Elektrostatische randvoorwaarden

The electric field across a homogenous medium is continuous, but while crossing a boundary between two media, it becomes discontinuous. The difference between the field components above and below the boundary gives the change in the field.

Consider a Gaussian pillbox on the surface boundary. Applying Gauss's law, the surface integral can be written as the sum of the integrals over individual faces.

If the thickness of the pillbox tends to zero, only the faces parallel to the boundary contribute. Simplifying this gives the discontinuity of the normal component of the electric field at any boundary.

For the tangential component, consider a thin loop on the boundary. The integration of the electric field over this closed path is zero. Neglecting the contributions along the thickness implies that the tangential component is always continuous across a boundary.

Combining these expressions, the field at the boundary can be written by defining a unit vector.

The line integral of the field from below to above the surface tends to zero, implying that potential is continuous across any boundary.