I am trying to calculate the electric field E induced in a vibrating cantilever of conductive material, oscillating in the field of a permanent magnet. However, I am having some difficulty getting pdsolve to work the way I want it to. I'm also not sure if the partial differential eqations I derived from Maxwell's equations are correct, or if the boundary conditions for the electric field in the cantilever are correct. Currently pdsolve gives me no solutions, which makes me think that either my PDEs or my BCs are not correct. It may be that I need to try some sort of numerical method as well. I am assuming that the z component of the electric field is just 0. My third PDE comes from setting the divergence of the electric field to 0. My first two PDEs come from the vector laplacian and its relation to the divergence and curl:
Laplacian * E = Div(E) -Curl(Curl(E))
The x and y components of this should be my first and second PDE, respectively. Note that in this equation the divergence of E is 0, and the curl of E is -dB/dt, where B is the magnetic field.
My boundary conditions are simply that the components of the electric field at the surface of the cantilever is always tangent to the surface.
I have tried various simplifications, such as setting the right hand side of the PDEs to 0, and still I don't get any solution.
My question: Are my PDEs and BCs sensible? And if so, what do I need to do with pdsolve to get a proper solution?