* Magnetic Scalar Potential * Magnetic Vector Potential. PPT No. 19


 Suzanna O’Neal’
 1 years ago
 Views:
Transcription
1 * Magnetic Scalar Potential * Magnetic Vector Potential PPT No. 19
2 Magnetic Potentials The Magnetic Potential is a method of representing the Magnetic field by using a quantity called Potential instead of the actual B vector field.
3 Magnetic Potentials Magnetic field can be related to a potential by two methods which give rise to two possible types of magnetic potentials used in different situations: 1. Magnetic Scalar Potential 2. Magnetic Vector Potential
4 A) Magnetic Scalar Potential In Electrostatics, electric field E is derivable from the electric potential V. V is a scalar quantity and easier to handle than E which is a vector quantity. In Magnetostatics, the quantity Magnetic scalar potential can be obtained using analogues relation
5 A) Magnetic Scalar Potential In regions of space in the absence of currents, the current density j =0 = 0 B is derivable from the gradient of a potential Therefore B can be expressed as the gradient of a scalar quantity φ m B =  φ m φ m is called as the Magnetic scalar potential.
6 A) Magnetic Scalar Potential The presence of a magnetic moment m creates a magnetic field B which is the gradient of some scalar field φ m. The divergence of the magnetic field B is zero,.b = 0 By definition, the divergence of the gradient of the scalar field is also zero, . φ m = 0 or 2 φ m = 0. The operator 2 is called the Laplacian and 2 φ m = 0 is the Laplace s equation.
7 A) Magnetic Scalar Potential 2 φ m = 0 Laplace s equation is valid only outside the magnetic sources and away from currents. Magnetic field can be calculated from the magnetic scalar potential using solutions of Laplace s equation.
8 The magnetic scalar potential is useful only in the region of space away from free currents. If J=0, then only magnetic flux density can be computed from the magnetic scalar potential The potential function which overcomes this limitation and is useful to compute B in region where J is present is. Magnetic Vector Potential
9 Magnetic fields are generated by steady (timeindependent) currents & satisfy Gauss Law Since the divergence of a curl is zero, B can be written as the curl of a vector A as
10 Any solenoidal vector field (e.g. B) in physics can always be written as the curl of some other vector field (A). The quantity A is known as the Magnetic Vector Potential.
11 {However, magnetic vector potential is not directly associated with work the way that scalar potential (e.g. Electric potential V) is associated with work} Work done against the electric field E is stored as electric potential energy U given in terms of electric dipole moment p and E as
12 The vector potential is defined to be consistent with Ampere s Circuital Law and It can be expressed in terms of either current i or current density j (i.e. the sources of magnetic field) as follows
13 However, A is Not uniquely defined by the above equation. Any function whose curl is zero, can be added to A, then the result would still be the same field B. e.g. If ψ, the Gradient of a scalar ψ is added to A x (A + ψ )= x A + x ψ = x A = B
14 To make A more specific/ unique, additional condition needs to be imposed on A. In Magnetostatics a convenient condition which makes calculations easier can be specified as. A = 0 (In Electrodynamics, this condition cannot be imposed)
15 The set of equations which uniquely define the vector potential A and also satisfy the fundamental equation of Gauss Law. B = 0 {the magnetic field is divergencefree}, are as follows
16 From Ampere s law Therefore the equation can be written as This equation is similar to Poisson's equation, the only difference is that A is a vector.
17 Each component (e.g. along x, y, z axes) of A must satisfy the differential equation of the type A unique solution to the above Poisson's equation can be found (By combining the solutions for components on x, y, z). It specifies the magnetic vector potential A generated by steady currents.
18 First A is determined using Poisson's equation then it is substituted in the equation Thus the field B produced by a steady current can be computed.
