Electromagnetism - Lecture 13
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Chapter 3 Electromagnetic Waves & Maxwell's Equations
Chapter 3 Electromagnetic Waves & Maxwell’s Equations Part I Maxwell’s Equations Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist. Famous equations published in 1861 Maxwell’s Equations: Integral Form Gauss's law Gauss's law for magnetism Faraday's law of induction (Maxwell–Faraday equation) Ampère's law (with Maxwell's addition) Maxwell’s Equations Relation of the speed of light and electric and magnetic vacuum constants 1 c = 2.99792458 108 [m/s] 0 0 permittivity of free space, also called the electric constant permeability of free space, also called the magnetic constant Gauss’s Law For any closed surface enclosing total charge Qin, the net electric flux through the surface is This result for the electric flux is known as Gauss’s Law. Magnetic Gauss’s Law The net magnetic flux through any closed surface is equal to zero: As of today there is no evidence of magnetic monopoles See: Phys.Rev.Lett.85:5292,2000 Ampère's Law The magnetic field in space around an electric current is proportional to the electric current which serves as its source: B ds 0I I is the total current inside the loop. ds B i1 Direction of integration i 3 i2 Faraday’s Law The change of magnetic flux in a loop will induce emf, i.e., electric field B E ds dA A t Lenz's Law Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated. Problem with Ampère's Law Maxwell realized that Ampere’s law is not valid when the current is discontinuous. -
Poynting Vector and Power Flow in Electromagnetic Fields
Poynting Vector and Power Flow in Electromagnetic Fields: Electromagnetic waves can transport energy as a result of their travelling or propagating characteristics. Starting from Maxwell's Equations: Together with the vector identity One can write In simple medium where and are constant, and Divergence theorem states, This equation is referred to as Poynting theorem and it states that the net power flowing out of a given volume is equal to the time rate of decrease in the energy stored within the volume minus the conduction losses. In the equation, the following term represents the rate of change of the stored energy in the electric and magnetic fields On the other hand, the power dissipation within the volume appears in the following form Hence the total decrease in power within the volume under consideration: Here (W/mt2) is called the Poynting vector and it represents the power density vector associated with the electromagnetic field. The integration of the Poynting vector over any closed surface gives the net power flowing out of the surface. Poynting vector for the time harmonic case: Using the convention, the instantaneous value of a quantity is the real part of the product of a phasor quantity and when is used as reference. Considering the following phasor: −푗훽푧 퐸⃗ (푧) = 푥̂퐸푥(푧) = 푥̂퐸0푒 The instantaneous field becomes: 푗푤푡 퐸⃗ (푧, 푡) = 푅푒{퐸⃗ (푧)푒 } = 푥̂퐸0푐표푠(휔푡 − 훽푧) when E0 is real. Let us consider two instantaneous quantities A and B such that where A and B are the phasor quantities. Therefore, Since A and B are periodic with period , the time average value of the product form AB, 푇 1 퐴퐵 = ∫ 퐴퐵푑푡 푎푣푒푟푎푔푒 푇 0 푇 1 퐴퐵 = ∫|퐴||퐵|푐표푠(휔푡 + 훼)푐표푠(휔푡 + 훽)푑푡 푎푣푒푟푎푔푒 푇 0 1 퐴퐵 = |퐴||퐵|푐표푠(훼 − 훽) 푎푣푒푟푎푔푒 2 For phasors, and , where * denotes complex conjugate. -
Electrodynamics of Topological Insulators
Electrodynamics of Topological Insulators Author: Michael Sammon Advisor: Professor Harsh Mathur Department of Physics Case Western Reserve University Cleveland, OH 44106-7079 May 2, 2014 0.1 Abstract Topological insulators are new metamaterials that have an insulating interior, but a conductive surface. The specific nature of this conducting surface causes a mixing of the electric and magnetic fields around these materials. This project, investigates this effect to deepen our understanding of the electrodynamics of topological insulators. The first part of the project focuses on the fields that arise when a current carrying wire is brought near a topological insulator slab and a cylindrical topological insulator. In both problems, the method of images was able to be used. The result were image electric and magnetic currents. These magnetic currents provide the manifestation of an effect predicted by Edward Witten for Axion particles. Though the physics is extremely different, the overall result is the same in which fields that seem to be generated by magnetic currents exist. The second part of the project begins an analysis of a topological insulator in constant electric and magnetic fields. It was found that both fields generate electric and magnetic dipole like responses from the topological insulator, however the electric field response within the material that arise from the applied fields align in opposite directions. Further investigation into the effect this has, as well as the overall force that the topological insulator experiences in these fields will be investigated this summer. 1 List of Figures 0.4.1 Dyon1 fields of a point charge near a Topological Insulator . -
Double Negative Dispersion Relations from Coated Plasmonic Rods∗
