DOCSLIB.ORG

Electromagnetism Course Notes (PHYS09060) 2017/18

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Electromagnetism Course Notes (PHYS09060) 2017/18 Electromagnetism Course Notes (PHYS09060) 2017/18 Andreas Hermann School of Physics and Astronomy The University of Edinburgh September 2017 Contents 1 Whistlestop tour of Vector Calculus 6 1.1 Gradient . 6 1.2 Divergence and Gauss' Theorem . 7 1.3 Curl and Stokes' Theorem . 7 1.4 Laplacian . 8 1.5 Useful Identities . 8 1.6 3D Taylor expansion . 9 1.7 Important Theorem . 9 2 Revision of Electrostatics 10 2.1 Charge Density . 10 2.2 Point charges and the δ-distribution . 10 2.3 Coulomb's Law . 11 2.4 Electric Field . 12 2.5 Gauss' law for E ...................................... 13 2.6 Electrostatic Potential . 14 3 Gauss' Law 16 3.1 Conductors and Insulators . 16 3.2 Gauss' Law in differential form . 16 3.3 Using Gauss' Law . 17 1 3.4 Important δ-function identity and point charges (see tutorial 1.3) . 19 4 Poisson's Equation and Images 20 4.1 Poisson's Equation . 20 4.2 Properties of Poisson's Equation . 20 4.3 A simple example: a hollow conductor . 22 4.4 The Method of Images . 22 5 Electric Dipoles and Multipoles 24 5.1 Field of an electric dipole . 24 5.2 Dipole interaction with external Electric Field . 25 5.3 Multipole expansions . 26 6 Electrostatic Energy and Capacitors 28 6.1 Electrostatic Energy of a general charge distribution . 28 6.2 Capacitors . 29 6.3 *Finite size disc capacitors . 31 7 Magnetic force, Currents and Biot Savart Law 32 7.1 Magnetic force . 32 7.2 Current density and current elements . 32 7.3 Biot Savart Law and Magnetic Fields . 34 7.4 Magnetic Force between Currents . 34 7.5 Long straight wires . 35 8 Divergence and Curl of B; Gauss' and Amp`ere'slaws 36 8.1 div B and Gauss' Law . 36 8.2 Magnetic Dipoles . 36 8.3 curl B and Amp`ere'sLaw . 38 9 Applications of Ampere's` Law; Magnetic Vector Potential 40 9.1 Applications of Amp`ere'sLaw . 40 9.2 The Magnetic Vector Potential . 41 9.3 Poisson's equation for the vector potential . 42 9.4 Summary of 'statics . 43 10 Electromotance and Faraday's Law 44 2 10.1 Sources of steady currents . 44 10.2 Induced emf . 44 10.3 Faraday's Law . 45 10.4 Connection to the Magnetic Vector Potential . 46 10.5 Lenz's Law . 47 11 Inductance 48 11.1 Examples of Induction . 48 11.2 Mutual Inductance . 49 11.3 Self-Inductance . 50 11.4 Energy Stored in Inductors . 50 12 The Displacement Current 52 12.1 Continuity equation . 52 12.2 The Displacement Current . 52 12.3 Capacitor Paradox and Resolution . 53 12.4 Maxwell's Equations . 54 12.5 Solution of Maxwell's Equations in Free Space . 55 13 Description of Electromagnetic Waves 56 13.1 Recap of wave equations . 56 13.2 Plane Waves . 57 13.3 Electromagnetic Plane Waves . 57 13.4 Linear (Plane) Polarisation . 59 13.5 Circular Polarisation . 59 14 Electromagnetic Energy and the Poynting Vector 60 14.1 Poynting's Theorem (Griffiths 8.1.2)........................... 60 14.2 Energy of Electromagnetic Waves . 61 14.3 Energy of discharging capacitor . 62 14.4 ∗Momentum of electromagnetic radiation . 63 15 Dielectric Materials 64 15.1 Overview . 64 15.2 Dielectric Materials . 64 15.3 Electric displacement vector and Gauss' law in media . 65 3 15.4 Linear Isotropic Homogeneous Media . 66 15.5 Example: Dielectric in a Capacitor . 67 16 Magnetic Media 68 16.1 Types of magnetism . 68 16.2 The Magnetization Vector . 68 16.3 Ampere's Law in media . 70 17 Summary of EM in media; boundary conditions on fields 72 17.1 Effect of Magnetic Materials on Inductance . 72 17.2 Electromagnetism in media: summary . 72 17.3 Energy densities and Poynting Vector . 73 17.4 Boundary conditions of fields . 74 18 Examples of continuity conditions; waves in media 76 18.1 Continuity conditions: examples . 76 18.2 Waves in media . 78 18.3 Waves in conductors . 79 19 Waves in Conductors 80 19.1 Skin Depth . 80 19.2 Good and poor conductors . 81 19.3 Phase relations of fields . 82 19.4 Intrinsic Impedance . 83 20 Waves at interfaces: normal incidence 84 20.1 Summary on plane waves and interfaces . 84 20.2 Interfaces between two dielectric media . 84 20.3 General waves at interfaces: normal incidence . 86 20.4 Reflection at Conducting Surface . 87 A Supplementary Material on Types of Magnetism 88 A.1 Diamagnetism . 88 A.2 Paramagnetism . 88 A.3 Ferromagnetism . 89 A.4 Effect of Magnetic Materials on Inductance . 89 4 B Reflection at boundaries: oblique incidence 91 B.1 General Angle of Incidence . 91 B.2 Brewster's Angle . 93 B.3 Total Internal Reflection . ..