19 Gauge Transformation According to Helmholtz's theorem a vector field is fully specified by its divergence and its curl. The curl of the vector potential A gives the magnetic field B via Eq. However, the divergence of A has no physical significance can be chosen freely as desired
20 According to the equation the magnetic field is invariant under the transformation In other words, the vector potential is undetermined to the gradient of a scalar field can be chosen as desired
21 The electric scalar potential is undetermined to an arbitrary additive constant, since the transformation leaves the electric field invariant in Equation The transformations and are examples of gauge transformations in Mathematics.
22 In electromagnetic theory, several "gauges" have been used to advantage depending on the specific types of calculations The choice of a particular function ψ or a particular constant c is referred to as a choice of the gauge.
23 The gauge can be fixed as desired. Usually it is chosen to make equations simplest possible. It is convenient to choose gauge for the scalar potential Ф such that Ф 0 at infinity. The gauge for A is chosen such that This particular choice is known as the Coulomb gauge
Electro Magnetic Fields
Electro Magnetic Fields Faheem Ahmed Khan, Assoc Prof. EEE Department, Ghousia College of Engineering, Ramanagaram EEE, GCE,Ramanagaram Page 1 of 50 Coulomb s Law and electric field intensity Experimental
More information* Biot Savart s Law Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.
* Biot Savart s Law Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying
More information73 The BiotSavart Law and the Magnetic Vector Potential
11/14/4 section 7_3 The BiotSavart Law blank.doc 1/1 73 The BiotSavart Law and the Magnetic ector Potential Reading Assignment: pp. 818 Q: Given some field B, how can we determine the source J that
More informationMagnetostatics II. Lecture 24: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay
Magnetostatics II Lecture 4: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Magnetic field due to a solenoid and a toroid A solenoid is essentially a long current loop
More information6 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 informationVII MAXWELL S EQUATIONS
VII MAXWELL S EQUATIONS 71 The story so far In this section we will summarise the understanding of electromagnetism which we have arrived at so far We know that there are two fields which must be considered,
More informationExample: The Electrostatic Fields of a Coaxial Line
/8/24 Example The Electorostatic Fields of a Coaxial Line / Example: The Electrostatic Fields of a Coaxial Line A common form of a transmission line is the coaxial cable. Outer Conductor a b ε + V  Inner
More informationÇANKAYA UNIVERSITY Faculty of Engineering and Architecture
ÇANKAYA UNIVERSITY Faculty of Engineering and Architecture Course Definition Form This form should be used for both a new elective or compulsory course being proposed and curricula development processes
More information6. In cylindrical coordinate system, the differential normal area along a z is calculated as: a) ds = ρ d dz b) ds = dρ dz c) ds = ρ d dρ d) ds = dρ d
Electrical Engineering Department Electromagnetics I (802323) G1 Dr. Mouaaz Nahas First Term (14361437), Second Exam Tuesday 07/02/1437 H االسم: الرقم الجامعي: Start from here Part A CLO 1: Students will
More informationChapter 5. Magnetostatics and Electromagnetic Induction
Chapter 5. Magnetostatics and Electromagnetic Induction 5.1 Magnetic Field of Steady Currents The Lorentz force law The magnetic force in a charge q, moving with velocity v in a magnetic field B in a magnetic
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 informationDIVERGENCE AND CURL THEOREMS
This document is stored in Documents/4C/Gausstokes.tex. with LaTex. Compile it November 29, 2014 Hans P. Paar DIVERGENCE AND CURL THEOREM 1 Introduction We discuss the theorems of Gauss and tokes also
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 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 informationdv div F = F da (1) D
Note I.6 1 4 October The Gauss Theorem The Gauss, or divergence, theorem states that, if is a connected threedimensional region in R 3 whose boundary is a closed, piecewise connected surface and F is
More informationThe force on a moving charged particle. A particle with charge Q moving with velocity v in a magnetic field B is subject to a force F mag = Q v B
Magnetostatics The force on a moving charged particle A particle with charge Q moving with velocity v in a magnetic field B is subject to a force F mag = Q v B If there is also an electric field E present,
More informationCoefficient of Potential and Capacitance
Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that
More information"  angle between l and a R
Magnetostatic Fields According to Coulomb s law, any distribution of stationary charge produces a static electric field (electrostatic field). The analogous equation to Coulomb s law for electric fields
