In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of timeharmonic electric current densities (sources) and the resulting electromagnetic fields in Maxwell's equations for timeinvariant 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 RayleighCarson 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 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.
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 is used to prove other theorems about electromagnetic systems, such as the symmetry of the mutualimpedance matrix, symmetries of the scattering matrix or Green's functions for use in boundaryelement and transfermatrix 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 eigenoperators).
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Specifically, suppose that one has a current density that produces an electric field and a magnetic field , where all three are periodic functions of time with angular frequency ω, and in particular they have timedependence exp( − iωt). Suppose that we similarly have a second current at the same frequency ω which (by itself) produces fields and . The Lorentz reciprocity theorem then states, under certain simple conditions on the materials of the medium described below, that for an arbitrary surface S enclosing a volume V:
Equivalently, in differential form (by the divergence theorem):
This general form is commonly simplified for a number of special cases. In particular, one usually assumes that and are localized (i.e. have compact support), and that there are no incoming waves from infinitely far away. In this case, if one integrates over all space then the surfaceintegral terms cancel (see below) and one obtains:
This result (along with the following simplifications) is sometimes called the RayleighCarson reciprocity theorem, after Lord Rayleigh's work on sound waves and an extension by John R. Carson (1924; 1930) to applications for radio frequency antennas. Often, one further simplifies this relation by considering pointlike 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 applied current in one wire multiplied by the resulting voltage across another and vice versa; see also below.
Another special case of the Lorentz reciprocity theorem applies when the volume V entirely contains both of the localized sources (or alternatively if V intersects neither of the sources). In this case:
Above, Lorentz reciprocity was phrased in terms of an externally applied current source and the resulting field. Often, especially for electrical networks, one instead prefers to think of an externally applied voltage and the resulting currents. The Lorentz reciprocity theorem describes this case as well, assuming ohmic materials (i.e. currents that respond linearly to the applied field) with a 3×3 conductivity matrix σ that is required to be symmetric, which is implied by the other conditions below. In order to properly describe this situation, one must carefully distinguish between the externally applied fields (from the driving voltages) and the total fields that result (King, 1963).
More specifically, the above only consisted of external "source" terms introduced into Maxwell's equations. We now denote this by to distinguish it from the total current produced by both the external source and by the resulting electric fields in the materials. If this external current is in a material with a conductivity σ, then it corresponds to an externally applied electric field where, by definition of σ:
Moreover, the electric field above only consisted of the response to this current, and did not include the "external" field . Therefore, we now denote the field from before as , where the total field is given by .
Now, the equation on the lefthand side of the Lorentz reciprocity theorem can be rewritten by moving the σ from the external current term to the response field terms , and also adding and subtracting a term, to obtain the external field multiplied by the total current :
For the limit of thin wires, this gives the product of the externally applied voltage (1) multiplied by the resulting total current (2) and vice versa. In particular, the RayleighCarson reciprocity theorem becomes a simple summation:
where V and I denote the (complex) amplitudes of the AC applied voltages and the resulting currents, respectively, in a set of circuit elements (indexed by n) for two possible sets of voltages V_{1} and V_{2}.
Most commonly, this is simplified further to the case where each system has a single voltage source V, at and . Then the theorem becomes simply : the current at position (1) from a voltage at (2) is identical to the current at (2) from the same voltage at (1).
The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator relating and at a fixed frequency (in linear media):
is usually a Hermitian operator under the inner product for vector fields and . (Technically, this unconjugated form is not a true inner product because it is not realvalued for complexvalued fields, but that is not a problem here. In this sense, the operator is not truly Hermitian but is rather complexsymmetric.) This is true whenever the permittivity ε and the magnetic permeability μ, at the given ω, are symmetric 3×3 matrices (symmetric rank2 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 with finite conductivity σ (which is included in ε via )—and because of this the reciprocity theorem does not require time reversal invariance. The condition of symmetric ε and μ matrices is almost always satisfied; see below for an exception.
For any Hermitian operator under an inner product , we have by definition, and the RayleighCarson reciprocity theorem is merely the vectorial version of this statement for this particular operator : that is, . 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 moregeneral surfaceintegral theorem above. In particular, the key fact is that, for vector fields and , integration by parts (or the divergence theorem) over a volume V enclosed by a surface S gives the identity:
This identity is then applied twice to to yield plus the surface term, giving the Lorentz reciprocity relation.
The cancellation of the surface terms on the righthand side of the Lorentz reciprocity theorem, for an integration over all space, is not entirely obvious but can be derived in a number of ways.
