# Reciprocity (electromagnetism)

Last updated
This page is about reciprocity theorems in classical electromagnetism. See also Reciprocity theorem (disambiguation) for unrelated reciprocity theorems, and Reciprocity (disambiguation) 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.

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in the vacuum, the "speed of light". Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.

In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their modes of vibration. Thus, the term "harmonic" is applied when one is considering functions with sinusoidal variations, or solutions of Laplace's equation and related concepts.

In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the charges at this point. In SI base units, the electric current density is measured in amperes per square metre.

## Contents

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.

Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the transformation equations underpinning Albert Einstein's theory of special relativity.

In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.

Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to visible light, which is the visible spectrum that is visible to the human eye and is responsible for the sense of sight. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), or 4.00 × 10−7 to 7.00 × 10−7 m, between the infrared and the ultraviolet. This wavelength means a frequency range of roughly 430–750 terahertz (THz).

Reciprocity is useful in optics, which (apart from quantum effects) can be expressed in terms of classical electromagnetism, but also in terms of radiometry.

Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

Radiometry is a set of techniques for measuring electromagnetic radiation, including visible light. Radiometric techniques in optics characterize the distribution of the radiation's power in space, as opposed to photometric techniques, which characterize the light's interaction with the human eye. Radiometry is distinct from quantum techniques such as photon counting.

There is also an analogous theorem in electrostatics, known as Green's reciprocity, relating the interchange of electric potential and electric charge density.

Electrostatics is a branch of physics that studies electric charges at rest.

An electric potential is the amount of work needed to move a unit of charge from a reference point to a specific point inside the field without producing an acceleration. Typically, the reference point is the Earth or a point at infinity, although any point can be used.

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).

In radio engineering, an antenna is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an electric current to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves. In reception, an antenna intercepts some of the power of a radio wave in order to produce an electric current at its terminals, that is applied to a receiver to be amplified. Antennas are essential components of all radio equipment.

In the field of antenna design the term radiation pattern refers to the directional (angular) dependence of the strength of the radio waves from the antenna or other source.

Impedance parameters or Z-parameters are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal (linearized) response of non-linear networks. They are members of a family of similar parameters used in electronic engineering, other examples being: S-parameters, Y-parameters, H-parameters, T-parameters or ABCD-parameters.

## Lorentz reciprocity

Specifically, suppose that one has a current density ${\displaystyle \mathbf {J} _{1}}$ that produces an electric field ${\displaystyle \mathbf {E} _{1}}$ and a magnetic field ${\displaystyle \mathbf {H} _{1}}$, where all three are periodic functions of time with angular frequency ω, and in particular they have time-dependence ${\displaystyle \exp(-i\omega t)}$. Suppose that we similarly have a second current ${\displaystyle \mathbf {J} _{2}}$ at the same frequency ω which (by itself) produces fields ${\displaystyle \mathbf {E} _{2}}$ and ${\displaystyle \mathbf {H} _{2}}$. 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:

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

In physics, angular frequencyω is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

${\displaystyle \int _{V}\left[\mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}\right]dV=\oint _{S}\left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\cdot \mathbf {dS} .}$

Equivalently, in differential form (by the divergence theorem):

${\displaystyle \mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}=\nabla \cdot \left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right].}$

This general form is commonly simplified for a number of special cases. In particular, one usually assumes that ${\displaystyle \mathbf {J} _{1}}$ and ${\displaystyle \mathbf {J} _{2}}$ 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:

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

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 John R. 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:

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

## 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 ${\displaystyle \mathbf {J} }$ above only consisted of external "source" terms introduced into Maxwell's equations. We now denote this by ${\displaystyle \mathbf {J} ^{(e)}}$ 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 ${\displaystyle \mathbf {E} ^{(e)}}$ where, by definition of σ:

${\displaystyle \mathbf {J} ^{(e)}=\sigma \mathbf {E} ^{(e)}.}$

Moreover, the electric field ${\displaystyle \mathbf {E} }$ above only consisted of the response to this current, and did not include the "external" field ${\displaystyle \mathbf {E} ^{(e)}}$. Therefore, we now denote the field from before as ${\displaystyle \mathbf {E} ^{(r)}}$, where the total field is given by ${\displaystyle \mathbf {E} =\mathbf {E} ^{(e)}+\mathbf {E} ^{(r)}}$.

