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Ohm’s law and electric circuits 欧姆定律和电路
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/OSO/9780198821915.003.0006
J. Pierrus
{"title":"Ohm’s law and electric circuits","authors":"J. Pierrus","doi":"10.1093/OSO/9780198821915.003.0006","DOIUrl":"https://doi.org/10.1093/OSO/9780198821915.003.0006","url":null,"abstract":"This chapter considers various simple dc and ac circuits which contain at least one active element (always a voltage source) and passive elements (resistors, capacitors and inductors) arranged in different combinations to form a bilateral network. The notions of complex voltage, complex current and complex impedance are introduced and then used in the ensuing analysis. Some standard ‘network theorems’ including Kirchhoff’s rules, the delta-star transformation, Thevenin’s theorem and the superposition theorem are employed. Included in the questions are circuits involving bridges, filters, audio amplifiers and transformers. Important topics such as series and parallel resonance in LRC circuits are also considered along the way. Much of the laborious algebra involved in manipulating the complex quantities above is avoided by relegating this task to Mathematica.","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134432388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some essential mathematics 一些基本的数学
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/oso/9780198821915.003.0001
J. Pierrus
{"title":"Some essential mathematics","authors":"J. Pierrus","doi":"10.1093/oso/9780198821915.003.0001","DOIUrl":"https://doi.org/10.1093/oso/9780198821915.003.0001","url":null,"abstract":"This chapter introduces most of the important mathematics that will be used repeatedly throughout this book. Whilst scalars like the electric potential Ф and the vector fields E and B are familiar quantities in electricity and magnetism, it is not always known that they are examples of a mathematical form termed a tensor. More sophisticated tensors are required in some descriptions and theories in electrodynamics. As tensor notation is compact and usually allows for a simple derivation of a result, the chapter begins with a series of questions involving Cartesian tensors. The results obtained here will facilitate the solution of many questions in subsequent chapters. Readers who are not familiar with tensors are advised to consult Appendix A before proceeding. Towards the end of this Appendix a ‘checklist for detecting errors when using tensor notation’ is given, which will hopefully provide some help for the uninitiated.","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"16 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114006524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The electromagnetic potentials 电磁势
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/OSO/9780198821915.003.0008
J. Pierrus
{"title":"The electromagnetic potentials","authors":"J. Pierrus","doi":"10.1093/OSO/9780198821915.003.0008","DOIUrl":"https://doi.org/10.1093/OSO/9780198821915.003.0008","url":null,"abstract":"This chapter expresses the fields E\u0000 (\u0000 r, t) and B\u0000 (\u0000 r, t) in terms of the electromagnetic potentials Ф(\u0000 r, t) and A\u0000 (\u0000 r, t), and shows that these potentials are defined only up to a gauge transformation. This leads the reader naturally to the Coulomb and Lorenz gauges which are usually encountered in textbooks. The inhomogeneous wave equations whose solutions are the retarded electromagnetic potentials are also considered, as well as the Lienard–Wiechert potentials for an arbitrarily moving point charge. A few questions are included in which the Lagrangian and Hamiltonian of a point charge are expressed in terms of Ф and A. The chapter concludes by deriving a multipole expansion for the dynamic vector potential which will provide the starting point in our treatment of electromagnetic radiation later on in Chapter 11","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133688687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some applications of Maxwell’s equations in matter 麦克斯韦方程组在物质中的一些应用
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/oso/9780198821915.003.0010
J. Pierrus
{"title":"Some applications of Maxwell’s equations in matter","authors":"J. Pierrus","doi":"10.1093/oso/9780198821915.003.0010","DOIUrl":"https://doi.org/10.1093/oso/9780198821915.003.0010","url":null,"abstract":"This chapter comprises questions of a miscellaneous nature. They mostly have little in common except that all processes are time-dependent and occur within matter. The first few questions introduce some important preliminaries. For example, modifying Maxwell’s equations to include the effect of matter. The behaviour of the electromagnetic field at the boundary between two media having different properties is an important topic. The matching conditions (as they are known) are derived from both the integral and differential forms of Maxwell’s equations. Certain specific examples then follow, including some simple applications involving conductors, dielectrics and tenuous electronic plasmas. Along the way, the connection between Maxwell’s electrodynamics and the laws of geometrical optics is demonstrated explicitly.","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127660438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Electromagnetism and special relativity 电磁学和狭义相对论
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/OSO/9780198821915.003.0012
J. Pierrus
{"title":"Electromagnetism and special relativity","authors":"J. Pierrus","doi":"10.1093/OSO/9780198821915.003.0012","DOIUrl":"https://doi.org/10.1093/OSO/9780198821915.003.0012","url":null,"abstract":"In 1905, when Einstein published his theory of special relativity, Maxwell’s work was already about forty years old. It is therefore both remarkable and ironic (recalling the old arguments about the aether being the ‘preferred’ reference frame for describing wave propagation) that classical electrodynamics turned out to be a relativistically correct theory. In this chapter, a range of questions in electromagnetism are considered as they relate to special relativity. In Questions 12.1–12.4 the behaviour of various physical quantities under Lorentz transformation is considered. This leads to the important concept of an invariant. Several of these are encountered, and used frequently throughout this chapter. Other topics considered include the transformationof E- and B-fields between inertial reference frames, the validity of Gauss’s law for an arbitrarily moving point charge (demonstrated numerically), the electromagnetic field tensor, Maxwell’s equations in covariant form and Larmor’s formula for a relativistic charge.","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114521766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The electrostatics of conductors 导体的静电学
