The Electrodynamics of Free and Bound Charge Electricity Generators usingImpressed Sources and the Modification to Maxwell’s Equations

Michael E. Tobar,1, ∗ Ben T. McAllister,1 and Maxim Goryachev1

1ARC Centre of Excellence for Engineered Quantum Systems and ARC Centre of Excellence for Dark Matter Particle Physics,Department of Physics, University of Western Australia,

35 Stirling Highway, Crawley, WA 6009, Australia.(Dated: January 8, 2021)

Electric generators convert external energy, such as mechanical, thermal, nuclear, chemical andso forth, into electricity and are the foundation of power station and energy harvesting opera-tion. Inevitably, the external source supplies a force per unit charge (commonly referred to as animpressed electric field) to free or bound charge, which produces AC electricity. In general, theexternal impressed force acts outside Maxwell’s equations and supplies a non-conservative electricaction generating an oscillating electrodynamic degree of freedom. In this work we analyze the elec-trodynamics of ideal free and bound charge electricity generators by introducing a time dependentpermanent polarization, which exists without any applied electric field, necessarily modifying theconstitutive relations and essential to oscillate free or bound charge in a lossless way. For bothcases, we show that Maxwell’s equations, and in particular Faraday’s law are modified, along withthe required boundary conditions through the addition of an effective impressed magnetic currentboundary source and the impressed electric field, related via the left hand rule. For the free chargecase, we highlight the example of an electromagnetic generator based on Lorentz force, where theimpressed force per unit charge that polarizes the conductor, comes from mechanical motion of freeelectrons due to the impressed velocity of the conductor relative to a stationary DC magnetic field.In contrast, the bound charge generator is simply an idealized permanently polarized bar electret,where the general case of a time dependent polarized electret is the underlying principle behindpiezoelectric nano-generators. In the open circuit state, both bound and free charge electricitygenerators are equivalent to idealized Hertzian dipoles, with the open circuit voltage equal to theinduced electromotive force (emf). Analyzing the short circuit responses, we show that the boundcharge electricity generator has a capacitive source impedance. In contrast, we show for the idealfree charge AC electricity generator, the back emf from the inductance of the loop that defines theshort circuit, directly cancels the source emf, so the voltage across the inductor is solely determinedby the magnetic current boundary source. Thus, we determine the magnetic current boundarysource best describes the output voltage of an AC generator, rather than the electric field.

I. INTRODUCTION

The most common form of electricity generation con-verts motive (or mechanical) energy into photonic or elec-tromagnetic energy described by Faraday’s law and isknown as an alternating current (AC) induction genera-tor. In general, the creation of photons or electricity viaFaraday’s law must come from another external energysource to drive the turbines into motion. The conserva-tion of energy is a fundamental law of physics, but onlyapplies to an isolated system. This law means that en-ergy cannot be created or destroyed, but only transferredfrom one form to another. For example, a nuclear reac-tion where mass is converted into other forms of energythrough E = mc2. In this case, if the other form of en-ergy is electricity, then we can call this device a nucleargenerator or battery. For example, a nuclear battery usesthe energy from the decay of a radioactive isotope to gen-erate electricity and can produces large DC electric fieldsand voltages of up to 10-100 kV [1]. The modern form ofthe nuclear battery is a micro-electromechanical system

∗ michael.tobar@uwa.edu.au

or MEMS device [2, 3], which can also be configured as anAC generator capable of generating radio frequencies of60−260MHz [4]. A more recent type of generator is thenanogenerator, which uses special materials that convertan external strain or motion to a time-dependent elec-tric polarization of a material independent of an appliedelectric field[5, 6]. Such a device can convert mechanicalenergy to electricity via a piezoelectric or triboelectriceffect [7–11], or temperature variations to electricity viaa pyroelectric effect [12–18].

When considering the generation of electricity onlyfrom the view point of the created electromagnetic de-gree of freedom, the creation of electromagnetic energyis a non-conservative process. Thus, when we considerthe electrodynamics in isolation to the whole system, thestandard Maxwell’s equations must be made more gen-eral to take into account the non-conservative processes.This requires the ability to add the impressed forces intoMaxwell’s equations, and we show in this work, that thiscan be achieved by generalising the constitutive relations.The modification of the constitutive relations inevitablyleads to a modification of Maxwell’s equations, in a simi-lar way to what happens when we consider the differencebetween Maxwell’s equations in Matter and in Vacuum.For example, in a dielectric medium, the applied electric

arX

iv:1

904.

0577

4v5

[ph

ysic

s.cl

ass-

ph]

7 J

an 2

021

2

field polarizes the material, which causes an opposing in-ternal electric force, which reduces the electric field inthe medium, which modifies Gauss’ Law.

The standard Maxwell’s equation in differential formand in dielectric and magnetic media are in general givenby (SI units),

~∇ · ~D = ρf , (1)

~∇× ~H − ∂ ~D

∂t= ~Jf , (2)

~∇ · ~B = 0, (3)

~∇× ~E +∂ ~B

∂t= 0, (4)

where

~D = ε0 ~E + ~P ~H = µ−10~B − ~M, (5)

and

ρb = −∇ · ~P ~Jb = ∇× ~M, (6)

Here ~E is the electric field intensity, ~H is the magnetic

field intensity, ~D is the electric flux density, ~B is the mag-

netic flux density, ~Jf is the free electric current density,~Jb is the bound electric current density, ρf is the free elec-tric charge density, ρb is the bound electric charge densityand ε0 and µ0 are the permittivity and permeability offree space.

When an electric field is applied to a dielectric mate-rial its molecules respond by forming microscopic elec-tric dipoles, which causes a distribution of bound charge

dependent on the applied field ~E, which acts to reducethe electric field in the dielectric. For a dielectric mate-rial the macroscopic polarization vector, ~P is related tothe average microscopic bound charge density by eqn.

(6). In a similar way when a ~B-field is supplied toa magnetic material, magnetic moments of the atoms

align to cause a macroscopic magnetization ~M relatedto the bound current given in eqn. (6). Thus in matter,Maxwell’s equations can be fully specified in terms of the~E and ~B fields along with the auxiliary fields ~D and ~Hgiven by eqns. (4), combined with the constitutive re-lations given by eqns. (5). However, in order to fullyapply these equations the general relationships betweenthe fields in eqns. (5) must be further specified, whichdepend on the material properties. In general this can bequite complex, as materials may be anisotropic, magneto-electric, piezoelectric, and so forth. The simplest formof the constitutive relations represents isotropic and lin-

ear materials such that ~D = ε0εr ~E and ~H = µ−10 µ−1

r~B,

where ~P = ε0χe0 ~E so εr = 1 + χe0 and ~M = χm0~H so

µr = 1+χm0. The free and bound currents and charges inMaxwell’s equations given above are not source terms forthe fields and do not add energy into the system. Theyeither propagate without loss due to the interaction withthe electromagnetic fields, or can describe a dissipative(or resistive) system where electromagnetic energy is lost,usually by conversion to heat (in this case the electric andmagnetic field phasors can become complex).

The non-electric energy sources (for example, nuclearenergy as discussed previously) capable of transmittingenergy and hence a force to electric charges are commonlyreferred to as “impressed” sources of the field. Theoreti-cally, they can be represented either as an ideal current orvoltage generator of electronic network theory [19]. Thusin electrodynamics, an impressed electric current needsto be added as a current source and hence will mod-ify Ampere’s law. In contrast, we show an impressedelectric voltage needs to be added as a magnetic cur-rent source, and hence will modify Faraday’s law. Thelater does not mean that magnetic monopole particlesexists, but is a consistent way to model boundary valueproblems when considering the electrodynamics of a non-conservative electricity generator [20–22]. This techniqueis more generally known as the Compensation Theorem[19, 23, 24]. We also note the impressed sources are notinfluenced by Maxwell’s equations because they repre-sent creation of electromagnetic energy from an exter-nal source. Two-potential theory is summarized in theappendix, which is a common way to include the non-conservative impressed terms in antenna theory.

