to a form known as Wal´en's equation. 3. Strength of a magnetic flux tube. A magnetic flux tube is the region enclosed by the surface determined by all magnetic field lines passing through a given material circuit Ct . The \strength" of a flux tube is defined as the circulation of the potential vector A along the material circuit Ct , viz. ξ def def 2 Φm(t) = A d α = A[α(ξ; t)] α0(ξ; t) d ξ ; (4) ICt · Zξ1 · where α = α(ξ; t) is the parametric expression of the circuit Ct . -
Turbulent Dynamos: Experiments, Nonlinear Saturation of the Magnetic field and field Reversals
Turbulent dynamos: Experiments, nonlinear saturation of the magnetic field and field reversals C. Gissinger & C. Guervilly November 17, 2008 1 Introduction Magnetic fields are present at almost all scales in the universe, from the Earth (roughly 0:5 G) to the Galaxy (10−6 G). It is commonly believed that these magnetic fields are generated by dynamo action i.e. by the turbulent flow of an electrically conducting fluid [12]. Despite this space and time disorganized flow, the magnetic field shows in general a coherent part at the largest scales. The question arising from this observation is the role of the mean flow: Cowling first proposed that the coherent magnetic field could be due to coherent large scale velocity field and this problem is still an open question. 2 MHD equations and dimensionless parameters The equations describing the evolution of a magnetohydrodynamical system are the equation of Navier-Stokes coupled to the induction equation: @v 1 + (v )v = π + ν∆v + f + (B )B ; (1) @t · r −∇ µρ · r @B = (v B) + η ∆B : (2) @t r × × where v is the solenoidal velocity field, B the solenoidal magnetic field, π the pressure r gradient in the fluid, ν the kinematic viscosity, f a forcing term, µ the magnetic permeabil- ity, ρ the fluid density and η the magnetic permeability. Dealing with the geodynamo, the minimal set of parameters for the outer core are : ρ : density of the fluid • µ : magnetic permeability • 54 ν : kinematic diffusivity • σ: conductivity • R : radius of the outer core • V : typical velocity • Ω : rotation rate • The problem involves 7 independent parameters and 4 fundamental units (length L, time T , mass M and electic current A). -
Polywell – a Path to Electrostatic Fusion
Polywell – A Path to Electrostatic Fusion Jaeyoung Park Energy Matter Conversion Corporation (EMC2) University of Wisconsin, October 1, 2014 1 Fusion vs. Solar Power For a 50 cm radius spherical IEC device - Area projection: πr2 = 7850 cm2 à 160 watt for same size solar panel Pfusion =17.6MeV × ∫ < συ >×(nDnT )dV For D-T: 160 Watt à 5.7x1013 n/s -16 3 <συ>max ~ 8x10 cm /s 11 -3 à <ne>~ 7x10 cm Debye length ~ 0.22 cm (at 60 keV) Radius/λD ~ 220 In comparison, 60 kV well over 50 cm 7 -3 (ne-ni) ~ 4x10 cm 2 200 W/m : available solar panel capacity 0D Analysis - No ion convergence case 2 Outline • Polywell Fusion: - Electrostatic Fusion + Magnetic Confinement • Lessons from WB-8 experiments • Recent Confinement Experiments at EMC2 • Future Work and Summary 3 Electrostatic Fusion Fusor polarity Contributions from Farnsworth, Hirsch, Elmore, Tuck, Watson and others Operating principles (virtual cathode type ) • e-beam (and/or grid) accelerates electrons into center • Injected electrons form a potential well • Potential well accelerates/confines ions Virtual cathode • Energetic ions generate fusion near the center polarity Attributes • No ion grid loss • Good ion confinement & ion acceleration • But loss of high energy electrons is too large 4 Polywell Fusion Combines two good ideas in fusion research: Bussard (1985) a) Electrostatic fusion: High energy electron beams form a potential well, which accelerates and confines ions b) High β magnetic cusp: High energy electron confinement in high β cusp: Bussard termed this as “wiffle-ball” (WB). + + + e- e- e- Potential Well: ion heating &confinement Polyhedral coil cusp: electron confinement 5 Wiffle-Ball (WB) vs.