More informationElectromagnetic Induction
Electromagnetic Induction Lecture 29: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Mutual Inductance In the last lecture, we enunciated the Faraday s law according to
More informationPOTENTIAL FORMULATIONS IN MAGNETICS APPLYING THE FINITE ELEMENT METHOD. Lecture notes
MIKLÓS KUCZMANN POTENTIAL FORMULATIONS IN MAGNETICS APPLYING THE FINITE ELEMENT METHOD Lecture notes Laboratory of Electromagnetic Fields Széchenyi István University Győr, Hungary 2009. Contents 1 Introduction
More informationPH585: Magnetic dipoles and so forth
PH585: Magnetic dipoles and so forth 1 Magnetic Moments Magnetic moments µ are analogous to dipole moments p in electrostatics. There are two sorts of magnetic dipoles we will consider: a dipole consisting
More informationPPT No. 26. Uniformly Magnetized Sphere in the External Magnetic Field. Electromagnets
PPT No. 26 Uniformly Magnetized Sphere in the External Magnetic Field Electromagnets Uniformly magnetized sphere in external magnetic field The Topic Uniformly magnetized sphere in external magnetic field,
More informationFaraday s Law & Maxwell s Equations (Griffiths Chapter 7: Sections 23) B t da = S
Dr. Alain Brizard Electromagnetic Theory I PY 3 Faraday s Law & Maxwell s Equations Griffiths Chapter 7: Sections 3 Electromagnetic Induction The flux rule states that a changing magnetic flux Φ B = S
More informationThe electronic Hamiltonian in an electromagnetic field
The electronic Hamiltonian in an electromagnetic field Trygve Helgaker Department of Chemistry, University of Oslo, P.O.B. 1033 Blindern, N0315 Oslo, Norway Poul Jørgensen and Jeppe Olsen Department of
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 informationNote: be careful not confuse the conductivity σ with the surface charge σ, or resistivity ρ with volume charge ρ.
SECTION 7 Electrodynamics This section (based on Chapter 7 of Griffiths) covers effects where there is a time dependence of the electric and magnetic fields, leading to Maxwell s equations. The topics
More informationPhysics 217: Electricity and Magnetism I
Physics 217: Electricity and Magnetism I Fall 2002 This semester we will explore electrostatics and magnetostatics the consequences of the laws discovered empirically by Coulomb, Gauss, Ampère and Faraday
More informationThen the second equation becomes ³ j
Magnetic vector potential When we derived the scalar electric potential we started with the relation r E = 0 to conclude that E could be written as the gradient of a scalar potential. That won t work for
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 informationElasticity Theory Basics
G22.3033002: 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 informationElectromagnetism  Lecture 8. Maxwell s Equations
Electromagnetism  Lecture 8 Maxwell s Equations Continuity Equation Displacement Current Modification to Ampère s Law Maxwell s Equations in Vacuo Solution of Maxwell s Equations Introduction to Electromagnetic
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 informationLecture 13. Magnetic Field, Magnetic Forces on Moving Charges. Outline:
Lecture 13. Magnetic Field, Magnetic Forces on Moving Charges. Outline: Intro to Magnetostatics. Magnetic Field Flux, Absence of Magnetic Monopoles. Force on charges moving in magnetic field. 1 Intro to
More informationSpecial Relativity and Electromagnetism Yannis PAPAPHILIPPOU CERN
Special Relativity and Electromagnetism Yannis PAPAPHILIPPOU CERN United States Particle Accelerator School, University of California  Santa Cruz, Santa Rosa, CA 14 th 18 th January 2008 1 Outline Notions
More information3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field
3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important
More informationSIO 229 Gravity and Geomagnetism: Class Description and Goals
SIO 229 Gravity and Geomagnetism: Class Description and Goals This graduate class provides an introduction to gravity and geomagnetism at a level suitable for advanced nonspecialists in geophysics. Topics
More informationDivergence and Curl. . Here we discuss some details of the divergence and curl. and the magnetic field B ( r,t)
Divergence and url Overview and Motivation: In the upcoming two lectures we will be discussing Maxwell's equations. These equations involve both the divergence and curl of two vector fields the electric
More informationarxiv:1111.4354v2 [physics.accph] 27 Oct 2014
Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.accph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic
More informationVECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors
Prof. S.M. Tobias Jan 2009 VECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors A scalar is a physical quantity with magnitude only A vector is a physical quantity with magnitude and direction A unit
More informationF = U. (1) We shall answer these questions by examining the dimensions n = 1,2,3 separately.
