The electromagnetism uniqueness theorem states the
uniqueness
Uniqueness is a state or condition wherein someone or something is unlike anything else in comparison, or is remarkable, or unusual. When used in relation to humans, it is often in relation to a person's personality, or some specific characterist ...
(but not necessarily the
existence
Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being.
Etymology
The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistentia' ...
) of a solution to
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
, if the
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s provided satisfy the following requirements:
# At
, the
initial
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
values of all fields (, , and ) everywhere (in the entire volume considered) is specified;
# For all times (of consideration), the component of either the
electric field or the
magnetic field tangential to the boundary surface (
or
, where
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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
*
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
*
Surface equivalence principle
*
Uniqueness theorem In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems ...
References
*
*
;Specific
Vector calculus
Physics theorems
Uniqueness theorems
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