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The Significance of the Poynting Vector

Frederick David Tombe,

Belfast, Northern Ireland,

United Kingdom,

sirius184@hotmail.com

27th January 2020

Abstract. The Poynting vector, S = E×H, represents the rate of flow of

electromagnetic energy per unit area per unit time. It appears in

Poynting’s theorem because of the involvement of Ampère’s circuital

law and Faraday’s law of time-varying electromagnetic induction. It

will now be investigated as to whether or not the Poynting vector has

any significance if the E field is an electrostatic field, or would it just

amount to multiplying apples and bananas?

Historical Background

I. A telegrapher’s equation linking the speed of light to electric signals

propagating along a conducting wire was first derived by German

physicist Gustav Kirchhoff in 1857 [1]. In Kirchhoff’s theory, it was

assumed that the energy travelled inside the conducting wires. Some

years later however, in 1883, English physicist John Henry Poynting

made a proposal regarding the transfer of energy in electric circuits.

Poynting proposed that at least some of the energy is actually transferred

through the space outside the conducting wires [2]. This idea was also

taken up around about the same time by English electrical engineer

Oliver Heaviside [3].

It was already known since the time of Faraday and Henry that

electrical energy can be transferred through the space between two

electric circuits in the case of electromagnetic induction, but Poynting

and Heaviside were now suggesting that in the case of electrical energy

that is applied directly to a circuit, that some of the energy travels through

the space in the immediate the vicinity of the conducting wires.

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Poynting’s Theorem

II. The derivation of Poynting’s Theorem in this section begins by

considering the equation of continuity as applied to the sum of two

energy density fields in space. One of these is the electromagnetic energy

density field, ½[εoEK∙EK + μoH∙H], which is sourced in a dynamic

magnetic field where EK = −∂A/∂t, and where the magnetic vector

potential, A, satisfies ∇×A = μoH. The other is the electrostatic energy

density field, ½εoES∙ES, which is sourced in an identifiable electric charge

and where ES = −∇φ. The two E∙E terms represent the potential energy

associated with stress in the all-pervading elastic solid which acts as the

medium for the propagation of light. This mathematical form is in the

likeness of the form used for the potential energy, ½kx2, that is stored in a

stretched mechanical spring, where E corresponds to k. See equation (8)

in section IV below, and also Part III in Maxwell’s 1861 paper “On

Physical Lines of Force” [4]. The electric permittivity, εo, is the inverse

of the elastic constant. Meanwhile, the H∙H term represents fine-grained

kinetic energy in the magnetic field in like manner to the familiar

mechanical term, ½mv2, where H corresponds to the speed, v. See Parts I

and II in Maxwell’s 1861 paper. The magnetic permeability, μo, is the

mass density term. The total energy density, w, is therefore,

w = ½εoEK2 + ½μoH2 + ½εoES2 (1)

Taking the partial time derivative of (1) leads to,

∂w/∂t = εoEK∙∂EK/∂t + μoH∙∂H/∂t + εoES∙∂ES /∂t (2)

The first term on the right-hand side of equation (2) contains

Maxwell’s displacement current, εo∂EK/∂t, as is used in electromagnetic

radiation, and we know from Ampère’s Circuital Law, as applied in

space, that,

∇×H = εo∂EK/∂t (3)

Regarding the second term on the right-hand side of equation (2), we

know from Faraday’s Law that,

∇×EK = −μo∂H/∂t (4)

Substituting equations (3) and (4) into equation (2) we get,

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∂w/∂t = EK∙∇×H − H∙∇×EK + εoES∙∂ES/∂t (5)

Hence, using the vector identity,

∂w/∂t = − ∇∙(EK×H) + εoES∙∂ES/∂t (6)

the Poynting vector, S, is then defined as,

S = EK×H (7)

The Significance of the Poynting Vector

III. By comparison with the equation for the continuity of charge, the

Poynting vector is analogous to electric current density J, hence it

represents the flow of energy per unit area per unit time. One might say

that the Poynting vector represents a current of electromagnetic energy

which comprises both an electric component and a magnetic component.

