Boundary conditions (EM)
The matching rules at an interface between two media: the normal component of D jumps by the free surface charge; the tangential component of E is continuous.
Definition
When the electric field crosses an interface between two media — vacuum to glass, glass to water, conductor to dielectric — its components behave very differently along the surface and across it. The two boundary conditions for static electric fields, derived from Gauss's law and the curl-free property of E, are the matching rules every interface problem starts from.
The normal component of D jumps by the free surface charge density: D_2⊥ − D_1⊥ = σ_f, where the subscripts label the two sides and ⊥ is the component along the outward normal from medium 1 to medium 2. If no free charge sits on the interface (the typical case at a vacuum-dielectric boundary), D⊥ is continuous. The tangential component of E is always continuous across any interface: E_2∥ = E_1∥. These two together pin down the kinks the field lines have to take when crossing from one medium to another, and they are why electric field lines that hit a dielectric surface refract toward or away from the normal — the same kind of bending that light undergoes when it crosses from air into glass.
The conductor case is the special limit. A conductor in electrostatic equilibrium has E = 0 in its bulk, so the tangential component of E just outside the surface must be zero too (otherwise charges would still be sliding along the surface), and the normal component is whatever Gauss's law requires from the local surface charge: E⊥ = σ/ε₀ pointing outward. These are the rules that make the method of images work, that explain why field lines hit a conductor at right angles, and that determine how a Faraday cage screens its interior.