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 symmetric 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 light by 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. Reciprocity is useful in optics, which (apart from quantum effects) can be expressed in terms of classical electromagnetism, but also in terms of radiometry. 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 impedance matrix and scattering matrix, symmetries of 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 J 1 {\displaystyle \mathbf {J} _{1}} that produces an electric field E 1 {\displaystyle \mathbf {E} _{1}} and a magnetic field H 1 , {\displaystyle \mathbf {H} _{1}\,,} where all three are periodic functions of time with angular frequency ω, and in particular they have time-dependence exp ( − i ω t ) . {\displaystyle \exp(-i\omega t)\,.} Suppose that we similarly have a second current J 2 {\displaystyle \mathbf {J} _{2}} at the same frequency ω which (by itself) produces fields E 2 {\displaystyle \mathbf {E} _{2}} and H 2 . {\displaystyle \mathbf {H} _{2}\,.} 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:
∫ V [ J 1 ⋅ E 2 − E 1 ⋅ J 2 ] d V = ∮ S [ E 1 × H 2 − E 2 × H 1 ] ⋅ d S . {\displaystyle \int _{V}\left[\mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}\right]\mathrm {d} V=\oint _{S}\left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\cdot \mathbf {\mathrm {d} S} \ .}
Equivalently, in differential form (by the divergence theorem):
J 1 ⋅ E 2 − E 1 ⋅ J 2 = ∇ ⋅ [ E 1 × H 2 − E 2 × H 1 ] . {\displaystyle \mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}=\nabla \cdot \left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\ .}
This general form is commonly simplified for a number of special cases. In particular, one usually assumes that J 1 {\displaystyle \ \mathbf {J} _{1}\ } and J 2 {\displaystyle \mathbf {J} _{2}} are localized (i.e. have compact support), and that there are no incoming waves from infinitely far away. In this case, if one integrates throughout space then the surface-integral terms cancel (see below) and one obtains:
∫ J 1 ⋅ E 2 d V = ∫ E 1 ⋅ J 2 d V . {\displaystyle \int \mathbf {J} _{1}\cdot \mathbf {E} _{2}\,\mathrm {d} V=\int \mathbf {E} _{1}\cdot \mathbf {J} _{2}\,\mathrm {d} V\ .}
This result (along with the following simplifications) is sometimes called the Rayleigh-Carson reciprocity theorem, after Lord Rayleigh's work on sound waves and an extension by Carson (1924; 1930) to applications for radio frequency antennas. Often, one further 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 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:
∮ S ( E 1 × H 2 ) ⋅ d S = ∮ S ( E 2 × H 1 ) ⋅ d S . {\displaystyle \ \oint _{S}(\mathbf {E} _{1}\times \mathbf {H} _{2})\cdot \mathbf {\mathrm {d} S} =\oint _{S}(\mathbf {E} _{2}\times \mathbf {H} _{1})\cdot \mathbf {\mathrm {d} S} \ .}
In practical problems, there are other more generalized forms of Lorentz and other reciprocity relations, in which, in addition to electric current density J {\displaystyle \ \mathbf {J} \ } , magnetic current density M {\displaystyle \ \mathbf {M} \ } is also used. These types of reciprocity relations are usually discussed in electrical engineering literature.
