Appendix A: Fundamentals of Electrostatics
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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 -
James Clerk Maxwell
James Clerk Maxwell JAMES CLERK MAXWELL Perspectives on his Life and Work Edited by raymond flood mark mccartney and andrew whitaker 3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Oxford University Press 2014 The moral rights of the authors have been asserted First Edition published in 2014 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2013942195 ISBN 978–0–19–966437–5 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. -
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”. -
Electricity Outline
Electricity Outline A. Electrostatics 1. Charge q is measured in coulombs 2. Three ways to charge something. Charge by: Friction, Conduction and Induction 3. Coulomb’s law for point or spherical charges: q1 q2 2 FE FE FE = kq1q2/r r 9 2 2 where k = 9.0 x 10 Nm /Coul 4. Electric field E = FE/qo (qo is a small, + test charge) F = qE Point or spherically symmetric charge distribution: E = kq/r2 E is constant above or below an ∞, charged sheet. 5. Faraday’s Electric Lines of Force rules: E 1. Lines originate on + and terminate on - _ 2. The E field vector is tangent to the line of force + 3. Electric field strength is proportional to line density 4. Lines are ┴ to conducting surfaces. 5. E = 0 inside a hollow or solid conductor 6. Electric potential difference (voltage): ΔV = W/qo We usually drop the Δ and just write V. Sometimes the voltage provided by a battery is know as the electromotive force (emf) ε 7. Potential Energy due to point charges or spherically symmetric charge distribution V= kQ/r 8. Equipotential surfaces are surfaces with constant voltage. The electric field vector is always to an equipotential surface. 9. Emax Air = 3,000,000 N/coul = 30,000 V/cm. If you exceed this value, you will create a conducting path by ripping e-s off air molecules. B. Capacitors 1. Capacitors A. C = q/V; unit of capacitance is the Farad = coul/volt B. For Parallel plate capacitors: V = E d C is proportional to A/d where A is the area of the plates and d is the plate separation. -
Advanced Magnetism and Electromagnetism
ELECTRONIC TECHNOLOGY SERIES ADVANCED MAGNETISM AND ELECTROMAGNETISM i/•. •, / .• ;· ... , -~-> . .... ,•.,'. ·' ,,. • _ , . ,·; . .:~ ~:\ :· ..~: '.· • ' ~. 1. .. • '. ~:;·. · ·!.. ., l• a publication $2 50 ADVANCED MAGNETISM AND ELECTROMAGNETISM Edited by Alexander Schure, Ph.D., Ed. D. - JOHN F. RIDER PUBLISHER, INC., NEW YORK London: CHAPMAN & HALL, LTD. Copyright December, 1959 by JOHN F. RIDER PUBLISHER, INC. All rights reserved. This book or parts thereof may not be reproduced in any form or in any language without permission of the publisher. Library of Congress Catalog Card No. 59-15913 Printed in the United States of America PREFACE The concepts of magnetism and electromagnetism form such an essential part of the study of electronic theory that the serious student of this field must have a complete understanding of these principles. The considerations relating to magnetic theory touch almost every aspect of electronic development. This book is the second of a two-volume treatment of the subject and continues the attention given to the major theoretical con siderations of magnetism, magnetic circuits and electromagnetism presented in the first volume of the series.• The mathematical techniques used in this volume remain rela tively simple but are sufficiently detailed and numerous to permit the interested student or technician extensive experience in typical computations. Greater weight is given to problem solutions. To ensure further a relatively complete coverage of the subject matter, attention is given to the presentation of sufficient information to outline the broad concepts adequately. Rather than attempting to cover a large body of less important material, the selected major topics are treated thoroughly. Attention is given to the typical practical situations and problems which relate to the subject matter being presented, so as to afford the reader an understanding of the applications of the principles he has learned. -
We Continue to Compare the Electrostatic and Magnetostatic Fields. the Electrostatic Field Is Conservative