The simplest argument would be that the fields goes to zero at infinity for a localized source, but this argument fails in the case of lossless media: in the absence of absorption, radiated fields decay inversely with distance, but the surface area of the integral increases with the square of distance, so the two rates balance one another in the integral.
Instead, it is common (e.g. King, 1963) to assume that the medium is homogeneous and isotropic sufficiently far away. In this case, the radiated field asymptotically takes the form of planewaves propagating radially outward (in the direction) with and where Z is the impedance of the surrounding medium. Then it follows that , which by a simple vector identity equals . Similarly, and the two terms cancel one another.
The above argument shows explicitly why the surface terms can cancel, but lacks generality. Alternatively, one can treat the case of lossless surrounding media by taking the limit as the losses (the imaginary part of ε) go to zero. For any nonzero loss, the fields decay exponentially with distance and the surface integral vanishes, regardless of whether the medium is homogeneous. Since the lefthand side of the Lorentz reciprocity theorem vanishes for integration over all space with any nonzero losses, it must also vanish in the limit as the losses go to zero. (Note that we implicitly assumed the standard boundary condition of zero incoming waves from infinity, because otherwise even an infinitesimal loss would eliminate the incoming waves and the limit would not give the lossless solution.)
The inverse of the operator , i.e. in (which requires a specification of the boundary conditions at infinity in a lossless system), has the same symmetry as and is essentially a Green's function convolution. So, another perspective on Lorentz reciprocity is that it reflects the fact that convolution with the electromagnetic Green's function is a complexsymmetric (or antiHermitian, below) linear operation under the appropriate conditions on ε and μ. More specifically, the Green's function can be written as giving the nth component of at from a point dipole current in the mth direction at (essentially, G gives the matrix elements of ), and RayleighCarson reciprocity is equivalent to the statement that . Unlike , it is not generally possible to give an explicit formula for the Green's function (except in special cases such as homogeneous media), but it is routinely computed by numerical methods.
One case in which ε is not a symmetric matrix is for magnetooptic materials, in which case the usual statement of Lorentz reciprocity does not hold (see below for a generalization, however). If we allow magnetooptic materials, but restrict ourselves to the situation where material absorption is negligible, then ε and μ are in general 3×3 complex Hermitian matrices. In this case the operator is Hermitian under the conjugated inner product , and a variant of the reciprocity theorem still holds:
where the sign changes come from the 1 / iω in the equation above, which makes the operator antiHermitian (neglecting surface terms). For the special case of , this gives a restatement of conservation of energy or Poynting's theorem (since here we have assumed lossless materials, unlike above): the timeaverage rate of work done by the current (given by the real part of ) is equal to the timeaverage outward flux of power (the integral of the Poynting vector). By the same token, however, the surface terms do not in general vanish if one integrates over all space for this reciprocity variant, so a RayleighCarson form does not hold without additional assumptions.
The fact that magnetooptic materials break RayleighCarson reciprocity is the key to devices such as Faraday isolators and circulators. A current on one side of a Faraday isolator produces a field on the other side but not viceversa.
For a combination of lossy and magnetooptic materials, and in general when the ε and μ tensors are neither symmetric nor Hermitian matrices, one can still obtain a generalized version of Lorentz reciprocity by considering and to exist in different systems.
In particular, if solve Maxwell's equations at ω for a system with materials , and solve Maxwell's equations at ω for a system with materials , where T denotes the transpose, then the equation of Lorentz reciprocity holds.
For nonlinear media, no reciprocity theorem generally holds. Reciprocity also does not generally apply for timevarying ("active") media; for example, when ε is modulated in time by some external process. (In both of these cases, the frequency ω is not generally a conserved quantity.)
A closely related reciprocity theorem was articulated independently by Y. A. Feld and C. T. Tai in 1992 and is known as FeldTai reciprocity or the FeldTai lemma. It relates two timeharmonic localized current sources and the resulting magnetic fields:
However, the FeldTai lemma is only valid under much more restrictive conditions than Lorentz reciprocity. It generally requires timeinvariant 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.
More precisely, FeldTai reciprocity requires the Hermitian (or rather, complexsymmetric) symmetry of the electromagnetic operators as above, but also relies on the assumption that the operator relating and is a constant scalar multiple of the operator relating and , 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.
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 φ_{1} denote the electric potential resulting from a total charge density ρ_{1}. The electric potential satisfies Poisson's equation, , where is the vacuum permittivity. Similarly, let φ_{2} denote the electric potential resulting from a total charge density ρ_{2}, satisfying . 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:
This theorem is easily proven from Green's second identity. Equivalently, it is the statement that , i.e. that is a Hermitian operator (as follows by integrating by parts twice).