In this paper we analyse the electrodynamics of freeand bound charge electricity generators using impressedsources. Importantly, we have shown that the impressedsources modify the constitutive relations, which essen-tially are the same as the force balance equations be-tween the external energy source and the electrodynamicgenerated degree of freedom. We show that this mod-ifies Maxwell’s equations with the addition of a free orbound charge macroscopic polarization, created withoutan applied electric field. Because the vector curl of thispolarization is non-zero, we have shown that Faraday’slaw must be modified. We then apply this technique toexplain the physics of some free charge and bound chargeelectricity generators. In particular we find that the def-inition of the voltage, Ve, of the electricity generator isbest described by, Ve = −Iim, where Iim is the total effec-tive impressed magnetic current at or near the boundary.This approach is similar to the approach recently under-taken to explain axion modifications to electrodynamics,where the axion mixes with a photon to acts as the exter-nal impressed force, which converts axions into electricity[25–27].

3

Figure 1: Illustration of the electric field generated in afree-charge ideal DC voltage source from an external

energy source. The associated delivered external force

per unit charge, ~fS , supplies the energy to seperate (andhence polarize) impressed free surface charges, σie at the

axial boundaries. This is equivalent to an impressed

electric field, ~Eie = ~fS , or polarization, ~P ie , as given byequations (7) and (8), and related to a magnetic current

at the radial boundary, Imenc, by the left hand rule.

The non-conservative nature means an electromotiveforce as calculate in eqn. (11), E = −Imenc =

¸P~Eie · d~l,

is generated, resulting in a voltage output that candrive an electric circuit. The separated free charges

then generate a conservative dipole electric field, ~E.

II. ELECTRODYNAMICS OF THEGENERATION OF ELECTRICITY FROM FREE

CHARGE

To analyse a free charge voltage generator at DC, orin the quasi-static limit, we can start with the equationsgiven in advanced electrodynamics text books such asGriffiths [28], where he shows that the total force per unit

charge, ~f involved in a free charge DC voltage source isgiven by,

~f = ~E + ~fS . (7)

Here ~fS is the force per unit charge, which supplies theenergy to seperate the charges and supply an electromo-tive force (emf) from an external energy source. Follow-

ing this a resulting electric field, ~E, is produced by theseparated charges. Harrington [20] presents essentiallythe same equation as Griffiths [28] for a general AC gen-erator, but using different terminology. Harrington con-siders the electric field more generally such that a total

field, ~ET ≡ ~f , which is consistent with the ~ET in the

appendix. Harrington also defines ~Eie ≡ ~fS as the sourceimpressed electric field. Here to be consistent with Grif-fith [28] and the left hand rule for the relation between

magnetic current and electric field, we have defined ~Eiein the opposite direction to Harrington, so that

~ET = ~E + ~Eie, (8)

The effective impressed field, ~Eie, is confined to the volt-age source and does not exist outside it.

In a DC battery where external energy is convertedto electromagnetic energy [29–31], the impressed electricfield allows the electrons to move in the opposite direc-tion to the electric field created by the electrons them-selves, even though the internal resistance inside an idealfree charge voltage source is near zero, with (ignoring

fringing) ~E ≈ − ~Eie and thus ~ET ≈ 0 (these values de-pend highly on aspect ratio and hence the fringing field).An example of such a voltage source is illustrated inFig.1. Given that the impressed force per unit charge~Eie is non-conservative, there is an effective impressedmagnetic current boundary source linked by the closedpath, as shown in Harrington and Balanis [20, 21], whichmeans for the DC case,

~∇× ~Eie = − ~J im, ~∇× ~E = 0, ~∇× ~ET = − ~J im. (9)

Then by Stokes’ theorem we can calculate the emf of thevoltage generator, E , by,

E = −Iimenc=

˛P

~Eie · d~l =

˛P

~ET · d~l, (10)

where the path encloses the effective impressed magneticcurrent at the radial boundary, given by,

Iimenc=

ˆS

~J im · d~a. (11)

Since ~Eie will apply a force per unit charge to seperatefree charge in the system, an impressed free charge dis-tribution, ρie will be created so that,

ε0∇ · ~Eie = ∇ · ~P ie = −ρie, (12)

effectively polarizing the free charge (creating a dipole),and allowing the definition of a permanent free charge

polarization of, ~P ie = ε0 ~Eie. For the situation in Fig.1,

the impressed free charges occur as surface charge at theends of the DC voltage source, given by,

σ± = ~P ie .n = ±σie. (13)

According to this definition, the ideal DC voltage source

will only have impressed free charge, and thus ε0∇ · ~E =ρie, as a result. The net result is the system has bothan electric vector and scalar potential and outside thevoltage source the electric field resembles a capacitor likedipole even though it is modelled as a perfect conduc-tor. The next step is to expand this technique to an ACelectricity generator.

4

A. Time-Dependent Free Charge ElectricityGeneration

The DC system described previously has no magneticfield component in the system and adding time depen-dence will potentially induce a magnetic field. First,we note that the impressed magnetic current boundarysource is divergenceless. This means from two potential

theory in the appendix, ρim = 0 and ~∇ · ~B = 0. Follow-ing this we can implement the quasi-static approximation

and calculate the ~B-field from the modified Ampere’slaw. To calculate this, we combine with the continuity

equation in eqn. (107) for the impressed free current, ~J ie,with eqn. (12) to obtain,

~J ie =∂ ~P ie∂t

= ε0∂ ~Eie∂t

. (14)

Assuming a lossless system then any other free current

besides the impressed current, ~J ie, must be divergence

free, so ∇ · ~Jf = 0 and ρf = 0 in eqn. (99) and (100) inthe appendix, then in the time varying case, the modified

Ampere’s law is given by, ~∇× ~B − ε0µ0∂ ~ET

∂t = ~Jf , with

a modified Gauss’ law of ~∇ · ~ET = 0. The next step ofthe quasi-static approximation is to calculate the created~E-field from the calculated ~B-field from Faraday’s law,

∇ × ~E + ∂ ~B∂t = 0, then combining with eqn. (9), the

modified Faraday’s law becomes, ~∇× ~ET + ∂ ~B∂t = − ~J im,

here the ∂ ~B∂t term can be considered as a displacement

magnetic current. The only part of Maxwell’s equations,which remains unmodified in this case is the magneticGauss’ law.

Thus, Maxwell’s equations in differential form may be

written in terms of ~Eie and ~E, by,

~∇ · ~E =ρieε0

and ~∇ · ~Eie = −ρie

ε0, (15)

~∇× ~B − ε0µ0∂ ~E

∂t= µ0

~J ie = ε0µ0∂ ~Eie∂t

, (16)

~∇ · ~B = 0, (17)

~∇× ~E +∂ ~B

∂t= 0 and ~∇× ~Eie = − ~J im. (18)

or ~ET by

~∇ · ~ET = 0, (19)

~∇× ~B − ε0µ0∂ ~ET∂t

= 0, (20)

~∇ · ~B = 0, (21)

~∇× ~ET +∂ ~B

∂t= − ~J im. (22)

Note, we set ~Jf = 0, for the open circuit no-load case.Following this, the integral forms of Maxwell’s equations

may be written as,

‹S

~E · d~a =Qieenc

ε0and

‹S

~Eie · d~a = −Qieenc

ε0, (23)

˛P

~B · d~l − µ0ε0d

dt

ˆS

( ~E + ~Eie) · d~a = 0 (24)

‹S

~B · d~a = 0, (25)

˛P

~E·d~l+ d

dt

ˆS

~B·d~a = 0 and

˛P

~Eie·d~l = −Iimenc(26)

We can also write eqn. (26) as,

˛P

~ET · d~l +d

dt

ˆS

~B · d~a = −Iimenc(27)

and (23) as

‹S

~ET · d~a = 0, (28)

In general from a circuit theory perspective, the struc-tures under consideration are considered as electricallysmall (quasi-static limit), no time delay exists betweensources and the rest of the circuit and the only loss occursthrough dissipation. From an antenna theory perspectivethese assumptions in general can be relaxed as time de-lays may be important. In this work we just consider thequasi-static limit relevant for antenna and circuit theoryin the limit that the structures are small compared to thewavelength.

