Page 89 - Electromagnetics Handbook

The useful relationships we seek occur when the first two bracketed quantities on the
right-hand side of the above expression are zero. Whether this is true depends not only
on the behavior of the fields, but on the properties of the medium at the point in question.
Though we have assumed that the sources of the field sets are independent, it is apparent
that they must share a similar time dependence in order for the terms within each of the
bracketed quantities to cancel. Of special interest is the case where the two sources are
both sinusoidal in time with identical frequencies, but with differing spatial distributions.
We shall consider this case in detail in § 4.10.2after we have discussed the properties of
the time harmonic field. Importantly, we will find that only certain characteristics of the
constitutive parameters allow cancellation of the bracketed terms; materials with these
characteristics are called reciprocal, and the fields they support are said to display the
property of reciprocity. To see what this property entails, we set the bracketed terms to
zero and integrate over a volume V to obtain

(E a × H b − E b × H a ) · dS = (J a · E b − J b · E a − J ma · H b + J mb · H a ) dV,
S V
which is the time-domain version of the Lorentz reciprocity theorem.
Two special cases of this theorem are important to us. If all sources lie outside S,we
have Lorentz’s lemma

(E a × H b − E b × H a ) · dS = 0.
S
This remarkable expression shows that a relationship exists between the fields produced
by completely independent sources, and is useful for establishing waveguide mode or-
thogonality for time harmonic fields. If sources reside within S but the surface integral
is equal to zero, we have

(J a · E b − J b · E a − J ma · H b + J mb · H a ) dV = 0.
V
This occurs when the surface is bounded by a special material (such as an impedance
sheet or a perfect conductor), or when the surface recedes to infinity; the expression is
useful for establishing the reciprocity conditions for networks and antennas. We shall
interpret it for time harmonic fields in § 4.10.2.

2.9.4 Similitude
A common approach in physical science involves the introduction of normalized vari-
ables to provide for scaling of problems along with a chance to identify certain physically
significant parameters. Similarity as a general principle can be traced back to the earliest
attempts to describe physical effects with mathematical equations, with serious study un-
dertaken by Galileo. Helmholtz introduced the first systematic investigation in 1873, and
the concept was rigorized by Reynolds ten years later [216]. Similitude is now considered
a fundamental guiding principle in the modeling of materials [199].
The process often begins with a consideration of the fundamental differential equations.
In electromagnetics we may introduce a set of dimensionless field and source variables

E, D, B, H, J, ρ, (2.266)
by setting
E = Ek E , B = Bk B , D = Dk D ,
H = Hk H , J = Jk J , ρ = ρk ρ . (2.267)

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