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 ${\displaystyle \mathbf {J} ^{(e)}}$ to the response field terms ${\displaystyle \mathbf {E} ^{(r)}}$, and also adding and subtracting a ${\displaystyle \sigma \mathbf {E} _{1}^{(e)}\mathbf {E} _{2}^{(e)}}$ term, to obtain the external field multiplied by the total current ${\displaystyle \mathbf {J} =\sigma \mathbf {E} }$:

{\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]dV\\={}&\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]dV\\={}&\int _{V}\left[\mathbf {E} _{1}^{(e)}\cdot \mathbf {J} _{2}-\mathbf {J} _{1}\cdot \mathbf {E} _{2}^{(e)}\right]dV.\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:

${\displaystyle \sum _{n}V_{1}^{(n)}I_{2}^{(n)}=\sum _{n}V_{2}^{(n)}I_{1}^{(n)}\!}$

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 ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$.

Most commonly, this is simplified further to the case where each system has a single voltage source V, at ${\displaystyle V_{1}^{(1)}=V}$ and ${\displaystyle V_{2}^{(2)}=V}$. Then the theorem becomes simply

${\displaystyle I_{1}^{(2)}=I_{2}^{(1)}}$

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 ${\displaystyle {\hat {O}}}$ relating ${\displaystyle \mathbf {J} }$ and ${\displaystyle \mathbf {E} }$ at a fixed frequency ${\displaystyle \omega }$ (in linear media):

${\displaystyle \mathbf {J} ={\frac {1}{i\omega }}\left[{\frac {1}{\mu }}\left(\nabla \times \nabla \times \right)-\;\omega ^{2}\varepsilon \right]\mathbf {E} \equiv {\hat {O}}\mathbf {E} }$

is usually a symmetric operator under the "inner product" ${\displaystyle (\mathbf {F} ,\mathbf {G} )=\int \mathbf {F} \cdot \mathbf {G} \,dV}$ for vector fields ${\displaystyle \mathbf {F} }$ and ${\displaystyle \mathbf {G} }$. (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 realcomplex values correspond to materials with losses, such as conductors with finite conductivity σ (which is included in ε via ${\displaystyle \varepsilon \rightarrow \varepsilon +i\sigma /\omega }$)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 ${\displaystyle {\hat {O}}}$ under an inner product ${\displaystyle (f,g)\!}$, we have ${\displaystyle (f,{\hat {O}}g)=({\hat {O}}f,g)}$ by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator ${\displaystyle \mathbf {J} ={\hat {O}}\mathbf {E} }$: that is, ${\displaystyle (\mathbf {E} _{1},{\hat {O}}\mathbf {E} _{2})=({\hat {O}}\mathbf {E} _{1},\mathbf {E} _{2})}$. 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 ${\displaystyle \mathbf {F} }$ and ${\displaystyle \mathbf {G} }$, integration by parts (or the divergence theorem) over a volume V enclosed by a surface S gives the identity:

${\displaystyle \int _{V}\mathbf {F} \cdot (\nabla \times \mathbf {G} )\,dV\equiv \int _{V}(\nabla \times \mathbf {F} )\cdot \mathbf {G} \,dV-\oint _{S}(\mathbf {F} \times \mathbf {G} )\cdot \mathbf {dA} .}$

This identity is then applied twice to ${\displaystyle (\mathbf {E} _{1},{\hat {O}}\mathbf {E} _{2})}$ to yield ${\displaystyle ({\hat {O}}\mathbf {E} _{1},\mathbf {E} _{2})}$ plus the surface term, giving the Lorentz reciprocity relation.