Lecture 24 Conservative forces in physics (cont d) Determining whether or not a force is conservative We have just examined some examples of conservative forces in R 2 and R 3. We now address the following
More informationA Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle
A Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle Page 1 of 15 Abstract: The wireless power transfer link between two coils is determined by the properties of the
More informationModule 3 : MAGNETIC FIELD Lecture 15 : Biot Savarts' Law
Module 3 : MAGNETIC FIELD Lecture 15 : Biot Savarts' Law Objectives In this lecture you will learn the following Study BiotSavart's law Calculate magnetic field of induction due to some simple current
More informationUsing contemporary education strategies and approaches to redesign Classical Electrodynamics
Using contemporary education strategies and approaches to redesign Classical Electrodynamics Wang Yue College of Applied Science Beijing University of Technology Beijing 100022 People s Republic of China
More informationModule 3 : Electromagnetism Lecture 13 : Magnetic Field
Module 3 : Electromagnetism Lecture 13 : Magnetic Field Objectives In this lecture you will learn the following Electric current is the source of magnetic field. When a charged particle is placed in an
More informationSyllabus for EE 341 Electromagnetic Fields and Waves Fall
Syllabus for EE 34 Electromagnetic Fields and Waves Fall 205206 Instructor: Prof. Dr. Erdem YAZGAN Office: 38 Phone: (032) 5850027 Email: erdem.yazgan@tedu.edu.tr Time Schedule: Monday (.00 2.50),
More informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road, New Delhi , Ph. : ,
1 E L E C T R O S TAT I C S 1. Define lines of forces and write down its properties. Draw the lines of force to represent (i) uniform electric field (ii) positive charge (iii) negative charge (iv) two
More informationPart A Electromagnetism
Part A Electromagnetism James Sparks sparks@maths.ox.ac.uk Hilary Term 2009 E = ρ ǫ 0 B = 0 E = B ( ) E B = µ 0 J + ǫ 0 Contents 0 Introduction ii 0.1 About these notes.................................
More informationChapter 22: The Electric Field. Read Chapter 22 Do Ch. 22 Questions 3, 5, 7, 9 Do Ch. 22 Problems 5, 19, 24
Chapter : The Electric Field Read Chapter Do Ch. Questions 3, 5, 7, 9 Do Ch. Problems 5, 19, 4 The Electric Field Replaces actionatadistance Instead of Q 1 exerting a force directly on Q at a distance,
More informationAP Physics C Chapter 23 Notes Yockers Faraday s Law, Inductance, and Maxwell s Equations
AP Physics C Chapter 3 Notes Yockers Faraday s aw, Inductance, and Maxwell s Equations Faraday s aw of Induction  induced current a metal wire moved in a uniform magnetic field  the charges (electrons)
More informationLecture 13. Magnetic Field, Magnetic Forces on Moving Charges. Outline:
Lecture 13. Magnetic Field, Magnetic Forces on Moving Charges. Outline: Intro to Magnetostatics. Magnetic Field Flux, Absence of Magnetic Monopoles. Force on charges moving in magnetic field. 1 Structure
More informationMultipole Theory in Electromagnetism
Multipole Theory in Electromagnetism Classical, quantum, and symmetry aspects, with applications R. E. RAAB О. L. DE LANGE School of Chemical and Physical Sciences, University of Natal, Pietermaritzburg,
More informationObjectives for the standardized exam
III. ELECTRICITY AND MAGNETISM A. Electrostatics 1. Charge and Coulomb s Law a) Students should understand the concept of electric charge, so they can: (1) Describe the types of charge and the attraction
More informationChapter 9. Electromagnetic Waves
Chapter 9. Electromagnetic Waves 9.2 Electromagnetic waves in Vacuum 9.2.1 The Wave Equation for E and B In Vacuum, no free charges and no currents 0, J 0, q 0, I 0 B E 0 B 0 E  t B 0 0 Let s derive the
More informationELECTROSTATICS. Ans: It is a fundamental property of matter which is responsible for all electrical effects
ELECTROSTATICS One Marks Questions with Answers: 1.What is an electric charge? Ans: It is a fundamental property of matter which is responsible for all electrical effects 2. Write the SI unit of charge?