B. Ideal Cylindrical AC Free Charge ElectricityGenerator

In the following we will analyse the electrodynamics ofan ideal zero resistance cylindrical AC voltage generator,as shown in Fig.2. We recognise that the net chargein the system is zero, and the charge will be present asa surface charge density, ±σie, at the upper and lowerboundaries of the cylinder. Furthermore, there will be anon-familiar boundary condition to be determined at the

radial boundary, since ~Eie must be contained within thecylinder.

For an ideal source in the quasi-static limit, we ignoreany source impedance, including inductance or capaci-tance, so the AC emf is similar to the DC case (eqn. 10),given by,

E = −Iimenc=

˛P

~Eie · d~l, (29)

and the surface charge on the end faces caused by the

5

Figure 2: Above, schematic of the cylindrical ideal freecharge AC electric generator of axial length, de and

radius re, with associated field and source terms. Forthe ideal AC generator of terminal voltage, Ve, the

impressed electric current density, ~J ie, flows up anddown the cylindrical axis as the terminal charges, ±σie,

oscillate in polarity. Below, the active near field of aHertzian dipole antenna resembles that of a fringing

field in a capacitor[32], and is equivalent to the abovefree charge AC electric generator.

impressed field, ~Eie, can be calculated to be,

σie = ~P ie · n = ε0 ~Eie · n, where ~P ie = σiez. (30)

Here n is the normal to the surface, which is equal to zon the top surface and -z on the bottom surface. Then,from eqn.(10) the emf generated in the quasi-static limitis calculated to be,

E = Eiede =σiedeε0

= Ve, (31)

which is similar to a voltage across a capacitor, we labelthis the free charge terminal voltage, Ve.

Next we consider the magnetic surface current per unitlength, which will be apparent at the radial boundary,(r = re) of the generator. This surface magnetic currentwill determine the parallel boundary condition, and canbe calculated from equation (29), using the left hand ruleto be,

~κim = −σie

ε0φ. (32)

From the integral equations (23)→(26) it is straightfor-ward to derive the boundary conditions. Here subscript“in” refers to inside the ideal generator and subscript“out” refers to outside the generator, while the subscript“⊥” refers to the perpendicular components of the fieldwith respect to a surface and the subscript “‖” refersto the parallel components of the fields with respect to

the surface. We also note that ~Ein = −σie

ε0z, and that

there is no electric surface current (only volume current).Thus, the boundary conditions on the axial surfaces forthe ideal voltage source become,

E⊥out = 0, (33)

~B⊥out = ~B⊥in = 0, (34)

and the boundary conditions on the radial surface gives,

~E‖out = −~κim × n = − ~Eie = −σ

ie

ε0z, (35)

~B‖out = ~B

‖in = 0, (36)

Applying the radial boundary condition, eqn.(35),

gives ~E‖out = −σ

ie

ε0z. This means the electric field just

outside the generator has maximum value on the radialboundary, despite having infinite conductance, similar tofringing in a capacitor. To calculate the magnetic fieldin the system we can start inside the voltage source andapply the modified Ampere’s law given by eqn. (24).

Effectively we find that the ~B-field caused by the dis-

placement current produced by the time varying ~E-field

is suppressed by the ~B-field caused by the time vary im-

pressed ~Eie-field (or impressed electrical current), so issmall within the electricity generator depending on theaspect ratio. Thus, with no load circuit attached to thegenerator, the solution is similar to that of a capacitorlike dipole, even though we consider an ideal conductiv-ity. In reality the generator will have an internal resis-

tance, so ~ET will be non-zero, and there will be finitefield to match on the axial boundary.

Assuming a harmonic surface charge density of theform, σie = σe0e

jω0t, then the terminal voltage may bedetermined to be,

Ve(t) = Ve0ejω0t =

σe0deε0

ejω0t. (37)

Likewise, the current inside the voltage source may bedetermined to be,

Iie(t) = Ie0ejω0t = jω0σe0πr

2eejω0t. (38)

The ratio of the terminal voltage to internal current can

be calculated to be Ve

Iie= 1

jω0Ceffwhere Ceff =

ε0πr2e

de

is similar to a capacitance and Ve lags Iie by π2 . Note,

6

this is not a capacitance, but just the phase relation-ship between the current and voltage of the generator,an active component where the impressed force main-tains the charge separation with no power dissipation, sothe internal current and terminal voltage must be outof phase (power factor of π

2 for no load). Such currentand emf generation with no load will create the near fieldof a Hertzian dipole antenna, which is reactive and ex-ists as stored energy, acting very much like the field of adynamically charging and discharging capacitor of Ceff ,with the dipole ‘ends’ acting as plates giving a fringingcapacitance. To dissipate or radiate power an effectiveresistive component in the model must be added, whichwould represent far field radiation or resistive dissipationwithin the generator.

C. Short circuit response of the ideal free chargeAC generator

Effectively the calculation in the previous section cal-culated the open circuit voltage of the AC generator. Inreality the generator has a source impedance, which willbe designed to have minimal effect, usually representedby a Thevenin or Norton equivalent circuit depending ifconfigured as a voltage or current source. To calculatethe Norton equivalent impedance, the short circuit cur-

rent needs to be calculated and ~Jf in eqn. (20) cannotbe set to zero and has to be reinstated on the right handside. For the ideal system, assuming a perfect conductor,the impedance will actually depend on how the voltageterminals are short circuited. Inevitably the short circuitforms a current loop as highlighted in Fig.3 and thuswill have an inductance which will limit the current flowfor an AC signal. At DC the finite conductivity of thegenerator and the metal in the short circuit, will givean effective resistance, which will limit the current flow.Combined with this inductance, the resistance will definethe time constant of the short circuit.

If we consider an idealized short circuit for the time-dependent case, with just the equivalent inductance ofthe loop, then as shown in Fig.3, the net force on theelectrons will be zero, as a back emf will be produced

from a Faraday induced electric field of, ~E = − ~Eie, so

that ~ET = 0. Because they have opposing values of curl,Faraday’s law becomes,

− ~Jm =∂ ~B

∂t, (39)

so that the voltage across the loop is given by,

Ve = −Iimenc=dΦBdt

, (40)

where ΦB is the magnetic flux. This is basically the

Figure 3: Exaggerated short circuit of an ACfree-charge voltage generator. Any short circuit will

form a loop, which will enclose the impressed magnetic

current on the boundary, ~J im(t). Initially, the electrons

will flow in the loop driven by, ~Eie. However, a back emfwill be generated, which directly opposes this field, so~ET = ~Eie + ~E = 0, and there is no net force on the

electrons. The net result is that the voltage supplied bythe generator, Ve, which exists across the effective

inductance, will be determined uniquely by the enclosedmagnetic current, given by, Ve = −Imenc

= dΦB

dt

= LdIscdt + IscdLdt , were ΦB is the linked magnetic flux of

the loop, and Isc is the short circuit current. Theinductance and current flow will be determined by

geometry of the loop by implementing Ampere’s law.

equation for a voltage across the inductor, where

Ve =dΦBdt

= LdIscdt

+ IscdL

dt. (41)

In this case, the free current flow in the loop is determinedby the inductance of the loop, calculable from the imple-

mentation of Ampere’s law with ~Jf of non-zero value.

Even though ~ET = 0, the voltage across the short circuitis defined uniquely by the magnetic current boundarysource, which best describes the output voltage of an ACgenerator, rather than the total electric field.

This concept also allows interpretation of Faraday ex-periments, which either rely on a Coulomb force from theseparation of charges opposing the impressed force (opencircuit case), or a back emf, from a Faraday force causedby the time rate of change of magnetic flux through aclosed loop, which also opposes the impressed force (shortcircuit case).

