1
$\begingroup$

So I have a problem which (in one case) leads me to the following vector Laplace equation:

$\nabla^2 \mathbf{A} = 0$

with $\mathbf{A}$ the magnetic vector potential, whereon I have imposed the Coulomb gauge $\nabla\cdot\mathbf{A} = 0$. The problem contains cylindrical symmetry, which is why I want to solve this in cylindrical coordinates. Boundary conditions are to follow from solving the homogeneous Helmholtz equation and demanding continuity at a cylindrical surface.
However, I need to solve the Laplace equation first, which I don't know how to do. If the coordinates were cartesian, the vector equation would just be equivalent to three scalar equations, but now it's not so easy. Does anyone feel like explaining to me how I would best go about solving this vector Laplace equation in cylindrical coordinates? Greatly appreciated!

  • 0
    @Fabian: I'll take a look at Jackson's book, thanks for the reference. But perhaps I should have said the external magnetic field is along the z-axis and the cylinder lies along the z-axis as well, so it's not really a waveguide problem. The physical context is that of a superconducting nanowire along z in an external magnetic field along z, which I am describing using the Ginzburg-Landau equations. This Laplace equation is derived from the second GL equation with the assumption that the order parameter is zero outside and constant inside the nanowire (and the regime is stationary).2012-12-08

0 Answers 0