More informationRelativistic Electromagnetism
Chapter 8 Relativistic Electromagnetism In which it is shown that electricity and magnetism can no more be separated than space and time. 8.1 Magnetism from Electricity Our starting point is the electric
More information* Selfinductance * Mutual inductance * Transformers. PPT No. 32
* Selfinductance * Mutual inductance * Transformers PPT No. 32 Inductance According to Faraday s Electromagnetic Induction law, induction of an electromotive force occurs in a circuit by varying the magnetic
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 36737 (Retyped for readability with same page breaks) ON AN EXPRESSION OF THE ELECTROMAGNETIC
More informationScalar and vector potentials, Helmholtz decomposition, and de Rham cohomology
Scalar and vector potentials,, and de Rham cohomology Alberto Valli Department of Mathematics, University of Trento, Italy A. Valli Potentials,, de Rham cohomology Outline 1 Introduction 2 3 4 5 A. Valli
More informationCLASSICAL ELECTRODYNAMICS AND THEORY OF RELATIVITY
arxiv:physics/0311011v1 [physics.edph] 4 Nov RUSSIAN FEDERAL COMMITTEE FOR HIGHER EDUCATION BASHKIR STATE UNIVERSITY SHARIPOV R. A. CLASSICAL ELECTRODYNAMICS AND THEORY OF RELATIVITY the manual Ufa 1997
More informationThe Charge to Mass Ratio (e/m) Ratio of the Electron. NOTE: You will make several sketches of magnetic fields during the lab.
The Charge to Mass Ratio (e/m) Ratio of the Electron NOTE: You will make several sketches of magnetic fields during the lab. Remember to include these sketches in your lab notebook as they will be part
More informationPhysics 221A Spring 2016 Appendix A Gaussian, SI and Other Systems of Units in Electromagnetic Theory
Copyright c 2016 by Robert G. Littlejohn Physics 221A Spring 2016 Appendix A Gaussian, SI and Other Systems of Units in Electromagnetic Theory 1. Introduction Most students are taught SI units in their
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 informationPHY 301: Mathematical Methods I Curvilinear Coordinate System (1012 Lectures)
PHY 301: Mathematical Methods I Curvilinear Coordinate System (1012 Lectures) Dr. Alok Kumar Department of Physical Sciences IISER, Bhopal Abstract The Curvilinear coordinates are the common name of
More informationMagnetic Field & Right Hand Rule. Academic Resource Center
Magnetic Field & Right Hand Rule Academic Resource Center Magnetic Fields And Right Hand Rules By: Anthony Ruth Magnetic Fields vs Electric Fields Magnetic fields are similar to electric fields, but they
More informationChapter 33. The Magnetic Field
Chapter 33. The Magnetic Field Digital information is stored on a hard disk as microscopic patches of magnetism. Just what is magnetism? How are magnetic fields created? What are their properties? These
More informationAP Physics C: Electricity and Magnetism: Syllabus 2
AP Physics C: Electricity and Magnetism: Syllabus 2 Scoring Components SC1 SC2 SC SC SC5 SC6 SC7 The course provides and provides instruction in electrostatics. The course provides and provides instruction
More informationChapter 29 Electromagnetic Induction
Chapter 29 Electromagnetic Induction  Induction Experiments  Faraday s Law  Lenz s Law  Motional Electromotive Force  Induced Electric Fields  Eddy Currents  Displacement Current and Maxwell s Equations
More informationSources of Magnetic Field: Summary
Sources of Magnetic Field: Summary Single Moving Charge (BiotSavart for a charge): Steady Current in a Wire (BiotSavart for current): Infinite Straight Wire: Direction is from the Right Hand Rule The
More informationElectromagnetism Laws and Equations
Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E and Dfields............................................. Electrostatic Force............................................2
More informationEðlisfræði 2, vor 2007
[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline
More informationThis chapter describes magnetostatics in a vacuum. By magnetostatics we, of course, don t mean that the charges are static but rather the magnetic
This chapter describes magnetostatics in a vacuum. By magnetostatics we, of course, don t mean that the charges are static but rather the magnetic fields, electric fields and currents are constant in time.
