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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
