Reciprocity (electromagnetism) Totally Explained

Everything about Reciprocity Electromagnetism totally explained

ť This page is about reciprocity theorems in classical electromagnetism. See also Reciprocity (mathematics) for unrelated reciprocity theorems, and Reciprocity for more general usages of the term.

In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities (sources) and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints. Reciprocity is closely related to the concept of Hermitian operators from linear algebra, applied to electromagnetism.
   Perhaps the most common and general such theorem is Lorentz reciprocity (and its various special cases such as Rayleigh-Carson reciprocity), named after work by Hendrik Lorentz in 1896 following analogous results regarding sound by Lord Rayleigh and Helmholtz (Potton, 2004). Loosely, it states that the relationship between an oscillating current and the resulting electric field is unchanged if one interchanges the points where the current is placed and where the field is measured. For the specific case of an electrical network, it's sometimes phrased as the statement that voltages and currents at different points in the network can be interchanged. More technically, it follows that the mutual impedance of a first circuit due to a second is the same as the mutual impedance of the second circuit due to the first.
   There is also an analogous theorem in electrostatics, known as Green's reciprocity, relating the interchange of electric potential and electric charge density.
   Forms of the reciprocity theorems are used in many electromagnetic applications, such as analyzing electrical networks and antenna systems. For example, reciprocity implies that antennas work equally well as transmitters or receivers, and specifically that an antenna's radiation and receiving patterns are identical. Reciprocity is also a basic lemma that's used to prove other theorems about electromagnetic systems, such as the symmetry of the mutual-impedance matrix, symmetries of the scattering matrix or Green's functions for use in boundary-element and transfer-matrix computational methods, as well as orthogonality properties of harmonic modes in waveguide systems (as an alternative to proving those properties directly from the symmetries of the eigen-operators).

Lorentz reciprocity

Specifically, suppose that one has a current density

mathbf/varepsilon)

, which is true when ε is a constant scalar multiple of μ (the two operators generally differ by an interchange of ε and μ). As above, one can also construct a more general formulation for integrals over a finite volume.

Green's reciprocity

Whereas the above reciprocity theorems were for oscillating fields, Green's reciprocity is an analogous theorem for electrostatics with a fixed distribution of electric charge (Panofsky and Phillips, 1962).
In particular, let

phi_1

denote the electric potential resulting from a total charge density

ho_1

. The electric potential satisfies Poisson's equation,

-
abla^2 phi_1 = 
ho_1 / varepsilon_0

, where

varepsilon_0

is the vacuum permittivity. Similarly, let

phi_2

denote the electric potential resulting from a total charge density

ho_2

, satisfying

-
abla^2 phi_2 = 
ho_2 / varepsilon_0

. In both cases, we assume that the charge distributions are localized, so that the potentials can be chosen to go to zero at infinity. Then, Green's reciprocity theorem states that, for integrals over all space:
ť

int 
ho_1 phi_2 dV = int 
ho_2 phi_1 dV.

This theorem is easily proven from Green's second identity. Equivalently, it's the statement that

int phi_2 (
abla^2 phi_1 ) dV = int phi_1 (
abla^2 phi_2 ) dV

, for example that

abla^2

is a Hermitian operator (as follows by integrating by parts twice).

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