Chapter 4. Electrostatics of Macroscopic Media
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Role of Dielectric Materials in Electrical Engineering B D Bhagat
ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 2, Issue 5, September 2013 Role of Dielectric Materials in Electrical Engineering B D Bhagat Abstract- In India commercially industrial consumer consume more quantity of electrical energy which is inductive load has lagging power factor. Drawback is that more current and power required. Capacitor improves the power factor. Commercially manufactured capacitors typically used solid dielectric materials with high permittivity .The most obvious advantages to using such dielectric materials is that it prevents the conducting plates the charges are stored on from coming into direct electrical contact. I. INTRODUCTION Dielectric materials are those which are used in condensers to store electrical energy e.g. for power factor improvement in single phase motors, in tube lights etc. Dielectric materials are essentially insulating materials. The function of an insulating material is to obstruct the flow of electric current while the function of dielectric is to store electrical energy. Thus, insulating materials and dielectric materials differ in their function. A. Electric Field Strength in a Dielectric Thus electric field strength in a dielectric is defined as the potential drop per unit length measured in volts/m. Electric field strength is also called as electric force. If a potential difference of V volts is maintained across the two metal plates say A1 and A2, held l meters apart, then Electric field strength= E= volts/m. B. Electric Flux in Dielectric It is assumed that one line of electric flux comes out from a positive charge of one coulomb and enters a negative charge of one coulombs. -
Electrostatics Vs Magnetostatics Electrostatics Magnetostatics
Electrostatics vs Magnetostatics Electrostatics Magnetostatics Stationary charges ⇒ Constant Electric Field Steady currents ⇒ Constant Magnetic Field Coulomb’s Law Biot-Savart’s Law 1 ̂ ̂ 4 4 (Inverse Square Law) (Inverse Square Law) Electric field is the negative gradient of the Magnetic field is the curl of magnetic vector electric scalar potential. potential. 1 ′ ′ ′ ′ 4 |′| 4 |′| Electric Scalar Potential Magnetic Vector Potential Three Poisson’s equations for solving Poisson’s equation for solving electric scalar magnetic vector potential potential. Discrete 2 Physical Dipole ′′′ Continuous Magnetic Dipole Moment Electric Dipole Moment 1 1 1 3 ∙̂̂ 3 ∙̂̂ 4 4 Electric field cause by an electric dipole Magnetic field cause by a magnetic dipole Torque on an electric dipole Torque on a magnetic dipole ∙ ∙ Electric force on an electric dipole Magnetic force on a magnetic dipole ∙ ∙ Electric Potential Energy Magnetic Potential Energy of an electric dipole of a magnetic dipole Electric Dipole Moment per unit volume Magnetic Dipole Moment per unit volume (Polarisation) (Magnetisation) ∙ Volume Bound Charge Density Volume Bound Current Density ∙ Surface Bound Charge Density Surface Bound Current Density Volume Charge Density Volume Current Density Net , Free , Bound Net , Free , Bound Volume Charge Volume Current Net , Free , Bound Net ,Free , Bound 1 = Electric field = Magnetic field = Electric Displacement = Auxiliary -
The Magnetic Moment of a Bar Magnet and the Horizontal Component of the Earth’S Magnetic Field
260 16-1 EXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH’S MAGNETIC FIELD I. THEORY The purpose of this experiment is to measure the magnetic moment μ of a bar magnet and the horizontal component BE of the earth's magnetic field. Since there are two unknown quantities, μ and BE, we need two independent equations containing the two unknowns. We will carry out two separate procedures: The first procedure will yield the ratio of the two unknowns; the second will yield the product. We will then solve the two equations simultaneously. The pole strength of a bar magnet may be determined by measuring the force F exerted on one pole of the magnet by an external magnetic field B0. The pole strength is then defined by p = F/B0 Note the similarity between this equation and q = F/E for electric charges. In Experiment 3 we learned that the magnitude of the magnetic field, B, due to a single magnetic pole varies as the inverse square of the distance from the pole. k′ p B = r 2 in which k' is defined to be 10-7 N/A2. Consider a bar magnet with poles a distance 2x apart. Consider also a point P, located a distance r from the center of the magnet, along a straight line which passes from the center of the magnet through the North pole. Assume that r is much larger than x. The resultant magnetic field Bm at P due to the magnet is the vector sum of a field BN directed away from the North pole, and a field BS directed toward the South pole. -