D. Example: A cylindrical ideal conductoroscillating in a DC magnetic field

In this example we use the impressed source techniqueto analyse the emf induced in a cylindrical ideal conduc-tor due to an external motional kinetic energy or appliedforce in the quasi-static limit. The conductor generateselectricity from its motion due to the impressed Lorentz

force, ~F ie , acting on free charge carriers in the bar as

7

shown in Fig.4 and given by,

~F ie = q~vi(t)× ~BDC = q ~Eie(t). (42)

For this case the impressed force per unit charge comesfrom a driven oscillating mechanical degree of freedominteracting with a DC magnetic field (DC photonic de-gree of freedom). This in turn creates an oscillating ACphotonic degree of freedom characterised by an oscillat-ing emf, E(t), and given by,

E(t) = Eie(t)de = deBDCvi(t) (43)

Assuming simple harmonic motion, ~x(t) = x0 sin(ω0t)x,then the impressed velocity vector may be written as,

~vi(t) = ω0x0 cos(ω0t)x. (44)

Then given that ~BDC = BDC y, the impressed Lorentzforce per unit charge becomes,

~Eie(t) = ω0BDCx0 cos(ω0t)z, (45)

creating an emf of,

E(t) = ω0BDCdex0 cos(ω0t). (46)

The Lorentz force also drives the motion of the chargesand hence the impressed internal electrical current withinthe cylinder of,

~Iie(t) = −ω20ε0πr

2eBDCx0 sin(ω0t)z. (47)

Note the current lags the voltage by π2 as expected. The

oscillating separation of ± charges creates an electricfield, which opposes the impressed force so the net forceon the oscillating charges is zero, which means there is nowork done in driving the internal electrical current. Thismeans that there is only reactive power in the electricaldegree of freedom as there is no load to dissipate ac-tive power. However, as a consequence of this oscillatingcurrent in the DC magnetic field, there is an additional

Lorentz force acting on the mass of the cylinder, ~FB(t),which is given by,

~FB(t) = de~Iie(t)× ~BDC , (48)

due to the mechanical motion. This force is in the samedirection, but out of phase with the velocity, given by,

~FB(t) = ω20ε0VeB2

DCx0 sin(ω0t)x, (49)

where Ve is the volume of the conductor. In the idealcase there is no work done unless a load is attached tothe generator. Of course the analysis ignores any realmaterial effects, such as finite conductivity, skin effect,kinetic inductance etc., which occur at high frequencies.

Thus, in the real case the oscillating conductor has aneffective impedance, including a source resistance due to

Figure 4: A cylindrical ideal conductor (orange) oftime-dependent velocity, ~vi(t), moving in the x

direction within a DC magnetic field of ~BDC orientatedin the y direction (black crosses). An emf is generated

through the induced Lorentz force per unit charge,~F ie

q = ~Eie = ~vi(t)× ~BDC = v0(t)BDC z in the z direction,

which is impressed on the free charges in the conductor.The separation of charges due to the impressed Lorentz

force creates a reverse electric field, ~E (green), with aneffective magnetic current boundary source, ~κim

(magenta).

DC conductivity, which will dominate at low frequencies,

this will be a source of damping as a component of, ~FB(t)would be in phase with the velocity, which also means acomponent of the current and voltage would be in phase.At higher frequencies the resistance will increase due tothe skin effect, and the cylinder will have an inductance.It is not our intention to analyse these effects in this pa-per, which will contribute to the source impedance of areal voltage source or generator. As discussed previously,

the cylindrical generator will also emit an ~E(t)-field out-side, which will be in the form of the near field of anoscillating Hertzian dipole.

III. GENERATION OF ELECTRICITY FROMBOUND CHARGE

A. The DC Electret

A DC bound charge voltage source is essentially abar electret. The ideal bar electret exhibits a perma-nent electrical dipolar field as shown in Fig.5, due to animpressed macroscopic polarization, P ib , and has over-all charge neutrality[33]. Some common ways to impressa polarization and make an electret, is to heat a polardielectric material under the influence of a large elec-tric field (thermo electret) [34] or through piezoelectric-ity (piezo electret) [7]. For the former, once cooled andremoved from the electric field a net polarization will bemaintained, while the later is maintained through the ap-plication of strain. Other forms of electrets include themagneto-electret and the magneto active electret [35] aswell as ferroelectric electrets [36–38]. The electret thus

8

Figure 5: From left to right, 3D sketch of the ~E, ~P ib

and~D fields inside and outside a cylindrical bar electret

(reproduced from the solution manual of [28]).

Assuming the impressed polarization, ~P ib, is constant

within the electret and along the cylindrical z−axis of

the bar in the positive direction. Note, ~P ib

is onlydefined inside the electret.

becomes a bound charge voltage source and is useful forsupplying DC bias reducing the requirement for high ex-ternal DC voltages [39, 40], and can supply a currentand be discharged in a similar way to a battery [41]. Infact batteries with solid electrolytes are not too dissim-ilar to a DC electret. For example, when a ferroelectricmaterial becomes permanently polarized, it undergoes aphase change, which corresponds to the crystal structurebreaking a certain symmetry under phase transition [42].Likewise a rechargeable battery with a solid electrolyteundergoes induced symmetry breaking of the electrolytestructure, such as polyhedron distortions upon chargeand discharge [43]. Some solid batteries even have elec-trolyte of perovskite structure [44, 45], similar to thestructure of common ferroelectric materials.

The bar electret as shown in Fig.5 is the electrostaticanalogue of the bar magnet. Here we assume a losslessdielectric media so ideally there is no free charge or cur-rent in the system and hence they are set to zero, in thiscase Maxwell’s equations, (1)→(5), become

~∇ · ~D = 0, (50)

~∇× ~E = 0, (51)

where

~D = ε0 ~E + ε0χe ~E + ~P ib = ε0εr ~E + ~P ib , (52)

for a linear dielectric material with a permanent polar-

ization, ~P ib , which is independent of the electric field, ~E.

1. Impressing a Source Term into static Maxwell’sEquations to Describe an Electret

Equations (50)→(52) seem incomplete, as there is alack of a source term on the right hand side of the equa-tions. For the analogue bar magnet, Maxwell’s equations

give ~∇· ~B = 0, ~∇× ~H = 0 and ~∇× ~B = µ0µr ~∇× ~MS for

a linear magnetic material with a permanent magnetiza-

tion ~MS . The vector ~MS is in actual fact an impressedsource as energy needs to be added by an external force topermanently magnetize the material, and we may iden-

tify an impressed bound current, ~J ib = ~∇ × ~MS so that

Ampere’s law becomes ~∇ × ~B = µ0µr ~Jib where the ef-

fective impressed bound current at the radial boundaryof the bar magnet sources the magnetic field. Thus, thefields in the bar magnet could also be represented in Fig.5

with the following substitutions ~B → ~D, ~H → ~E and~MS → ~P ib . Now we can identify how static Maxwell’s

equations need to be generalised to be able to describea system with a permanent polarization as an impressedsource, that is to take the curl of eqn. (52) and combine

with eqn. (51) to give ~∇× ~D = ~∇× ~P ib . One can iden-tify an impressed magnetic bound (subscriptmb) current,~Jmb, boundary source at the radial boundary of the elec-tret, from,

~∇× ~P ib = −ε0εr ~J imb (53)

so that a modified Faraday’s law becomes,

~∇× ~D = −ε0εr ~J imb (left hand rule). (54)

There is no free charge in this system so the divergence

of ~D is zero. However, the divergence of ~E will be nonzero due to the separation of bound charge, which hasin general two components, ρbχe , due to the dielectricsusceptibility of the material and, ρbPi , the bound charge

driven by the AC impressed polarization vector, ~P ie(t), sothat,

ε0~∇ · ~E = ρbχe+ ρbPi , (55)

and

ε0εr ~∇ · ~E = −~∇ · ~P ib = ρbPi . (56)

In a similar way to the free charge voltage source, theforces in the bound charge system may be defined us-ing equation (8) [20, 28]. Thus, the total force per unit

charge, ~ET acting on the bound charges is given by,

~ET = ~E + ~Eib, (57)

Accordingly the impressed electric field acting on bound

charge, ~Eib, may be identified to be related to the perma-

nent source polarization ~P ib by,

~Eib = ~P ib/(ε0εr), (58)

and then if we multiply equation (57) through by thepermittivity, ε0εr, we obtain exactly eqn. (52) (where~D = ε0εr ~ET ), the well known constitutive relation be-

tween the ~D-field, ~E-field and ~P -field in the electret.Thus, equation (57), which balances the forces in the

9

voltage source is essentially on the same footing as a con-stitutive relationship between fields given by eqn. (52).In this case the modified static Faraday’s law may bewritten as,

~∇× ~ET = − ~J imb, (59)

which is equivalent to eqn. (54). Thus as emphasized inthe appendix, the DC voltage source has two componentsof force per unit charge in the system. The impressed ex-

ternal force per unit charge, ~Eib, with an electric vectorpotential, which generates an emf, and applies the force

to seperate the ± charges, and the electric field, ~E, whichis sourced by the separated ± bound charges, which ex-hibits a scalar electric potential.

