# E&M clarification

1. Apr 30, 2008

### hartlw

This is a review I wrote for a text-book on Amazon but it really applies to all the E&M books I have seen. I put it here in case its useful to a student struggling with E&M.

This book is typical of most E&M text books on a purely theoretical level. They present E&M as a collection of symbols and rules for manipulating them. The development is full of incomplete, unintelligible statements with gaps in between, without any indication that these occur. I find this at best inconsiderate, and at worst nasty and arrogant.

In chapt 1 "X" "stumbles" across the problem of integrating over the origin with 1/r^2 in the integrand and then uses this to define the dirac delta function. In chapt 2 the problem of integration with 1/r^2 (and 1/r) is ignored and Gauss' law (divergence theorm) is "derived" by a combination of poor intuition (field lines improperly explained) and half-baked math, finishing with, "Evidently the flux through any surface enclosing the charge is q/eo." No one could possibly "get" this, but they could accept it, depending on their style of learning. "Double Vectors" are introduced later in the book. Wow! Cutting edge science? No, just the old, still very useful, dyadics with a new name.

I am reminded of books on Windows Server. The words are there but somehow they don't make sense. Then you look at the authors who it turns out are in marketing or sales.

To the good, honest, students who genuinely want to understand and learn E&M. Don't be intimidated by the E&M textbooks written for quantum physics (?), or the other reviews. The presentation is incomplete and often unintelligible, and the underlying message of the authors and the reviewers seems to be "I'm smart, and you are dumb if you don't understand this." Somehow struggle through the course knowing that the lack of understanding is not your fault. If after the course by some miracle you are still interested in E&M, teach yourself.

I appreciate that there is a school of thought that doesn't really care where the equations come from or what they mean, that just wants to get E&M out of the way and get on with quantum physics or partial differential equations, ie, learn the language of E&M without the grammar. You will get the words in this book.

A major problem with E&M textbooks is they use vector calculus with total disregard of the content, so that the results don't make sense.

I will try to fill in some of the math gaps. E&M here is the study of continuous charge and current distributions.

Math Prerequisites:
Intuitive notion of continuity, convergence, partial derivatives (lim [f(x+e,y,z)-f(x,y,z)]/e as e-> ifinity), Definitions of U & E (potential and electric field vector) as volume integrations over charge distributions. A vector is continuous and differentiable if its components are. E&M integrals are improper at 1/r and 1/r^2 when r=0. Because U is an improper integral inside V, you can't assume E = delU. To get this write U and then delU by taking del under the integral sign. This is the same as formula for E. But you have to prove this is OK because normally you can't differentiate under the integral sign if integral is improper.

E&M

U & E due to a volume distribution of piecewise continuous charge rho in the bounded volume V exist at points of V and ARE CONTINUOUS THROUGHOUT SPACE. U is everywhere differentiable and E=delU THROUGHOUT SPACE.

Where U and E are continuous, U has continuous second derivatives (E has continuous first derivatives) and then from the divergence theorem:
del^2U = -4pirho.
del^2U is discontinuous at the boundaries because it has different values on either side.

A lighter requirement for del^2U to exist at an interior point of V is that U be piecewise continuous and satisfy a Hoelder condition (believe me, you don't want to go there).

Similar theorems apply for surface charge.

Model Problem in Electrostatics: Charged conducting sphere of radius a:
a) U=constant 0=<r<=a, del^2U=0 r>a
b) U everywhere continuous
c) first order derivatives everywhere continuous except at r=a where dU/dn+ - dU/dn- = -4pisigma where sigma is surface charge density.
d) rU -> E as r becomes infinite.

c) is from Gauss' theorem and pillbox. I don't get d).

Appendix

Given a vector field with components X,Y,Z and Normal region N:
Divergence Theorem. Assume X,Y,Z and first partial derivatives are contiuous within and on the boundary of N.
Extension: X,Y,Z are continuous in the region R and on its boundary, and R can be broken up into a finite number of regions for which divergence theorem holds, and in each of which X,Y,Z have derivatives which are continuous, the boundary included. This means that as P approaches the boundary from one of the partial regions, each derivative approaches a limit, and that these limits together with the values in the interior form a continuous function. The limits, however, need not be the same as P approaches a common boundary of two partial regions from two sides.

Stokes" Theorem
X,Y,Z and their partial derivatives contiuous in a region of space with S in its interior. The surface S is two-sided, and can be resolved into a finite number of normal surface elements. The functions X,Y,Z are continuous at all points of S, and their partial derivatives are continuous at all points of the normal surface elements into which S is divided. (See above)

References:
Phillips, Vector Analysis, 1933, pg 122 and on.
Kellog, Foundations of Potential Theory, 1929, pg 126 and on.

Don't bother trying to learn the proofs- they are impossible if you are not a born mathematician. They are not hard, but you have to keep in mind the steps as you go along which is difficult if you can't associate an image with them (remember them, what do you register in your mind if you can't recall the printed image?). Also, the algebra of limits gets monstrous.

With that said, you can forget the above and read an E&M book knowing that it's not you, its the book, and learn the language and hopefully figure out a few problems.

2. May 2, 2008

### pam

You could name, author, and level of the book to save us the trouble.