Neutron Electric Dipole Moment from Beyond the Standard Model on the Lattice
Introduction Quark EDM Dirac Equation Operator Mixing Lattice Results Conclusions Neutron Electric Dipole Moment from Beyond the Standard Model on the lattice Tanmoy Bhattacharya Los Alamos National Laboratory 2019 Lattice Workshop for US-Japan Intensity Frontier Incubation Mar 27, 2019 Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Standard Model CP Violation Quark EDM Experimental situation Dirac Equation Effective Field Theory Operator Mixing BSM Operators Lattice Results Impact on BSM physics Conclusions Form Factors Introduction Standard Model CP Violation Two sources of CP violation in the Standard Model. Complex phase in CKM quark mixing matrix. Too small to explain baryon asymmetry Gives a tiny (∼ 10−32 e-cm) contribution to nEDM CP-violating mass term and effective ΘGG˜ interaction related to QCD instantons Effects suppressed at high energies −10 nEDM limits constrain Θ . 10 Contributions from beyond the standard model Needed to explain baryogenesis May have large contribution to EDM Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Standard Model CP Violation Quark EDM Experimental situation Dirac Equation Effective Field Theory Operator Mixing BSM Operators Lattice Results Impact on BSM physics Conclusions Form Factors Introduction 4 Experimental situation 44 Experimental landscape ExperimentalExperimental landscape landscape nEDM upper limits (90% C.L.) 4 nEDMnEDM upper upper limits limits (90% (90% C.L.) C.L.) 10−19 19 19 Beam System Current SM 1010− −20 −20−20 BeamBraggBeam scattering SystemSystem -
Dielectric Permittivity Model for Polymer–Filler Composite Materials by the Example of Ni- and Graphite-Filled Composites for High-Frequency Absorbing Coatings
coatings Article Dielectric Permittivity Model for Polymer–Filler Composite Materials by the Example of Ni- and Graphite-Filled Composites for High-Frequency Absorbing Coatings Artem Prokopchuk 1,*, Ivan Zozulia 1,*, Yurii Didenko 2 , Dmytro Tatarchuk 2 , Henning Heuer 1,3 and Yuriy Poplavko 2 1 Institute of Electronic Packaging Technology, Technische Universität Dresden, 01069 Dresden, Germany; [email protected] 2 Department of Microelectronics, National Technical University of Ukraine, 03056 Kiev, Ukraine; [email protected] (Y.D.); [email protected] (D.T.); [email protected] (Y.P.) 3 Department of Systems for Testing and Analysis, Fraunhofer Institute for Ceramic Technologies and Systems IKTS, 01109 Dresden, Germany * Correspondence: [email protected] (A.P.); [email protected] (I.Z.); Tel.: +49-3514-633-6426 (A.P. & I.Z.) Abstract: The suppression of unnecessary radio-electronic noise and the protection of electronic devices from electromagnetic interference by the use of pliable highly microwave radiation absorbing composite materials based on polymers or rubbers filled with conductive and magnetic fillers have been proposed. Since the working frequency bands of electronic devices and systems are rapidly expanding up to the millimeter wave range, the capabilities of absorbing and shielding composites should be evaluated for increasing operating frequency. The point is that the absorption capacity of conductive and magnetic fillers essentially decreases as the frequency increases. Therefore, this Citation: Prokopchuk, A.; Zozulia, I.; paper is devoted to the absorbing capabilities of composites filled with high-loss dielectric fillers, in Didenko, Y.; Tatarchuk, D.; Heuer, H.; which absorption significantly increases as frequency rises, and it is possible to achieve the maximum Poplavko, Y. -
Review of Electrostatics and Magenetostatics
Review of electrostatics and magenetostatics January 12, 2016 1 Electrostatics 1.1 Coulomb’s law and the electric field Starting from Coulomb’s law for the force produced by a charge Q at the origin on a charge q at x, qQ F (x) = 2 x^ 4π0 jxj where x^ is a unit vector pointing from Q toward q. We may generalize this to let the source charge Q be at an arbitrary postion x0 by writing the distance between the charges as jx − x0j and the unit vector from Qto q as x − x0 jx − x0j Then Coulomb’s law becomes qQ x − x0 x − x0 F (x) = 2 0 4π0 jx − xij jx − x j Define the electric field as the force per unit charge at any given position, F (x) E (x) ≡ q Q x − x0 = 3 4π0 jx − x0j We think of the electric field as existing at each point in space, so that any charge q placed at x experiences a force qE (x). Since Coulomb’s law is linear in the charges, the electric field for multiple charges is just the sum of the fields from each, n X qi x − xi E (x) = 4π 3 i=1 0 jx − xij Knowing the electric field is equivalent to knowing Coulomb’s law. To formulate the equivalent of Coulomb’s law for a continuous distribution of charge, we introduce the charge density, ρ (x). We can define this as the total charge per unit volume for a volume centered at the position x, in the limit as the volume becomes “small”. -