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/oso/9780198821915.003.0003
J. Pierrus
{"title":"The electrostatics of conductors","authors":"J. Pierrus","doi":"10.1093/oso/9780198821915.003.0003","DOIUrl":"https://doi.org/10.1093/oso/9780198821915.003.0003","url":null,"abstract":"This chapter begins by proving some important properties of (i) conductors in electrostatic equilibrium, and (ii) harmonic functions. These results underpin most of the remaining questions of Chapter 3. The coefficients of capacitance for an arbitrary arrangement of conductors are introduced at an early stage, and numerical calculations then follow in a number of subsequent questions. Some important techniques (both analytical and numerical) for finding solutions to Laplace’s equation are considered. These include: the Fourier method, the relaxation method, themethod of images and the method of conformal transformation. All of these are discussed in some detail, and with appropriate examples.","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125201206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Electromagnetic fields and waves in vacuum 真空中的电磁场和波
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/OSO/9780198821915.003.0007
J. Pierrus
{"title":"Electromagnetic fields and waves in vacuum","authors":"J. Pierrus","doi":"10.1093/OSO/9780198821915.003.0007","DOIUrl":"https://doi.org/10.1093/OSO/9780198821915.003.0007","url":null,"abstract":"In previous chapters four experimental laws of electromagnetism were encountered: Gauss’s law in electrostatics, Gauss’s law in magnetism, Faraday’s law and Ampere’s law. Now, in this chapter, these laws are generalized where appropriate to include the time-dependent charge and current densities ρ‎(\u0000 r, t) and J\u0000 (\u0000 r, t) respectively. The result is a set of four coupled differential equations—known as Maxwell’s equations— which provide the foundation upon which the theory of classical electrodynamics is based. One of the most important aspects which emerges from Maxwell’s theory is the prediction of electromagnetic waves, and an entire spectrum of electromagnetic radiation. Some of the properties of these waves travelling in unbounded vacuum are considered, as well as their polarization states, energy and momentum conservation in the electromagnetic field and also applications to wave guides and transmission lines.","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133843378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 14
Static electric fields in vacuum 真空中的静电场
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/OSO/9780198821915.003.0002
J. Pierrus
{"title":"Static electric fields in vacuum","authors":"J. Pierrus","doi":"10.1093/OSO/9780198821915.003.0002","DOIUrl":"https://doi.org/10.1093/OSO/9780198821915.003.0002","url":null,"abstract":"This chapter begins using Coulomb’s law to derive Maxwell’s electrostatic equations for a vacuum. In doing this, the integral forms of the electrostatic potential Ф and field E are obtained. These results are then used to determine Ф and E for various charge distributions possessing some symmetry: either via Gauss’s law or by directly integrating a known charge density over a line, surface or volume. Applications which require the use of computer algebra software (Mathematica) are included. A multipole expansion of the potential Ф leads to the various multipolemoments of a static charge distribution. Examples which deal with important properties like origin independence are presented. A range of questions and their solutions, not usually encountered in standard textbooks, appear in this chapter.","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132936252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Static magnetic fields in vacuum 真空中的静态磁场
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/OSO/9780198821915.003.0004
J. Pierrus
{"title":"Static magnetic fields in vacuum","authors":"J. Pierrus","doi":"10.1093/OSO/9780198821915.003.0004","DOIUrl":"https://doi.org/10.1093/OSO/9780198821915.003.0004","url":null,"abstract":"Wherever possible, an attempt has been made to structure this chapter along similar lines to Chapter 2 (its electrostatic counterpart). Maxwell’s magnetostatic equations are derived from Ampere’s experimental law of force. These results, along with the Biot–Savart law, are then used to determine the magnetic field B arising from various stationary current distributions. The magnetic vector potential A emerges naturally during our discussion, and it features prominently in questions throughout the remainder of this book. Also mentioned is the magnetic scalar potential. Although of lesser theoretical significance than the vector potential, the magnetic scalar potential can sometimes be an effective problem-solving device. Some examples of this are provided. This chapter concludes by making a multipole expansion of A and introducing the magnetic multipole moments of a bounded distribution of stationary currents. Several applications involving magnetic dipoles and magnetic quadrupoles are given.","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114451184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Static electric and magnetic fields in matter 物质中的静电和磁场
Solved Problems in Classical Electromagnetism Pub Date : 2018-08-02 DOI: 10.1093/OSO/9780198821915.003.0009
J. Pierrus
{"title":"Static electric and magnetic fields in matter","authors":"J. Pierrus","doi":"10.1093/OSO/9780198821915.003.0009","DOIUrl":"https://doi.org/10.1093/OSO/9780198821915.003.0009","url":null,"abstract":"The electric and magnetic fields at an arbitrary point inside matter fluctuate rapidly both in space and time. It is therefore necessary, in the chapter’s introduction, tomake the important distinction between microscopic and macroscopic fields. It is assumed throughout that the electric dipole approximation is valid, and several important properties which apply to matter in bulk are described early on. Along the way, it is convenient to define two new fields D and H, and these appear in a number of questions. Another feature which emerges during this treatment includes the concept of bound charge and current densities, and how these quantities are incorporated intoMaxwell’s inhomogeneous equations. Although Chapter 9 deals exclusively with static fields, it forms an important backdrop to Chapter 10 where some general timedependent aspects of macroscopic electromagnetism are considered.","PeriodicalId":184566,"journal":{"name":"Solved Problems in Classical Electromagnetism","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122122749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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