On the expanded Maxwell’s equations for moving …

1 Zhong Lin WangBeijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences, Beijing 101400, PR ChinaSchool of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245, USAThe conventional maxwell s equations are for media whose boundaries and volumes arefixed. But forcases that involve moving media and time-dependent configuration, the equations have to beexpanded. Here, starting from the integral form of the maxwell s equations for general cases, wefirstderived the expanded maxwell s equations in differential form by assuming that the medium ismoving as a rigid translation object.

2 Secondly, the expanded maxwell s equations are furtherdeveloped with including the polarization density termPsin displacement vector owing toelectrostatic charges on medium surfaces as produced by effect such as triboelectrification, based onwhich thefirst principle theory for the triboelectric nanogenerators (TENGs) is developed. Theexpanded equations are the most comprehensive governing equations including both electromagneticinteraction and power generation as well as their coupling. Thirdly, general approaches are presentedfor solving the expanded maxwell s equations using vector and scalar potentials as well as perturbationtheory, so that the scheme for numerical calculations is set.

3 Finally, we investigated the conservation ofenergy as governed by the expanded maxwell s equations , and derived the general approach forcalculating the displacement current@@tPsfor the output power of TENGs. The current theory is generaland it may impact the electromagnetic wave generation and interaction (reflection) with movingtrain/car,flight jets, missiles, comet, and even galaxy stars if observed from : expanded maxwell s equations ; Displacement current; moving charged media; Triboelectric nanogeneratorIntroductionMaxwell s equations are probably the top #1 equations for thefield of physics, which have huge importance in fundamentalscience and practical technologies[1].

4 Starting from experimen-tally observed physics laws, such as Faraday s electromagneticinduction law, Ampere s law, maxwell s equations unified theelectricity and magnetism, which later inspires the advocatingof unifying the four forces in nature. The electromagnetic wavetheory and the wireless communication technologies establishedbased on maxwell s equations are the foundation of moderntelecommunication. Just like many others, we learnt maxwell sequations mainly through standard text books that does notspecifically devote much text for introducing the backgroundand assumptions made for deriving the maxwell s equations . Inthe famous Jackson s book on Classical Electrodynamics , onlya couple of pages were devoted to thefirst introduction of dis-placement current, which is then fully integrated in the Max-well s equations without in-depth exploration[2].

5 This isprobably the reason that the main objective of developing theMaxwell s equations was for the purpose of studying the behav-ior of electromagnetic wave and its interaction with , the concept of displacement current has beenapplied for quantifying the power output of piezoelectric andTENG, Triboelectric the expanded maxwell s equations formoving charged media system Generaltheory, mathematical solutions andapplications in TENGE-mail address:Wang, 2021 Elsevier Ltd. All rights TodaydVolume xxx, Number xxdxxxx 2021 RESEARCHRESEARCHP lease cite this article in press as: Wang, Materials Today, (2021), nanogenerators (TENGs)[3 5], which is emerging asa new technology for fully utilizing high entropy energy[6],which is the energy that is widely distributed in our living envi-ronment with low quality, low amplitude and even low fre-quency.

6 TENG wasfirst invented in 2012, and it has four basicworking modes: the contact-separation mode, lateral slidingmode, single-electrode mode, and free-standing mode (seeFig. 1)[7,8]. TENG has a broad application as micro-nano powersource, self-powered sensors, blue energy and high voltagesources, covering area from medical science, wearable electron-ics,flexible electronics, security, human machine interfacesand even environmental science[7,9 15]. Let s take thecontact-separation mode TENG shown inFig. 1a as an the mechanical pull-press force acting in vertical direc-tion, the two dielectric layers are periodically contacted and sep-arated. The two surfaces have opposite electrostatic chargesowing to contact electrification effect.

7 A change in spatial distri-bution of the media, surface electrostatic charge density, as wellas the distance between the two electrodes, results in a variationof electricfield in space, which is a form of displacement currentthat generates an output conduction current across the load con-nected between the two electrodes. As a general case, the mediaboundaries here do vary with time, and we need to derive theMaxwell s equations for moving charged added into the induction equation the term thatdescribed the induction due to the motion of the medium[16].Hertz systematically extended maxwell s theory for movingmedia[17]but his equations were valid only for conductorsand needed to be expanded on the cases of dielectrics and emptyspace.

8 Minkowski derived electrodynamic equations for movingmedia using the principle of relativity[18]. However, the devel-opment of electrodynamics for moving media was almost inter-rupted due to the appearance of theory of relativity. In recentyears, the interest on the study of electrodynamics of movingmedia has been revived[19,20]. There are a number of studiesabout the maxwell s equations for moving media/bodies with afocus on the scattering, reflection and transmission of electro-magnetic waves from moving media[21 23]. But most of thesestudies are mainly focused on the establishment of thefirst prin-ciple equations without much progress proposed for analyticalsolutions of the equations and their practical this paper, starting from the integral form of the maxwell sequations, wefirst derived the standard differential form of theMaxwell s equations by assuming that the media volumes andsurfaces/interfaces arefixed.

9 Secondly, we derived the expandedMaxwell s equations by assuming that the medium is moving asa rigid translation. It is important to point out the differencesbetween our expansion of maxwell s equations presented hereFIGURE 1 Schematics showing the four modes of tribielectric nanogenerators based on coupling effect of triboelectrification and electrostatic induction,for effectivelyharvesting high entropy energy in our living TodaydVolume xxx, Number xxdxxxx 20212 Please cite this article in press as: Wang, Materials Today, (2021), special relativity. Special relativity is the theory of how dif-ferent observers, moving at constant velocity with respect to oneanother, report their experience of the same physical event.

10 Ourtheory is about the observation of the electromagnetic behaviorof the system in a stationary coordination frame when some ofthe charged media are moving at different speeds, and theremay have interaction and/or charges/current exchange betweenthe media that are at rest and in moving (see the case inFig. 1forexample). Thirdly, we expanded the equations for including thecases in which the media surfaces have electrostatic charges thatwere produced by triboelectrification or piezoelectric effect. Anaddition of a polarization termPsin the displacement vectorthat arises from the surface electrostatic charges is the most logicapproach for dealing with thefirst principle theory of the equations are the most comprehensive governing equa-tions including both electromagnetic interaction and power gen-eration.