Untitled Document

10 October 2003

More On Fields

News from MAST

We have been working on getting grants these last few weeks. This is why we have not been getting many columns in. Other members of MAST may begin contributing columns in the near future, if they have the time. The most immediate will be an image/signal processing column by James Firmiss, if he can find the time; it is his busy season for work.

The Field Due to a Force

Recall from the column, "Outstanding in Their Fields," that a field permeates all of the region of space under study. We imagine that the field can be represented by a set of arrows pointing in the direction of the field and whose length represent the field magnitude at that point. The gravitational field for some imaginary particle is determined by the formula

An important thing to remember about this is that we have imaginary particles, each with their own position vector. We call these imaginary particles test particles. Here is the gravitational field of a central object whose mass is ten times that of the test particles.

[Graphics:Images/index_3.gif]

This is always an attractive force. In space, such a field would look like this,

[Graphics:Images/index_4.gif]

We can make similar plots for the electric field,

where we have a number of test particles that are often called test charges. If the charges are opposite charges, then the field will look much like that of a gravitational field. If the field is generated by like charges, we have

[Graphics:Images/index_6.gif]

or in space,

[Graphics:Images/index_7.gif]

Theory Challenge from Last Time

The task was to determine a number of line integrals. The first expression is

whose line integral is

If we make the assumption that we are interested in a circular path, for 0 to , then we integrate,

Realizing that the limits of this integral form two halves of a circle we can split the integral into two intervals

&nbs ... ϕ(x^1) x^1 + Overscript[e,^]^2∫_ (2 π)^(4 π) ϕ(x^1) x^2

&nbs ... ;) sin x^2 + y^2x + Overscript[e,^]^2∫_ (2 π)^(4 π) sin x^2 + y^2y

$FormBox[RowBox[{ ... , +, RowBox[{6.28319, , sin, , x^2}]}], )}], Overscript[e,^]^2}]}]}]}], TraditionalForm]$

$FormBox[RowBox[{ ... ,^]^2}], , +, RowBox[{6.28319, , sin, , x^2, Overscript[e,^]^2}]}]}]}], TraditionalForm]$

$FormBox[RowBox[{ ... Overscript[e,^]^1}], +, RowBox[{578.784, , Overscript[e,^]^2, }]}]}]}], TraditionalForm]$

$FormBox[RowBox[{ ... 664, , sin, , x^2}], +, 578.784}], )}], , Overscript[e,^]^2, }]}]}]}], TraditionalForm]$

$FormBox[RowBox[{ ... RowBox[{12.5664, , sin, , x^2}]}], )}], , Overscript[e,^]^2 .}]}]}]}], TraditionalForm]$

Being that this was so much work, I will continue the rest of the integrals over the next two weeks.

Theory Challenge

Invent three functions (one scalar and two vector) of the standard Cartesian coordinates and find their path integrals . I will use this one,

$F(x, y, z) = {x^2, y, z},$

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 (October 10, 2003)