Abstract: In electrodynamics, Poynting’s theorem is a statement of conservationof energy for the electromagnetic field in the form of a partial differentialequation.
In this paper, Poynting’s theorem is re-examined since Poynting’s theorem follows from a deeper and moregeneral result: the conservation of the electromagnetic stress-energy tensor derived from Noether’s theorem.Keywords: Poynting’s theorem, Poynting vector, energy flux, electricallongitudinal field, 1. IntroductionDue tothe British physicist John Henry Poynting1,2, Poynting’s theorem is analogousto the work-energy theorem in classical mechanics and mathematically similar tothe continuity equation.In the case of DC circuit, electrons aredriven by an electric potential difference in the wire, the potential energy isthen converted to kinetic energy. Poyntingvector is just a mathematical definition. Since it does not form real energyflow, there is no transmission of electromagnetic energy flow from the exteriorto the interior metal wire. Poynting’stheorem is re-examined since Poynting’stheorem follows from a deeper and more general result: the conservation of the electromagnetic stress-energy tensor T?? derived from Noether’s theorem.
2. OnPoynting’s theoremAsimple derivation of Poynting’s theorem is as follows 3-6,According to Ampere loop law, ? (1)where jc is the conduction current density. Using the dot product of Eto both sides of Eq.(7), then ?? ? (2)Using vector operation, we get ? ? (3)Substituting (3) into (2), we simplifying ontain (4)According to Faraday’s law: ?weget (5)For the dielectric constant ?and magneticpermeability , (6)Substituting (6) into (5)?weget ?7?Defineing?thisis the Poynting vector(energy flow), where ?•S is the divergence of the Poyntingvector and jc•E is the rate at which the fields dowork on a charged object (jcis the current density corresponding to the motion of charge, E is the electric field and • is thedot product).
The second term on the right hand side of Eq.(7) is the density ofelectromagnetic energy. Thus, Poynting’stheorem is the statement of localconservation of energy in classical electrodynamics. 3. Energy Flow in DC CircuitIn the case of DC circuit, electrons aredriven by an electric potential difference in the wire, the potential energy isthen converted to kinetic energy. In this case, Poynting vector is just a mathematical definition. Since it does not formreal energy flow, there is no transmission of electromagnetic energy flow fromthe exterior to the interior metal wire. DC power is transmitted entirelywithin the metal wire.
Theelectrostatic field is longitudinal field. In the DC circuit, the electricalfield in the metal wire is also longitudinal field. Although these two electricfields have some similarities, in fact, there is a big difference. To simplifythe discussion, a typical DC circuit has a battery, and the battery connects tothe load resistor by metal wires. From textbooksof electromagnetism, the tangential (longitudinal) components of electric fieldstrength along the external and internal surfaces of the metal wire are equalto each other. Even in a DC circuit, electric energy is transmitted via thespace around the metal wire, and then input to the load resistor.
7 (SeeFigure 1)Fig. 1, Electric field and energy flow around themetal wireInFigure 1, E is the electric field and S is the energy flow, i.e. the Poynting vector.Indeed,in the case of a DC circuit, theboundary condition for the electric fields at the above wire boundary is notclear. Since the electrostatic field can be shielded with a metal cover, we havemade a few tests. In the DC circuit, we connect a current meter in series. a).
weshield the battery with a metal cover; b). we shield the resistor with a metal cover;c). we use a coaxial cable to shield inside wire. The results are as follows:the reading of the current meter is essentially the same (with accuracy of 4digits). These tests show that in the DC circuit, the distribution of outsideelectric field (including the external electric field near the interface of themetal wire) has essentially no effect to the electric field inside the metalwire. 8Since DCelectric field is associated with the conduction current, which has an importantdifference from the electrostatic field.
It shows that Poynting vector is justa mathematical definition in the DC case. There may be no flow ofelectromagnetic energy is transferred from outside wire to inside wire. This deduction is different from the conclusionin the textbook of electromagnetism. 4.
Rethinking of Poynting’s TheoremWhen Poynting’s theorem was derived, the theory ofrelativity has not yet occurred. Therefore, the above derivation from Eq.(2) toEq.
(7) do not meet 4D tensor algorithm from the theory of relativity.Poynting’s theorem follows from a deeper and moregeneral result: the conservation of the electromagnetic stress-energy tensor T?? derived from Noether’s theorem with a contributionfrom the electromagnetic field. It is the statement ??T0?=0written out explicitly in terms of electromagnetic fields. The other three conserved currents Ta? (a = 1, 2, 3) are electromagnetic momenta. Comparing Eq.(7) with 4D divergence ??T0?=0,Its explicit form is just a special case ofEq.(7) where the conduction current density jc is zero. Indeed, when we take the dot product of Eto both sides of Eq.
(7), we have to consider where is the local tiny point that wetake the differential operation. If this local tiny point is out ofthe wire, then the conduction current density jc is zero, and also jc•E =0 in the free space. On the other side, If this local tiny point is insidethe wire, the conduction current density jc may be different in different locations, and also jc•E may be different in different locations.If we integrate Eq.(7), the area for integration can only be inside the wire. The above analysis and discussionare our Rethinking of Poynting’s Theorem.
The author wouldlike to thank Professor Chu Jun-hao for his helpful discussions References1. J. H. Poynting, “On the Transfer of Energy in theElectromagnetic Field”, Philosophical Transactions of the Royal Society of London, 175, 343–361, 1884.2. J.
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Fan?EnergyFlow and the Speed of Electric Field in DC Circuit, 2014; 1(3): 24-28, publishedonline Sept. 30, 2014 (http://www.aascit.org/journal/ijees)