Electrodynamics in 1 and 2 Spatial Dimensions 1 Problem 2 Solution
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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. -
Charges and Fields of a Conductor • in Electrostatic Equilibrium, Free Charges Inside a Conductor Do Not Move
Charges and fields of a conductor • In electrostatic equilibrium, free charges inside a conductor do not move. Thus, E = 0 everywhere in the interior of a conductor. • Since E = 0 inside, there are no net charges anywhere in the interior. Net charges can only be on the surface(s). The electric field must be perpendicular to the surface just outside a conductor, since, otherwise, there would be currents flowing along the surface. Gauss’s Law: Qualitative Statement . Form any closed surface around charges . Count the number of electric field lines coming through the surface, those outward as positive and inward as negative. Then the net number of lines is proportional to the net charges enclosed in the surface. Uniformly charged conductor shell: Inside E = 0 inside • By symmetry, the electric field must only depend on r and is along a radial line everywhere. • Apply Gauss’s law to the blue surface , we get E = 0. •The charge on the inner surface of the conductor must also be zero since E = 0 inside a conductor. Discontinuity in E 5A-12 Gauss' Law: Charge Within a Conductor 5A-12 Gauss' Law: Charge Within a Conductor Electric Potential Energy and Electric Potential • The electrostatic force is a conservative force, which means we can define an electrostatic potential energy. – We can therefore define electric potential or voltage. .Two parallel metal plates containing equal but opposite charges produce a uniform electric field between the plates. .This arrangement is an example of a capacitor, a device to store charge. • A positive test charge placed in the uniform electric field will experience an electrostatic force in the direction of the electric field. -
Maxwell's Equations
Maxwell’s Equations Matt Hansen May 20, 2004 1 Contents 1 Introduction 3 2 The basics 3 2.1 Static charges . 3 2.2 Moving charges . 4 2.3 Magnetism . 4 2.4 Vector operations . 5 2.5 Calculus . 6 2.6 Flux . 6 3 History 7 4 Maxwell’s Equations 8 4.1 Maxwell’s Equations . 8 4.2 Gauss’ law for electricity . 8 4.3 Gauss’ law for magnetism . 10 4.4 Faraday’s law . 11 4.5 Ampere-Maxwell law . 13 5 Conclusion 14 2 1 Introduction If asked, most people outside a physics department would not be able to identify Maxwell’s equations, nor would they be able to state that they dealt with electricity and magnetism. However, Maxwell’s equations have many very important implications in the life of a modern person, so much so that people use devices that function off the principles in Maxwell’s equations every day without even knowing it. 2 The basics 2.1 Static charges In order to understand Maxwell’s equations, it is necessary to understand some basic things about electricity and magnetism first. Static electricity is easy to understand, in that it is just a charge which, as its name implies, does not move until it is given the chance to “escape” to the ground. Amounts of charge are measured in coulombs, abbreviated C. 1C is an extraordi- nary amount of charge, chosen rather arbitrarily to be the charge carried by 6.41418 · 1018 electrons. The symbol for charge in equations is q, sometimes with a subscript like q1 or qenc. -
Electro Magnetic Fields Lecture Notes B.Tech
ELECTRO MAGNETIC FIELDS LECTURE NOTES B.TECH (II YEAR – I SEM) (2019-20) Prepared by: M.KUMARA SWAMY., Asst.Prof Department of Electrical & Electronics Engineering MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY (Autonomous Institution – UGC, Govt. of India) Recognized under 2(f) and 12 (B) of UGC ACT 1956 (Affiliated to JNTUH, Hyderabad, Approved by AICTE - Accredited by NBA & NAAC – ‘A’ Grade - ISO 9001:2015 Certified) Maisammaguda, Dhulapally (Post Via. Kompally), Secunderabad – 500100, Telangana State, India ELECTRO MAGNETIC FIELDS Objectives: • To introduce the concepts of electric field, magnetic field. • Applications of electric and magnetic fields in the development of the theory for power transmission lines and electrical machines. UNIT – I Electrostatics: Electrostatic Fields – Coulomb’s Law – Electric Field Intensity (EFI) – EFI due to a line and a surface charge – Work done in moving