Revisiting the magnetic analogue of the DC electret (a

permanent magnet), where, ~∇× ~B = µ0µr ~Jib, introduces

a bound (or Amperian) current [46]. In this example thebound current is an effective impressed electrical current,and there is no real current flow, especially this is high-lighted by the fact that a permanent magnet is usuallynot a good conductor. The collective alignment of spins,which creates this bound current occurs due to an im-pressed force magnetising the material (in this case amagnetomotive force). This bound current is treated ex-actly like a real current when analysing macroscopic elec-trodynamic equations, like in magnetic circuit analysis.For example, the bound electrical current in a bar mag-net acts as a source term for the fields of the macroscopicmagnetic dipole. In a similar way, the magnetic currentdefined and introduced in eqns. (53), (54) and (59) isnot a real magnetic current, and of course it cannot beas monopoles as far as we know do not exist. This term isan effective magnetic current that defines the boundaryconditions, and in analogy can be used when analysingmacroscopic electrodynamic equations, like in electric cir-cuit analysis. Similarly, the bound magnetic current ina bar electret acts as a source term for the fields of themacroscopic electric dipole. Another interesting point isthat the free charge voltage source discussed previouslycan be also thought of as a macroscopic dipole, with afree charge polarization, with a similar magnetic currentboundary source.

To calculate the fields in the DC electret, a numer-ical calculation is necessary, similar to that of a cylin-drical bar magnet. However, to calculate the circuitproperties of the voltage source one just need to im-plement the integral form of equation (59), which gives

E = −Iimb = 1ε0εr

¸P~P ib · d~l independent of the electric

field since¸P~E ·d~l = 0. The other parameter that is nec-

essary to calculate is the source impedance, which willjust be the capacitance of the electret with the sourcepolarization removed. To first order this capacitance

may be calculated assuming a constant ~E field withinthe dielectric, which ignores fringing, and assumes an as-pect ratio of a thin polarized plate. Nevertheless, thecalculation uncertainties in this case is with the source

impedance and not the emf. However, in a general sys-tem one might expect a numeric calculation is necessaryif the polarization and electric fields cannot be assumedconstant, similar to a bar magnet.

B. AC Generation of Electricity with TimeVarying Bound Charge

Next, we generalise the concept of the electret to a time

varying impressed permanent polarization, ~P ib (t), alongthe lines to what was undertaken in the work of ZhongLin Wang [6]. This system describes an AC electricitygenerator based on oscillating bound charge. The start-ing point is to consider Maxwell’s equations for a linearisotropic dielectric media with a time varying permanentpolarization and an impressed magnetic current bound-ary source, in a similar way to the static version discussedpreviously (also see appendix). First we consider the con-

tributions to Gauss’ Law for the three fields, ~D(t), ~E(t)

and ~P ib (t), which is modified in the same way as the DCcase,

~∇ · ~D = 0, ~∇ · ~E =ρbPi

ε0εrand ~∇ · ~P ib = −ρbPi . (60)

Next, Ampere’s law is modified along the lines as firstsuggested by Wang [6],

~∇× ~B − µ0ε0εr∂ ~E

∂t− µ0

∂ ~P ib∂t

= 0. (61)

The magnetic Gauss’ law remains unmodified, whileFaraday’s law is modified so,

1

ε0εr~∇× ~D +

∂ ~B

∂t= − ~J imb, (62)

with

~∇× ~E +∂ ~B

∂t= 0 and ~∇× ~P ib = −ε0εr ~J imb. (63)

The last term in equation (62) is the impressed magneticcurrent, which acts as a source of the system, to createan AC voltage output. The modified Maxwell’s equationswith impressed sources may also be written in terms of

the total force per unit charge, ~ET = ~E + ~Eib, for thetime-dependent electret (also see the appendix) as,

~∇ · ~ET = 0 (64)

~∇× ~B − µ0ε0εr∂ ~ET∂t

= 0 (65)

~∇ · ~B = 0 (66)

~∇× ~ET +∂ ~B

∂t= − ~J imb, (67)

10

Here the ∂ ~B∂t term in eqn.(62) and (67) can be identified

as the magnetic displacement current. In this system, the

effective magnetic current source term, ~J imb exists on theradial boundary of the electret, and drives the impressed

electric filed, ~Eib by the left hand rule and also sets theboundary condition for the parallel components of thefields on the radial boundary.

C. Boundary Conditions

The boundary conditions of the fields on the normaland parallel surfaces of the electret can be calculatedfrom the integral version of equations (60)→(63), whichare given by,

‹S

~D · d~a = 0, (68)

‹S

~E · d~a =QibPi

ε0εr,

‹S

~P ib · d~a = −QibPi, (69)

˛P

~B · d~l − µ0d

dt

ˆS

~D · d~a = 0 (70)

‹S

~B · d~a = 0, (71)

1

ε0εr

˛P

~D · d~l +d

dt

ˆS

~B · d~a = −Iimb, (72)

˛P

~E ·d~l+ d

dt

ˆS

~B ·d~a = 0,1

ε0εr

˛P

~P ib ·d~l = −Iimb (73)

From these integral equations it is straightforwardto derive the modified boundary conditions given in(74)→(79). Here subscript “in” refers to inside the barelectret and subscript “out” refers to outside the bar elec-tret.,

D⊥out = D⊥in, (74)

εoutE⊥out − εinE⊥in = σbPi , εinE

i⊥bin = P i⊥bin = σbPi (75)

~B‖out = ~B

‖in, (76)

B⊥out = B⊥in. (77)

~D‖out − ~D

‖in = −ε0εr~κimb × n, (78)

~E‖out = ~E

‖in,

~Ei‖bin

=~P‖bin

εin= −~κimb × n, (79)

In the following we show how the values of the elec-tromagnetic fields, output voltage and magnetic currentmay be calculated with the aid of the constitutive rela-tions and boundary conditions that define an ideal elec-tret.

1. Axial Boundaries

Assuming the impressed polarization is constant of the

form ~P ib = P ib z = ε0εrEibz, at the top and bottom axial

boundaries, as shown in Fig.6, we can use eqn.(74) to

obtain a relationship between ~E and ~Eib, where

ε0 ~Eout · n = ε0εr( ~Ein + ~Eib) · n (80)

To take the analysis a step further one really needs a nu-meric solution for the general case, as another equation is

needed to solve the problem. In general, ~Eout is non zero

as highlighted in Fig.5, and ~Ein points in the opposite

direction to ~Eout and ~Eib and in general | ~Eib| ≥ | ~Ein|. In

our example we have assumed a constant ~P ib along thez-axis, this is actually an approximation for a thin plate,

where we ignore fringing, which means ~Eout ≈ 0 so that~Ein ≈ − ~Eib. This limit gives as a simple way of calculat-ing the Thevenin equivalent circuit for an electret.

Matching this condition gives the following relation be-tween vectors at the axial boundary,

~Eout = 0 and ~Ein = − ~Eib. (81)

so that ~D = 0 above and below the axial boundaries. Tocalculate the impressed bound surface current, ±σib atthe upper and lower axial end faces respectively, we use

σib = ~P ib · n = ε0εrEib, (82)

as n = z, while on the lower axial boundary the value isnegative as n = −z.

