Page 175 - Maxwell House

POYNTING's THEOREM 155

3.4.3 Lorentz’s Reciprocity Theorem

Let us consider a simple engineering problem. You just built and delivered to your customer a
transmitting antenna that met such antenna requirements as power handling, gain, radiation
pattern, bandwidth, and impedance. Later the client
changed his mind and decided to use the same
antenna as a receiving one. How could you convince
him that your antenna will satisfy (with some
reservations for ultra-wide band applications) the
same requirements in receiving mode? You have the
excellent tool: educate him/her about the meaning of
Lorentz’s Reciprocity Theorem. This theorem is
based on the conservation energy law in the same
way as Poynting’s theorem.

Let us assume that within a linear and isotropic
Figure 3.4.3 Source configuration medium there is two separate domain and as
1 2
shown in Figure 3.4.3 enclosed inside the larger
current of identical
domain . Each of these detached domains includes the electric 1,2
frequencies. According to Maxwell’s equations, each set of currents generates the electric
1,2
and magnetic fields inside the domain . Consequently, we can write these equations in

1,2
the same system (3.44) that we have used for Poynting’s theorem proof
∘ � x = − () 2 ∘ � x = − () 1 (3.100)
2
2
0
0
1
1
∘ x = () + ∘ x = () +
2
1
1
1
0
2
2
0
2
1
Dotting all Maxwell’s equations and then combining the products into one equity we have
)
( ∘ x − ∘ x ) − ( ∘ x − ∘ x ) = ( ∘ 2 − ∘ 1
1
1
2
1
2
1
1
2
2
2
(3.101)
The vector identity in Appendix, ∘ x − ∘ x = ∘ ( x ) lets cut (3.101) to
3
∘ ( x − x ) = ( ∘ − ∘ ) [W/m ] (3.102)
1
2
1
1
1
2
2
2
This equity defines Lorentz’s reciprocity theorem in differential form and means some point-
to-point power balance in W/m for the fields created by two independent sources. Integrating
3
(3.102) over the domain and applying the divergence theorem we have Lorentz’s reciprocity
theorem in integral form
∯ ( x − x ) ∘ = ∫ ∘ − ∫ ∘ (3.103)
2 1 1 2 2 1 2 1 2 1
Here A is the closed surface bounding the domain . Lorentz’s reciprocity theorem (3.103) is
the starting point for multiple engineering applications. Let us study some of them. First of all,