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][2]

1. At ${\displaystyle t=0}$, 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 (${\displaystyle {\hat {n}}\times \mathbf {E} }$ or ${\displaystyle {\hat {n}}\times \mathbf {H} }$, where ${\displaystyle {\hat {n}}}$ 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

1. Smith, Glenn S. (1997-08-13). An Introduction to Classical Electromagnetic Radiation. Cambridge University Press. ISBN 9780521586986.
2. "2.8: Uniqueness Theorem". Physics LibreTexts. 2020-05-11. Retrieved 2022-12-11.