Electromagnetism uniqueness theorem explained

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 (and) everywhere (in the entire volume considered) is specified;

1. For all times (of consideration), the component of either the electric field or the magnetic field tangential to the boundary surface (

\hat n x E or \hat n x 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.

See also

• Maxwell's equations
• Green's function
• Surface equivalence principle
• Uniqueness theorem

References

• Book: L.D. Landau, E.M. Lifshitz . 1975 . The Classical Theory of Fields . 4th . 2 . . 978-0-7506-2768-9.
• Book: J. D. Jackson . 1998 . Classical Electrodynamics . 3rd . . 978-0-471-30932-1.

Specific

1. Book: Smith, Glenn S.. An Introduction to Classical Electromagnetic Radiation. 1997-08-13. Cambridge University Press. 9780521586986. en.
2. Web site: 2020-05-11 . 2.8: Uniqueness Theorem . 2022-12-11 . Physics LibreTexts . en.