A Student's Guide to Maxwell's Equations

Citation

Fleisch, D. (2008). A Student's Guide to Maxwell's Equations. Cambridge University Press.

Chapter Summary

A short book of about 135 pages: one chapter for each of the four Maxwell equations, a fifth chapter that combines them into the wave equation, and an appendix on Maxwell's equations in matter. In the publisher's description, each equation is explained in plain language, symbol by symbol, in both its integral and differential forms. Chapters 1–4 follow the same plan: the integral form is taken apart term by term, applied to examples, and then rebuilt as the differential form, which is applied in turn. The author's website adds hints and full solutions to every problem in the book, audio podcasts for each module of the four equation chapters, three-dimensional models of the field sketches, and a review of coordinate systems that was left out of the book to keep it short.

Chapter 1: Gauss's law for electric fields

The longest chapter, and the one where most of the vector calculus is introduced. The integral form leads through the electric field, the dot product, the unit normal vector, the normal component of the field, the surface integral, the flux of a vector field, the electric flux through a closed surface, the enclosed charge and the permittivity of free space, followed by applications. The differential form then introduces nabla (the del operator) and the divergence, applied to the electric field.

Chapter 2: Gauss's law for magnetic fields

The same structure, much shorter, because the vector tools from Chapter 1 are already in place: the magnetic field, the magnetic flux through a closed surface, and the divergence of the magnetic field, each with applications in integral and differential form.

Chapter 3: Faraday's law

The integral form introduces the induced electric field, the line integral and the path integral of a vector field, the circulation of the electric field, the rate of change of magnetic flux and Lenz's law. The differential form introduces the curl ("del cross") and the curl of the electric field.

Chapter 4: The Ampere-Maxwell law

The circulation of the magnetic field, the permeability of free space, the enclosed electric current and the rate of change of electric flux, then the curl of the magnetic field, the electric current density and the displacement current density, with applications of both forms.

Chapter 5: From Maxwell's Equations to the wave equation

The book's synthesis. It sets out the divergence theorem, Stokes' theorem, the gradient and some useful vector identities, and uses them to derive the wave equation, the basis of the electromagnetic theory of light. The author's corrections page shows that the chapter also discusses Poisson's equation as a route to the electrostatic field.

Appendix: Maxwell's Equations in matter

A short treatment of how the equations are modified inside materials, followed by suggestions for further reading.

Key Concepts

Chapters 1–2: Gauss's laws

• Flux: the amount of a vector field passing through a surface, computed as a surface integral of the field's normal component.
• Enclosed charge: the source of net electric flux through a closed surface, linked to the flux by the permittivity of free space.
• Divergence: the del operator applied with a dot product, measuring how much a field spreads out from a point.
• No net magnetic flux: through any closed surface, the content of Gauss's law for magnetic fields.

Chapter 3: Faraday's law

• Line integral and circulation: the integral of a field's component along a path, and its value around a closed loop.
• Induced electric field: an electric field produced by a changing magnetic flux.
• Lenz's law: the induced effects oppose the change in flux that produces them.
• Curl: the del operator applied with a cross product, measuring the circulation of a field per unit area.

Chapters 4–5: The Ampere-Maxwell law and waves

• Displacement current: Maxwell's term for the changing electric flux, a source of magnetic field alongside conduction current.
• Current density: current per unit area, used in the differential form.
• Divergence and Stokes' theorems: the results that link integral and differential forms of the equations.
• Wave equation: the equation derived from Maxwell's equations that describes electromagnetic waves.

Critical Analysis

Strengths

• One equation at a time, both forms. The publisher describes plain-language explanations of each symbol, and the section headings confirm the method: each chapter moves from the integral form, term by term, to the differential form, with applications of each.
• Vector calculus taught as it is needed. Dot product, surface integral, flux, divergence, line integral and curl appear exactly where each equation requires them, so the book works as a first introduction to these operators in a physical setting.
• Strong online support. The author's website gives a series of hints and then a full solution for every problem in the book (39 problems for Chapters 1–4), audio walk-throughs of each module and three-dimensional field models.
• Short and focused. At about 135 pages, it can be read alongside a course without displacing the main text.

Limitations

• A guide, not a course. It covers only the four equations and how they combine; electrostatics in depth, potentials, circuits, radiation, special relativity and boundary-value methods are left to the main textbook. The appendix on matter is brief.
• Coordinate systems left out. The author notes that the review of rectangular, cylindrical and spherical coordinates was moved to the website to keep the book short; students who need it must download it.
• Errata. The author's corrections page lists errors fixed across printings, including the misnamed Poisson's equation in Chapter 5 and a correction to Equation 5.12 still listed as pending; readers of early printings should check it.

Where It Fits in the Course

Fleisch's guide is a companion, not a main text, in this electromagnetism list. For a first calculus-based course it pairs naturally with OpenStax's University Physics Volume 2, whose electricity and magnetism units build from Coulomb's law to electromagnetic waves, and with Purcell and Morin's Electricity and Magnetism, a more physically searching first serious course. Read the matching Fleisch chapter when each law first appears in the main text.

It is also useful as preparation for, or revision alongside, David Griffiths's Introduction to Electrodynamics, a comprehensive advanced undergraduate text, where the divergence and curl of the fields are used from the start. Jackson's Classical Electrodynamics (graduate level) and Bittencourt's Fundamentals of Plasma Physics are much later steps; Fleisch's book can still serve as a quick reminder of what each equation means before tackling them.

How to Use This Book

The book suits students in a first electromagnetism course who have some calculus and a first-year physics course behind them, and anyone who can manipulate Maxwell's equations but wants to understand what each term means. Readers unsure of cylindrical and spherical coordinates should download the author's coordinate-system review first.

• Core: Chapters 1 and 3, which introduce the key vector operations (flux and divergence, circulation and curl) used again in Chapters 2 and 4.
• Then: Chapters 2 and 4, which are quicker once the tools are familiar, and Chapter 5, the reward for the earlier work.
• Optional: the appendix on matter, best read when the main course reaches dielectrics and magnetic materials.

For each chapter, read the integral form first, try the applications, then the differential form. Work the problems with the website's hints before looking at the full solutions, and listen to the podcast for a module whenever a step is unclear.