Magnetostatics with FORTRAN and Mathematica
Aleksandar Donev - Dr. Phillip Duxbury
Due Friday 10th Nov - two weeks
During the next two weeks you will develop
a Mathematica code and a FORTRAN code to find the
magnetic field generated by a current ring.
PROBLEM 1
Consider current flowing in a circular ring of radius R.
Set R=1 and define the product: Permeability *current/4*pi = 1.
Now we will use the Biot-Savart law to find the magnetic field
due to this current ring centered at (0,0,0) and lying in the
x-y plane.
(a) Find a general integral expression for the
magnetic field at an arbitrary point (x,y,z).
(b) Evaluate your expression analytically (using Mathematica)
for the special case (0,0,z). What is the magnetic field at
(0,0,0)?
(c) For general (x,0,z), the integrals are called
elliptic integrals and must be done numerically. Mathematica
knows how to evaluate them.
Using the package Graphics`PlotField`, plot the x and z components
of the magnetic field in the x-z plane.
In case you have forgotten how to use the Biot-Savart law(haven't we all at some point!), here is ADDITIONAL HELP in setting up the problem.
Doing the integrals using Fortran 90
Numerical analysis is a very broad area
which spans most of the areas of applications of computers,
from mathematics to engineering and data analysis. In computational
physics, the emphasis is on large scale number crunching and
this is an area in which Mathematica is not competitive, however it
takes some work to use understand and design efficient code.
Some basic areas of numerical analysis are:
(i) Roots, derivatives, integrals.
(ii) Linear algebra: Equations, eigenvalues.
(iii) Partial differential equations.
(iv) Curve fitting.
In this worksheet you will write a Fortran 90 code to find the
value of an integral using the Trapezoid rule.
Read the introduction to numerical integration given at the site
(Sections 4.0,4.1).
PROBLEM 2. Now repeat the calculation 1(c) by writing a Fortran 90 code to evaluate the integral using the trapezoid rule. Use the trapezoid rule formula from Section 4.1 of numerical recipes. You can use the code given in Section 4.2 as a base, but convert it to Fortran 90 syntax, and write your own comments. Structure your code so that the trapezoid rule is placed in a separate module containing a function. Illustrate the convergence of the trapezoid rule by plotting the value of the integral as a function of the number of elements used in the integration, for a selected value of (x,0,z), with neither x or z non-zero(use xmgrace for this). Check that Mathematica gives the same result. Now use the vector plot facility (in FunGraphics) you used in workshseet 7 to plot the x-z vectors of the magnetic field.
Try to write this code on your own. However Aleksandar has develop some additional suggestions which you are welcome to look at.
About this document ...
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The translation was initiated by Phil Duxbury on 2000-10-27
Phil Duxbury
2000-10-27