# What does it take to understand Maxwell's equations?

Assume I want to learn math and physics enough to reach a level where I understand Maxwell's equations (The terms and reasoning in the equations I.e. why they "work"). What would I have to learn in order to have the tools I need to make sense out of it?

I'm kindof looking for a road map which I can use to get started and know what to focus on. The fields are pretty big so some pruning would be very helpful (if possible)

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I changed the tags because the way this is phrased, it isn't actually asking about books, rather it's asking about an academic progression leading up to basic electromagnetism. More like this question than this one. – David Z Sep 30 '11 at 16:39
It takes coffee, and usually a little vodka. – Colin K Nov 20 '11 at 21:05

Math

1. High school mathematics, including algebra, pre-calculus, and basic solid geometry
2. Single-variable calculus, both differentiation and integration
3. Calculus of functions of several variables, including volume integrals and surface integrals
4. Differential equations
5. Vector algebra, including dot products and cross products
6. Vector calculus, including line integrals, divergence, gradient, curl, and the Laplacian.

I'll second the recommendation for "Div, Grad, Curl", but that's starting at step 6. Silvanus Thompson's Calculus Made Easy is good for step 2. James Nearing's Mathematical Tools for Physicists, especially chapters 1,4,6,8, 10 and 13 should be helpful for steps 3 - 6. The various Schaum's outlines might be sufficient if you need to review the high school stuff in step 1.

Physics

1. Basic mechanics - Newton's laws (technically, you don't need this, but any book you read about electromagnetism will assume you know it)
2. High school electromagnetism - basics of charge, current, electric and magnetic fields, Coulomb's law, Ampere's Law, Lenz' Law, the right-hand rule, and displacement currents.
3. (optional) Special Relativity - Lorentz transformations, the interval, four-vectors (optional because you could choose to learn it concurrently)

That's it - the physics prerequisites are brief. However, it takes quite a bit of time and effort to get used to basic physics if you haven't done so already. In theory you could skip the physics prereqs, but you'd probably have a hard time relating the equations you were learning to the real world.

Volume 1 of The Feynman Lectures on Physics is a good source for the physics prereqs, although you'll probably need to supplement with practice problems from another source.

There is a book called A Student's Guide to Maxwell's Equations that I have heard recommended highly. However, I haven't read it. When you're ready for electromagnetism with Maxwell's equations, I can second recommendations for Purcell and Griffiths.

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I advise to start with Purcell's "Electricity and Magnetism", then read Griffiths' "Introduction to electrodynamics". As for mathematics, in order to understand Maxwell's equations you need to know vector calculus, not to mention differential equations.

Edit: The review of vector analysis can be found in the first chapter of Griffiths' book.

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Course based on Purcell's E&M. – Dale Jun 16 '13 at 23:13

Best book to get started on Vector Calculus: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus

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I learned undergraduate electrostatics from Wangsness' Electromagnetic Fields. The text is one of the clearest and most comprehensible I've seen at this level and I highly recommend it. The first chapter covers only vector calculus and provides an excellent basis for the necessary math. We got to Maxwell's equations at the end of the first semester so you should expect to spend at least half a year to get to that level.

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Most books have a detailed section on the level of the text, with the assumed background. Pick up a book on the subject, and read this section carefully to understand the prerequisites. This section is usually in the forward, before the actual text begins. There are frequently course maps in these sections as well, to guide instructors/self taught students on what the author/publisher feels would be an appropriate setup for teaching/learning the material.

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By far the greatest book I have ever had the pleasure of reading and listening to, is a recent book by Dan Fleisch.

A Student's Guide to Maxwell's Equations - The Podcast

A Student's Guide to Maxwell's Equations - The Book

A Student's Guide to Maxwell's Equations - The Pdf

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I really don't want to second-guess Mark Eichenlaub's excellent answer, because I agree that he methodically goes through the exact learning path towards the full mastery of Maxwell's Equations. I just want to remark that in practise, for the great majority of things that people do with Maxwell's Equations, you don't actually have to literally solve Maxwell's Equations.

All my life I have had to solve problems in electromagnetic theory, and I've never actually written a partial differential equation and solved for the boundary conditions. What I am familiar with is the plane wave solutions of Maxwell's Equations. Most of what I do is patching plane wave solutions, or superpositions of such solutions, into the physical situation of interest. I have some understanding of the physics that is going on at the boundaries, and that is also essential. But I'm never actually doing vector calculus.

It might get a bit sketchy if I was looking for near-field solution in cylindrical or spherical geometry, but how often are you doing that? More often you can deduce what you are looking for based on the far-field behavior, and up to a phase factor that basically reduces to directed plane waves with some obvious scaling corrections for distance and direction.

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