We continue to compare the electrostatic and magnetostatic fields. The electrostatic field is conservative: This allows us to define the potential V: dl because a b is independent of the path. If a vector field has no curl (i.e., is conservative), it must be something's gradient. Gravity is conservative. Therefore you do see water flowing in such a loop without a pump in the physical world. For the magnetic field, If a vector field has no divergence (i.e., is solenoidal), it must be something's curl. In other words, the curl of a vector field has zero divergence. Let’s use another physical context to help you understand this math: Ampère’s law J ds 0 Kirchhoff's current law (KCL) S Since , we can define a vector field A such that Notice that for a given B, A is not unique. For example, if then , because Similarly, for the electrostatic field, the scalar potential V is not unique: If then You have the freedom to choose the reference (Ampère’s law) Going through the math, you will get Here is what means: Just notation. Notice that is a vector. Still remember what means for a scalar field? From a previous lecture: Recall that the choice for A is not unique. It turns out that we can always choose A such that (Ampère’s law) The choice for A is not unique. We choose A such that Here is what means: Notice that is a vector. Thus this is actually three equations: Recall the definition of for a scalar field from a previous lecture: Poisson’s equation for the magnetic field is actually three equations: Compare Poisson’s equation for the magnetic field with that for the electrostatic field: Given J, you can solve A, from which you get B by Given , you can solve V, from which you get E by Exams (Test 2 & Final) problems will not involve the vector potential. -
A Selection of New Arrivals September 2017
A selection of new arrivals September 2017 Rare and important books & manuscripts in science and medicine, by Christian Westergaard. Flæsketorvet 68 – 1711 København V – Denmark Cell: (+45)27628014 www.sophiararebooks.com AMPERE, Andre-Marie. Mémoire. INSCRIBED BY AMPÈRE TO FARADAY AMPÈRE, André-Marie. Mémoire sur l’action mutuelle d’un conducteur voltaïque et d’un aimant. Offprint from Nouveaux Mémoires de l’Académie royale des sciences et belles-lettres de Bruxelles, tome IV, 1827. Bound with 18 other pamphlets (listed below). [Colophon:] Brussels: Hayez, Imprimeur de l’Académie Royale, 1827. $38,000 4to (265 x 205 mm). Contemporary quarter-cloth and plain boards (very worn and broken, with most of the spine missing), entirely unrestored. Preserved in a custom cloth box. First edition of the very rare offprint, with the most desirable imaginable provenance: this copy is inscribed by Ampère to Michael Faraday. It thus links the two great founders of electromagnetism, following its discovery by Hans Christian Oersted (1777-1851) in April 1820. The discovery by Ampère (1775-1836), late in the same year, of the force acting between current-carrying conductors was followed a year later by Faraday’s (1791-1867) first great discovery, that of electromagnetic rotation, the first conversion of electrical into mechanical energy. This development was a challenge to Ampère’s mathematically formulated explanation of electromagnetism as a manifestation of currents of electrical fluids surrounding ‘electrodynamic’ molecules; indeed, Faraday directly criticised Ampère’s theory, preferring his own explanation in terms of ‘lines of force’ (which had to wait for James Clerk Maxwell (1831-79) for a precise mathematical formulation). -
Roadmap on Transformation Optics 5 Martin Mccall 1,*, John B Pendry 1, Vincenzo Galdi 2, Yun Lai 3, S
