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:
At t = 0 {\displaystyle t=0} , the initial values of all fields (E, H, B and D) everywhere (in the entire volume considered) is specified; For all times (of consideration), the component of either the electric field E or the magnetic field H tangential to the boundary surface ( n ^ × E {\displaystyle {\hat {n}}\times \mathbf {E} } or n ^ × H {\displaystyle {\hat {n}}\times \mathbf {H} } , where n ^ {\displaystyle {\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.
See also Maxwell's equations Green's function Surface equivalence principle Uniqueness theorem
References L.D. Landau, E.M. Lifshitz (1975). The Classical Theory of Fields. Vol. 2 (4th ed.). Butterworth–Heinemann. ISBN 978-0-7506-2768-9. J. D. Jackson (1998). Classical Electrodynamics (3rd ed.). John Wiley & Sons. ISBN 978-0-471-30932-1. Specific