Abstract
We address to the Poynting theorem for the bound (velocity-dependent) electromagnetic field, and demonstrate that the standard expressions for the electromagnetic energy flux and related field momentum, in general, come into the contradiction with the relativistic transformation of four-vector of total energy–momentum. We show that this inconsistency stems from the incorrect application of Poynting theorem to a system of discrete point-like charges, when the terms of self-interaction in the product \({\varvec{j}} \cdot {\varvec{E}}\) (where the current density \({\varvec{j}}\) and bound electric field \({\varvec{E}}\) are generated by the same source charge) are exogenously omitted. Implementing a transformation of the Poynting theorem to the form, where the terms of self-interaction are eliminated via Maxwell equations and vector calculus in a mathematically rigorous way (Kholmetskii et al., Phys Scr 83:055406, 2011), we obtained a novel expression for field momentum, which is fully compatible with the Lorentz transformation for total energy–momentum. The results obtained are discussed along with the novel expression for the electromagnetic energy–momentum tensor.
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Notes
Below, in Sect. 3, we clarify the limits of applicability of this approximation.
Here we stress that the well-known (and unambiguous) definition of EM momentum (10) results from the standard expression for the EM energy–momentum tensor (1). We also mention that the Poynting theorem (7) [i.e. Eq. (2) at \(\nu =0]\) itself does not, in general, uniquely defines the flux density \({\varvec{S}}\) of EM field and the related field momentum density \({\varvec{p}}_{EM}={\varvec{S}}/c^{2}\), due to some freedom in the choice of \({\varvec{S}}\), maintaining the divergence \(\nabla \cdot {\varvec{S}}\) unchanged. At the same time, one should stress that any modification of \({\varvec{S}}\) in the framework of the indicated limitation must be carried out along with the appropriate modification of EM energy–momentum tensor (1), which was never proposed to the moment before our paper [20], see Sect. 3.
In general, the magnetic field of the plates is not equal to zero in the frame K near the capacitor’s boundary, where the leakage electric field exists. However, the straightforward estimation shows that the component of momentum of such a leakage field is much less than (7), and cannot resolve the issue. Below we show that the contribution of momentum of leakage EM field to the total momentum of capacitor in the frame K has the order of magnitude (\(v/c)^{4}\).
This is, in general, not the case for the Coulomb gauge \(( {\nabla \cdot {\varvec{A}}=0})\), where manipulations with the tensor (53) occur more complicated; at the same time, the motional and continuity equations occur the same, like in the Lorenz gauge.
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We thank the anonymous referee for the important points he raised, which had been useful for the improvement of the final version of the paper.
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Kholmetskii, A., Missevitch, O. & Yarman, T. Poynting Theorem, Relativistic Transformation of Total Energy–Momentum and Electromagnetic Energy–Momentum Tensor. Found Phys 46, 236–261 (2016). https://doi.org/10.1007/s10701-015-9963-9
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DOI: https://doi.org/10.1007/s10701-015-9963-9