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 This example considers the problem of determining the magnetic vector

 potential A around a coil.


 According to Maxwell's equations,  

       curl H = J  

       div B = 0  

       B = mu*H


 where B is the manetic flux density

       H is the magnetic field strength

       J is the electric current density

 and  mu is the magnetic permeability of the material.


 The magnetic vector potential A is related to B by  

       B = curl A  


       curl( (1/mu)*curl A ) = J


 This equation is usually supplmented with the Coulomb Gauge condition  

       div A = 0.


 In the axisymmetric case, the current is assumed to flow only in the

 azimuthal direction, and only the azimuthal component of the vector

 potential is present.  Henceforth, we will simply refer to this component as A.


 The Coulomb Gauge is identically satisfied, and the PDE to be solved in this

 model takes the form  

       curl((1/mu)*curl (A)) = J(x,y)     in the domain

                          A  = g(x,y)     on the boundary.


 The magnetic induction B takes the simple form  

       B = (-dz(A), 0, dr(A)+A/r)


 and the magnetic field is given by  

       H = (-dz(A)/mu, 0, (dr(A)+A/r)/mu)


 Expanding the equation in cylindrical geometry results in the final equation,  

       dz(dz(A)/mu) + dr((dr(A)+A/r)/mu) = -J


 The interpretation of the natural boundary condition becomes  

       Natural(A) = n X H


 where n is the outward surface-normal unit vector.


 Across boundaries between regions of different material properties, the

 continuity of (n X H) assumed by the Galerkin solver implies that the

 tangential component of H is continuous, as required by the physics.



 In this simple test problem, we consider a circular coil whose axis of

 rotation lies along the X-axis. We bound the coil by a distant spherical

 surface at which we specify a boundary condition (n X H) = 0.

 At the axis, we use a Dirichlet boundary condition A=0.


 The source J is zero everywhere except in the coil, where it is defined

 arbitrarily as "10".  The user should verify that the prescribed values

 of J are dimensionally consistent with the units of his own problem.







  { Cylindrical coordinates, with cylinder axis along Cartesian X direction }  




   Aphi       { the azimuthal component of the  vector potential }  



   mu = 1             { the permeability }  

   rmu = 1/mu  

   J = 0               { the source defaults to zero }  

   current = 10       { the source value in the coil }  

   Bz = dr(r*Aphi)/r  


initial values  

   Aphi = 2           { unimportant unless mu varies with H }  



  { FlexPDE expands CURL in proper coordinates }  

   Aphi : curl(rmu*curl(Aphi)) = J  



  region 1  


    value(Aphi) = 0       { specify A=0 along axis }  

      line to (10,0)  

    natural(Aphi) = 0     { H<dot>n = 0 on distant sphere }  

      arc(center=0,0) angle 180 to close  


  region 2  

     J = current           { override source value in the coil }  

    start (-0.25,1)  

      line to (0.25,1) to (0.25,1.5) to (-0.25,1.5) to close  



  contour(Bz) zoom(-2,0,4,4) as 'FLUX DENSITY B'  

  contour(Aphi) as 'Potential'  




  contour(Bz) as 'FLUX DENSITY B'  

  contour(Bz) zoom(-2,0,4,4) as 'FLUX DENSITY B'  


      from (0,0) to (0,1) as 'Bz'  

  vector(dr(Aphi)+Aphi/r,-dz(Aphi)) as 'FLUX DENSITY B'  

  vector(dr(Aphi)+Aphi/r,-dz(Aphi)) zoom(-2,0,4,4) as 'FLUX DENSITY B'  

  contour(Aphi) as 'MAGNETIC POTENTIAL'  

  contour(Aphi) zoom(-2,0,4,4) as 'MAGNETIC POTENTIAL'  

  surface(Aphi) as 'MAGNETIC POTENTIAL' viewpoint (-1,1,30)