The question still arises however as to what these two components

actually mean in real terms. Reducing it all to the hydrodynamics of the

fundamental electric fluid (or aether) from which everything is made, it

will be proposed that the electric force term, EK, represents potential

energy, and more specifically hydrostatic aether pressure, while the

magnetic term, H, represents kinetic energy, and more specifically aether

flow. Hence the two terms are related to each other through Bernoulli’s

Principle, and it is proposed that Faraday’s law relates to the conversion

between pressure and flow in a sea of tiny aether vortices that fills all of

space. Transverse pressure, EK, in a vortex gives way to angular

acceleration, ∂H/∂t, where H represents the vorticity of the vortex. The

rate of flow of the aether, weighted for its hydrostatic pressure would

represent the rate of flow of total electromagnetic energy in the same way

that electric current density, J, is the product ρv.

Electromagnetic radiation would therefore appear to be a complex

electric current that flows through space, and when it strikes a conducting

wire, the component that strikes the wire at right angles, channels into a

simple conduction current, J, which then flows along the wire. This is

like the case of convectively induced electromagnetic induction where a

current is induced in a conducting wire that moves at right angles to a

magnetic field.

It is proposed that the electromagnetic Poynting vector represents the

power density of a complex electric current undergoing a fine-grained

vortex flow of electric fluid through a dense sea of rotating electron-

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positron dipoles [5], [6], [7], [8]. These dipolar vortices will have a vorticity,

H, equivalent to the magnetic intensity, where H = 2ω. This current is

continually flowing between neighbouring rotating electron-positron

dipolar vortices, each which constitutes a tiny electric circuit in its own

right. This complex electric current is Maxwell’s Displacement Current.

When emitted from an alternating current source, EK and H will be out of

phase by ninety degrees due to Bernoulli’s Principle. See Appendix I.

The Poynting vector therefore applies to wireless radiation providing

that we can isolate H from that of the already existing background

magnetic field [9]. In the case of AC transformers, the Poynting vector

would apply to the energy that leaves the primary circuit, flows through

space, and enters the secondary circuit.

The Electrostatic Component

IV. The electrostatic component, ES, is not included in the Poynting

vector as derived and defined in Section II above. In J.H. Poynting’s

original derivation [2], ES is present, but it should not have been.

Poynting’s own derivation begins as if he has ignored EK altogether. He

starts by splitting the energy terms at equation (1) (in his own paper) into

electrostatic and electromagnetic energy components. But in the

electromagnetic energy component, he uses only H and he omits EK. That

is his first mistake. Then at equation (5) (in his own paper), he produces

Maxwell’s electromotive force equation. This was originally equation

(77) in Maxwell’s 1861 paper [4], and later equation (D) in the original

listing of eight Maxwell’s equations in his 1865 paper “A Dynamical

Theory of the Electromagnetic Field” [10]. J.H. Poynting then substitutes

all three of Maxwell’s EMF terms into what had been exclusively an

electrostatic energy component. This meant that he had added the

electrostatic term into itself, along with the time-varying electromagnetic

term EK, and along with the convective electromagnetic term, μov×H,

which ultimately was not involved in the Poynting vector. Next, he

segregates both non-convective terms, ES and EK, from the convective

term, and from then on, he treats the two non-convective terms as a single

bundled entity.

By comparison with the less cumbersome derivation in Section II

above, it would be like as if J.H. Poynting had used a single potential

energy term in equation (1) in the form ½ εo(ES + EK)2. This would only

be legitimate if ES and EK were two mutually orthogonal vectors, but in

general they are not. The electrostatic field, ES, can be superimposed at

any angle to the electromagnetic field EK. In a transmission line, the two

are orthogonal, but this still doesn’t legitimize the presence of ES in the

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Poynting vector, because ES is not involved in the constituent equations

(3) and (4).

So, what about the electrostatic term, εoES∙∂ES/∂t, which is the last

term on the right-hand side of equation (2)? It relates to a time-varying

electrostatic field, such as we would find in the vicinity of a charging

capacitor. The term εoES∙∂ES/∂t relates to linear polarization in a

dielectric, where Maxwell’s fifth equation, the electric elasticity equation,

applies as in,

D = −εoES (8)

and hence,

εoES∙∂ES/∂t = −ES.JD (9)

where D is the electric displacement vector and where,

JD = ∂D/∂t (10)

is the original displacement current as derived by Maxwell in Part III

of his 1861 paper [4], [11], [12]. Although Maxwell originally used dielectric

polarization in order to derive displacement current in 1861, when he

came to apply it to the derivation of the electromagnetic wave equation in

his 1865 paper, [10], he was now using EK in the displacement current, [12].

