# Essential Electromagnetism

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## About the book

### Description

Essential Electromagnetism provides a concise introduction to this fundamental topic. Starting with forces on charges, it takes a logical step-by-step progression through electrostatics and magnetostatics, both in empty space and in matter. The book goes into sufficient detail to explain the important concepts using clear explanations as well as numerous diagrams, examples and problems. The notorious “grad, div and curl” feature regularly in electromagnetism, and so a straightforward introduction to vectors and the vector calculus needed in this book is included in the Appendices.

### Preface

“Essential Electromagnetism” and “Essential Electrodynamics” (also to be published by Ventus) are intended to be resources for students taking electromagnetism courses while pursuing undergraduate studies in physics and engineering. Due to limited space available in this series, it is not possible to go into the material in great depth, so I have attempted to encapsulate what I consider to be the essentials. This book does not aim to replace existing textbooks on these topics of which there are many excellent examples, several of which are listed in the bibliography. Nevertheless, if appropriately supplemented, this book and “Essential Electrodynamics” could together serve as a textbook for 2nd and 3rd year electromagnetism courses at Australian and British universities, or for junior/senior level electromagnetism courses at American universities/colleges.

The book assumes a working knowledge of partial differential equations, vectors and vector calculus as would normally be acquired in mathematics courses taken by physics and engineering students. However, very brief introductions to vectors, vector calculus and index notation are given in the appendices. Some of the mathematical derivations have been relegated to the appendices, and some of those are carried out using index notation which is briefly introduced in Appendix D, but elsewhere in the book manipulation of equations involving vector differential calculus is done using standard vector calculus identities (also given in the appendices). The presentation of the subject material in this book is conventional, starting with forces between charges, electric field, Coulomb’s law, electric flux, Gauss’ law and the electrostatic potential. Chapter 2 is on Poisson’s and Laplace’s equations, and their solution. The method of images is applied to simple examples in plane, cylindrical and spherical geometry. Laplace’s equation is solved analytically in Cartesian coordinates for the cases where the boundaries are orthogonal planes, and in spherical coordinates where the boundary surface is a sphere; these being the most commonly-encountered problems involving Laplace’s equation. Solution of Laplace’s equation in cylindrical coordinates is not included, but may be found in most applied mathematics texts. A numerical finite-difference method is described for solving Laplace’s equation. This method is applicable whether or not the boundary conditions allow analytic solutions.

Chapter 3 is on the multipole expansion of the electrostatic potential, up to and including the quadrupole term. Chapter 4 is on macroscopic and microscopic dielectric theory, starting with the polarisation field, polarisation charges, Gauss’ law and the displacement field, susceptibility and permittivity, and the boundary conditions on the electric and displacement fields. Orientational polarisability of molecules, electronic polarisability of non-polar molecules and ionic polarisability of crystals are discussed. Finally, the Clausius-Mossotti formula which connects the microscopic and macroscopic theories is derived.

Chapter 5 deals with the magnetic field, its causes, magnetic forces and the magnetic flux. The magnetic vector potential is derived from the Biot-Savart law, and a derivation of Ampère’s law follows, together with examples of its use in solving problems in magnetostatics. Finally the multipole expansion of the vector potential up to the dipole term is performed, and magnetic dipoles are discussed together with their magnetic field and the torques and forces they experience in a magnetic field.

Chapter 6 focuses on the magnetism of materials, starting with the magnetisation field, magnetisation currents and their inclusion in Ampère’s law, and the introduction of the auxiliary field H. Next, susceptibility and permeability are defined, and the boundary conditions on the magnetic and auxiliary fields are derived. Finally, the causes of magnetisation are discussed, namely orientational polarisability of atomic magnetic dipoles associated with unpaired electron spins causing paramagnetism, and the quantum mechanical exchange interaction in some materials giving rise to unpaired electron spins lining up in the same direction as in neighbouring atoms, and leading to ferromagnetism.

### Content

Preface

- Electrostatics
- Electric charge, field and flux
- The electrostatic potential

- Poisson’s and Laplace’s equations
- Poisson’s equation
- Method of images
- Laplace’s equation
- Laplace’s equation in spherical coordinates
- Finite-difference method for Laplace’s equation

- Multipole expansion for localised charge distribution
- Point charge on z axis
- The monopole moment
- The dipole moment
- The quadrupole moment tensor
- Electric dipoles
- Multipole expansion in spherical coordinates

- Macroscopic and microscopic dielectric theory
- Dielectrics
- Macroscopic dielectric theory
- Gauss’ law and the electric displacement
- Boundary conditions on V , E and D
- Microscopic dielectric theory

- Magnetic field and vector potential
- Magnetism
- The magnetic field and forces
- The magnetic vector potential
- Ampère’s law
- Magnetic flux
- Multipole expansion of vector potential

- Magnetism of materials
- Magnetic materials
- Magnetisation field
- Magnetisation currents and Ampère’s law
- Ampère’s law in magnetised materials
- Magnetic susceptibility and permeability
- Paramagnetism
- Ferromagnetism

Bibliography

Appendices

### About the Author

Raymond Protheroe obtained his PhD in 1978 from Durham University, U.K., for a thesis on simulation of showers of energetic sub-atomic particles in the atmosphere produced by high energy cosmic rays (the highest energy particles in nature). He then spent the next three years in the U.S. as a NAS/NRC Fellow at NASA's Goddard Space Flight Center where he worked on the propagation and origin of cosmic rays. In 1983 Protheroe moved to the University of Adelaide in Australia on a Queen Elizabeth II Fellowship to work on cosmic rays and ground-based gamma-ray astronomy, and was appointed Associate Professor/Reader in 1998. He was elected Fellow of the RAS (1979), IoP (1984), AIP (1990), ASA (1996), and has been Vice-Chair of Commission C4 (Cosmic Rays) of the IUPAP (2002-2005) and a member of the IAU since 1986.

As an educator, Professor Protheroe has taught introductory physics, undergraduate courses in optics, electromagnetism, quantum mechanics, astrophysics and relativity and cosmology, and at honours (graduate level) classical electrodynamics and high energy astrophysics. An example of his commitment to fully understanding and explaining the physics being taught led to his writing a paper on a fundamental aspect of electromagnetism which was inconsistent between textbooks at the time (“The Transient Magnetic Field Outside an Infinite Solenoid” by R. J. Protheroe and D. Koks, 1996, American J. Physics, 64, 1389).

Dr Protheroe's research has ranged widely from topics such as cosmic ray acceleration, energetic particle interactions in terrestrial, astrophysical and cosmological environments to predicting fluxes of cosmic rays, radio to gamma-ray emission and/or neutrinos from pulsars, supernovae, supernova remnants, our Milky Way galaxy and active galactic nuclei. Together with radio-astronomer colleagues, he recently instigated projects using radio telescopes to do neutrino astronomy with large radio telescopes and the Moon as the neutrino target. Protheroe has given numerous invited talks at international conferences, been awarded prizes including the inaugural international Shakti P. Duggal Prize at the 19th International Cosmic Ray Conference held in La Jolla. Protheroe's research has led to well over 300 publications including more than 140 articles on his research in peer-reviewed scientific journals. Lists of his publications can be found at:

Dr Protheroe's publications on SAO/NASA Astrophysics Data System