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.
Forms of the reciprocity theorems are used in many electromagnetic applications, such as analyzing electrical networks and antenna systems.
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 this case, if one integrates throughout space then the surface-integral terms cancel (see below) and one obtains: 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.
In this case: In practical problems, there are another more generalized forms of Lorentz and other reciprocity relations, in which, in addition to electric current density
[2][3][4][5][6][7] Above, Lorentz reciprocity was phrased in terms of an externally applied current source and the resulting field.
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.
to distinguish it from the total current produced by both the external source and by the resulting electric fields in the materials.
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
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.
[8] (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.
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
We shall prove a general form of the electromagnetic reciprocity theorem due to Lorenz which states that fields
This form is valid for general media, but in the common case of linear, isotropic, time-invariant materials, ε is a scalar independent of time.
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.
[10] 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.
In this case, the radiated field asymptotically takes the form of planewaves propagating radially outward (in the
For any nonzero loss, the fields decay exponentially with distance and the surface integral vanishes, regardless of whether the medium is homogeneous.
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.
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.
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
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.
Reciprocity also does not generally apply for time-varying ("active") media; for example, when ε is modulated in time by some external process.
It relates two time-harmonic localized current sources and the resulting magnetic fields: 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.
Apart from quantal effects, classical theory covers near-, middle-, and far-field electric and magnetic phenomena with arbitrary time courses.
Historically, in 1849, Sir George Stokes stated his optical reversion principle without attending to polarization.
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.