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 1.1 Introduction
    1.1 Introduction This chapter is concerned primarily with establishing formulas for the electromagnetic field vectors E and H in terms of all the sources causing these radiating fields, but at points far removed from the sources. The collection of sources is called an antenna and the formulas to be derived form the basis for what is generally referred to as antenna pattern analysis and synthesis. A natural division into two types of antennas will emerge as the analysis develops. There are radiators, such as dipoles and helices, on which the current dis- tribution can be hypothesized with good accuracy; for these, one set of formulas will prove useful. But there are other radiators, such as slots and horns, for which an estimation of the actual current distribution is exceedingly difficult, but for which the close-in fields can be described quite accurately. In such cases it is possible to replace the actual sources, for purposes of field calculation, with equivalent sources that properly terminate the close-in fields. This procedure leads to an alternate set of formulas, useful for antennas of this type. The chapter begins with a brief review of relevant electromagnetic theory, including an inductive establishment of the retarded potential functions. This is fol- lowed by a rigorous derivation of the Stratton-Chu integrals (based on a vector Green's theorem), which give the fields at any point within a volume Kin terms of the sources within Fand the field values on the surfaces S that bound V. This formulation possesses the virtue that it applies to either type of antenna, or to a hybrid mix of the two.
    [Show full text]
  • Module 2: Fundamental Behavior of Electrical Systems 2.0 Introduction
    Module 2: Fundamental Behavior of Electrical Systems 2.0 Introduction All electrical systems, at the most fundamental level, obey Maxwell's equations and the postulates of electromagnetics. Under certain circumstances, approximations can be made that allow simpler methods of analysis, such as circuit theory, to be employed. However, the problems associated with EMC usually involve departures from these approximations. Therefore, a review of fundamental concepts is the logical starting point for a proper understanding of electromagnetic compatibility. 2.1 Fundamental quantities and electrical dimensions speed of light, permittivity and permeability in free space The speed of light in free space has been measured through very precise experiments, and extremely accurate values are known (299,792,458 m/s). For most purposes, however, the approximation c 3x108 m/s is sufficiently accurate. In the International System of units (SI units), the constant known as the permeability of free space is defined to be 7 0 4 ×10 H/m. The constant permittivity of free space is then derived through the relationship 1 c o µo and is usually expressed in SI units as 1 1 8.854×10 12 ×10 9 F/m. 0 2 36 c µo Both the permittivity and permeability of free space have been repeatedly verified through experiment. wavelength in lossless media Wavelength is defined as the distance between adjacent equiphase points on a wave. For an electromagnetic wave propagating in a lossless medium, this is given by 2-2 v f where f is frequency. In free space v=c, and in a simple lossless medium other than free space, the velocity of propagation is v 1 µ , where r 0 and µ µr µ0 .