Reciprocity for electrical networks
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 J {\displaystyle \ \mathbf {J} \ } above only consisted of external "source" terms introduced into Maxwell's equations. We now denote this by J ( e ) {\displaystyle \ \mathbf {J} ^{(e)}\ } 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 E ( e ) {\displaystyle \ \mathbf {E} ^{(e)}\ } where, by definition of σ:
J ( e ) = σ E ( e ) . {\displaystyle \ \mathbf {J} ^{(e)}=\sigma \mathbf {E} ^{(e)}\ .}
Moreover, the electric field E {\displaystyle \mathbf {E} } above only consisted of the response to this current, and did not include the "external" field E ( e ) . {\displaystyle \ \mathbf {E} ^{(e)}\ .} Therefore, we now denote the field from before as E ( r ) , {\displaystyle \ \mathbf {E} ^{(r)}\ ,} where the total field is given by E = E ( e ) + E ( r ) . {\displaystyle \ \mathbf {E} =\mathbf {E} ^{(e)}+\mathbf {E} ^{(r)}\ .} Now, the equation on the left-hand side of the Lorentz reciprocity theorem can be rewritten by moving the σ from the external current term J ( e ) {\displaystyle \mathbf {J} ^{(e)}} to the response field terms E ( r ) , {\displaystyle \ \mathbf {E} ^{(r)}\ ,} and also adding and subtracting a σ E 1 ( e ) E 2 ( e ) {\displaystyle \ \sigma \mathbf {E} _{1}^{(e)}\mathbf {E} _{2}^{(e)}\ } term, to obtain the external field multiplied by the total current J = σ E : {\displaystyle \ \mathbf {J} =\sigma \mathbf {E} \ :}
∫ V [ J 1 ( e ) ⋅ E 2 ( r ) − E 1 ( r ) ⋅ J 2 ( e ) ] d V =
∫ V [ σ E 1 ( e ) ⋅ ( E 2 ( r ) + E 2 ( e ) ) − ( E 1 ( r ) + E 1 ( e ) ) ⋅ σ E 2 ( e ) ] d V =
∫ V [ E 1 ( e ) ⋅ J 2 − J 1 ⋅ E 2 ( e ) ] d V . {\displaystyle {\begin{aligned}&\int _{V}\left[\mathbf {J} _{1}^{(e)}\cdot \mathbf {E} _{2}^{(r)}-\mathbf {E} _{1}^{(r)}\cdot \mathbf {J} _{2}^{(e)}\right]\operatorname {d} V\\={}&\int _{V}\left[\sigma \mathbf {E} _{1}^{(e)}\cdot \left(\mathbf {E} _{2}^{(r)}+\mathbf {E} _{2}^{(e)}\right)-\left(\mathbf {E} _{1}^{(r)}+\mathbf {E} _{1}^{(e)}\right)\cdot \sigma \mathbf {E} _{2}^{(e)}\right]\operatorname {d} V\\={}&\int _{V}\left[\mathbf {E} _{1}^{(e)}\cdot \mathbf {J} _{2}-\mathbf {J} _{1}\cdot \mathbf {E} _{2}^{(e)}\right]\operatorname {d} V\ .\end{aligned}}}
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 Rayleigh-Carson reciprocity theorem becomes a simple summation:
∑ n V 1 ( n ) I 2 ( n ) = ∑ n V 2 ( n ) I 1 ( n ) {\displaystyle \ \sum _{n}{\mathcal {V}}_{1}^{(n)}I_{2}^{(n)}=\sum _{n}{\mathcal {V}}_{2}^{(n)}I_{1}^{(n)}}
where V {\displaystyle \ {\mathcal {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 {\displaystyle \ {\mathcal {V}}_{1}\ } and V 2 . {\displaystyle \ {\mathcal {V}}_{2}\ .} Most commonly, this is simplified further to the case where each system has a single voltage source V s , {\displaystyle \ {\mathcal {V}}_{\text{s}}\ ,} at V 1 ( 1 ) = V s {\displaystyle \ {\mathcal {V}}_{1}^{(1)}={\mathcal {V}}_{\text{s}}\ } and V 2 ( 2 ) = V s . {\displaystyle \ {\mathcal {V}}_{2}^{(2)}={\mathcal {V}}_{\text{s}}\ .} Then the theorem becomes simply
I 1 ( 2 ) = I 2 ( 1 ) {\displaystyle I_{1}^{(2)}=I_{2}^{(1)}}
or in words:
The current at position (1) from a voltage at (2) is identical to the current at (2) from the same voltage at (1).