Meanwhile, equation (6) now becomes,

∇∙S + JD.ES = −∂w/∂t (11)

Conclusion - Multiplying Apples and Bananas?

V. Ultimately, the product E×H is a power product similar in principle to

the product VI (voltage times current) in circuit electricity. It is a measure

of the rate of change of potential energy into kinetic energy in a

hydrodynamical system. The E term is a force term which drives the

aethereal electric current that is intrinsic to EM radiation, and hence it

also drives the changing magnetic field, ∂H/∂t.

Poynting’s theorem deals with the dynamic state, and the Poynting

vector applies to the rate of energy flow in wireless radiation where

Faraday’s law of time-varying EM induction is involved. The theorem

also applies to charging and discharging capacitors, and to linear

polarization current in a dielectric.

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There is also the convective state to be discussed. A transmission line

pulse, [13], involves what would have been a linear polarization field

propagating at close to the speed of light* in the space between two

conducting wires. But because the pulse is travelling at close to the speed

of light, any would-be electrostatic field arising from the pulse’s electric

charge will have converted, nearly totally, into a magnetic field. And

since the pulse is travelling on its own momentum, with no time-varying

electromagnetic induction involved, we cannot use a Poynting vector in

this context.

Finally, if we were to form the product ES×H outside of Poynting’s

theorem in connection with a stationary electrostatic field superimposed

upon the magnetic field of a stationary bar magnet, the product would

indicate a non-zero value even though there is no actual flow of energy

occurring. The electrostatic field in this case is not driving the magnetic

field and so we could safely say that it would be a case of multiplying

apples and bananas. It would be like multiplying the force of a motorcar

engine in London, England, with the speed of a motorcar in Cape Town,

South Africa.

* See Appendix II – The Speed of Light

Appendix I

(Electromagnetic Waves)

A diagram in Maxwell’s 1873 publication “A Treatise on Electricity and

Magnetism” [14], indicates that Maxwell believed that the electric

displacement, D, and the magnetic force, H, in an electromagnetic wave

are mutually perpendicular to each other, as well as being in phase with

each other in time. On page 389 of the Treatise, under the heading “Plane

Waves”, Maxwell begins the analysis with the magnetic induction

equation, ∇×A = B, and he identifies the magnetic induction vector, B,

with magnetic disturbance. At equation (14), Maxwell writes Ampère’s

Circuital Law as μJ = ∇×B = −∇2A, and he identifies the electric current

density, J, with electric disturbance. The magnetic disturbance and the

electric disturbance will therefore be mutually perpendicular and in time-

phase with each other. Equation (15), J = ∂D/∂t, where D is the electric

displacement, tells us that if J and D obey a sinusoidal relationship in

time, then they will be out of phase with each other in time by ninety

degrees. Since B = μH, where H is the magnetic force, it follows then

that the magnetic force and the electric displacement will be out of phase

in time by ninety degrees.

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Fig. 66 on page 390 however shows H and D to be in time-phase

with each other, and so this would appear to be an error. While

Maxwell’s plane wave solutions at equation (20) in his Treatise were in

the electromagnetic momentum, A, modern textbooks provide similar

solutions in H and EK, where D = εEK, and where EK is the electromotive

force induced by time-varying electromagnetic induction, as per the

Maxwell-Faraday equation, ∇× EK = −∂B/∂t. These sinusoidal solutions

are used to prove that H and EK are in time-phase with each other.

However, these sinusoidal solutions ignore the full three-dimensional

physical interrelationships between A, H, and EK within the context of

the vortices through which they were initially defined. They ignore the

fact that an EM wave involves a chain reaction of precessing vortices, in

which the energy is exchanged between neighbouring vortices when H is

pointing along the direction of wave propagation. See “Wireless

Telegraphy Beyond the Near Magnetic Field” [9]. The textbook solutions

on the other hand only consider the projection of H perpendicular to the

direction of propagation where it appears to have reached its maximum

magnitude at the same moment in time when EK reaches its maximum

magnitude. In actual fact though, H reaches its absolute maximum

magnitude when it has rotated downwards parallel to the direction of

propagation.

Appendix II

(The Speed of Light)

The correspondence between the speed of an electrical signal along a

wire on the one hand, and the speed of light on the other hand, is based

largely on aether hydrodynamics, on the principle that electric current is

primarily an aethereal fluid that flows between positive particles

(sources) and negative particles (sinks), and at an average speed in the

order of the speed of light.