The polarization current density may be calculated

from the time rate of change of eqn. (82) to be, ~JP =∂ ~P i

b

∂t = ε0εr∂Ei

b

∂t z, and if it has a cross sectional area ofAb the effective polarization current through the voltagesource is given by,

IP = Ab∂P ib∂t

= Ab∂σib∂t

= Abε0εr∂Eib∂t

(83)

2. Radial Boundaries

On the radial boundary we can determine that the im-pressed surface magnetic current density from equation

11

Figure 6: Top left and right show respectively theNorton and Thevenin equivalent circuits, for the ideal

AC bar electret of cross sectional area Ab and length dbshown underneath. The associated impressed force per

unit charge ~Eib(t) supplies the force to seperate thebound charges σib resulting in an electret with a time

varying permanent polarisation of ~P ib (t) = P ib (t)z, whichcan be modelled as an impressed magnetic currentboundary source, ~κimb(t). The Thevenin equivalent

circuit is pictorially shown with the open circuitvoltage, Vb(t) = E(t) = −Iimb, and effective source

impedance, Zb, which is calculated in the text and isequivalent to the capacitance of the electret without a

permanent polarization.

(79) to be,

~κim = −Eibin φ = − σibε0εr

φ, (84)

and the electric fields, ~E to be,

~Ein = ~Eout = −Eibz = − σibε0εr

z (85)

This is of similar form to the free charge voltage source,

with σie ≡σib

εr, because both have ~ET = 0 above and

below the axial boundary. This also means that the total

displacement current, ~JD = 0, above and below the axial

boundary and hence there will be no ~B field within thevoltage source.

3. Equivalent Circuit

To calculate the emf we need to integrate around the

radial boundary, with ∂ ~B∂t = 0 and therefore the emf may

be calculated from,

E = −Iimb =

˛P

~Eib · d~l, (86)

where Iimb =´S~J im · d~a, is the enclosed magnetic cur-

rent. Defining the length of the voltage source as db, theinduced emf is given by,

E = Eibdb =σibdbε0εr

(87)

Note, ~JD = 0 is an approximation, and the value willdepend on the aspect ratio, thus for any finite electret

there will be in fact a finite ~D field (and hence finite ~ETfield) above the electret, as shown in Fig.5. For thesecases a small magnetic field will be generated within theelectret due to the time dependence.

Now assuming the charge density oscillates harmon-ically such that σib = σb0e

jω0 , we can calculate theThevenin equivalent voltage, where the open circuit volt-age across the electret is equivalent to the emf calculatedin eqn. (87) to give,

Vb =σb0dbε0εr

ejω0t (88)

Also by short circuiting the electret, a free charge willoscillate in the short circuit wire equivalent to the po-

larization current, as the ~Ein field is shorted to zero, theNorton equivalent current source can be calculated to be,

Ib = Ab∂σib∂t

= jω0Abejω0t. (89)

Ignoring any small resistive or inductive effects, thesource impedance may be calculated to be,

Zb =VbIb

=1

jω0CbCb =

ε0εrAbdb

. (90)

Note, the impedance can also be calculated by settingthe voltage source to zero (or permanent polarization,P ib = 0), and calculating the capacitance of the left overdielectric, which is equivalent to eqn. (90). This systemis a Hertzian dipole with similar characteristics to thefree charge system.

D. Example: A piezoelectric nano generator(PENG)

One way to generate the time varying electret as dis-cussed is through piezoelectricity, which has become animportant way to undertake energy harvesting [47]. Inthis example we consider a direct piezoelectric effectwhere polarization in certain materials can also be in-duced by mechanical loads as shown in Fig.7. An exter-nal impressed load (force, stress or strain) if time har-

12

Figure 7: Exaggerated diagram of an impressed

mechanical load, ~F i(t), distorting a cubic structure

(left) to induce an impressed polarization, ~P ib (t) (right),which is the principle of PENG.

monic, will cause a time harmonic variation of impressed

polarization, ~P ib (t). Thus, the piezoelectric effect is ex-plained through the coupling of electric and elastic phe-nomena, and thus in general is a tensorial theory com-bining continuum mechanics and electrodynamics leadingto many complex effects beyond the scope of our discus-sion, see standard text books such as [47–49]. For thepurposes of this work, we assume a basic mathematicalformulation[50], where dimensional effects are small andthe process is linear and in one dimension along the z-axis. For this ideal example the constitutive relationsare,

P ibz = dpe × T = dpe × cpe × S = epe × S (91)

Dz = P ibz + ε0εrEz. (92)

Here, dpe is the piezoelectric strain coefficient, T is thestress to which piezoelectric material is subjected, cpe isthe elastic constant relating the generated stress, T , andthe applied strain, S, (T = cpe×S) spe is the compliancecoefficient, which relates the deformation produced bythe application of a stress (S = spe × T ), and epe is thepiezoelectric stress constant. In this case equation (92) isof the same form as eqn. (52), where the magnitude andtime dependence of the electret polarization is dependenton the external mechanical load.

A general theory for potential and fields has been de-veloped for PENG in [6, 51], which focuses on the genera-

tion of the displacement current, ~JD = ∂Dz

∂t = ε0εr∂Ez

∂t +∂P i

bz

∂t . However, in the no-load situation, with a flat as-

pect ratio, we have shown in the last section that ~JD ≈ 0,

with the open circuit voltage, Voc = E =P i

bzdb

ε0εr=

σibdbε0εr

,calculable from the modified Faraday’s law, given byequations (86) and (87), which is consistent with that

derived in [6]. This calculation highlights, when the ter-minals are short circuited, the free current that flows will

be equal to the polarization current, IP = Ab∂P i

bz

∂t , drivenby the induced emf, E(t) as indicated by the Norton andThevenin equivalent circuits shown in Fig.6. Note it isgenerally assumed that the external strains do not sig-nificantly perturb the dimensions[6], so to fist order thesource impedance shown in Fig.6 is constant. Also, anyinductive or resistive short circuit properties will be smallcompared to the source capacitance.

Typical parameters for an AC electret based on vibra-tional energy include; 1) Based on charged resonantlyvibrating cantilevers with vibrations of 0.1g(1m/s2), canharvest up to 30µW per gram of mobile mass, with asource capacitance between 1− 8pF [52]; 2) A vibration-driven polymer energy harvester[53, 54], obtained outputpower as large as 100µW at 30Hz and 0.15g acceleration.3) The flexible triboelectric generator[10] has attainedan electrical output peak voltage of 3.3V and current of0.6mA with a peak power density of 10.4mW/cm3. Forsuch devices, output impedances are capacitive and typ-ically of the order of 100 MΩ.

IV. CONCLUSION

We have explored the electrodynamics of bound andfree charge electricity generators and voltage sources.The external input to the system was represented by animpressed force per unit charge, which converts the exter-nal energy into electromagnetic energy and may be con-sidered as a non-conservative electric field vector, or emfper unit length, with an electric vector potential. Thesource term is necessarily impressed into Maxwell’s equa-tions as an effective magnetic current boundary source,which sources the resulting charge distribution and emfproduced by the generator, resulting in a modificationof Faraday’s law, the constitutive relations and henceMaxwell’s equations.

ACKNOWLEDGEMENTS

This work was funded by the Australian ResearchCouncil Centre of Excellence for Engineered QuantumSystems, CE170100009 and Centre of Excellence for DarkMatter Particle Physics, CE200100008. We also thankProfessor David Griffiths for allowing the reproductionof his figures and we thank Professor Ian McArthur forhis analysis and comments on the manuscript.

REFERENCES

13

[1] J. H. Coleman, “Radioisotopic high-potential, low-current sources,” Nucleonics 11, 42–25 (1953).

[2] Amit Lal and James Blanchard, “Nuclear batteries thedaintiest dynamos,” IEEE Spectrum September, 36 –41 (2004).

[3] Hui Li, Amit Lal, James Blanchard, and Douglass Hen-derson, “Self-reciprocating radioisotope-powered can-tilever,” Journal of Applied Physics 92, 1122–1127(2002), https://doi.org/10.1063/1.1479755.

[4] H. Li, A Lal, J. Blanchard, and D. Henderson, “Self-reciprocating radioisotope-powered cantilever,” in Trans-ducers ’01 Eurosensors XV, edited by E. Obermeier(2001).