MULTISCALE MODEL. SIMUL. c 2013 Society for Industrial and Applied Mathematics Vol. 11, No. 1, pp. 192–212 DOUBLE NEGATIVE DISPERSION RELATIONS FROM COATED PLASMONIC RODS∗ YUE CHEN† AND ROBERT LIPTON‡ Abstract. A metamaterial with frequency dependent double negative effective properties is constructed from a subwavelength periodic array of coated rods. Explicit power series are developed for the dispersion relation and associated Bloch wave solutions. The expansion parameter is the ratio of the length scale of the periodic lattice to the wavelength. Direct numerical simulations for finite size period cells show that the leading order term in the power series for the dispersion relation is a good predictor of the dispersive behavior of the metamaterial. Key words. metamaterials, dispersion relations, Bloch waves, simulations AMS subject classifications. 35Q60, 68U20, 78A48, 78M40 DOI. 10.1137/120864702 1. Introduction. Metamaterials are artificial materials designed to have elec- tromagnetic properties not generally found in nature. One contemporary area of research explores novel subwavelength constructions that deliver metamaterials with both a negative bulk dielectric constant and bulk magnetic permeability across cer- tain frequency intervals. These double negative materials are promising materials for the creation of negative index superlenses that overcome the small diffraction limit and have great potential in applications such as biomedical imaging, optical lithography, and data storage. The early work of Veselago [39] identified novel ef- fects associated with hypothetical materials for which both the dielectric constant and magnetic permeability are simultaneously negative. Such double negative media support electromagnetic wave propagation in which the phase velocity is antiparallel to the direction of energy flow and other unusual electromagnetic effects, such as the reversal of the Doppler effect and Cerenkov radiation. -
Electromagnetic Energy
Physics 142 Electromagnetic Enmergy Page !1 Electromagnetic Energy A child of five can understand this; send someone to fetch a child of five. — Groucho Marx Energy in the fields can move from place to place We have discussed the energy in an electrostatic field, such as that stored in a capacitor. We found that this energy can be thought of as distributed in space, with an energy per unit volume (energy density) at each point in space. We obtained a formula for this 1 2 quantity, ! ue = 2 ε0E (if there are no dielectric materials). This formula applies to any E- field. We also found that magnetic fields possess energy distributed in space, described 2 by the magnetic energy density ! um = B /2µ0 . If there are both electric and magnetic fields, the total electromagnetic energy density is the sum of ! ue and ! um . These specify how much electromagnetic field energy there is at any point in space. But we have not yet considered how this energy moves from place to place. Consider the energy flow in a flashlight. We know that energy moves from the battery to the bulb, where it is converted into heat and light. It is tempting to assume that this energy flows through the conductors, like water in a pipe. But if we look carefully we find that the electromagnetic energy density in the conductors is much too small to account for the amount of energy in transit. Nearly all of the energy gets to the bulb by flowing through space near the conductors. In a sense they guide the energy but do not carry much of it. -
Modulation of Crystal and Electronic Structures in Topological Insulators by Rare-Earth Doping
Modulation of crystal and electronic structures in topological insulators by rare-earth doping Zengji Yue*, Weiyao Zhao, David Cortie, Zhi Li, Guangsai Yang and Xiaolin Wang* 1. Institute for Superconducting & Electronic Materials, Australian Institute of Innovative Materials, University of Wollongong, Wollongong, NSW 2500, Australia 2. ARC Centre for Future Low-Energy Electronics Technologies (FLEET), University of Wollongong, Wollongong, NSW 2500, Australia Email: [email protected]; [email protected]; Abstract We study magnetotransport in a rare-earth-doped topological insulator, Sm0.1Sb1.9Te3 single crystals, under magnetic fields up to 14 T. It is found that that the crystals exhibit Shubnikov– de Haas (SdH) oscillations in their magneto-transport behaviour at low temperatures and high magnetic fields. The SdH oscillations result from the mixed contributions of bulk and surface states. We also investigate the SdH oscillations in different orientations of the magnetic field, which reveals a three-dimensional Fermi surface topology. By fitting the oscillatory resistance with the Lifshitz-Kosevich theory, we draw a Landau-level fan diagram that displays the expected nontrivial phase. In addition, the density functional theory calculations shows that Sm doping changes the crystal structure and electronic structure compared with pure Sb2Te3. This work demonstrates that rare earth doping is an effective way to manipulate the Fermi surface of topological insulators. Our results hold potential for the realization of exotic topological effects -
Negative Refractive Index in Artificial Metamaterials