a point charge in an electrostatic field – Electric Potential – Properties of potential function – Potential gradient – Gauss’s law – Application of Gauss’s Law – Maxwell’s first law, div ( D )=ρv – Laplace’s and Poison’s equations . Electric dipole – Dipole moment – potential and EFI due to an electric dipole. UNIT – II Dielectrics & Capacitance: Behavior of conductors in an electric field – Conductors and Insulators – Electric field inside a dielectric material – polarization – Dielectric – Conductor and Dielectric – Dielectric boundary conditions – Capacitance – Capacitance of parallel plates – spherical co‐axial capacitors. Current density – conduction and Convection current densities – Ohm’s law in point form – Equation of continuity UNIT – III Magneto Statics: Static magnetic fields – Biot‐Savart’s law – Magnetic field intensity (MFI) – MFI due to a straight current carrying filament – MFI due to circular, square and solenoid current Carrying wire – Relation between magnetic flux and magnetic flux density – Maxwell’s second Equation, div(B)=0, Ampere’s Law & Applications: Ampere’s circuital law and its applications viz. -
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). -
Electromagnetic Fields and Energy
MIT OpenCourseWare http://ocw.mit.edu Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. Please use the following citation format: Haus, Hermann A., and James R. Melcher, Electromagnetic Fields and Energy. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed [Date]). License: Creative Commons Attribution-NonCommercial-Share Alike. Also available from Prentice-Hall: Englewood Cliffs, NJ, 1989. ISBN: 9780132490207. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms 8 MAGNETOQUASISTATIC FIELDS: SUPERPOSITION INTEGRAL AND BOUNDARY VALUE POINTS OF VIEW 8.0 INTRODUCTION MQS Fields: Superposition Integral and Boundary Value Views We now follow the study of electroquasistatics with that of magnetoquasistat ics. In terms of the flow of ideas summarized in Fig. 1.0.1, we have completed the EQS column to the left. Starting from the top of the MQS column on the right, recall from Chap. 3 that the laws of primary interest are Amp`ere’s law (with the displacement current density neglected) and the magnetic flux continuity law (Table 3.6.1). � × H = J (1) � · µoH = 0 (2) These laws have associated with them continuity conditions at interfaces. If the in terface carries a surface current density K, then the continuity condition associated with (1) is (1.4.16) n × (Ha − Hb) = K (3) and the continuity condition associated with (2) is (1.7.6). a b n · (µoH − µoH ) = 0 (4) In the absence of magnetizable materials, these laws determine the magnetic field intensity H given its source, the current density J. -
Retarded Potentials and Radiation
Retarded Potentials and Radiation Siddhartha Sinha Joint Astronomy Programme student Department of Physics Indian Institute of Science Bangalore. December 10, 2003 1 Abstract The transition from the study of electrostatics and magnetostatics to the study of accelarating charges and changing currents ,leads to ra- diation of electromagnetic waves from the source distributions.These come as a direct consequence of particular solutions of the inhomoge- neous wave equations satisfied by the scalar and vector potentials. 2 1 MAXWELL'S EQUATIONS The set of four equations r · D = 4πρ (1) r · B = 0 (2) 1 @B r × E + = 0 (3) c @t 4π 1 @D r × H = J + (4) c c @t known as the Maxwell equations ,forms the basis of all classical elec- tromagnetic phenomena.Maxwell's equations,along with the Lorentz force equation and Newton's laws of motion provide a complete description of the classical dynamics of interacting charged particles and electromagnetic fields.Gaussian units are employed in writing the Maxwell equations. These equations consist of a set of coupled first order linear partial differen- tial equations relating the various components of the electric and magnetic fields.Note that Eqns (2) and (3) are homogeneous while Eqns (1) and (4) are inhomogeneous. The homogeneous equations can be used to define scalar and vector potentials.Thus Eqn(2) yields B = r × A (5) where A is the vector potential. Substituting the value of B from Eqn(5) in Eqn(3) we get 1 @A r × (E + = 0 (6) c @t It follows that 1 @A E + = −∇φ (7) c @t where φ is the scalar potential. -
