Boundary conditions of the Electric field of a conducting transmission line
If you assume the conductive elements are perfect (\$\rho=0\$), then the boundary condition is that the E field tangent to the surface goes to 0. This is often called "perfect conductor boundary conditions".

If you want to model a real conductive material (\$\rho &gt; 0\$), then you will have to model the fields and currents inside the conductive region also. The boundary condition will be that the tangential component of \$\vec{E}\$ is continuous across the boundary.