1 Negative Refractive Index in Artificial Metamaterials A. N. Grigorenko Department of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK We discuss optical constants in artificial metamaterials showing negative magnetic permeability and electric permittivity and suggest a simple formula for the refractive index of a general optical medium. Using effective field theory, we calculate effective permeability and the refractive index of nanofabricated media composed of pairs of identical gold nano-pillars with magnetic response in the visible spectrum. PACS: 73.20.Mf, 41.20.Jb, 42.70.Qs 2 The refractive index of an optical medium, n, can be found from the relation n2 = εμ , where ε is medium’s electric permittivity and μ is magnetic permeability.1 There are two branches of the square root producing n of different signs, but only one of these branches is actually permitted by causality.2 It was conventionally assumed that this branch coincides with the principal square root n = εμ .1,3 However, in 1968 Veselago4 suggested that there are materials in which the causal refractive index may be given by another branch of the root n =− εμ . These materials, referred to as left- handed (LHM) or negative index materials, possess unique electromagnetic properties and promise novel optical devices, including a perfect lens.4-6 The interest in LHM moved from theory to practice and attracted a great deal of attention after the first experimental realization of LHM by Smith et al.7, which was based on artificial metallic structures -
Plasma Waves
Plasma Waves S.M.Lea January 2007 1 General considerations To consider the different possible normal modes of a plasma, we will usually begin by assuming that there is an equilibrium in which the plasma parameters such as density and magnetic field are uniform and constant in time. We will then look at small perturbations away from this equilibrium, and investigate the time and space dependence of those perturbations. The usual notation is to label the equilibrium quantities with a subscript 0, e.g. n0, and the pertrubed quantities with a subscript 1, eg n1. Then the assumption of small perturbations is n /n 1. When the perturbations are small, we can generally ignore j 1 0j ¿ squares and higher powers of these quantities, thus obtaining a set of linear equations for the unknowns. These linear equations may be Fourier transformed in both space and time, thus reducing the differential equations to a set of algebraic equations. Equivalently, we may assume that each perturbed quantity has the mathematical form n = n exp i~k ~x iωt (1) 1 ¢ ¡ where the real part is implicitly assumed. Th³is form descri´bes a wave. The amplitude n is in ~ general complex, allowing for a non•zero phase constant φ0. The vector k, called the wave vector, gives both the direction of propagation of the wave and the wavelength: k = 2π/λ; ω is the angular frequency. There is a relation between ω and ~k that is determined by the physical properties of the system. The function ω ~k is called the dispersion relation for the wave. -
Dispersion Relation & Index Ellipsoids
8/27/2020 Advanced Electromagnetics: 21st Century Electromagnetics Dispersion Relation & Index Ellipsoids Lecture Outline • Dispersion relation • Dispersion surfaces • Index ellipsoids Slide 2 1 8/27/2020 Dispersion Relation Slide 3 The Wave Vector The wave vector (wave momentum) is a vector quantity that conveys two pieces of information: 1. Wavelength and Refractive Index –The magnitude of the wave vector conveys the spatial period (i.e. wavelength) of the wave inside the material. When the frequency is known, the magnitude conveys the material’s refractive index n (more to be said later). 22 n k 0 free space wavelength 0 2. Direction –The direction of the wave is perpendicular to the wave fronts (more to be said later). ˆ kkakbkcabcˆˆ Slide 4 2 8/27/2020 The Dispersion Relation The dispersion relation for a material relates the wave vector to frequency . Essentially, it sets a rule for the values of as a function of direction and frequency. For an ordinary linear, homogeneous and isotropic (LHI) material, the dispersion relation is: 2 222n kkkabc c0 222 kkkabc 2 This can also be written as: 2 kk000 nc0 Slide 5 How to Derive the Dispersion Relation (1 of 2) The wave equation in a linear homogeneous anisotropic material is: 2 Assume no magnetic response i. e. 1 . Ek 00 r E 0 The solution to this equation is still a plane wave, but the allowed values for (modes) are more complicated. jk r ˆ E Ee00 E Eaabcˆˆ Eb Ec Substituting this solution into the wave equation leads to the following relation: 2 kk E000r0 kE k E 0 This equation has the form: abcˆˆ ˆ 0 Each (•••) term has the form: EEEabc 0 Each vector component must be set to zero independently. -
Poynting's Theorem and the Wave Equation