More informationElectric Flux. Phys 122 Lecture 4 G. Rybka
Electric Flux Phys 122 Lecture 4 G. Rybka Electric Field Distribution Summary Dipole ~ 1 / R 3 Point Charge ~ 1 / R 2 Infinite Line of Charge ~ 1 / R Outline Electric field acting on charges Defining Electric
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationChapter 4 Parabolic Equations
161 Chapter 4 Parabolic Equations Partial differential equations occur in abundance in a variety of areas from engineering, mathematical biology and physics. In this chapter we will concentrate upon the
More informationMicrowaves. Microwaves in the Electromagnetic Spectrum (300 MHz GHz)
Microwaves Microwaves in the Electromagnetic Spectrum (300 MHz  300 GHz) ELF Extremely Low Frequency 330 Hz SLF Super Low Frequency 30300 Hz ULF Ultra Low Frequency 300 Hz  3 khz VLF Very Low Frequency
More informationCONSERVATION LAWS. See Figures 2 and 1.
CONSERVATION LAWS 1. Multivariable calculus 1.1. Divergence theorem (of Gauss). This states that the volume integral in of the divergence of the vectorvalued function F is equal to the total flux of F
More informationElectromagnetic interactionsi. 1.H.Hutchinson
Electromagnetic interactionsi 1.H.Hutchinson Chapter 1 Maxwell's Equations and Electromagnetic Fields 1.1 Introduction 1.l.1 Maxwell's Equations (1865) The governing equations of electromagnetism P V.E=
More informationDifferentiation of vectors
Chapter 4 Differentiation of vectors 4.1 Vectorvalued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where
More informationAPPLICATIONS OF TENSOR ANALYSIS
APPLICATIONS OF TENSOR ANALYSIS (formerly titled: Applications of the Absolute Differential Calculus) by A J McCONNELL Dover Publications, Inc, Neiv York CONTENTS PART I ALGEBRAIC PRELIMINARIES/ CHAPTER
More informationReduction of the NavierStocks Equation to the Natural ThreeVelocity Form
ISSN: 3590040 Vol. 2 Issue 4, April  205 Reduction of the NavierStocks Equation to the Natural ThreeVelocity Form Alexei M. Frolov Department of Applied Mathematics University of Western Ontario, London,
More informationScalars, Vectors and Tensors
Scalars, Vectors and Tensors A scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. Examples are hydrostatic pressure and temperature. A vector
More informationGauss Formulation of the gravitational forces
Chapter 1 Gauss Formulation of the gravitational forces 1.1 ome theoretical background We have seen in class the Newton s formulation of the gravitational law. Often it is interesting to describe a conservative
More informationElectrical impedance  Wikipedia, the free encyclopedia
Electrical impedance From Wikipedia, the free encyclopedia Electrical impedance, or simply impedance, describes a measure of opposition to a sinusoidal alternating current (AC). Electrical impedance extends
More informationName: Lab Partner: Section: The purpose of this lab is to study induction. Faraday s law of induction and Lenz s law will be explored.
