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05 March 2004

Continuous Charge Distributions

by George Hrabovsky, President of MAST

News from MAST

It is good to get back to the column after a nearly four-month hiatus.  

You will recall that in October I was talking about electric fields. I will continue that. I intend to cover basic field theory, then move on to quantum mechanics, then statistical mechanics, and finally some relativity. 

I don't know what I will write about after that, but I can worry about that when the time comes. There is a lot of ground to cover between now and then.

Continuous Charge Distributions

Despite the fact that all matter is composed of atoms, or discrete lumps of smaller matter, it is advisable to treat the matter of our common experience as being smooth (without the lumps and holes caused by the atoms).  Such a smooth distribution is called continuous

Matter that obeys this continuity is called a continuum . This can be applied to distributions of charge, too.  A metal cylinder that is charged is easier to treat as a continuous set of charges instead of an impossibly large collection of point charges

Let's assume we are studying a charge distribution in a wire. We can approximate the wire as a line. Recall that before we added together all of the charges to determine the electric field.  We do the same now, but we add together the charges at each significant interval on the line (this is called the line element ).  

If we think about this long enough, we can see this is just the process of taking the line integral. Instead of having discrete quantities of charge (symbolized as q_i), we have a function that describes the charge within a line element at a given position vector r called l (r). Indeed, one of the problems of the study of electric fields (also called electrostatics) is the invention of functions to describe these charge distributions.

To describe the electric fields of such a distribution of charge, instead of writing

E = Underoverscript[?, i = 1, arg3] (k q_i)/r_i^2Overscript[r,^] _i,

we write

E (r) = k?_Cl (r)/r^2Overscript[r,^] ?l .

We will study the ramifications of this kind of thing next time.

Theory Challenge from Last Time

The task was to determine line integrals of the expression

F(x, y, z) = {y z, x z, x y} .

Since this is a vector there are two line integrals

?_CF · ?r = ?_C (y zOverscript[e,^] _1 + x z Overscript[e,^] _2 + x y Overscript[e,^] _3) · ?r

and

?_CF??r = ?_C (y zOverscript[e,^] _1 + x z Overscript[e,^] _2 + x y Overscript[e,^] _3) ? ?r

If we evaluate the first integral, we have

?_CF · ?r = ?_C (y zOverscript[e,^] _1 + x z Overscript[e,^] _2 + x y Overscript[e,^] _3) · ?r

                   = ?y z?x + ?x z ?y + ?x y ?z

                   = x y z + x y z + x y z = 3 x y z .

The second integral is

?_CF??r = ?_C (y zOverscript[e,^] _1 + x z Overscript[e,^] _2 + x y Overscript[e,^] _3) ? ?r

               &nbs ... y ?x - y z?z) Overscript[e,^] _2, (y z?y - x z ?x) Overscript[e,^] _3

               &nbs ... x - y z?z) Overscript[e,^] _2, ?_C (y z?y - x z ?x) Overscript[e,^] _3}

               &nbs ... #63308;z) Overscript[e,^] _2, (?_Cy z?y - ?_Cx z ?x) Overscript[e,^] _3]

               &nbs ... ipt[e,^] _1, ((x^2y)/2 - (y z^2)/2) Overscript[e,^] _2, ((y^2 z)/2 - (x^2z)/2) Overscript[e,^] _3]

               &nbs ... )/2Overscript[e,^] _1, (y (x^2 - z^2))/2Overscript[e,^] _2, (z (y^2 - x^2))/2Overscript[e,^] _3] .

Theory Challenge

How would you integrate a function over a surface?

Sources That I Like

Books:

Robert C. Wrede (1963), Introduction to Vector and Tensor Analysis, John Wiley and Sons (Republished by Dover Publications in 1973). This is my all-time favorite book on vectors.

Edward M. Purcell (1985), Electricity and Magnetism, McGraw-Hill (Volume 2 of the famous and excellent Berkeley Physics Course). This is a very good introduction to field theory.

David J. Griffiths (1999), Introduction to Electrodynamics, Prentice-Hall. An extremely thorough treatment of field theory applied mostly to electrodynamics. Covers vector and tensor analysis along with special functions, complex analysis (conformal mappings), and relativity.

Online:

This first web page is pretty nice for getting a feel for vector fields.

http://www.math.duke.edu/education/ccp/materials/mvcalc/vfield/

A more advanced treatment can be found here,

http://vishnu.mth.uct.ac.za/omei/a-calculus/chap4/node2.html

Created by Mathematica  (February 26, 2004).