    [Show full text]
  • Retarded Potential & Lienard-Wiechert Potentials
    Retarded potential & Lienard-Wiechert potentials: By: Chandan Kumar Department of Physics SNS College Jehanabad Dated: 28/07/2020 Retarded potential: in the Lorentz gauge If there is no time dependence… In the general case, we need to consider the wave equation with the source term. First, we introduce the retarded time: The retarded time and represents the delay between the EM wave (photon) emission and the moment at which it reaches the observer. The retarded potentials correspond to electromagnetic fields of a time-varying current or charge distribution. The retardation between cause and effect is thereby essential; e.g., the signal takes a finite time, corresponding to the velocity of light, to propagate from the source point of the field to the point where an effect is produced or measured. Retarded potentials otherwise Consider a thought experiment in which a charge q appears at position r0 at time t=t1, persists for a while, and ​ ​ ​ ​ ​ ​ then disappears at time t2. What is the electric field generated by such a ​ ​ charge? (no currents) We can now appreciate the essential difference between time-dependent electromagnetism and the action at a distance laws of Coulomb and Biot & Savart. In the latter theories, the field-lines act rather like rigid wires attached to charges (or circulating around currents). If the charges (or currents) move then so do the field-lines, leading inevitably to unphysical action at a distance type behavior. In the time-dependent theory, charges act rather like water sprinklers: i.e., they spray out the Coulomb field in all directions at the speed of light.
    [Show full text]
  • Retarded Potentials and the Radiative Arrow of Time
    IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 28 (2007) 1131–1143 doi:10.1088/0143-0807/28/6/010 Retarded potentials and the radiative arrow of time ADBoozer Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA E-mail: [email protected] Received 3 July 2007, in final form 14 August 2007 Published 28 September 2007 Online at stacks.iop.org/EJP/28/1131 Abstract Maxwell’s equations admit both retarded and advanced solutions, but only the retarded solutions seem to characterize the radiative phenomena that actually occur in nature. The reason for this, however, is not usually explained in electrodynamics textbooks. We justify the use of the retarded solutions by considering a simple toy model, which allows us to avoid the technical complexities of electrodynamics and focus on the physical concepts. In addition, we explain how the use of the retarded solutions is related to a number of fundamental issues in physics, such as time-reversal invariance and the radiative arrow of time. 1. Introduction Maxwell’s equations admit both retarded and advanced solutions, but only the retarded solutions seem to characterize the radiative phenomena that actually occur in nature. The reason for this, however, is not explained in most electrodynamics textbooks; instead, it is simply assumed that the retarded solutions are the only ones that are relevant. In this paper, we justify the use of the retarded solutions by considering a simple toy model, which allows us to avoid the technical complexities of electrodynamics and focus on the physical concepts. In addition, our discussion connects the use of the retarded solutions to a number of fundamental issues in physics, such as time-reversal invariance and the radiative arrow of time.
    [Show full text]
  • Maxwell-Lorentz Without Self Interactions
    Maxwell-Lorentz without self interactions: Conservation of energy and momentum. Jonathan Gratus1,2,3 August 20, 2021 1 Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom, 2 The Cockcroft Institute, Sci-Tech Daresbury, Daresbury WA4 4AD, United Kingdom, 3 https://orcid.org/0000-0003-1597-6084 Abstract Since a classical charged point particle radiates energy and momentum it is argued that there must be a radiation reaction force. Here we present an action for the Maxwell-Lorentz without self interactions model, where each particle only responds to the fields of the other charged particles. The corresponding stress-energy tensor automatically conserves energy and momentum in Minkowski and other appropriate spacetimes. 1 Introduction Over the last century there has been significant scientific work debating how elementary charged particles respond to their own electromagnetic field, and the corresponding question of the electromagnetic mass [1–6]. The standard conclusion is that a charged particle obeys the Abraham–Lorentz–Dirac force. This has well known pathologies. There exist runaway solutions where particles accelerate forever without a force [7]. The run-away solutions can be avoided by using the critical submanifold [8], but this leads to pre-acceleration, where a particle moves before a force is applied. Alternative approaches include using a delay equation [3], the Eliezer–Ford–O’Connell equation [9], considering the Landau-Lipsitz equation as fundamental (and not an approximation) [10], and replacing Maxwell’s equation with Born-Infield [11] or Bopp-Podolski [12–14]. For practical purposes it has often been simpler to ignore any radiation reaction and assume that each particle responds via the Lorentz force to the external electromagnetic field and the fields of all the other charged particles.
