A surface is a mathematical object. It has units of area. It may be finite or infinite. It may be open or closed. It has no thickness. It is the mathematical object over which a two dimensional integral is evaluated. Examples include a spherical shell, a cylindrical shell, and a plane.
A boundary is a physical object that is represented by a surface. It is closed and is idealized as having no thickness. It separates all of space into two volumes: inside the boundary and outside the boundary. In boundary value problems, the boundary can have surface charge on it. Information about the surface charge can be given as the surface charge density or the potential at every location on the boundary. Sometimes the boundary is referred to as a boundary surface.
Boundary conditions describe the behavior of the electric field and how it changes immediately above and immediately below a localized surface. A localized surface is just a very small patch of a much larger boundary. The component of the electric field parallel to the boundary surface is always continuous (i.e. it is the same). The component of the electric field perpendicular to the boundary surface is discontinuous if the boundary surface has charge on it. In this case, the perpendicular component of the electric field immediately above and below the boundary surface points in opposite directions. This is all summarized in these equations below for the case of free space (vacuum on either side of the boundary):
( \vec{E}_2 - \vec{E}_1 ) \cdot n_{1 \rightarrow 2} = \sigma/\epsilon_0
( \vec{E}_2 - \vec{E}_1 ) \times n_{1 \rightarrow 2} = 0
where:
- \vec{E}_2 is the electric field immediately above the boundary surface
- \vec{E}_1 is the electric field immediately below the boundary surface
- \sigma is the surface charge density on the boundary surface
- \epsilon_0 is the electric constant
- n_{1 \rightarrow 2} is unit normal vector to the boundary surface that points from below the surface to above
Sometimes, we use the phrase boundary conditions to also include the given boundary values. This is sloppy and confusing to the beginner, but they are two very different things. Boundary values are the given inputs to a problem and are explicit and implicit. An example of an explicit boundary value is the potential or surface charge density on the surface of spherical or cylindrical shell. An implicit boundary value is the behavior of the observable, most commonly the electric potential, at the origin and at some location infinitely far away from the origin. See this discussion for a deeper dive into the value or behavior at infinity:
Dirichlet boundary conditions are when the value of some function is given on a surface. Neumann boundary conditions are when the value of the derivative of a function is given on some surface. For electrostatic problems, a unique solution for the potential can be found using one or the other, but not both.