9 AP-C Electric Potential, Energy and Capacitance
#9 AP-C Electric Potential, Energy and Capacitance AP-C Objectives (from College Board Learning Objectives for AP Physics) 1. Electric potential due to point charges a. Determine the electric potential in the vicinity of one or more point charges. b. Calculate the electrical work done on a charge or use conservation of energy to determine the speed of a charge that moves through a specified potential difference. c. Determine the direction and approximate magnitude of the electric field at various positions given a sketch of equipotentials. d. Calculate the potential difference between two points in a uniform electric field, and state which point is at the higher potential. e. Calculate how much work is required to move a test charge from one location to another in the field of fixed point charges. f. Calculate the electrostatic potential energy of a system of two or more point charges, and calculate how much work is require to establish the charge system. g. Use integration to determine the electric potential difference between two points on a line, given electric field strength as a function of position on that line. h. State the relationship between field and potential, and define and apply the concept of a conservative electric field. 2. Electric potential due to other charge distributions a. Calculate the electric potential on the axis of a uniformly charged disk. b. Derive expressions for electric potential as a function of position for uniformly charged wires, parallel charged plates, coaxial cylinders, and concentric spheres. 3. Conductors a. Understand the nature of electric fields and electric potential in and around conductors. -
13. Maxwell's Equations and EM Waves. Hunt (1991), Chaps 5 & 6 A
13. Maxwell's Equations and EM Waves. Hunt (1991), Chaps 5 & 6 A. The Energy of an Electromagnetic Field. • 1880s revision of Maxwell: Guiding principle = concept of energy flow. • Evidence for energy flow through seemingly empty space: ! induced currents ! air-core transformers, condensers. • But: Where is this energy located? Two equivalent expressions for electromagnetic energy of steady current: ½A J ½µH2 • A = vector potential, J = current • µ = permeability, H = magnetic force. density. • Suggests energy located outside • Suggests energy located in conductor. conductor in magnetic field. 13. Maxwell's Equations and EM Waves. Hunt (1991), Chaps 5 & 6 A. The Energy of an Electromagnetic Field. • 1880s revision of Maxwell: Guiding principle = concept of energy flow. • Evidence for energy flow through seemingly empty space: ! induced currents ! air-core transformers, condensers. • But: Where is this energy located? Two equivalent expressions for electrostatic energy: ½qψ# ½εE2 • q = charge, ψ = electric potential. • ε = permittivity, E = electric force. • Suggests energy located in charged • Suggests energy located outside object. charged object in electric field. • Maxwell: Treated potentials A, ψ as fundamental quantities. Poynting's account of energy flux • Research project (1884): "How does the energy about an electric current pass from point to point -- that is, by what paths and acording to what law does it travel from the part of the circuit where it is first recognizable as electric and John Poynting magnetic, to the parts where it is changed into heat and other forms?" (1852-1914) • Solution: The energy flux at each point in space is encoded in a vector S given by S = E × H. -
Classical Electromagnetism
Classical Electromagnetism Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Maxwell’s Equations 7 1.1 Introduction . .................................. 7 1.2 Maxwell’sEquations................................ 7 1.3 ScalarandVectorPotentials............................. 8 1.4 DiracDeltaFunction................................ 9 1.5 Three-DimensionalDiracDeltaFunction...................... 9 1.6 Solution of Inhomogeneous Wave Equation . .................... 10 1.7 RetardedPotentials................................. 16 1.8 RetardedFields................................... 17 1.9 ElectromagneticEnergyConservation....................... 19 1.10 ElectromagneticMomentumConservation..................... 20 1.11 Exercises....................................... 22 2 Electrostatic Fields 25 2.1 Introduction . .................................. 25 2.2 Laplace’s Equation . ........................... 25 2.3 Poisson’sEquation.................................. 26 2.4 Coulomb’sLaw................................... 27 2.5 ElectricScalarPotential............................... 28 2.6 ElectrostaticEnergy................................. 29 2.7 ElectricDipoles................................... 33 2.8 ChargeSheetsandDipoleSheets.......................... 34 2.9 Green’sTheorem.................................. 37 2.10 Boundary Value Problems . ........................... 40 2.11 DirichletGreen’sFunctionforSphericalSurface.................. 43 2.12 Exercises....................................... 46 3 Potential Theory -