Conditions and proof of Lorenz reciprocity using Maxwell's equations and vector operations [1]

We shall prove a general form of the electromagnetic reciprocity theorem due to Lorenz which states that fields ${\displaystyle \mathbf {E} _{1},\mathbf {H} _{1}}$ and ${\displaystyle \mathbf {E} _{2},\mathbf {H} _{2}}$ generated by two different sinusoidal current densities respectively ${\displaystyle \mathbf {J} _{1}}$ and ${\displaystyle \mathbf {J} _{2}}$ of the same frequency, satisfy the condition ${\displaystyle \int _{V}\left[\mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}\right]dV=\oint _{S}\left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\cdot \mathbf {dS} .}$

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:

${\displaystyle {\begin{array}{ccc}\nabla \times \mathbf {E} &=&-{\frac {\partial \mathbf {B} }{\partial t}},\\\nabla \times \mathbf {H} &=&\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}.\end{array}}}$

Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case:

${\displaystyle {\begin{array}{ccc}\nabla \times \mathbf {E} &=&-j\omega \mathbf {B} ,\\\nabla \times \mathbf {H} &=&\mathbf {J} +j\omega \mathbf {D} .\end{array}}}$

It must be recognized that the symbols in the equations of this article represent the complex multipliers of ${\displaystyle e^{j\omega t}}$, giving the in-phase and out-of-phase parts with respect to the chosen reference. The complex vector multipliers of ${\displaystyle e^{j\omega t}}$ may be called vector phasors by analogy to the complex scalar quantities which are commonly referred to as phasors.

An equivalence of vector operations shows that

${\displaystyle \mathbf {H} \cdot (\nabla \times \mathbf {E} )-\mathbf {E} \cdot (\nabla \times \mathbf {H} )=\nabla \cdot (\mathbf {E} \times \mathbf {H} )}$ for every vectors ${\displaystyle \mathbf {E} }$ and ${\displaystyle \mathbf {H} }$.

If we apply this equivalence to ${\displaystyle \mathbf {E} _{1}}$ and ${\displaystyle \mathbf {H} _{2}}$ we get:

${\displaystyle \mathbf {H} _{2}\cdot (\nabla \times \mathbf {E} _{1})-\mathbf {E} _{1}\cdot (\nabla \times \mathbf {H} _{2})=\nabla \cdot (\mathbf {E} _{1}\times \mathbf {H} _{2})}$.

If products in the Time-Periodic equations are taken as indicated by this last equivalence, and added,

${\displaystyle -\mathbf {H} _{2}\cdot j\omega \mathbf {B} _{1}-\mathbf {E} _{1}\cdot j\omega \mathbf {D} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}=\nabla \cdot (\mathbf {E} _{1}\times \mathbf {H} _{2})}$.

This now may be integrated over the volume of concern,

${\displaystyle \int _{V}(\mathbf {H} _{2}\cdot j\omega \mathbf {B} _{1}+\mathbf {E} _{1}\cdot j\omega \mathbf {D} _{2}+\mathbf {E} _{1}\mathbf {J} _{2})dV=-\int _{V}\nabla \cdot (\mathbf {E} _{1}\times \mathbf {H} _{2})dV}$.

From the divergence theorem the volume integral of ${\displaystyle div(\mathbf {E} _{1}\times \mathbf {H} _{2})}$ equals the surface integral of ${\displaystyle \mathbf {E} _{1}\times \mathbf {H} _{2}}$ over the boundary.

${\displaystyle \int _{V}(\mathbf {H} _{2}\cdot j\omega \mathbf {B} _{1}+\mathbf {E} _{1}\cdot j\omega \mathbf {D} _{2}+\mathbf {E} _{1}\cdot \mathbf {J} _{2})dV=-\oint _{S}(\mathbf {E} _{1}\times \mathbf {H} _{2})\cdot {\widehat {dS}}}$.

This form is valid for general media, but in the common case of linear, isotropic, time-invariant materials, ${\displaystyle \epsilon }$ is a scalar independent of time. Then generally as physical magnitudes ${\displaystyle \mathbf {D} =\epsilon \mathbf {E} }$ and ${\displaystyle \mathbf {B} =\mu \mathbf {H} }$.

Last equation then becomes

${\displaystyle \int _{V}(\mathbf {H} _{2}\cdot j\omega \mu \mathbf {H} _{1}+\mathbf {E} _{1}\cdot j\omega \epsilon \mathbf {E} _{2}+\mathbf {E} _{1}\cdot \mathbf {J} _{2})dV=-\oint _{S}(\mathbf {E} _{1}\times \mathbf {H} _{2})\cdot {\widehat {dS}}}$.

