Talk:Reciprocity (electromagnetism)

Hey, I need application of reciprocity and compensation theorem can any one please help me???

Reciprocity: Victor Rumsey's "Reaction Concept"
A discussion of Reciprocity should mention Vic Rumsey's Reaction Concept. Rumsey used Lorenz/Carson reciprocity relationships, developing a very powerful tool for analyzing scattering, antenna impedance, effects of obstacles, problems, etc, and showing that variational formulas result from application of the Reaction. See "Reaction Concept in Electromagnetic Theory", V. H. Rumsey, Physical Review, Vol 94, #6, 1954, pp. 1483-1491 and "A Short Way of Solving Advanced Problems in electromagnetic Fields and Other Linear Systems", IEEE Transactions on antennas and Propagation, Jan 1963, pp 73-86. Also, Roger Harrington in his book "Time-Harmonic Electromagnetic Fields", McGraw-Hill, 1961 references Rumsey's work and uses it in many examples on scattering, etc.

All wave phenomena matching the conditions...
There was brief mention of "analogous" results for acoustics, but it would be nice to have an expansion of that. The most general result on the page has to do with Helmholtz' theorem and when combined with Green's identity, that applies to ALL wave phenomena (not just E&M) which match the boundary conditions (linear water waves, gravity waves, acoustics, etc.). If I have time, I'll point that out. Unfortunately, it'll probably require radically expanding the whole article, and certainly renaming it to "Reciprocity (wave phenomena)". If someone else can do the necessary editing, that'd be great, too. Smoo222 (talk) 16:00, 30 July 2009 (UTC)


 * Yes, you can derive similar results for any wave equation, because all wave equations have a similar self-adjoint algebraic structure. (I wrote some notes on this for a class here.)  Acoustic waves in gas or fluids are a rather simple case because they are described by a scalar wave equation; acoustic waves in solids are rather complicated Lamé-Navier equations (more complicated than electromagnetism).


 * I should out that the the Green's reciprocity describe on the page is not nearly general enough to describe all wave phenomena, not even all acoustic waves, since it only covers the case of scalar waves in homogeneous media (the only case many people learn in school when studying wave equations, but arguably the least interesting case in practice). The power of reciprocity is that it works for very general inhomogeneous media as well, providing a useful theorem even in cases that are impossible to solve analytically.


 * However, there is a huge amount of published work on the conditions and consequences of reciprocity for the case of electromagnetism, and as a result the electromagnetic case probably deserves its own article even if it is a sub-article of a more general article (if any is ever written).  Note also that we have to be cautious of original research in presenting  generalizations, although there are certainly published articles on reciprocity for acoustic waves (mainly scalar waves from a cursory look, but including inhomogeneous media).  — Steven G. Johnson (talk) 01:31, 31 July 2009 (UTC)


 * Thank you for that link to your class notes, Steven. They look valuable for my own study, and I might want to snag some of the material for future teaching (I shall certainly ask permission if that's the case!).
 * In reference to the general, non-scalar, non-homogeneous reciprocity theorems, I can do no better than to refer people to Newton's Scattering Theory of Waves and Particles, Pierce's Acoustics: An Introduction to its Physical Principles and Applications (his mention of historical derivations is particularly helpful), and Morse and Feshbach's Methods of Theoretical Physics. In light of the general derivations in these works, I wouldn't be too worried about original research technicalities (so long as the references are properly attributed, of course). Smoo222 (talk) 02:16, 31 July 2009 (UTC)

relation between time reversal symmetry and reciprocity
I just did a journal club on different methods used for on chip optical diodes. I must say I'm confuse about the relation between time reversal symmetry and reciprocity. Does reciprocity always imply time reversal symmetry? I think I understand the two are identical in the basic two port example? All the new work on PT symmetry and reciprocity just adds to my confusion. Any advice from an expert is appreciated. — Preceding unsigned comment added by Eranus (talk • contribs) 08:23, 12 December 2011 (UTC)
 * Reciprocity does not imply time-reversal symmetry. An electrical network which includes resistors may be reciprocal, but does not have time-reversal symmetry - no applied signal can induce the resistors to turn heat back into electrical energy. The article states this (for EM fields) in the Conditions and proof of Lorentz reciprocity section. --catslash (talk) 00:17, 13 December 2011 (UTC)

Reference is required
The proof of reciprocity is very interesting. It seems a modern proof. could you offer a old proof or give a list of references. It is very interesting to us to know who contributed to notice that reciprocity has some limitations and need to be extended. The following formula is very interesting, it seems very useful. It is better to offer a list of reference who contributers to it.