[5] Zhong Lin Wang, Tao Jiang, and Liang Xu, “Toward theblue energy dream by triboelectric nanogenerator net-works,” Nano Energy 39, 9 – 23 (2017).

[6] Zhong Lin Wang, “On maxwell’s displacement currentfor energy and sensors: the origin of nanogenerators,”Materials Today 20, 74 – 82 (2017).

[7] G. M. Sessler, P. Pondrom, and X. Zhang, “Stackedand folded piezoelectrets for vibration-based energyharvesting,” Phase Transitions 89, 667–677 (2016),https://doi.org/10.1080/01411594.2016.1202408.

[8] Zhong Lin Wang and Jinhui Song, “Piezoelectric nano-generators based on zinc oxide nanowire arrays,” Science312, 242–246 (2006).

[9] Rusen Yang, Yong Qin, Liming Dai, and Zhong LinWang, “Power generation with laterally packaged piezo-electric fine wires,” Nature Nanotechnology 4, 34–39(2009).

[10] Feng-Ru Fan, Zhong-Qun Tian, and Zhong Lin Wang,“Flexible triboelectric generator,” Nano Energy 1, 328 –334 (2012).

[11] Zhong Lin Wang, “Triboelectric nanogenerators asnew energy technology for self-powered systemsand as active mechanical and chemical sensors,”ACS Nano 7, 9533–9557 (2013), pMID: 24079963,https://doi.org/10.1021/nn404614z.

[12] Hao Xue, Quan Yang, Dingyi Wang, Weijian Luo, Wen-qian Wang, Mushun Lin, Dingli Liang, and Qiming Luo,“A wearable pyroelectric nanogenerator and self-poweredbreathing sensor,” Nano Energy 38, 147 – 154 (2017).

[13] Y. Yang, W. Guo, K.C. Pradel, G. Zhu, Y. Zhou,Y. Zhang, Y. Hu, L. Lin, and Z.L. Wang, “Pyro-electric nanogenerators for harvesting thermoelectric en-ergy,” Nano Letters 12, 2833–2838 (2012), cited By 384.

[14] Y.J. Ko, D.Y. Kim, S.S. Won, C.W. Ahn, I.W. Kim,A.I. Kingon, S.-H. Kim, J.-H. Ko, and J.H. Jung, “Flex-ible pb(zr0.52ti0.48)o3 films for a hybrid piezoelectric-pyroelectric nanogenerator under harsh environments,”ACS Applied Materials and Interfaces 8, 6504–6511(2016), cited By 32.

[15] Y. Yang, J.H. Jung, B.K. Yun, F. Zhang, K.C. Pradel,W. Guo, and Z.L. Wang, “Flexible pyroelectric nanogen-erators using a composite structure of lead-free knbo3nanowires,” Advanced Materials 24, 5357–5362 (2012),cited By 145.

[16] Y. Zi, L. Lin, J. Wang, S. Wang, J. Chen, X. Fan,P.-K. Yang, F. Yi, and Z.L. Wang, “Triboelectric-pyroelectric-piezoelectric hybrid cell for high-efficiencyenergy-harvesting and self-powered sensing,” Advanced

Materials 27, 2340–2347 (2015).[17] J.-H. Lee, K.Y. Lee, M.K. Gupta, T.Y. Kim, D.-Y. Lee,

J. Oh, C. Ryu, W.J. Yoo, C.-Y. Kang, S.-J. Yoon, J.-B.Yoo, and S.-W. Kim, “Highly stretchable piezoelectric-pyroelectric hybrid nanogenerator,” Advanced Materials26, 765–769 (2014), cited By 290.

[18] T. Park, J. Na, B. Kim, Y. Kim, H. Shin, and E. Kim,“Photothermally activated pyroelectric polymer films forharvesting of solar heat with a hybrid energy cell struc-ture,” ACS Nano 9, 11830–11839 (2015), cited By 46.

[19] B. D. Popovic, “Electromagnetic field theorems,” IEEProceedings A - Physical Science, Measurement and In-strumentation, Management and Education - Reviews128, 47–63 (1981).

[20] Roger E. Harrington, Introduction to ElectromagneticEngineering, 2nd ed. (Dover Publications, Inc., 31 East2nd Street, Mineola, NY 11501, 2012).

[21] Constantine A Balanis, Advanced Engineering Electro-magnetics (John Wiley,, 2012).

[22] Edward Conrad Jordan and Kieth G. Balmain, “Electro-magnetic waves and radiating systems,” (Prentice Hall,Inc., 1968) Chap. 13, 2nd ed.

[23] G. D.Monteath, “Application of the compensation the-orem to certain radiation and propagation problems,”Proc. IEE 98 Pt. IV, 23–30 (1951).

[24] Martin Stumpf, Electromagnetic Reciprocity in AntennaTheory, edited by Tariq Samad (IEEE Press, 2018).

[25] Michael E. Tobar, Ben T. McAllister, and Maxim Gory-achev, “Modified axion electrodynamics as impressedelectromagnetic sources through oscillating backgroundpolarization and magnetization,” Physics of the DarkUniverse 26, 100339 (2019).

[26] Michael E. Tobar, Ben T. McAllister, and Maxim Gory-achev, “Broadband electrical action sensing techniqueswith conducting wires for low-mass dark matter axion de-tection,” Physics of the Dark Universe 30, 100624 (2020).

[27] ChunJun Cao and Ariel Zhitnitsky, “Axion detectionvia topological casimir effect,” Phys. Rev. D 96, 015013(2017).

[28] David J Griffiths, Introduction to Electrodynamics, 3rded. (Prentice Hall, Upper Saddle River, New Jersey07458, 1999).

[29] Dana Roberts, “How batteries work: A gravitational ana-log,” American Journal of Physics 51, 829–831 (1983),https://doi.org/10.1119/1.13128.

[30] Ralph Baierlein, “The elusive chemical potential,”American Journal of Physics 69, 423–434 (2001),https://doi.org/10.1119/1.1336839.

[31] Wayne M. Saslow, “Voltaic cells for physicists:Two surface pumps and an internal resistance,”American Journal of Physics 67, 574–583 (1999),https://doi.org/10.1119/1.19327.

[32] Sean Victor Hum, Ideal (Hertzian) Dipole, Vol. ece422(University of Toronto, 2020).

[33] G. M. Sessler, Electrets (Springer-Verlag, New York,Berlin, Heidelberg, 1987).

[34] Oleg D. Jefimenko and David K. Walker, “Elec-trets,” The Physics Teacher 18, 651–659 (1980),https://doi.org/10.1119/1.2340651.

[35] G J Monkman, D Sindersberger, A Diermeier, andN Prem, “The magnetoactive electret,” Smart Materials

14

and Structures 26, 075010 (2017).[36] Haruhiko Asanuma, Hiroyuki Oguchi, Motoaki Hara,

Ryo Yoshida, and Hiroki Kuwano, “Ferroelectric dipoleelectrets for output power enhancement in electrostaticvibration energy harvesters,” Applied Physics Letters103, 162901 (2013), https://doi.org/10.1063/1.4824831.

[37] Ingrid Graz and Axel Mellinger, “Polymer electrets andferroelectrets as eaps: Fundamentals,” in Electromechan-ically Active Polymers: A Concise Reference, editedby Federico Carpi (Springer International Publishing,Cham, 2016) pp. 1–10.

[38] Chaoying Wan and Christopher Rhys Bowen,“Multiscale-structuring of polyvinylidene fluoridefor energy harvesting: the impact of molecular-, micro-and macro-structure,” J. Mater. Chem. A 5, 3091–3128(2017).

[39] Chikako Sano, Manabu Ataka, Gen Hashiguchi, andHiroshi Toshiyoshi, “An electret-augmented low-voltagemems electrostatic out-of-plane actuator for acoustictransducer applications,” Micromachines 11, 267 (2020).

[40] C. Jean-Mistral, T. Vu Cong, and A. Sylvestre,“Advances for dielectric elastomer generators: Re-placement of high voltage supply by electret,”Applied Physics Letters 101, 162901 (2012),https://doi.org/10.1063/1.4761949.

