Boy of Age 16 Asks Me about Relativity, etc.8. "I Can't Get Gauss's Flux Theorem"

Gauss's portrait published in Astronomische Nachrichten 1828.
By Siegfried Detlev Bendixen [Public domain],
from Wikimedia Commons.

A friend of mine on Twitter, Aaron (a pseudonym), is an overseas, 16-year old boy, who seriously admires Albert Einstein and wants to become a physicist. He continually writes me (Ted, also a pseudonym) questions about the theory of relativity and related topics, and I am sending answers. In this series of blog posts, those questions and answers are reproduced with modifications. I am not an expert in the fields of physics related to relativity. So, my answers might contain errors. If you find any error, please do not hesitate to write a comment for the benefit, not only of the boy and me, but also of other readers.

Aaron: I have a problem with Maxwell's equations. We don't learn them at school, and there're a lot of math. I'm not ready for that level of math. So, I can't get Gauss's flux theorem. But I want to understand it.

Ted: You are quite ahead of school lessons. Gauss's flux theorem was formulated by Carl Friedrich Gauss in 1835 and is also known as Gauss's law. I learned it only at university.

Concepts of physics can be learned to some extent without the use of math. For example, Gauss's law is stated by words as follows: The electric flux through any closed surface is proportional to the enclosed electric charge (quoted in Ref. 1 from Ref. 2).

However, true understanding of working and beauty of physics and practical application of physical laws require mathematics. Especially, if you want to become a physicist, early study of essential mathematics together with physics would be useful. Gauss's law can be expressed either by the equation of the integral form or by the equation of the differential form. The equivalence of the two equations can be shown by the use of the divergence theorem in vector calculus. Doesn't this sound interesting? (See, for example, Ref. 1.)

In The Feynman Lectures on Physics (Ref. 3), Gauss's law appears in Chapter 4 of Volume II after necessary mathematical preparation of "Vector Integral Calculus" in Chapter 3. Why don't you learn by the use of that fine textbook of Feynman? If you want to understand Maxwell's equations before studying Feynman's hefty books, however, Daniel Fleisch's A Student's Guide to Maxwell's Equations (Ref. 4) might be useful. Fleisch also explains the equations after giving the preparation of necessary mathematics.

References

1. "Gauss's law," Wikipedia: The Free Encyclopedia (5 January 2012 at 13:53).
2. Raymond A. Serway, Physics for Scientists and Engineers with Modern Physics, 4th edition (1996).
3. Richard P. Feynman, edited by Robert B. Leighton and Matthew Sands, The Feynman Lectures on Physics, Boxed set: The New Millennium Edition (Basic Books, 2011).
4. Daniel A. Fleisch, A Student's Guide to Maxwell's Equations (Cambridge University Press, 2008).

(Originally written on March 5–12, 2011)