Metamaterials and the Landau–Lifshitz Permeability Argument: Large Permittivity Begets High-Frequency Magnetism
Metamaterials and the Landau–Lifshitz permeability argument: Large permittivity begets high-frequency magnetism Roberto Merlin1 Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040 Edited by Federico Capasso, Harvard University, Cambridge, MA, and approved December 4, 2008 (received for review August 26, 2008) Homogeneous composites, or metamaterials, made of dielectric or resonators, have led to a large body of literature devoted to metallic particles are known to show magnetic properties that con- metamaterials magnetism covering the range from microwave to tradict arguments by Landau and Lifshitz [Landau LD, Lifshitz EM optical frequencies (12–16). (1960) Electrodynamics of Continuous Media (Pergamon, Oxford, UK), Although the magnetic behavior of metamaterials undoubt- p 251], indicating that the magnetization and, thus, the permeability, edly conforms to Maxwell’s equations, the reason why artificial loses its meaning at relatively low frequencies. Here, we show that systems do better than nature is not well understood. Claims of these arguments do not apply to composites made of substances with ͌ ͌ strong magnetic activity are seemingly at odds with the fact that, Im S ϾϾ /ഞ or Re S ϳ /ഞ (S and ഞ are the complex permittivity ϾϾ ഞ other than magnetically ordered substances, magnetism in na- and the characteristic length of the particles, and is the ture is a rather weak phenomenon at ambient temperature.* vacuum wavelength). Our general analysis is supported by studies Moreover, high-frequency magnetism ostensibly contradicts of split rings, one of the most common constituents of electro- well-known arguments by Landau and Lifshitz that the magne- magnetic metamaterials, and spherical inclusions. -
Section 14: Dielectric Properties of Insulators
Physics 927 E.Y.Tsymbal Section 14: Dielectric properties of insulators The central quantity in the physics of dielectrics is the polarization of the material P. The polarization P is defined as the dipole moment p per unit volume. The dipole moment of a system of charges is given by = p qiri (1) i where ri is the position vector of charge qi. The value of the sum is independent of the choice of the origin of system, provided that the system in neutral. The simplest case of an electric dipole is a system consisting of a positive and negative charge, so that the dipole moment is equal to qa, where a is a vector connecting the two charges (from negative to positive). The electric field produces by the dipole moment at distances much larger than a is given by (in CGS units) 3(p⋅r)r − r 2p E(r) = (2) r5 According to electrostatics the electric field E is related to a scalar potential as follows E = −∇ϕ , (3) which gives the potential p ⋅r ϕ(r) = (4) r3 When solving electrostatics problem in many cases it is more convenient to work with the potential rather than the field. A dielectric acquires a polarization due to an applied electric field. This polarization is the consequence of redistribution of charges inside the dielectric. From the macroscopic point of view the dielectric can be considered as a material with no net charges in the interior of the material and induced negative and positive charges on the left and right surfaces of the dielectric. -
Chapter 16 – Electrostatics-I
Chapter 16 Electrostatics I Electrostatics – NOT Really Electrodynamics Electric Charge – Some history •Historically people knew of electrostatic effects •Hair attracted to amber rubbed on clothes •People could generate “sparks” •Recorded in ancient Greek history •600 BC Thales of Miletus notes effects •1600 AD - William Gilbert coins Latin term electricus from Greek ηλεκτρον (elektron) – Greek term for Amber •1660 Otto von Guericke – builds electrostatic generator •1675 Robert Boyle – show charge effects work in vacuum •1729 Stephen Gray – discusses insulators and conductors •1730 C. F. du Fay – proposes two types of charges – can cancel •Glass rubbed with silk – glass charged with “vitreous electricity” •Amber rubbed with fur – Amber charged with “resinous electricity” A little more history • 1750 Ben Franklin proposes “vitreous” and “resinous” electricity are the same ‘electricity fluid” under different “pressures” • He labels them “positive” and “negative” electricity • Proposaes “conservation of charge” • June 15 1752(?) Franklin flies kite and “collects” electricity • 1839 Michael Faraday proposes “electricity” is all from two opposite types of “charges” • We call “positive” the charge left on glass rubbed with silk • Today we would say ‘electrons” are rubbed off the glass Torsion Balance • Charles-Augustin de Coulomb - 1777 Used to measure force from electric charges and to measure force from gravity = - - “Hooks law” for fibers (recall F = -kx for springs) General Equation with damping - angle I – moment of inertia C – damping -