[41] Bernhard Gross and R. J. de Moraes, “Polarization of theelectret,” The Journal of Chemical Physics 37, 710–713(1962), https://doi.org/10.1063/1.1733151.

[42] Zijun C. Zhao, Maxim Goryachev, Jerzy Krupka, andMichael E. Tobar, “Emergence of dielectric anisotropyof crystalline strontium titanate due to temperature-dependent phase transitions,” arXiv:2008.07088 [cond-mat.mtrl-sci] (2020).

[43] X.Z. Liu, Z.X. Tang, Q.H. Li, Q.H. Zhang, X.Q. Yu,and L. Gu, “Symmetry-induced emergent electrochemi-cal properties for rechargeable batteries,” Cell ReportsPhysical Science 1, 100066 (2020).

[44] Henghui Xu, Po-Hsiu Chien, Jianjian Shi, YutaoLi, Nan Wu, Yuanyue Liu, Yan-Yan Hu, andJohn B. Goodenough, “High-performance all-solid-state batteries enabled by salt bonding to perovskitein poly(ethylene oxide),” Proceedings of the Na-tional Academy of Sciences 116, 18815–18821 (2019),https://www.pnas.org/content/116/38/18815.full.pdf.

[45] Shahab Ahmad, Chandramohan George, David J.Beesley, Jeremy J. Baumberg, and Michael De Volder,“Photo-rechargeable organo-halide perovskite batteries,”Nano Letters, Nano Letters 18, 1856–1862 (2018).

[46] C Birch, “The amperian current model of magnetisationand the prolate spheroid,” European Journal of Physics6, 180–182 (1985).

[47] Alper Erturk and Daniel J. Inman, PIEZOELECTRICENERGY HARVESTING (John Wiley and Sons, Ltd,2011).

[48] T Ikeda, Fundamentals of Piezoelectricity (Oxford Uni-versity Press, New York, 1996).

[49] Jiashi Yang, An Introduction to the Theory of Piezoelec-tricity (Springer Nature Switzerland, 2018).

[50] R.S. Dahiya and M. Valle, Robotic Tactile Sensing(Springer Science+Business Media Dordrecht, 2013).

[51] Zhong Lin Wang, “On the first principle theory of nano-generators from maxwell’s equations,” Nano Energy 68,104272 (2020).

[52] S Boisseau, G Despesse, T Ricart, E Defay, andA Sylvestre, “Cantilever-based electret energy har-vesters,” Smart Materials and Structures 20, 105013(2011).

[53] Y. Suzuki, “Electret based vibration energy harvester forsensor network,” in 2015 Transducers - 2015 18th In-ternational Conference on Solid-State Sensors, Actuatorsand Microsystems (TRANSDUCERS) (2015) pp. 43–46.

[54] Kimiaki Kashiwagi, Kuniko Okano, Tatsuya Miyajima,Yoichi Sera, Noriko Tanabe, Yoshitomi Morizawa, andYuji Suzuki, “Nano-cluster-enhanced high-performanceperfluoro-polymer electrets for energy harvesting,” Jour-nal of Micromechanics and Microengineering 21, 125016(2011).

[55] A. N. Kudryavtsev and S. I. Trashkeev, “Formalism oftwo potentials for the numerical solution of maxwell’sequations,” Computational Mathematics and Mathemat-ical Physics 53, 1653–1663 (2013).

[56] N. Cabibbo and E. Ferrari, “Quantum electrodynamicswith dirac monopoles,” Il Nuovo Cimento (1955-1965)23, 1147–1154 (1962).

[57] Ole Keller, “Electrodynamics with magnetic monopoles:Photon wave mechanical theory,” Phys. Rev. A 98,052112 (2018).

[58] Andreas Asker, Axion Electrodynamics and MeasurableEffects in Topological Insulators (Kaerstads University,2018).

[59] Michael E. Tobar, Raymond Y. Chiao, and Maxim Gory-achev, “Dual aharanov-bohm berry phase due to the gen-eration of electricity through permanent bound and freecharge polarization,” arXiv:2101.00945 [physics.class-ph](2021).

V. APPENDIX A: TWO POTENTIALFORMULATION

The general two potential formulation of impressedcurrent and fields has been discussed in detail in standardtext books on Electrical Engineering [20–22, 55]. Thetwo potential formulation is used in electrodynamics tomodel electricity generation in circuit and antenna the-ory, when there is conversion of external energy into elec-tromagnetic energy through non conservative processesas discussed in the main body of this paper. It has alsobeen used to describe duality in electrodynamics and ax-ion electrodynamics[25, 56–58], and was recently appliedto electricity generation [59].

Using superposition, we can consider the electric andmagnetic current sources separately. So setting the mag-netic sources to zero, the electric and magnetic fields may

be written in terms of the magnetic vector potential, ~A,and the electric scalar potential, φ,

~EA = −∇φ− ∂ ~A∂t

~BA = ∇× ~A.(93)

Then by setting the electric sources to zero the electricand magnetic fields may be written in terms of the elec-

tric vector potential, ~C, and the magnetic scalar poten-

15

tial, φm,

~EC = − 1ε0∇× ~C

~BC = −µ0∇φm − µ0∂ ~C∂t

(94)

The total electric and magnetic fields may be calcu-lated using the principle of superposition and are givenby [20, 21];

~ETot = ~EA + ~EC = −∇φ− ∂ ~A

∂t− 1

ε0∇× ~C (95)

~BTot = ~BA + ~BC = −µ0∇φm − µ0∂ ~C

∂t+∇× ~A. (96)

Considering the electric field given by equation (95), inthe quasi-static limit we can ignore the time-dependentterms and the main source terms are due to the chargedistributions defined by the electric charge and the effec-tive magnetic current, with the electric vector potentialgiven by [20, 21],

~C(~r, t) =ε04π

ˆΩ

~J im(~r′, t′

)|~r − ~r′|

d3~r′. (97)

and the electric scalar potential given by,

φ(~r, t) =1

4πε0

ˆΩ

ρ(~r′, t′

)|~r − ~r′|

d3~r′. (98)

Here ~C and φ at point ~r and time t is calculated frommagnetic current and charge distribution at distant posi-tion ~r′ at an earlier time t′ = t−

∣∣~r − ~r′∣∣ /c (known as theretarded time). The location ~r′ is a source point withinvolume Ω that contains the magnetic current distribu-tion. The integration variable, d3~r′, is a volume elementaround position r′. In a similar way the magnetic po-tentials may be written in terms of the electric currentdensity and magnetic charge density, however, as suchwork focuses on voltage sources, we do not give the for-mulae but refer the reader to Refs [20, 21].

The two potential formulation also means we can sep-

arate Maxwell’s equations into two parts given by,

ε0~∇ · ~EA = ρie + ρf , (99)

~∇× ~BA − ε0µ0∂ ~EA∂t

= µ0( ~J ie + ~Jf ), (100)

~∇ · ~BA = 0, (101)

~∇× ~EA +∂ ~BA∂t

= 0, (102)

for the electric sources, and

~∇ · ~EC = 0, (103)

~∇× ~BC − ε0µ0∂ ~EC∂t

= 0, (104)

~∇ · ~BC = ρim, (105)

~∇× ~EC +∂ ~BC∂t

= − ~J im, (106)

for the magnetic sources. Here, the impressed sources,

ρie, ~Jie, ρ

im and ~J im, in equations (99)→(106) can only ex-

ist due to an impressed external source exciting the sys-tem. Also, in general, there may be some free charge and

current in the system, ρf and ~Jf respectively. Becausemagnetic monopoles do not exist, the effective magnetic

current, ~J im, and the effective magnetic charge terms,ρim can only exist as impressed sources. Equations (103)to (106) are the dual representation of equations (99)to (102). Due to the conservation of charge, impressedsource currents and charges must satisfy the continuityequations,

∂ρie∂t

= −~∇ · ~J ie, and∂ρf∂t

= −~∇ · ~Jf (107)

∂ρim∂t

= −~∇ · ~J im, (108)

which completes Maxwell’s equations with impressedsources, which describe how electromagnetic energy orelectricity can be generated from an external impressedenergy source.