References

[1] Kirchhoff, G.R., “On the Motion of Electricity in Wires”, Philosophical

Magazine, Volume XIII, Fourth Series, pages 393-412 (1857)

Pages 280-282 in this link,

https://www.ifi.unicamp.br/~assis/Weber-Kohlrausch(2003).pdf

[2] Poynting, J.H., “On the Transfer of Energy in the Electromagnetic Field”,

Philos. Trans. Roy. Soc. London 175, pp. 343-361(1884)

https://en.wikisource.org/wiki/On_the_Transfer_of_Energy_in_the_Electromagnetic_

Field

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[3] Heaviside, O., “The Induction of Currents in Cores”, The Electrician, Volume

13, pp. 133-4, 21st June 1884

[4] Maxwell, J.C., “On Physical Lines of Force”, Philosophical Magazine, Volume

XXI, Fourth Series, London, (1861)

http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf

[5] Lodge, Sir Oliver, “Ether (in physics)”, Encyclopaedia Britannica,

Fourteenth Edition, Volume 8, Pages 751-755, (1937)

http://gsjournal.net/Science-

Journals/Historical%20PapersMechanics%20/%20Electrodynamics/Download/4105

In relation to the speed of light, “The most probable surmise or guess at present is

that the ether is a perfectly incompressible continuous fluid, in a state of fine-

grained vortex motion, circulating with that same enormous speed. For it has been

partly, though as yet incompletely, shown that such a vortex fluid would transmit

waves of the same general nature as light waves— i.e., periodic disturbances across

the line of propagation—and would transmit them at a rate of the same order of

magnitude as the vortex or circulation speed”

[6] Tombe, F.D., “The Double Helix Theory of the Magnetic Field” (2006)

Galilean Electrodynamics, Volume 24, Number 2, p.34, (March/April 2013)

https://www.researchgate.net/publication/295010637_The_Double_Helix_Theory_of

_the_Magnetic_Field

[7] Tombe, F.D., “The Double Helix and the Electron-Positron Aether” (2017)

https://www.researchgate.net/publication/319914395_The_Double_Helix_and_the_El

ectron-Positron_Aether

[8] Tombe, F.D., “The Positronium Orbit in the Electron-Positron Sea” (2020)

https://www.researchgate.net/publication/338816847_The_Positronium_Orbit_in_the

_Electron-Positron_Sea

[9] Tombe, F.D., “Wireless Radiation Beyond the Near Magnetic Field” (2019)

https://www.researchgate.net/publication/335169091_Wireless_Radiation_Beyond_th

e_Near_Magnetic_Field

[10] Maxwell, J.C., “A Dynamical Theory of the Electromagnetic Field”, Philos.

Trans. Roy. Soc. London 155, pp. 459-512 (1865). Abstract: Proceedings of the Royal

Society of London 13, pp. 531--536 (1864). Maxwell presents his eight original

equations in Part III under the heading ‘General Equations of the Electromagnetic

Field’ which begins on page 480. Maxwell’s derivation of the electromagnetic wave

equation is found in Part VI entitled ‘Electromagnetic Theory of Light’ which begins

on page 497.

http://www.zpenergy.com/downloads/Maxwell_1864_3.pdf

[11] Tombe, F.D., “Maxwell’s Displacement Current and Capacitors” (2020)

https://www.researchgate.net/publication/338669407_Maxwell's_Displacement_Curr

ent_and_Capacitors

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[12] Tombe, F.D., “Maxwell’s Displacement Current” (2019)

https://www.researchgate.net/publication/335276838_Maxwell's_Displacement_Curr

ent

[13] Tombe, F.D., “Cable Telegraphy and Poynting’s Theorem” (2019) Section IV

https://www.researchgate.net/publication/334654102_Cable_Telegraphy_and_Poynti

ng's_Theorem

[14] Maxwell, J.C., “A Treatise on Electricity and Magnetism” Volume II, Chapter

XX, pp. 389-390 (1873)

https://www.equipes.lps.u-psud.fr/Montambaux/histoire-physique/Maxwell-2.pdf

https://en.wikisource.org/wiki/A_Treatise_on_Electricity_and_Magnetism

25th September 2022 amendment