Abstract
Rumsey proposed the concept of reaction to describe the interaction between two sets of time-harmonic electromagnetic sources through the fields produced by sources, and expressed the Lorentz Reciprocity Theorem in a concise form of < 1,2 > = < 2,1 > , in 1954. Later, Lindell et al. extended the Rumsey’s reaction using the differential form, which contains two terms. The spatial term is the traditional Rumsey’s reaction term, which represents the electromagnetic power density reaction (scalar reaction), and the time-related term is the electromagnetic force density reaction (vector reaction). In this paper, the corresponding unified space and time form of the reciprocal equation is derived for Lindell’s reaction term by the differential form of the electromagnetic field equation, under the assumption that there is only electric source. However, the special form of the reciprocity equation < 1,2 > = ± < 2,1 > does not exist. Thus, we derived a new unified space and time equation of energy–momentum reciprocity and its special form. From this, a new generalized reaction is defined.
Similar content being viewed by others
Data Availability Statement
This manuscript has associated data in a data repository [Authors’ comment: The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.]
References
I.V. Lindell, A. Sihvola, Rumsey’s reaction concept generalized. Prog. Electromagn. Res. Lett. 89, 1–6 (2020)
I.V. Lindell, A. Sihvola, Errata to “Rumsey’s reaction concept generalized.” Prog. Electromagn. Res. Lett. 89, 1–6 (2020)
I.V. Lindell, Differential Forms in Electromagnetics (Wiley and IEEE Press, Hoboken NJ, 2004)
G. Liu, Y. Li, J. Liu, A mutual momentum theorem for electromagnetic field. IEEE Antennas Wirel. Propag. Lett. 19(12), 2159–2161 (2020). https://doi.org/10.1109/LAWP.2020.3025614
G. Liu, J. Liu, Y. Li, Generalized reciprocity theorem for electromagnetic field (Science Press, Beijing, 2020)
G. Liu, J. Liu, Y. Yang, Y. Li, The generalization of Rumsey’s reaction concept. Eur. Phys. J. Plus 137, 1081 (2022). https://doi.org/10.1140/epjp/s13360-022-03290-6
Liu GQ, Liu J, Li YY. 2022b. The Reciprocity Theorems for Momentum and Angular Momentum for Electromagnetic Field. The proceedings of the 16th Annual Conference of China Electrotechnical Society. Lecture Notes in Electrical Engineering, vol 891. Springer, Singapore. https://doi.org/10.1007/978-981-19-1532-1_106
V.H. Rumsey, Reaction concept in electromagnetic theory”. Phys. Rev. 94(6), 1483–1491 (1954)
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant No 52007182, 51937010, and Scientific Research Fund Projects of Institute of Electrical Engineering and Advanced Electromagnetic Drive Technology.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Liu, G., Liu, J. & Li, Y. The differential forms of Rumsey reaction and the corresponding reciprocal equations in the unified space and time form. Eur. Phys. J. Plus 138, 203 (2023). https://doi.org/10.1140/epjp/s13360-023-03791-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-023-03791-y