In an exactly analogous way we get for vectors ${\displaystyle \mathbf {E} _{2}}$ and ${\displaystyle \mathbf {H} _{1}}$ the following expression:

${\displaystyle \int _{V}(\mathbf {H} _{1}\cdot j\omega \mu \mathbf {H} _{2}+\mathbf {E} _{2}\cdot j\omega \epsilon \mathbf {E} _{1}+\mathbf {E} _{2}\cdot \mathbf {J} _{1})dV=-\oint _{S}(\mathbf {E} _{2}\times \mathbf {H} _{1})\cdot {\widehat {dS}}}$.

Subtracting the two last equations by members we get

${\displaystyle \int _{V}\left[\mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}\right]dV=\oint _{S}\left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\cdot \mathbf {dS} .}$

and equivalently in differential form

${\displaystyle \mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}=\nabla \cdot \left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]}$ q.e.d.

### Surface-term cancellation

The cancellation of the surface terms on the right-hand 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 general argument comes from a straightforward application of the divergence theorem: For localized sources, one can choose the bounding surface ${\displaystyle S}$ such that it contains all sources. This bounding surface is also a bounding surface (reversing the unit-normal vector) for the complementary region of space going out to infinity, ${\displaystyle V_{c}}$, containing no sources. The reciprocity relation thus still holds: ${\displaystyle \int _{V_{c}}\left[\mathbf {J} _{1}\cdot \mathbf {E} _{2}-\mathbf {E} _{1}\cdot \mathbf {J} _{2}\right]dV=-\oint _{S}\left[\mathbf {E} _{1}\times \mathbf {H} _{2}-\mathbf {E} _{2}\times \mathbf {H} _{1}\right]\cdot \mathbf {dS} ,}$ with the replacement ${\displaystyle V\rightarrow V_{c}}$ and a negative sign for the unit normal. The left-hand side of the expression is zero, because there are no sources in ${\displaystyle V_{c}}$, and thus the right-hand side is zero as well. [2]

Another simple 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 ${\displaystyle {\hat {\mathbf {r} }}}$ direction) with ${\displaystyle {\hat {\mathbf {r} }}\cdot \mathbf {E} =0}$ and ${\displaystyle \mathbf {H} ={\hat {\mathbf {r} }}\times \mathbf {E} /Z}$ where Z is the impedance ${\displaystyle {\sqrt {\mu /\epsilon }}}$ of the surrounding medium. Then it follows that ${\displaystyle \mathbf {E} _{1}\times \mathbf {H} _{2}=\mathbf {E} _{1}\times {\hat {\mathbf {r} }}\times \mathbf {E} _{2}/Z}$, which by a simple vector identity equals ${\displaystyle {\hat {\mathbf {r} }}(\mathbf {E} _{1}\cdot \mathbf {E} _{2})/Z}$. Similarly, ${\displaystyle \mathbf {E} _{2}\times \mathbf {H} _{1}={\hat {\mathbf {r} }}(\mathbf {E} _{2}\cdot \mathbf {E} _{1})/Z}$ 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 left-hand side of the Lorentz reciprocity theorem vanishes for integration over all space with any non-zero 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.)

### Reciprocity and the Green's function

The inverse of the operator ${\displaystyle {\hat {O}}}$, i.e. in ${\displaystyle \mathbf {E} ={\hat {O}}^{-1}\mathbf {J} }$ (which requires a specification of the boundary conditions at infinity in a lossless system), has the same symmetry as ${\displaystyle {\hat {O}}}$ 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 complex-symmetric (or anti-Hermitian, below) linear operation under the appropriate conditions on ε and μ. More specifically, the Green's function can be written as ${\displaystyle G_{nm}(\mathbf {x} ',\mathbf {x} )}$ giving the n-th component of ${\displaystyle \mathbf {E} }$ at ${\displaystyle \mathbf {x} '}$ from a point dipole current in the m-th direction at ${\displaystyle \mathbf {x} }$ (essentially, ${\displaystyle G}$ gives the matrix elements of ${\displaystyle {\hat {O}}^{-1}}$), and Rayleigh-Carson reciprocity is equivalent to the statement that ${\displaystyle G_{nm}(\mathbf {x} ',\mathbf {x} )=G_{mn}(\mathbf {x} ,\mathbf {x} ')}$. Unlike ${\displaystyle {\hat {O}}}$, 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.

