Force-Free Magnetic-Field Solution in Toroidal Coordinates

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Abstract
A method is presented for the solution of the force-free magnetic-field boundary-value problem in toroidal coordinates. An exact solution utilizing the Stokes' current function is given for the free space surrounding the toroid. The solution of the force-free equations within the current-carrying region of the toroid, however, requires the use of approximate methods. A suitable approximate method is developed which, also in accordance with physical considerations, exactly satisfies the requirement that the divergence of the current be zero. Galerkin's method is chosen for use in this approximation and use is made of the flexibility of this method to reduce the complexity of the approximation process. A sample calculation using this approximation method and the results of this calculation are described. This approximation method is found to give an accurate solution to the boundary-value problem with a minimum of effort. An equation describing the forces still present on the surface of the toroid is developed and the limitation imposed by this equation on the reduction of these forces in the boundary-value problem is noted.

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