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26 March 2004

Curves of Charge

by George E. Hrabovsky, President MAST

News from MAST

MAST is offering advanced severe weather spotter training at no cost!  We are also offering severe weather base operations training at no cost.  See our website at www.madscitech.org for details.

A Note of Apology

It is a little bit embarrassing, but I need to apologize for an egregious error I made some time ago.  I wrote that the outer product and the dyadic were two different things.  I was confusing the outer product with the vector product (also called the cross product).  I realize that for some reason I had transposed the ideas in my mind.  In fact the dyadic is the outer product and not the cross product.  Sorry!

Charged Curves

Last time we explored how you can represent a continuous charge distribution along a line.  That line need not be straight, so long as we can describe the curve mathematically we can express a charge distribution along it.  In the Mathematics Corner from both last week and this week I discuss plane curves, I would recommend that you go ahead and read those columns now, then return to this one.

By now you should realize that any curve is a type of vector field, it is just a one dimensional field.  Now, when we consider a curve on a plane (say a sheet of paper) we have embedded the curve in a two-dimensional space.  If we have a curve in space, we have embedded the curve into a three dimensional space.

If we have a line of charge in one dimension, the electric field becomes,

Let us assume that we have a unit of charge per unit of length, the electric field is then,

If, on the other hand, the density of charge is given by a nonlinear function,

Where .  We need to determine the value of this function, and we cannot integrate it directly.  We can approximate this by taking a power series expansion.  In this case, by trial and error I am choosing to expand the function to fifteenth order,

Then the electric field becomes,

We can plot this to see how the electric field changes.

The Frenet Frame

We can establish a coordinate system for curves.  The first component is based on the arc length of the curve (see this week's Mathematics Corner),

This is called the unit tangent vector.

The second component is given by,

Now, since we have established in a previous Mathematics Corner, that curvature is,

for some parametric curve and some complex structure , we can manipulate this to get (here the thick lines represent ||),

So,

This is called the unit normal vector.

The third component is given as,

and is called the binormal vector.

The set of all three vectors is called the Frenet-Serret formulas, or just the Frenet frame, of the curve after the mathematicians F. Frenet and J. A. Serret.

We will examine how this applies to electric fields next time.

Theory Challenge from Last Time

If we have a function described on a surface,

we integrate it first in the direction and then in the direction.

This is called a double integral.

Theory Challenge

How would you integrate a vector 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  (March 25, 2004)