Page 1 of 59 AUTHOR SUBMITTED MANUSCRIPT - draft 1 2 3 4 Roadmap on Transformation Optics 5 Martin McCall 1,*, John B Pendry 1, Vincenzo Galdi 2, Yun Lai 3, S. A. R. Horsley 4, Jensen Li 5, Jian 6 Zhu 5, Rhiannon C Mitchell-Thomas 4, Oscar Quevedo-Teruel 6, Philippe Tassin 7, Vincent Ginis 8, 7 9 9 9 6 10 8 Enrica Martini , Gabriele Minatti , Stefano Maci , Mahsa Ebrahimpouri , Yang Hao , Paul Kinsler 11 11,12 13 14 15 9 , Jonathan Gratus , Joseph M Lukens , Andrew M Weiner , Ulf Leonhardt , Igor I. 10 Smolyaninov 16, Vera N. Smolyaninova 17, Robert T. Thompson 18, Martin Wegener 18, Muamer Kadic 11 18 and Steven A. Cummer 19 12 13 14 Affiliations 15 1 16 Imperial College London, Blackett Laboratory, Department of Physics, Prince Consort Road, 17 London SW7 2AZ, United Kingdom 18 19 2 Field & Waves Lab, Department of Engineering, University of Sannio, I-82100 Benevento, Italy 20 21 3 College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano 22 Science and Technology, Soochow University, Suzhou 215006, China 23 24 4 University of Exeter, Department of Physics and Astronomy, Stocker Road, Exeter, EX4 4QL United 25 26 Kingdom 27 5 28 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, 29 United Kingdom 30 31 6 KTH Royal Institute of Technology, SE-10044, Stockholm, Sweden 32 33 7 Department of Physics, Chalmers University , SE-412 96 Göteborg, Sweden 34 35 8 Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussel, Belgium 36 37 9 Dipartimento di Ingegneria dell'Informazione e Scienze Matematiche, University of Siena, Via Roma, 38 39 56 53100 Siena, Italy 40 10 41 School of Electronic Engineering and Computer Science, Queen Mary University of London, 42 London E1 4FZ, United Kingdom 43 44 11 Physics Department, Lancaster University, Lancaster LA1 4 YB, United Kingdom 45 46 12 Cockcroft Institute, Sci-Tech Daresbury, Daresbury WA4 4AD, United Kingdom. -
Equipotential Surfaces and Their Properties 7
OSBINCBSE.COM OSBINCBSE.COM osbincbse.com ELECTROSTATICS - I – Electrostatic Force 1. Frictional Electricity 2. Properties of Electric Charges 3. Coulomb’s Law 4. Coulomb’s Law in Vector Form 5. Units of Charge 6. Relative Permittivity or Dielectric Constant 7. Continuous Charge Distribution i) Linear Charge Density ii) Surface Charge Density iii) Volume Charge Density OSBINCBSE.COM OSBINCBSE.COM OSBINCBSE.COM osbincbse.com OSBINCBSE.COM Frictional Electricity: Frictional electricity is the electricity produced and transfer of electrons from one body to other. + + + + + + + + + + + + by rubbing two suitable bodies Electrons in glass are loosely bound in it than the glass and silk are rubbed together,Glass the comparative from glass get transferred to silk.Silk As a result, glass becomes positively charged and s charged. Electrons in fur are loosely bound in it than the e . ebonite and fur are rubbed together, the comparati + + + - - - - - - - - - - + + + + from fur get transferred to ebonite. + + + As a result, ebonite becomes negatively charged and Ebonite charged. electrons in silk. So,Flannel when ly loosely bound electrons ilk becomes negatively lectrons in ebonite. So, when vely loosely bound electrons fur becomes positively OSBINCBSE.COM OSBINCBSE.COM osbincbse.com It is very important to note that the electrification of the body (whether positive or negative) is due to transfer of electrons from one body to another. i.e. If the electrons are transferred from a body, then the deficiency of electrons makes the body positive. If the electrons are gained by a body, then the excess of electrons makes the body negative. If the two bodies from the following list are rubbed, then the body appearing early in the list is positively charges whereas the latter is negatively charged. -
Electrostatics
Ben Franklin, 1750 “Kite Experiment” – link between lightening and sparks Electrostatics “electrical” fire from the clouds Greeks noticed when they rubbed things against amber an invisible force of attraction occurred. Greek word for amber is elektron, which today represents the word “electrical” Electrostatics Study of electrical charges that can be collected and held in one place Energy at rest- nonmoving electric charge Involves: 1) electric charges, 2) forces between, & 3) their behavior in material An object that exhibits electrical interaction after rubbing is said to be charged Static electricity- electricity caused by friction Charged objects will eventually return to their neutral state Electric Charge “Charge” is a property of subatomic particles. Facts about charge: There are 2 types: positive (protons) and negative (electrons) Matter contains both types of charge ( + & -) occur in pairs, friction separates the two LIKE charges REPEL and OPPOSITE charges ATTRACT Charges are symbolic of fluids in that they can be in 2 states: STATIC or DYNAMIC. Microscopic View of