
:''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 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. 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 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). == Lorentz reciprocity == Specifically, suppose that one has a current density $\backslash mathbf\_1$ that produces an electric field $\backslash mathbf\_1$ and a magnetic field $\backslash mathbf\_1$, where all three are periodic functions of time with angular frequency ω, and in particular they have timedependence $\backslash exp(i\backslash omega\; t)$. Suppose that we similarly have a second current $\backslash mathbf\_2$ at the same frequency ω which (by itself) produces fields $\backslash mathbf\_2$ and $\backslash mathbf\_2$. 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'': :$\backslash int\_V\; \backslash left(\backslash mathbf\_1\; \backslash cdot\; \backslash mathbf\_2\; \; \backslash mathbf\_1\; \backslash cdot\; \backslash mathbf\_2\; \backslash right)\; dV\; =\; \backslash oint\_S\; \backslash left(\backslash mathbf\_1\; \backslash times\; \backslash mathbf\_2\; \; \backslash mathbf\_2\; \backslash times\; \backslash mathbf\_1\; \backslash right)\; \backslash cdot\; \backslash mathbf\; .$ Equivalently, in differential form (by the divergence theorem): :$\backslash mathbf\_1\; \backslash cdot\; \backslash mathbf\_2\; \; \backslash mathbf\_1\; \backslash cdot\; \backslash mathbf\_2\; =\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash mathbf\_1\; \backslash times\; \backslash mathbf\_2\; \; \backslash mathbf\_2\; \backslash times\; \backslash mathbf\_1\; \backslash right)\; .$ This general form is commonly simplified for a number of special cases. In particular, one usually assumes that $\backslash mathbf\_1$ and $\backslash mathbf\_2$ are localized (i.e. have compact support), and that there are no incoming waves from infinitely far away. In this case, if one integrates over all space then the surfaceintegral terms cancel (see below) and one obtains: :$\backslash int\; \backslash mathbf\_1\; \backslash cdot\; \backslash mathbf\_2\; \backslash ,\; dV\; =\; \backslash int\; \backslash mathbf\_1\; \backslash cdot\; \backslash mathbf\_2\; \backslash ,\; dV.$ 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: :$\backslash oint\_S\; (\backslash mathbf\_1\; \backslash times\; \backslash mathbf\_2)\; \backslash cdot\; \backslash mathbf\; =\; \backslash oint\_S\; (\backslash mathbf\_2\; \backslash times\; \backslash mathbf\_1)\; \backslash cdot\; \backslash mathbf.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Reciprocity (electromagnetism)」の詳細全文を読む スポンサード リンク