### Lossless magneto-optic materials

One case in which ε is not a symmetric matrix is for magneto-optic materials, in which case the usual statement of Lorentz reciprocity does not hold (see below for a generalization, however). If we allow magneto-optic 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 ${\displaystyle {\frac {1}{\mu }}\left(\nabla \times \nabla \times \right)-{\frac {\omega ^{2}}{c^{2}}}\varepsilon }$ is Hermitian under the conjugated inner product ${\displaystyle (\mathbf {F} ,\mathbf {G} )=\int \mathbf {F} ^{*}\cdot \mathbf {G} \,dV}$, and a variant of the reciprocity theorem[ citation needed ] still holds:

${\displaystyle -\int _{V}\left[\mathbf {J} _{1}^{*}\cdot \mathbf {E} _{2}+\mathbf {E} _{1}^{*}\cdot \mathbf {J} _{2}\right]dV=\oint _{S}\left[\mathbf {E} _{1}^{*}\times \mathbf {H} _{2}+\mathbf {E} _{2}\times \mathbf {H} _{1}^{*}\right]\cdot \mathbf {dA} }$

where the sign changes come from the ${\displaystyle 1/i\omega }$ in the equation above, which makes the operator ${\displaystyle {\hat {O}}}$ anti-Hermitian (neglecting surface terms). For the special case of ${\displaystyle \mathbf {J} _{1}=\mathbf {J} _{2}}$, this gives a re-statement of conservation of energy or Poynting's theorem (since here we have assumed lossless materials, unlike above): the time-average rate of work done by the current (given by the real part of ${\displaystyle -\mathbf {J} ^{*}\cdot \mathbf {E} }$) is equal to the time-average 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 Rayleigh-Carson form does not hold without additional assumptions.

The fact that magneto-optic materials break Rayleigh-Carson 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 vice versa.

### Generalization to non-symmetric materials

For a combination of lossy and magneto-optic 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 ${\displaystyle (\mathbf {J} _{1},\mathbf {E} _{1})}$ and ${\displaystyle (\mathbf {J} _{2},\mathbf {E} _{2})}$ to exist in different systems.

In particular, if ${\displaystyle (\mathbf {J} _{1},\mathbf {E} _{1})}$ satisfy Maxwell's equations at ω for a system with materials ${\displaystyle (\varepsilon _{1},\mu _{1})}$, and ${\displaystyle (\mathbf {J} _{2},\mathbf {E} _{2})}$ satisfy Maxwell's equations at ω for a system with materials ${\displaystyle \left(\varepsilon _{1}^{T},\mu _{1}^{T}\right)}$, where T denotes the transpose, then the equation of Lorentz reciprocity holds. This can be further generalized to bi-anisotropic materials by transposing the full 6×6 susceptibility tensor. [3]

### Exceptions to reciprocity

For nonlinear media, no reciprocity theorem generally holds. Reciprocity also does not generally apply for time-varying ("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.)

## 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:

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

However, the Feld-Tai lemma is only valid under much more restrictive conditions than Lorentz reciprocity. It generally requires time-invariant 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, Feld-Tai reciprocity requires the Hermitian (or rather, complex-symmetric) symmetry of the electromagnetic operators as above, but also relies on the assumption that the operator relating ${\displaystyle \mathbf {E} }$ and ${\displaystyle i\omega \mathbf {J} }$ is a constant scalar multiple of the operator relating ${\displaystyle \mathbf {H} }$ and ${\displaystyle \nabla \times (\mathbf {J} /\varepsilon )}$, 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.

## Optical reciprocity in radiometric terms

Apart from quantal effects, classical theory covers near-, middle-, and far-field electric and magnetic phenomena with arbitrary time courses. Optics refers to far-field nearly-sinusoidal oscillatory electromagnetic effects. Instead of paired electric and magnetic variables, optics, including optical reciprocity, can be expressed in polarization-paired radiometric variables, such as spectral radiance, traditionally called specific intensity.