The Retarded Potential of a Non-Homogeneous Wave Equation: Introductory Analysis Through the Green Functions
EDUCATION Revista Mexicana de F´ısica E 64 (2018) 26–38 JANUARY–JUNE 2018 The retarded potential of a non-homogeneous wave equation: introductory analysis through the Green functions A. Tellez-Qui´ nones˜ a;¤, J.C. Valdiviezo-Navarroa, A. Salazar-Garibaya, and A.A. Lopez-Caloca´ b aCONACYT-Centro de Investigacion´ en Geograf´ıa y Geomatica,´ Ing. J.L. Tamayo, A.C. (Unidad Merida),´ Carretera Sierra Papacal-Chuburna Puerto Km 5, Sierra Papacal-Yucatan,´ 97302, Mexico.´ bCentro de Investigacion´ en Geograf´ıa y Geomatica,´ Ing. J.L. Tamayo, A.C., Contoy 137-Lomas de Padierna, Tlalpan-Ciudad de Mexico,´ 14240, Mexico.´ e-mail: [email protected] Received 17 August 2017; accepted 11 September 2017 The retarded potential, a solution of the non-homogeneous wave equation, is a subject of particular interest in many physics and engineering applications. Examples of such applications may be the problem of solving the wave equation involved in the emission and reception of a signal in a synthetic aperture radar (SAR), scattering and backscattering, and general electrodynamics for media free of magnetic charges. However, the construction of this potential solution is based on the theory of distributions, a topic that requires special care and time to be understood with mathematical rigor. Thus, the goal of this study is to provide an introductory analysis, with a medium level of formalism, on the construction of this potential solution and the handling of Green functions represented by sequences of well-behaved approximating functions. Keywords: Mathematical methods in physics; mathematics; diffraction theory; backscattering; radar. PACS: 01.30.Rr; 02.30.Em; 41.20.Jb 1. -
Topic 2.3: Electric and Magnetic Fields
TOPIC 2.3: ELECTRIC AND MAGNETIC FIELDS S4P-2-13 Compare and contrast the inverse square nature of gravitational and electric fields. S4P-2-14 State Coulomb’s Law and solve problems for more than one electric force acting on a charge. Include: one and two dimensions S4P-2-15 Illustrate, using diagrams, how the charge distribution on two oppositely charged parallel plates results in a uniform field. S4P-2-16 Derive an equation for the electric potential energy between two oppositely ∆ charged parallel plates (Ee = qE d). S4P-2-17 Describe electric potential as the electric potential energy per unit charge. S4P-2-18 Identify the unit of electric potential as the volt. S4P-2-19 Define electric potential difference (voltage) and express the electric field between two oppositely charged parallel plates in terms of voltage and the separation ⎛ ∆V ⎞ between the plates ⎜ε = ⎟. ⎝ d ⎠ S4P-2-20 Solve problems for charges moving between or through parallel plates. S4P-2-21 Use hand rules to describe the directional relationships between electric and magnetic fields and moving charges. S4P-2-22 Describe qualitatively various technologies that use electric and magnetic fields. Examples: electromagnetic devices (such as a solenoid, motor, bell, or relay), cathode ray tube, mass spectrometer, antenna Topic 2: Fields • SENIOR 4 PHYSICS GENERAL LEARNING OUTCOME SPECIFIC LEARNING OUTCOME CONNECTION S4P-2-13: Compare and contrast the Students will… inverse square nature of Describe and appreciate the gravitational and electric fields. similarity and diversity of forms, functions, and patterns within the natural and constructed world (GLO E1) SUGGESTIONS FOR INSTRUCTION Entry Level Knowledge Notes to the Teacher The definition of gravitational fields around a point When examining the gravitational field of a mass, mass it is useful to view the mass as a point mass, irrespective of its size.