Magnetic Fields
Welcome Back to Physics 1308 Magnetic Fields Sir Joseph John Thomson 18 December 1856 – 30 August 1940 Physics 1308: General Physics II - Professor Jodi Cooley Announcements • Assignments for Tuesday, October 30th: - Reading: Chapter 29.1 - 29.3 - Watch Videos: - https://youtu.be/5Dyfr9QQOkE — Lecture 17 - The Biot-Savart Law - https://youtu.be/0hDdcXrrn94 — Lecture 17 - The Solenoid • Homework 9 Assigned - due before class on Tuesday, October 30th. Physics 1308: General Physics II - Professor Jodi Cooley Physics 1308: General Physics II - Professor Jodi Cooley Review Question 1 Consider the two rectangular areas shown with a point P located at the midpoint between the two areas. The rectangular area on the left contains a bar magnet with the south pole near point P. The rectangle on the right is initially empty. How will the magnetic field at P change, if at all, when a second bar magnet is placed on the right rectangle with its north pole near point P? A) The direction of the magnetic field will not change, but its magnitude will decrease. B) The direction of the magnetic field will not change, but its magnitude will increase. C) The magnetic field at P will be zero tesla. D) The direction of the magnetic field will change and its magnitude will increase. E) The direction of the magnetic field will change and its magnitude will decrease. Physics 1308: General Physics II - Professor Jodi Cooley Review Question 2 An electron traveling due east in a region that contains only a magnetic field experiences a vertically downward force, toward the surface of the earth. -
Physics 112: Classical Electromagnetism, Fall 2013 Birefringence Notes
Physics 112: Classical Electromagnetism, Fall 2013 Birefringence Notes 1 A tensor susceptibility? The electrons bound within, and binding, the atoms of a dielectric crystal are not uniformly dis- tributed, but are restricted in their motion by the potentials which confine them. In response to an applied electric field, they may therefore move a greater or lesser distance, depending upon the strength of their confinement in the field direction. As a result, the induced polarization varies not only with the strength of the applied field, but also with its direction. The susceptibility{ and properties which depend upon it, such as the refractive index{ are therefore anisotropic, and cannot be characterized by a single value. The scalar electric susceptibility, χe, is defined to be the coefficient which relates the value of the ~ ~ local electric field, Eloc, to the local value of the polarization, P : ~ ~ P = χe0Elocal: (1) As we discussed in the last seminar we can `promote' this relation to a tensor relation with χe ! (χe)ij. In this case the dielectric constant is also a tensor and takes the form ij = [δij + (χe)ij] 0: (2) Figure 1: A cartoon showing how the electron is held by anisotropic springs{ causing an electric susceptibility which is different when the electric field is pointing in different directions. How does this happen in practice? In Fig. 1 we can see that in a crystal, for instance, the electrons will in general be held in bonds which are not spherically symmetric{ i.e., they are anisotropic. 1 Therefore, it will be easier to polarize the material in certain directions than it is in others. -
APPENDICES 206 Appendices
AAPPENDICES 206 Appendices CONTENTS A.1 Units 207-208 A.2 Abbreviations 209 SUMMARY A description is given of the units used in this thesis, and a list of frequently used abbreviations with the corresponding term is given. Units Description of units used in this thesis and conversion factors for A.1 transformation into other units The formulas and properties presented in this thesis are reported in atomic units unless explicitly noted otherwise; the exceptions to this rule are energies, which are most frequently reported in kcal/mol, and distances that are normally reported in Å. In the atomic units system, four frequently used quantities (Planck’s constant h divided by 2! [h], mass of electron [me], electron charge [e], and vacuum permittivity [4!e0]) are set explicitly to 1 in the formulas, making these more simple to read. For instance, the Schrödinger equation for the hydrogen atom is in SI units: È 2 e2 ˘ Í - h —2 - ˙ f = E f (1) ÎÍ 2me 4pe0r ˚˙ In atomic units, it looks like: È 1 1˘ - —2 - f = E f (2) ÎÍ 2 r ˚˙ Before a quantity can be used in the atomic units equations, it has to be transformed from SI units into atomic units; the same is true for the quantities obtained from the equations, which can be transformed from atomic units into SI units. For instance, the solution of equation (2) for the ground state of the hydrogen atom gives an energy of –0.5 atomic units (Hartree), which can be converted into other units quite simply by multiplying with the appropriate conversion factor (see table A.1.1).