This monograph explores classical electrodynamics from a geometrical perspective with a clear visual presentation throughout. Featuring over 200 figures, readers will delve into the definitions, properties, and uses of directed quantities in classical field theory. With an emphasis on both mathematical and electrodynamic concepts, the author’s illustrative approach will help readers understand the critical role directed quantities play in physics and mathematics.
Chapters are organized so that they gradually scale in complexity, and carefully guide readers through important topics. The first three chapters introduce directed quantities in three dimensions with and without the metric, as well as the development of the algebra and analysis of directed quantities. Chapters four through seven then focus on electrodynamics without the metric, such as the premetric case, waves, and fully covariant four-dimensional electrodynamics. Complementing the book’s careful structure, exercises are included throughout for readers seeking further opportunities to practice the material.
Directed Quantities in Electrodynamics will appeal to students, lecturers, and researchers of electromagnetism. It is particularly suitable as a supplement to standard textbooks on electrodynamics.
Author(s): Bernard Jancewicz
Edition: 1
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 319
Tags: Multivectors, Exterior Forms, Grassmann Algebra, Electrodynamics, Space-Time
Preface
Introduction
Contents
1 Directed Quantities
1.1 Linear Space of Vectors Without a Scalar Product
1.2 Multivectors
1.3 Exterior Forms
1.4 Premetric Physical Quantities
2 Linear Spaces of Directed Quantities and Their Transformations
2.1 Linear Operations and Linear Spaces
2.2 Transformations of Bases
2.3 Euclidean Space and Reduction of the Number of Directed Quantities
3 Algebra and Analysis of Directed Quantities
3.1 Grassmann Algebras
3.1.1 Extended Grassmann Algebra of Multivectors
3.1.2 Extended Grassmann Algebra of Exterior Forms
3.2 Contractions
3.3 Linear Operators as Tensors
3.4 Unit System
3.5 Differential Forms
3.6 Exterior Derivative
4 Selected Problems of Electrodynamics
4.1 Premetric Electrodynamics
4.2 Static Solutions in an Anisotropic Dielectric Medium
4.3 Static Solutions in an Anisotropic Magnetic Medium
5 Electromagnetic Waves
5.1 Phase of a Wave
5.2 General Waves
5.2.1 Premetric Description
5.2.2 Utilizing Scalar Products
5.2.3 Examples
5.2.4 Semiplane Waves
5.3 Plane Wave
5.3.1 Eigenwaves
5.3.2 Relations Between Eigenwaves
5.3.3 Densities of Energy, Momentum and Energy Flux
6 Interfaces
6.1 Directed Quantities of a Plane
6.1.1 From 3D to 2D
6.2 Conditions on an Interface
6.3 Fields of Infinite Plates with Charges or Currents
6.3.1 Electric Field of a Uniformly Charged Plate
6.3.2 Electric Field of Two Plates—Electric Capacitor
6.3.3 Magnetic Field of a Plate with Uniform Surface Current
6.3.4 Magnetic Field of Two Plates—Magnetic Capacitor
6.4 Electromagnetic Stress Tensor
6.4.1 Electric Stress Tensor
6.4.2 Magnetic Stress Tensor
6.5 Reflection and Refraction of Plane Waves
7 Electromagnetism in Space-Time
7.1 Directed Quantities in Four Dimensions
7.2 Premetric Electrodynamics in Space-Time
7.3 Energy and Momentum of the Electromagnetic Field
7.4 Plane Wave
7.5 Scalar Product in Space-Time
7.5.1 Elements of Particle Mechanics
7.6 Constitutive Relation
Appendix: Energy-Momentum Tensor in Coordinates
References
