Reciprocity (electromagnetism)

The Lorentz reciprocity theorem, sometimes called the Lorentz lemma or simply the reciprocity theorem, is a statement relating time-harmonic electric current densities (sources) and the resulting electric fields in Maxwell's equations for linear media (under certain constraints). Loosely, it states that the relationship between the current and the resulting field is unchanged if one interchanges the points where the current is placed and where the field is measured. It is closely related to the concept of Hermitian operators from linear algebra, applied to classical electromagnetism. It is named after work by Hendrik Lorentz in 1896, following analogous results regarding sound by Lord Rayleigh and Helmholtz.

For the specific case of an electrical network, it is 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.

Specifically, suppose that one has a current density $\mathbf {J} _{1}$ that produces an electric field $\mathbf {E} _{1}$, where both $\mathbf {J} _{1}$ and $\mathbf {E} _{1}$ are periodic functions of time with angular frequency ω, i.e. they have time-dependence $\exp(-i\omega t)$. Moreover, we assume that $\mathbf {J} _{1}$ is localized (i.e. it has compact support) so that $\mathbf {E} _{1}$ goes to zero at infinity. Suppose that we similarly have a second localized current $\mathbf {J} _{2}$ at the same frequency ω which (by itself) produces a second field $\mathbf {E} _{2}$. Then, the statement of the reciprocity theorem, under certain simple conditions on the materials of the medium described below, is that:

$\int \mathbf {J} _{1}\cdot \mathbf {E} _{2}dV=\int \mathbf {E} _{1}\cdot \mathbf {J} _{2}dV$

Often, one simplifies this relation by considering point-like dipole sources, in which case the integrals disappear and one simply has the product of the electric field with the corresponding dipole moments of the currents. Or, for wires of negligible thickness, one obtains the current in one wire multiplied by the voltage across another and vice versa. This simplified case, and also the above equation, are also called the Rayleigh-Carson reciprocity theorem, after Lord Rayleigh's work on sound waves an an extension by John R. Carson to applications for radio frequency antennas.

Another form of the Lorentz reciprocity theorem, derived from the same considerations, states that if one performs a surface integral over a surface S that does not contain either of the sources (or entirely contains both of the sources), then:

$\oint _{S}(\mathbf {E} _{1}\times \mathbf {H} _{2})\cdot \mathbf {dA} =\oint _{S}(\mathbf {E} _{2}\times \mathbf {H} _{1})\cdot \mathbf {dA}$

where $\mathbf {H}$ is the magnetic field produced by the corresponding source.

Conditions for reciprocity

The reciprocity theorem is simply a statement of the fact that the linear operator relating $\mathbf {J}$ and $\mathbf {E}$ at a fixed frequency (in linear media):

$\left(\nabla \times {\frac {1}{\mu }}\nabla \times -{\frac {\omega ^{2}}{c^{2}}}\varepsilon \right)\mathbf {E} =i{\frac {\omega }{c}}\mathbf {J}$

is generally a Hermitian operator under the inner product $(\mathbf {F} ,\mathbf {G} )=\int \mathbf {F} \cdot \mathbf {G} \,dV$ for vector fields $\mathbf {F}$ and $\mathbf {G}$. (Technically, this unconjugated form is not a true inner product because it is not positive-definite for complex-valued fields, but that is not a problem here.) This is true whenever the dielectric function ε and the magnetic permeability μ, at the given ω, are symmetric 3×3 matrices (symmetric rank-2 tensors) — this includes the common case where they are scalars (for isotropic media), of course. They need not be real—complex values correspond to materials with losses, such as conductors. This condition is almost always satisfied.

(For any Hermitian operator ${\hat {O}}$, we have $(f,{\hat {O}}g)=({\hat {O}}f,g)$ by definition, and the reciprocity theorem is merely the vectorial version of this statement for this particular operator $\mathbf {J} ={\hat {O}}\mathbf {E}$. The Hermitian property of the operator here can be derived by integration by parts. For a finite integration volume, the surface terms from this integration by parts yield the surface-integral theorem above.)

One case in which ε is not a symmetric matrix is for magneto-optic materials. For a magneto-optic material in which absorption losses can be neglected, ε is a 3×3 complex Hermitian matrix. In this case the operator $\nabla \times {\frac {1}{\mu }}\nabla \times -(\omega ^{2}/c^{2})\varepsilon$ is Hermitian under the conjugated inner product $(\mathbf {F} ,\mathbf {G} )=\int \mathbf {F} ^{*}\cdot \mathbf {G} \,dV$ for vector fields $\mathbf {F}$ and $\mathbf {G}$, and a version of the reciprocity theorem still holds:

$\int \mathbf {J} _{1}^{*}\cdot \mathbf {E} _{2}dV=-\int \mathbf {E} _{1}^{*}\cdot \mathbf {J} _{2}dV$

where the minus sign comes from the $i\omega$ on the right-hand side of the equation above, which makes the operator relating $\mathbf {J}$ and $\mathbf {E}$ anti-Hermitian.

For lossy magneto-optic materials or for nonlinear media, however, no reciprocity theorem generally holds.

Feld-Tai reciprocity

A closely related reciprocity theorem was articulated independently by Y. A. Feld and C. T. Tai in 1992 and is known as Feld-Tai reciprocity or the Feld-Tai lemma. It relates two time-harmonic localized current sources and the resulting magnetic fields:

$\int \mathbf {J} _{1}\cdot \mathbf {H} _{2}dV=\int \mathbf {H} _{1}\cdot \mathbf {J} _{2}dV$

However, the Feld-Tai lemma is only valid under much more restrictive conditions than Lorentz reciprocity. It generally requires linear media with an isotropic homogeneous impedance, i.e. a constant scalar μ/ε ratio, with the possible exception of regions of perfectly conducting material.

References

• L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley: Reading, MA, 1960).
• H. A. Lorentz, "The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light," Amsterdammer Akademie der Wetenschappen 4 p. 176 (1896).
• R. J. Potton, "Reciprocity in optics," Reports on Progress in Physics 67, 717-754 (2004). (A review article on the history of this topic.)
• Ya. N. Feld, "On the quadratic lemma in electrodynamics," Sov. Phys—Dokl. 37, 235-236 (1992).
• C.-T. Tai, "Complementary reciprocity theorems in electromagnetic theory," IEEE Trans. Antennas Prop. 40 (6), 675-681 (1992).