Discontinuities in the electromagnetic field occur primarily at point charges and along boundaries, particularly in superconductors. Superconductors, which can carry infinite current, require a unique treatment beyond simply setting resistance to zero. The London equations provide an effective theoretical framework for understanding these phenomena. Notably, the jump in the normal component of the electric field across conducting surfaces is quantified by the equation ##E_{n1}-E_{n2}=\sigma/\epsilon_0##, where ##\sigma## represents surface charge density.
PREREQUISITESPhysicists, electrical engineers, and students studying electromagnetism and superconductivity will benefit from this discussion.
In practice the resistance of the conductor allows the wave to slightly penetrate. I think the problem in general is that the wavelength is finite and so there is no discontinuity at the microscopic level. For instance, total internal reflection in a prism is accompanied by reactive fields in the air behind the reflecting surface.vanhees71 said:One example are jumps of the normal component of the electric field along conducting surfaces, carrying a surface charge density. The jump is ##E_{n1}-E_{n2}=\sigma/\epsilon_0##.
Superconductors must be treated differently. They cannot be described by simply making the resistance 0 (or the electric conductivity to ##\infty##). A nice effective theory is the London theory:
https://en.wikipedia.org/wiki/London_equations