Chapter 18: Poynting’s Theorem and the Wave Equation Chapter Learning Objectives: After completing this chapter the student will be able to: Use Poynting’s theorem to determine the direction and magnitude of power flow in an electromagnetic system. Use Maxwell’s Equations to derive a general homogeneous wave equation for the electric and magnetic field. Derive a simplified wave equation assuming propagation in a vacuum and an electric field polarized in only one direction. Use Maxwell’s Equations to derive the speed of light in a vacuum. You can watch the video associated with this chapter at the following link: Historical Perspective: John Henry Poynting (1852-1914) was an English physicist who did work in electromagnetic energy flow, elasticity, and astronomy. He coined the term “Greenhouse Effect.” Both the Poynting Vector and Poynting’s Theorem are named in his honor. Photo credit: https://upload.wikimedia.org/wikipedia/commons/5/5f/John_Henry_Poynting.jpg, [Public domain], via Wikimedia Commons. 1 18.1 Poynting’s Theorem With Maxwell’s Equations, we now have the tools necessary to derive Poynting’s Theorem, which will allow us to perform many useful calculations involving the direction of power flow in electromagnetic fields. We will begin with Faraday’s Law, and we will take the dot product of H with both sides: (Copy of Equation 16.24) (Equation 18.1) Next, we will start with Ampere’s Law and will take the dot product of E with both sides: (Copy of Equation 17.13) (Equation 18.2) Now, let’s subtract both sides of Equation 18.2 from both sides of Equation 18.1: (Equation 18.3) We can now apply the following mathematical identity to the left side of Equation 18.3: (Equation 18.4) This substitution yields: (Equation 18.5) Distributing the E across the right side gives: (Equation 18.6) 2 Now let’s concentrate on the first time on the right side. -
7 Plasmonics
7 Plasmonics Highlights of this chapter: In this chapter we introduce the concept of surface plasmon polaritons (SPP). We discuss various types of SPP and explain excitation methods. Finally, di®erent recent research topics and applications related to SPP are introduced. 7.1 Introduction Long before scientists have started to investigate the optical properties of metal nanostructures, they have been used by artists to generate brilliant colors in glass artefacts and artwork, where the inclusion of gold nanoparticles of di®erent size into the glass creates a multitude of colors. Famous examples are the Lycurgus cup (Roman empire, 4th century AD), which has a green color when observing in reflecting light, while it shines in red in transmitting light conditions, and church window glasses. Figure 172: Left: Lycurgus cup, right: color windows made by Marc Chagall, St. Stephans Church in Mainz Today, the electromagnetic properties of metal{dielectric interfaces undergo a steadily increasing interest in science, dating back in the works of Gustav Mie (1908) and Rufus Ritchie (1957) on small metal particles and flat surfaces. This is further moti- vated by the development of improved nano-fabrication techniques, such as electron beam lithographie or ion beam milling, and by modern characterization techniques, such as near ¯eld microscopy. Todays applications of surface plasmonics include the utilization of metal nanostructures used as nano-antennas for optical probes in biology and chemistry, the implementation of sub-wavelength waveguides, or the development of e±cient solar cells. 208 7.2 Electro-magnetics in metals and on metal surfaces 7.2.1 Basics The interaction of metals with electro-magnetic ¯elds can be completely described within the frame of classical Maxwell equations: r ¢ D = ½ (316) r ¢ B = 0 (317) r £ E = ¡@B=@t (318) r £ H = J + @D=@t; (319) which connects the macroscopic ¯elds (dielectric displacement D, electric ¯eld E, magnetic ¯eld H and magnetic induction B) with an external charge density ½ and current density J. -
Chapter 19/ Optical Properties
Chapter 19 /Optical Properties The four notched and transpar- ent rods shown in this photograph demonstrate the phenomenon of photoelasticity. When elastically deformed, the optical properties (e.g., index of refraction) of a photoelastic specimen become anisotropic. Using a special optical system and polarized light, the stress distribution within the speci- men may be deduced from inter- ference fringes that are produced. These fringes within the four photoelastic specimens shown in the photograph indicate how the stress concentration and distribu- tion change with notch geometry for an axial tensile stress. (Photo- graph courtesy of Measurements Group, Inc., Raleigh, North Carolina.) Why Study the Optical Properties of Materials? When materials are exposed to electromagnetic radia- materials, we note that the performance of optical tion, it is sometimes important to be able to predict fibers is increased by introducing a gradual variation and alter their responses. This is possible when we are of the index of refraction (i.e., a graded index) at the familiar with their optical properties, and understand outer surface of the fiber. This is accomplished by the mechanisms responsible for their optical behaviors. the addition of specific impurities in controlled For example, in Section 19.14 on optical fiber concentrations. 766 Learning Objectives After careful study of this chapter you should be able to do the following: 1. Compute the energy of a photon given its fre- 5. Describe the mechanism of photon absorption quency and the value of Planck’s constant. for (a) high-purity insulators and semiconduc- 2. Briefly describe electronic polarization that re- tors, and (b) insulators and semiconductors that sults from electromagnetic radiation-atomic in- contain electrically active defects.