Electromagnetism uniqueness theorem

The electromagnetism uniqueness theorem states the uniqueness (but not necessarily the existence) of a solution to Maxwell's equations, if the boundary conditions provided satisfy the following requirements:


 * 1) At $$t=0$$, the initial values of all fields ($E$, $H$, $B$ and $D$) everywhere (in the entire volume considered) is specified;
 * 2) For all times (of consideration), the component of either the electric field $E$ or the magnetic field $H$ tangential to the boundary surface ($$\hat n \times \mathbf{E}$$ or $$\hat n \times \mathbf{H}$$, where $$\hat n$$ is the normal vector at a point on the boundary surface) is specified.

Note that this theorem must not be misunderstood as that providing boundary conditions (or the field solution itself) uniquely fixes a source distribution, when the source distribution is outside of the volume specified in the initial condition. One example is that the field outside a uniformly charged sphere may also be produced by a point charge placed at the center of the sphere instead, i.e. the source needed to produce such field at a boundary outside the sphere is not unique.