Chapter 8 Induction  Faraday s Law Name: Lab Partner: Section: 8.1 Purpose The purpose of this lab is to study induction. Faraday s law of induction and Lenz s law will be explored. 8.2 Introduction It
More informationPHYS 1444 Section 003. Lecture #6. Chapter 21. Chapter 22 Gauss s Law. Electric Dipoles. Electric Flux. Thursday, Sept. 8, 2011 Dr.
PHYS 1444 Section 003 Chapter 21 Lecture #6 Dr. Jaehoon Electric Dipoles Chapter 22 Gauss s Law Electric Flux 1 Quiz #2 Thursday, Sept. 15 Beginning of the class Announcements Covers: CH21.1 through what
More informationSCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA.
Pocket book of Electrical Engineering Formulas Content 1. Elementary Algebra and Geometry 1. Fundamental Properties (real numbers) 1 2. Exponents 2 3. Fractional Exponents 2 4. Irrational Exponents 2 5.
More informationElectric field outside a parallel plate capacitor
Electric field outside a parallel plate capacitor G. W. Parker a) Department of Physics, North Carolina State University, Raleigh, North Carolina 769580 Received 7 September 00; accepted 3 January 00
More information(b) Draw the direction of for the (b) Draw the the direction of for the
2. An electric dipole consists of 2A. A magnetic dipole consists of a positive charge +Q at one end of a bar magnet with a north pole at one an insulating rod of length d and a end and a south pole at
More informationChapter 19 Magnetism Magnets Poles of a magnet are the ends where objects are most strongly attracted Two poles, called north and south Like poles
Chapter 19 Magnetism Magnets Poles of a magnet are the ends where objects are most strongly attracted Two poles, called north and south Like poles repel each other and unlike poles attract each other Similar
More informationarxiv:physics/0106088v1 [physics.edph] 27 Jun 2001
Introducing Time Dependence into the Static Maxwell Equations arxiv:physics/0106088v1 [physics.edph] 27 Jun 2001 Avraham Gal Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
More informationDerivation of the relativistic momentum and relativistic equation of motion from Newton s second law and Minkowskian spacetime geometry
Apeiron, Vol. 15, No. 3, July 2008 206 Derivation of the relativistic momentum and relativistic equation of motion from Newton s second law and Minkowskian spacetime geometry Krzysztof Rȩbilas Zak lad
More informationEðlisfræði 2, vor 2007
[ Assignment View ] [ Print ] Eðlisfræði 2, vor 2007 30. Inductance Assignment is due at 2:00am on Wednesday, March 14, 2007 Credit for problems submitted late will decrease to 0% after the deadline has
More informationHermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)
CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system
More informationTransformed E&M I homework. Divergence and Curl of B (Ampereʼs Law) (Griffiths Chapter 5)
Transformed E&M I homework Divergence and Curl of B (Ampereʼs Law) (Griffiths Chapter 5) Divergence and curl of B (Connections between E and B, Ampere s Law) Question 1. B of cylinder with hole Pollack
More informationDirichlet forms methods for error calculus and sensitivity analysis
Dirichlet forms methods for error calculus and sensitivity analysis Nicolas BOULEAU, Osaka university, november 2004 These lectures propose tools for studying sensitivity of models to scalar or functional
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special DistributionsVI Today, I am going to introduce
More informationChap 21. Electromagnetic Induction
Chap 21. Electromagnetic Induction Sec. 1  Magnetic field Magnetic fields are produced by electric currents: They can be macroscopic currents in wires. They can be microscopic currents ex: with electrons
More informationMoving Charge in Magnetic Field
Chapter 1 Moving Charge in Magnetic Field Day 1 Introduction Two bar magnets attract when opposite poles (N and S, or and N) are next to each other The bar magnets repel when like poles (N and N, or S
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 informationGradient, Divergence and Curl in Curvilinear Coordinates
Gradient, Divergence and Curl in Curvilinear Coordinates Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems.
More information