Page 158 - Maxwell House

138 Chapter 3

since ( = 0, ) = ( = 0, ) = 0. Therefore, according to (3.61) 3 () ≤ 0 at any
3
3
moment of time > 0. But this declaration contradicts (3.12) and (3.13) showing that the
energy conserved in EM fields must be positive or does not exist at all. If so, the inequity (3.61)
can only be satisfied if () ≡ 0 and (, ) = (, ) ≡ 0 at any moment of time ≥
3 3 3
0 that negates (3.56).
This “ab absurdo “ proves the Uniqueness Theorem as we intended to show: the fields
10
produced by sources within a limited and time-invariant region are unique as long as these
fields meet prescribed initial conditions and their tangential components satisfy given
conditions on the bounding surface. Therefore, both initial and boundary conditions must be
specified when our goal to get the unique solution of Maxwell’s equations in the space-time
domain. The initial conditions are represented by the constraints that the EM fields must satisfy
at a given time, while boundary conditions are, in general, restrictions that the EM fields must
meet over certain surfaces of the three-dimensional space.

Following the same path, it is possible to extend the uniqueness theorem to the problem with
the impedance boundary conditions: the projections of (, )| and (, )| are
∈ 1 ∈ 1
related at each point of as (, ) = () (, )| ∈ 1 , where () ≥ 0. The term

1
“impedance” in boundary conditions comes from the fact that the given parameter () is
measured in Ohms
() = (, )/ (, ) [(V/m) / (A/m) = V/A = Ohm]

It is worth noting the fundamental role of tangential components of electromagnetic fields.
Eventually, any 3D vector field can be assembled from six orthogonal components: four
tangential (two electric and two magnetic) and two normal (one for electric and one for
magnetic). Therefore, the normal components at the bounding surface are dependable values
and could be foreseen if Maxwell’s equations have been solved.

We will come back to boundary condition several times later because there are many different
sets of them can be formulated depending on the problem to be solved. For example, the
boundary could be artificial like the absorbing boundary representing the rest of the universe or
the periodic boundary conditions. Another boundary conditions may describe fields at artificial
walls simulating the geometrical and field symmetry.

3.2.3 Uniqueness Theorem in Space-Frequency Domain

Maxwell’s equations have a unique solution { (, ), (, )} at each point of if
1

1. The field sources and their distributions are given at each point of ,
1
2. The volume is occupied by dissipative medium meaning that (, ) ≠ 0 or/and
′′

1
(, ) ≠ 0 at each point ∈ ,
′′

1
10 It is borrowed from Latin and used to state that an assertion in an argument is false due to its absurdity.