NP 5.02: Magnetic Fields in Cylindrical Geometry


This problem deals with Jackson's problem 5.8. Then end result is to show the fieldlines of the B field through and around the cylindrical object.

Complete Statement of the Problem

Solution to Problem

I began by coping the Bland Starter Program in directory:

      ~bland/public_html/courses/703/hw/np0502/blines.pro

to my own directory. The graph depicts a solid (a=0) cylinder of iron in a magnetic field.

Now I modify the parameter mu in order to illustrate the property of magnetic field lines (which are perpendicular to the surface of the material) near the surface of the permeable material. Here's a few instructive plots:

  • when mu=0.5

  • when mu=1.0 Notice that mu is set to 1.0 which is what the mu is set to outside the cylinder (in air). There for the graph should look as though there is no cylinder there - which is what is graphed.

  • when mu=3.0

  • when mu=10.0

    It seems that in order that the field lines be perpendicular to the surface, mu needs to be close to one. Choose mu=1.1

    Part b deals with modification of the program to inclued a hole of radius a in the center of the cylinder. Therefore, phiIII needs to be rewritten as:

      phiIII (rho > b)= - b0*rho*cos(theta) + cos(theta)/rho*b0*b^2
                     * (1. - (2.*(b^2*(mu+1)+a^2*(mu-1))/(b^2*(mu+1)^2-a^2*(mu+1))))
    

    Phi needs to be rewritten to:

           phiI (rho < a) = -4.*mu*b0/((mu+1)^2 - a^2/b^2*(mu-1)^2))*rho*cos(theta)
    

    And an additional phi, phiII, needs to be added:

           phiII = -2.*(mu+1)*b0/((mu+1)^2-a^2/b^2*(mu-1)^2))*cos(theta)*rho
                   -2.*(mu-1)*b0/((mu+1)^2/a^2 - (mu-1)^2/b^2)*cos(theta)/rho
    

    Now the program has been transformed from determining the magnetic field lines of a solid cylinder of iron to determining the magnetic field lines of a solid cylinder of iron with a hole of radius a. Upon running my program, I realized that something is not correct. Maybe I just need to adjust something ie the array size but I've tried everything I can think of at the moment with no results. At the moment, I will continue on as thought everything was fine.

    Here's a few instructive plots illustrating the most interesting properties of the solution.

  • when a=0.1 and b=1.0

  • when a=0.5 and b=1.0

  • when a=0.9 and b=1.0

    If you are intested in checking out the code...

    Well, there's no more time left to analyze the situation. I'm sure the problem is something simple that I've over looked.

    Here's the updated code. It works for small a's but not large a's.