    [Show full text]
  • Interactions and Laws Petr K Řen © 2018-2019 [email protected]
    Interactions and laws Petr K řen © 2018-2019 [email protected] Abstract : The finite speed of interactions is a sufficient requirement for description of relativistic effects without the interpretation of metric deformations. An interpretation using the aether is also possible. However, deduced laws will be always conventional. Keywords : gravity, retarded potential, speed of light, refractive index, anisotropy, laws The speeds of electromagnetic and weak interactions (neutrino) are probably linked, as it was shown in [1]. However, it does not mean that this speed is a constant that does not depend on parameters. These interactions can be explained by retarded potentials with a speed of interaction that “depends” on near masses. I.e. it is modified by the gravitational potential (permittivity modification) using a factor like GM /rc2. The space (the unit metre) is defined as x=ct from the proper time (the unit second). The SI definition fixes value of the speed of light in vacuum (void). The relativistic (observer-based) interpretation of effects is that the space is (more precisely: “the space looks like”) deformed in the presence of mass (and fixing the gravity as a fundamental interaction). However, alternative explanation is that the space is flat (i.e. with infinite radius of curvature) and the medium for interaction is modified. I.e. it is weaker through a decrease of the fine-structure “constant” (that is also effectively modified in the high energy physics and it was previously interpreted as permittivity that also plays role as an interaction strength in the Coulomb's law), then it causes slower motion in atoms and thus slower clock (“time”) ticking that is real and independent on an observer.
    [Show full text]
  • 611: Electromagnetic Theory II
    611: Electromagnetic Theory II CONTENTS Special relativity; Lorentz covariance of Maxwell equations • Scalar and vector potentials, and gauge invariance • Relativistic motion of charged particles • Action principle for electromagnetism; energy-momentum tensor • Electromagnetic waves; waveguides • Fields due to moving charges • Radiation from accelerating charges • Antennae • Radiation reaction • Magnetic monopoles, duality, Yang-Mills theory • Contents 1 Electrodynamics and Special Relativity 4 1.1 Introduction.................................... 4 1.2 TheLorentzTransformation. ... 7 1.3 An interlude on 3-vectors and suffix notation . ...... 9 1.4 4-vectorsand4-tensors. 15 1.5 Lorentztensors .................................. 21 1.6 Propertimeand4-velocity. 23 2 Electrodynamics and Maxwell’s Equations 25 2.1 Naturalunits ................................... 25 2.2 Gauge potentials and gauge invariance . ...... 26 2.3 Maxwell’s equations in 4-tensor notation . ........ 28 2.4 Lorentz transformation of E~ and B~ ....................... 34 2.5 TheLorentzforce................................. 35 2.6 Action principle for charged particles . ........ 36 2.7 Gauge invariance of the action . 40 2.8 Canonical momentum, and Hamiltonian . 41 3 Particle Motion in Static Electromagnetic Fields 42 3.1 Description in terms of potentials . ...... 42 3.2 Particle motion in static uniform E~ and B~ fields ............... 44 4 Action Principle for Electrodynamics 50 4.1 Invariants of the electromagnetic field . ....... 50 4.2 Actionforelectrodynamics . 53 4.3 Inclusionofsources.............................. 55 4.4 Energydensityandenergyflux . 58 4.5 Energy-momentumtensor . 60 4.6 Energy-momentum tensor for the electromagnetic field . .......... 66 4.7 Inclusion of massive charged particles . ....... 69 5 Coulomb’s Law 71 5.1 Potential of a point charge . 71 5.2 Electrostaticenergy ............................. 73 1 5.3 Fieldofauniformlymovingcharge . 74 5.4 Motion of a charge in a Coulomb potential .
    [Show full text]