Conditions and proof of Lorentz reciprocity The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator O ^ {\displaystyle \operatorname {\hat {O}} } relating J {\displaystyle \mathbf {J} } and E {\displaystyle \mathbf {E} } at a fixed frequency ω {\displaystyle \omega } (in linear media):
J = O ^ E {\displaystyle \mathbf {J} =\operatorname {\hat {O}} \mathbf {E} }
where
O ^ E ≡ 1 i ω [ 1 μ ( ∇ × ∇ × ) − ω 2 ε ] E {\displaystyle \operatorname {\hat {O}} \mathbf {E} \equiv {\frac {1}{i\omega }}\left[{\frac {1}{\mu }}\left(\nabla \times \nabla \times \right)-\;\omega ^{2}\varepsilon \right]\mathbf {E} }
is usually a symmetric operator under the "inner product" ( F , G ) = ∫ F ⋅ G d V {\textstyle (\mathbf {F} ,\mathbf {G} )=\int \mathbf {F} \cdot \mathbf {G} \,\mathrm {d} V} for vector fields F {\displaystyle \mathbf {F} } and G . {\displaystyle \mathbf {G} \ .}
(Technically, this unconjugated form is not a true inner product because it is not real-valued for complex-valued fields, but that is not a problem here. In this sense, the operator is not truly Hermitian but is rather complex-symmetric.) This is true whenever the permittivity 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 with finite conductivity σ (which is included in ε via ε → ε + i σ / ω {\displaystyle \varepsilon \rightarrow \varepsilon +i\sigma /\omega \ } ) – 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 O ^ {\displaystyle \operatorname {\hat {O}} } under an inner product ( f , g ) {\displaystyle (f,g)\!} , we have ( f , O ^ g ) = ( O ^ f , g ) {\displaystyle (f,\operatorname {\hat {O}} g)=(\operatorname {\hat {O}} f,g)} by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator J = O ^ E : {\displaystyle \mathbf {J} =\operatorname {\hat {O}} \mathbf {E} \ :} that is, ( E 1 , O ^ E 2 ) = ( O ^ E 1 , E 2 ) . {\displaystyle (\mathbf {E} _{1},\operatorname {\hat {O}} \mathbf {E} _{2})=(\operatorname {\hat {O}} \mathbf {E} _{1},\mathbf {E} _{2})\ .} 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 more-general surface-integral theorem above. In particular, the key fact is that, for vector fields F {\displaystyle \mathbf {F} } and G , {\displaystyle \mathbf {G} \ ,} integration by parts (or the divergence theorem) over a volume V enclosed by a surface S gives the identity:
∫ V F ⋅ ( ∇ × G ) d V ≡ ∫ V ( ∇ × F ) ⋅ G d V − ∮ S ( F × G ) ⋅ d A . {\displaystyle \int _{V}\mathbf {F} \cdot (\nabla \times \mathbf {G} )\,\mathrm {d} V\equiv \int _{V}(\nabla \times \mathbf {F} )\cdot \mathbf {G} \,\mathrm {d} V-\oint _{S}(\mathbf {F} \times \mathbf {G} )\cdot \mathrm {d} \mathbf {A} \ .}
This identity is then applied twice to ( E 1 , O ^ E 2 ) {\displaystyle (\mathbf {E} _{1},\operatorname {\hat {O}} \mathbf {E} _{2})} to yield ( O ^ E 1 , E 2 ) {\displaystyle (\operatorname {\hat {O}} \mathbf {E} _{1},\mathbf {E} _{2})} plus the surface term, giving the Lorentz reciprocity relation.
Conditions and proof of Lorenz reciprocity using Maxwell's equations and vector operations We shall prove a general form of the electromagnetic reciprocity theorem due to Lorenz which states that fields E 1 , H 1 {\displaystyle \mathbf {E} _{1},\mathbf {H} _{1}} and E 2 , H 2 {\displaystyle \mathbf {E} _{2},\mathbf {H} _{2}} generated by two different sinusoidal current densities respectively J 1 {\displaystyle \mathbf {J} _{1}} and J 2 {\displaystyle \mathbf {J} _{2}} of the same frequency, satisfy the condition
∫ V [ J 1 ⋅ E 2 − E 1 ⋅ J 2 ] d V = ∮ S [ E 1 × H 2 − E 2 × H 1 ] ⋅ d S . {\displaystyle \int _{V}\left[\mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}\right]\mathrm {d} V=\oint _{S}\left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\cdot \mathbf {\mathrm {d} S} .} Let us take a region in which dielectric constant and permeability may be functions of position but not of time. Maxwell's equations, written in terms of the total fields, currents and charges of the region describe the electromagnetic behavior of the region. The two curl equations are:
∇ × E = − ∂ ∂ t B , ∇ × H = J + ∂ ∂ t D . {\displaystyle {\begin{array}{ccc}\nabla \times \mathbf {E} &=&-{\frac {\partial }{\partial t}}\mathbf {B} \ ,\\\nabla \times \mathbf {H} &=&\mathbf {J} +{\frac {\partial }{\partial t}}\mathbf {D} \ .\end{array}}}
Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case:
∇ × E = − j ω B , ∇ × H = J + j ω D . {\displaystyle {\begin{array}{ccc}\nabla \times \mathbf {E} &=&-j\omega \mathbf {B} \ ,\\\nabla \times \mathbf {H} &=&\mathbf {J} +j\omega \mathbf {D} \ .\end{array}}}
It must be recognized that the symbols in the equations of this article represent the complex multipliers of