Charge Electrical charges exist w/in atoms 1890, J.J. Thomson discovered the light, negatively charged particles of matter- Electrons 1909-1911 Ernest Rutherford, discovered atoms have a massive, positively charged center- Nucleus Positive nucleus balances out the negative electrons to make atom neutral Can remove electrons w/addition of energy, make atom positively charged or add to make them negatively charged ( called Ions) (+ cations) (- anions) Triboelectric Series Ranks materials according to their tendency to give up their electrons Material exchange electrons by the process of rubbing two objects Electric Charge – The specifics •The symbol for CHARGE is “q” •The SI unit is the COULOMB(C), named after Charles Coulomb •If we are talking about a SINGLE charged particle such as 1 electron or 1 proton we are referring to an ELEMENTARY charge and often use, Some important constants: e , to symbolize this. -
Lecture 6 Basics of Magnetostatics
Lecture 6 Basics of Magnetostatics Today’s topics 1. New topic – magnetostatics 2. Present a parallel development as we did for electrostatics 3. Derive basic relations from a fundamental force law 4. Derive the analogs of E and G 5. Derive the analog of Gauss’ theorem 6. Derive the integral formulation of magnetostatics 7. Derive the differential form of magnetostatics 8. A few simple problems General comments 1. What is magnetostatics? The study of DC magnetic fields that arise from the motion of charged particles. 2. Since charged particles are present, does this mean we also must have an electric field? No!! Electrons can flow through ions in such a way that current flows but there is still no net charge, and hence no electric field. This is the situation in magnetostatics 3. Magnetostatics is more complicated than electrostatics. 4. Single magnetic charges do not exist in nature. The simplest basic element in magnetostatics is the magnetic dipole, which is equivalent to two, closely paired equal and opposite charges in electrostatics. In magnetostatics the dipole forms from a small circular loop of current. 5. Magnetic geometries are also more complicated than in electrostatics. The vector nature of the magnetic field is less intuitive. 1 The basic force of magnetostatics 1. Recall electrostatics. qq rr = 12 1 2 F12 3 4QF0 rr12 2. In magnetostatics there is an equivalent empirical law which we take as a postulate qq vv×× ¯( rr ) = 12N 0 1212¢± F12 3 4Q rr12 3. The force is proportional to 1/r 2 . = 4. The force is perpendicular to v1 : Fv12¸ 1 0 . -
Contributions of Maxwell to Electromagnetism
GENERAL I ARTICLE Contributions of Maxwell to Electromagnetism P V Panat Maxwell, one of the greatest physicists of the nineteenth century, was the founder ofa consistent theory ofelectromag netism. However, it must be noted that significant discov eries and intelligent efforts of Coulomb, Volta, Ampere, Oersted, Faraday, Gauss, Poisson, Helmholtz and others preceded the work of Maxwell, enhancing a partial under P V Panat is Professor of standing of the connection between electricity and magne Physics at the Pune tism. Maxwell, by sheer logic and physical understanding University, His main of the earlier discoveries completed the unification of elec research interests are tricity and magnetism. The aim of this article is to describe condensed matter theory and quantum optics. Maxwell's contribution to electricity and magnetism. Introduction Historically, the phenomenon of magnetism was known at least around the 11th century. Electrical charges were discovered in the mid 17th century. Since then, for a long time, these phenom ena were studied separately since no connection between them could be seen except by analogies. It was Oersted's discovery, in July 1820, that a voltaic current-carrying wire produces a mag netic field, which connected electricity and magnetism. An other significant step towards establishing this connection was taken by Faraday in 1831 by discovering the law of induction. Besides, Faraday was perhaps the first proponent of, what we call in modern language, field. His ideas of the 'lines of force' and the 'tube of force' are akin to the modern idea of a field and flux, respectively. Maxwell was backed by the efforts of these and other pioneers.