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24 October 2003

Fields of More Than One Charge

by George Hrabovsky, President of MAST

News from MAST

The search for funding continues. Our goal since last April (before LabRats) has been to get an afterschool science program set up for area high schools.  We require outside funding for low-income students to attend. Our goal is to have a monthly theme for hands-on science project meetings, and students pay $60 per month - similar to a karate school. We need low income support so that the local school board will be able to directly cooperate.

The Field of Two Charged Particles

From last time we wrote the expression for the electric field as,

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

For two particles we have n = 2, or

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

     = (k q_1)/r_1^2Overscript[r,^] _1 + (k q_2)/r_2^2Overscript[r,^] _2

Let us say that particle 1 and particle 2 have equal, but opposite charges, and both are equidistant from the origin of our coordinate plane.  The field might look something like this,

[Graphics:Images/index_5.gif]

If they have the same charges then it would look like this,

[Graphics:Images/index_6.gif]

If particle 1 has ten times the charge of particle 2 (and opposite charges), we have,

[Graphics:Images/index_7.gif]

or with like charges.

[Graphics:Images/index_8.gif]

Theory Challenge from Last Time

The task was to determine line integrals of the expression

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

Since this is a vector there are two line integrals

∫_CF · r = ∫_C (x^2Overscript[e,^] _1 + y Overscript[e,^] _2 + z Overscript[e,^] _3) · r

and

∫_CFr = ∫_C (x^2Overscript[e,^] _1 + y Overscript[e,^] _2 + z Overscript[e,^] _3)  r

If we evaluate the first integral, we have

∫_CF · r = ∫_C (x^2Overscript[e,^] _1 + y Overscript[e,^] _2 + z Overscript[e,^] _3) · r

                   = ∫x^2x + ∫y y + ∫z z

                   = x^3/3 + y^2/2 + z^2/2 .

The second integral is

∫_CFr = ∫_C (x^2Overscript[e,^] _1 + y Overscript[e,^] _2 + z Overscript[e,^] _3)  r

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

               &nbs ... 8;x - x^2z) Overscript[e,^] _2, ∫_C (x^2y - y x) Overscript[e,^] _3}

               &nbs ... 2z) Overscript[e,^] _2, (∫_Cx^2y - ∫_Cy x) Overscript[e,^] _3]

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

                    =[0, z ( x - x^2) Overscript[e,^] _2, y(x^2 - x) Overscript[e,^] _3] .

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

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 24, 2003)