Electromagnetism uniqueness theorem

{{short description|Providing boundary conditions for Maxwell's equations uniquely fixes a solution}}

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:{{Cite book|url=https://books.google.com/books?id=nZZu85Gq_D8C&dq=electromagnetism+uniqueness+theorem&pg=PA65|title=An Introduction to Classical Electromagnetic Radiation|last=Smith|first=Glenn S.|date=1997-08-13|publisher=Cambridge University Press|isbn=9780521586986|language=en}}{{Cite web |date=2020-05-11 |title=2.8: Uniqueness Theorem |url=https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_(Staelin)/02%3A_Introduction_to_Electrodynamics/2.08%3A_Uniqueness_theorem |access-date=2022-12-11 |website=Physics LibreTexts |language=en}}

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