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:[1][2]

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 (n^×𝐄 or n^×𝐇, where 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. 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

1. ↑ Smith, Glenn S. (1997-08-13) (in en). An Introduction to Classical Electromagnetic Radiation. Cambridge University Press. ISBN 9780521586986. https://books.google.com/books?id=nZZu85Gq_D8C&dq=electromagnetism+uniqueness+theorem&pg=PA65.
2. ↑ "2.8: Uniqueness Theorem" (in en). 2020-05-11. https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_(Staelin)/02:_Introduction_to_Electrodynamics/2.08:_Uniqueness_theorem.

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