In 1856, Hermann von Helmholtz wrote:

"A ray of light proceeding from point A arrives at point B after suffering any number of refractions, reflections, &c. At point A let any two perpendicular planes a1, a2 be taken in the direction of the ray; and let the vibrations of the ray be divided into two parts, one in each of these planes. Take like planes b1, b2 in the ray at point B; then the following proposition may be demonstrated. If when the quantity of light J polarized in the plane a1 proceeds from A in the direction of the given ray, that part K thereof of light polarized in b1 arrives at B, then, conversely, if the quantity of light J polarized in b1 proceeds from B, the same quantity of light K polarized in a1 will arrive at A." [4]

This is sometimes called the Helmholtz reciprocity (or reversion) principle. [5] [6] [7] [8] [9] [10] When the wave propagates through a material acted upon by an applied magnetic field, reciprocity can be broken so this principle will not apply. [4] When there are moving objects in the path of the ray, the principle may be entirely inapplicable. Historically, in 1849, Sir George Stokes stated his optical reversion principle without attending to polarization. [11] [12] [13]

Like the principles of thermodynamics, this principle is reliable enough to use as a check on the correct performance of experiments, in contrast with the usual situation in which the experiments are tests of a proposed law. [14] [15]

The most extremely simple statement of the principle is 'if I can see you, then you can see me'.

The principle was used by Gustav Kirchhoff in his derivation of his law of thermal radiation and by Max Planck in his analysis of his law of thermal radiation.

For ray-tracing global illumination algorithms, incoming and outgoing light can be considered as reversals of each other, without affecting the bidirectional reflectance distribution function (BRDF) outcome. [15]

## 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 ${\displaystyle \phi _{1}}$ denote the electric potential resulting from a total charge density ${\displaystyle \rho _{1}}$. The electric potential satisfies Poisson's equation, ${\displaystyle -\nabla ^{2}\phi _{1}=\rho _{1}/\varepsilon _{0}}$, where ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity. Similarly, let ${\displaystyle \phi _{2}}$ denote the electric potential resulting from a total charge density ${\displaystyle \rho _{2}}$, satisfying ${\displaystyle -\nabla ^{2}\phi _{2}=\rho _{2}/\varepsilon _{0}}$. 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:

${\displaystyle \int \rho _{1}\phi _{2}dV=\int \rho _{2}\phi _{1}dV.}$

This theorem is easily proven from Green's second identity. Equivalently, it is the statement that ${\displaystyle \int \phi _{2}(\nabla ^{2}\phi _{1})dV=\int \phi _{1}(\nabla ^{2}\phi _{2})dV}$, i.e. that ${\displaystyle \nabla ^{2}}$ is a Hermitian operator (as follows by integrating by parts twice).

## Related Research Articles

In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of

In physics, the Poynting vector represents the directional energy flux of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m2). It is named after its discoverer John Henry Poynting who first derived it in 1884. Oliver Heaviside also discovered it independently.

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is named after George Green, though its first proof is due to Bernhard Riemann and is the two-dimensional special case of the more general Kelvin–Stokes theorem.

"A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.

In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's theorem.

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz.

In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field, in the form of a partial differential equation, due to the British physicist John Henry Poynting. Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic field to the work done on a charge distribution, through energy flux.

The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, or simply vector potential, A; and the magnetic scalar potentialψ. Both quantities can be used in certain circumstances to calculate the magnetic field B.

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

The electric-field integral equation is a relationship that allows the calculation of an electric field (E) generated by an electric current distribution (J).

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

The Maxwell stress tensor is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impossibly difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

Lorentz–Heaviside units constitute a system of units within CGS, named from Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that the electric constant ε0 and magnetic constant µ0 do not appear, having been incorporated implicitly into the unit system and electromagnetic equations. Lorentz–Heaviside units may be regarded as normalizing ε0 = 1 and µ0 = 1, while at the same time revising Maxwell's equations to use the speed of light c instead.

In continuum mechanics, a compatible deformation tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.

In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. These concepts were introduced by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.

In optics, the Ewald–Oseen extinction theorem, sometimes referred to as just "extinction theorem", is a theorem that underlies the common understanding of scattering. It is named after Paul Peter Ewald and Carl Wilhelm Oseen, who proved the theorem in crystalline and isotropic media, respectively, in 1916 and 1915. Originally, the theorem applied to scattering by an isotropic dielectric objects in free space. The scope of the theorem was greatly extended to encompass a wide variety of bianisotropic media.