 * $$- \int_V \left[ \mathbf{J}_1^* \cdot \mathbf{E}_2 + \mathbf{E}_1^* \cdot \mathbf{J}_2 \right] dV = \oint_S \left[ \mathbf{E}_1^* \times \mathbf{H}_2 + \mathbf{E}_2 \times \mathbf{H}_1^* \right] \cdot \mathbf{dA}$$

Imrecons (talk) 09:22, 6 June 2014 (UTC)


 * This approach to reciprocity is briefly discussed in this book; although the discussion of reciprocity is very brief, mainly in an appendix, it follows from a more general discussion of Hermitian operators in chapter 2. — Steven G. Johnson (talk) 18:44, 6 June 2014 (UTC)

More Descriptions of Applications
It would be very much useful to actually see how the theorem is applied in practical applicartions. One such important and recent application is to allow a new type of full duplex of receive and transmit in RF devices, such as mobile phones, WiFi and now also LiFi. Referenced here: http://www.svmi.com/new-full-duplex-radio-chip-transmits-and-receives-wireless-signals-at-once/ and here: http://spectrum.ieee.org/tech-talk/telecom/wireless/new-full-duplex-radio-chip-transmits-and-receives-wireless-signals-at-once However, this is beyond my abilities to describe and explain. Jahibadkaret (talk) 05:56, 18 April 2016 (UTC)

Hermitian matrices
A section of the article starts thus:

Conditions and proof of Lorentz reciprocity

————————————

The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator $\hat{O}$ relating $\mathbf{J}$ and $\mathbf{E}$ at a fixed frequency $\omega$ (in linear media):


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

is usually a Hermitian operator under the inner product $(\mathbf{F}, \mathbf{G}) = \int \mathbf{F} \cdot \mathbf{G} \, dV$ for vector fields $\mathbf{F}$ and $\mathbf{G}$. (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&times;3 matrices (symmetric rank-2 tensors) &mdash; this includes the common case where they are scalars (for isotropic media), of course. They need not be real&mdash;complex values correspond to materials with losses, such as conductors with finite conductivity σ (which is included in ε via $\varepsilon \rightarrow \varepsilon + i\sigma/\omega$)&mdash;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.

May I ask, could this wording be simplified without loss of meaning by removing the mention of Hermitian operators and instead directly using complex-symmetric matrices? Then the text could read:

The Lorentz reciprocity theorem holds when the linear operator $\hat{O}$ relating $\mathbf{J}$ and $\mathbf{E}$ at a fixed frequency $\omega$ (in linear media), defined by


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

is expressible as a complex-symmetric matrix. This is true whenever the permittivity ε and the magnetic permeability μ, at the given ω, are symmetric 3&times;3 matrices (symmetric rank-2 tensors) &mdash; this includes the common case where they are scalars (for isotropic media), of course. They need not be real&mdash;complex values correspond to materials with losses, such as conductors with finite conductivity σ (which is included in ε via $\varepsilon \rightarrow \varepsilon + i\sigma/\omega$)&mdash;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.

Other simpler expressions would follow.Chjoaygame (talk) 06:44, 20 April 2016 (UTC)

I think the derivation in ″Conditions and proof of Lorentz reciprocity" is wrong. In Landau/Lifshitz, "Electrodynamics of Continuous Media", the current in the derivation is an external current. This means that the current in the reciprocity relation causes the electric field but the electric field does not cause the current. Therefore, there is no relation $$\mathbf{j} = \hat{O}\mathbf{E}$$. Tostro (talk) 08:58, 26 September 2016 (UTC)


 * $$\mathbf{J} = \hat{O}\mathbf{E}$$ is not a statement about cause and effect, it is simply the statement that there is a linear equation relating current and field. If it makes you feel better, write it as $$\mathbf{E} = \hat{O}^{-1}\mathbf{J}$$ where the inverse operator $$\hat{O}^{-1}$$ is essentially integration with a Green's function. — Steven G. Johnson (talk) 21:16, 17 May 2018 (UTC)

You should tie this article to mutual inductance because it comes up in introductory college courses
I stumbled upon this article while writing an an equation sheet for OpenStax University Physics. I presume the Reciprocity Theorem explains the equality of mutual inductance (M12=M21) as encountered in first-year physics and engineering textbooks. Please let me know if you ever make that more explicit. Your article is currently linked via a permalink to the word "Likewise" in my formula sheet. Explaining reciprocity in this context would greatly enhance the utility of this article for my formula sheet and for the community at large, IMHO.--Guy vandegrift (talk) 03:45, 2 March 2018 (UTC)