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Essential Electromagnetism: Solutions

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Lingua:  English
This book contains solutions to problems at the end of each chapter of “Essential Electromagnetism” by Raymond Protheroe.
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This book contains solutions to problems at the end of each chapter of “Essential Electromagnetism” by Raymond Protheroe, available separately at, which provides a concise introduction to this fundamental topic and aims to encapsulate the essentials of electromagnetism. 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.

This book gives the solutions to the exercises at the end of each chapter of my book “Essential Electromagnetism” (also published by Ventus). I recommend that you attempt a particular exercise after reading the relevant chapter, and before looking at the solutions published here. Often there is more than one way to solve a problem, and obviously one should use any valid method that gets the result with the least effort. Usually this means looking for symmetry in the problem – for example from the information given can we say that from symmetry arguments the field we need to derive can only be pointing in a certain direction. If so, we only need to calculate the component of the field in that direction, or we may be able to use Gauss’ law or Ampère’s law to enable us to write down the result. In some of these exercise solutions the simplest route to the solution is deliberately not taken in order to illustrate other methods of solving a problem, but in these cases the simpler method is pointed out.

The solutions to the exercise problems for Each chapter of “Essential Electromagnetism” are presented here in the corresponding chapters of “Essential Electromagnetism - Solutions”.

I hope you find these exercises useful. If you find typos or errors I would appreciate you letting me know. Suggestions for improvement are also welcome – please email them to me at

Raymond J. Protheroe, January 2013

School of Chemistry & Physics, The University of Adelaide, Australia

  1. Electrostatics
  2. Poisson’s and Laplace’s equations
  3. Multipole expansion for localised charge distribution
  4. Macroscopic and microscopic dielectric theory
  5. Magnetic field and vector potential
  6. Magnetism of materials

Raymond John Protheroe

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

Dr Protheroe's publications on INSPIRE