The electrokinematics theorem connects the velocity and the charge of carriers moving inside an arbitrary volume to the currents, voltages and power on its surface through an arbitrary irrotational vector. Since it contains, as a particular application, the Ramo-Shockley theorem, the electrokinematics theorem is also known as Ramo-Shockly-Pellegrini theorem.

In mathematical physics, the Gordon decomposition of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.

## References

• L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley: Reading, MA, 1960). §89.
• Ronold W. P. King, Fundamental Electromagnetic Theory (Dover: New York, 1963). §IV.21.
• C. Altman and K. Such, Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics (Kluwer: Dordrecht, 1991).
• 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 Wetenschappen4 p. 176 (1896).
• R. J. Potton, "Reciprocity in optics," Reports on Progress in Physics67, 717-754 (2004). (A review article on the history of this topic.)
• J. R. Carson, "A generalization of reciprocal theorem," Bell System Technical Journal3 (3), 393-399 (1924). Also J. R. Carson, "The reciprocal energy theorem," ibid. 9 (4), 325-331 (1930).
• Ya. N. Feld, "On the quadratic lemma in electrodynamics," Sov. PhysDokl.37, 235-236 (1992).
• C.-T. Tai, "Complementary reciprocity theorems in electromagnetic theory," IEEE Trans. Antennas Prop.40 (6), 675-681 (1992).
• Wolfgang K. H. Panofsky and Melba Phillips, Classical Electricity and Magnetism (Addison-Wesley: Reading, MA, 1962).

## Citations

1. Ramo, Whinnery, Van Duzer: Fields and Waves in Communication Electronics, Wiley International Edition (1965)
2. Jin, J.-M. (2015). Theory and Computation of Electromagnetic Fields, 2nd edition, John Wiley & Sons, Inc., ISBN   978-1-119-10804-7, page 103.
3. Jin Au Kong, Theorems of bianisotropic media, Proceedings of the IEEE vol. 60, no. 9, pp. 1036–1046 (1972).
4. Helmholtz, H. von (1856). Handbuch der physiologischen Optik, first edition, Leopold Voss, Leipzig, volume 1, page 169, cited by Planck. Translation here based on that by Guthrie, F., Phil. Mag. Series 4, 20:2–21. Second printing (1867) at
5. Minnaert, M. (1941). The reciprocity principle in lunar photometry, Astrophysical Journal93: 403-410.
6. Chandrasekhar, S. (1950). Radiative Transfer, Oxford University Press, Oxford, pages 20-21, 171-177, 182.
7. Tingwaldt, C.P. (1952). Über das Helmholtzsche Reziprozitätsgesetz in der Optik, Optik, 9(6): 248-253.
8. Levi, L. (1968). Applied Optics: A Guide to Optical System Design, 2 volumes, Wiley, New York, volume 1, page 84.
9. Clarke, F.J.J., Parry, D.J. (1985). Helmholtz reciprocity: its validity and application to reflectometry, Lighting Research & Technology, 17(1): 1-11.
10. Born, M., Wolf, E. (1999). Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th edition, Cambridge University Press, ISBN   0-521-64222-1, page 423.
11. Stokes, G.G. (1849). On the perfect blackness of the central spot in Newton's rings, and on the verification of Fresnel's formulae for the intensities of reflected and refracted rays, Cambridge and Dublin Mathematical Journal, new series, 4: 1-14.
12. Mahan, A.I. (1943). A mathematical proof of Stokes' reversibility principle, J. Opt. Soc. Am., 33(11): 621-626.
13. Lekner, J. (1987). Theory of Reflection of Electromagnetic and Particle Waves, Martinus Nijhoff, Dordrecht, ISBN   90-247-3418-5, pages 33-37.
14. Rayleigh, Lord (1900). On the law of reciprocity in diffuse reflection, Phil. Mag. series 5, 49: 324-325.
15. Hapke, B. (1993). Theory of Reflectance and Emittance Spectroscopy, Cambridge University Press, Cambridge UK, ISBN   0-521-30789-9, Section 10C, pages 263-264.