However, the interface conditions for the electromagnetic field vectors can be derived from the integral forms of Maxwell's equations.
Two of our sides are infinitesimally small, leaving only After dividing by l, and rearranging, This argument works for any tangential direction.
The difference in electric field dotted into any tangential vector is zero, meaning only the components of
parallel to the normal vector can change between mediums.
This can be deduced by using Gauss's law and similar reasoning as above.
If there is no surface charge on the interface, the normal component of D is continuous.
Therefore, the tangential component of H is discontinuous across the interface by an amount equal to the magnitude of the surface current density.
This is done by assuming conditions at the boundaries which are physically correct and numerically solvable in finite time.
The most usual and simple example is a fully reflecting (electric wall) boundary - the outer medium is considered as a perfect conductor.
In some cases, it is more complicated: for example, the reflection-less (i.e. open) boundaries are simulated as perfectly matched layer or magnetic wall that do not resume to a single interface.