Posted by ADSW1243 on
URL: http://forum.featool.com/Vector-Poisson-Equation-for-a-3D-Current-Source-tp338p346.html

So this is an magnetostatics/quasistatics problem in the presence of source currents without any permanent magnetic material, the full equation I'd like to model is:

full_eq.png,

As a set of PDEs then this is:
anisi-pdes.png

where A is the magnetic vector potential, mu = mu(r) & v = 1/mu, is a spatially varying permativity and J is a vector describing current density. Note in this form A can vary by any arbitrary scalar function without physically affecting the physical fields and from reading solving it can be very difficult. Overcoming this then..


Assuming mu(r) = mu_o throughout the entire domain and using the coulomb gauge del . A =0 the following vector identity holds:
vector_ident.PNG


This reduces the equation to:
vector_poisson.PNG

Which forms 3 (separable?) PDEs of the form:
1_comp.PNG,
where Az is the component of vector A along the Z axis, and Jz = the component of current density along the same axis.

(I would have followed your suggestion, but I can barely read my own handwriting xD)