Volume 7, Issue 3, March – 2022 International Journal of Innovative Science and Research Technology

ISSN No:-2456-2165

Why Poynting's Theorem P = Ex H is

Quite Valid for DC Circuits
Dr. Ismail Abbas
Senior lecturer at MTC, Cairo University

Abstract:- Poynting's vector theorem P=ExH is one of Moreover, it should be noted that Maxwell's equations
the universal laws of physics that applies to are valid in all inertial frames and are the first equations in
electromagnetic fields in AC and DC circuits. A rigorous physics compatible with the laws of special relativity
analysis of two arbitrary cases of DC and AC circuit [1,2,3].
electromagnetic fields shows that Poynting's law P=ExH
applies to both station ary and time-varying The set of four Maxwell's equations in differential
electromagnetic fields. form is composed of two divergences and two curls as
follows:
Keeping the generality, we analyze two simple cases  Div E= Rho/∈
of time-varying and stationary fields of a regular  Div B= 0
cylindrical wire carrying direct or alternating current  Curl E = (partial) d (B/dt)
where in both cases the electromagnetic energy flow  Curl B= μ J + (partial) d (D/dt)
calculations validate the hypothesis that the theorem of
Poynting is absolute. Moreover, the interpretation of the In normal conventions [2,3].
results also suggests that the photons of light beams or
electromagnetic field bundles cannot live forever, they Where J is the conduction current density due to
do not have an infinite but a finite lifespan because they motion of free charges, Rho is the volumetric electric charge
can be created or annihilated during their interaction density and D = Eps E is the displacement vector while its
with free or bound charges such as electrons. partial time derivative is the displacement electric current
density as proposed by Maxwell.
I. INTRODUCTION
μ is free space permeability
There are at least two incompatible theories, classical
EM theory based on Maxwell's equations, where wave μ =4 Pi E-7 Kg m C^-2
energy is continuous and quantum theory QM, where the
energy of EM waves is essentially discrete or quantized in [μ] = [M L T^0 Q^-2]
photons.
∈ is Permittivity of free space
A. Soit depends on the theory you use to analyze the
∈=(E-9/36 Pi ) Kg^-1m^-3s^2C^2
problem.
In this article, we follow classical EM theory combined [∈] = [M ^-1L^-3 T^2 Q^2]
with Einstein's special theory of relativity similar to a
previous article explained in ref. [1,2] to arrive at a rigid C. It should be mentioned that equation 4 by itself implies
conclusion. that E and H must be mutually perpendicular when E
and H emanate from the same electric charge.
First, the Poynting vector P is mathematically and This property is particularly useful for calculating the
physically correct and represents the energy flux per unit pointing vector P = ExH for DC and AC circuits, as
area per unit time for electromagnetic fields in 4D space discussed later in Section II.
which is the subject of this article.
Considering, for example, vacuum or any medium with
However, in the present study, we discuss the case no free charge (Rho = 0), then equation 4 reduces to,
where the time variation of electromagnetic fields is small μ. Curl H = ∈. (partial) d/dt E . . . . (5)
enough, which means that we limit our study to the Poynting
vector in non-radiative electrical circuits. Since the LHS of Eq 5 is a vector orthogonal to H and
the RHS of Eq 5 is a vector in the direction of E since t is a
B. Some might simply expect that Poynting's vector formula scalar. then E is perpendicular to H for the dependent fields
for energy flow in EM fields would only apply to time- coming from the same Rho source.
varying fields, but in fact it is also valid for
electromagnetic fields of DC circuits. Assuming that Maxwell's four equations are universal
The classical theory of electromagnetic fields is imperial laws and a valid relativistic invariant for time-
mathematically and physically based on Maxwell's dependent and stationary fields, then we should expect the
equations which are a set of four equations, described in Poynting vector,
integral or differential form. They form the theoretical basis
for describing classical E&H fields in electromagnetism. P=ExH . . . . . . .(6)

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Volume 7, Issue 3, March – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
Is also an imperial law that is relativistic invariant for The classical Gaussian divergence theorem and
time-dependent and stationary fields. Stokes' curl theorem are fundamental theorems of
calculus that have an essential place in the theory of
The important question here is how it represents both electromagnetic fields.
the stationary and time-dependent flux of electromagnetic If we apply the first theorem keeping in mind the
power in amplitude and direction that is the subject of this orthogonality property of the E&H fields, we
article. conclude that the total electromagnetic power flux
P(tot) through the entire surface of the wire is given
II. THEORY by,
In order to show that the Poynting vector P is P (tot) = P x total wire passage area A
mathematically and physically correct and represents the In other words,
energy flux per unit area per unit time for electromagnetic P (tot) = E x H x 2 π a L . . . . . (A4)
fields in 3D configuration space, we validate it in both DC Obviously, the traverse area A = 2π. a.L
and AC circuits arbitrarily non-radiating without loss of When we substitute the values of E&Hon the wire
generality. surface given by equations A1 and A2 in the
More precisely, we consider the conservation of expression A4 we get,
electromagnetic energy and momentum at the surface of a P(tot) = IV / (2πa L) * 2 Pi a L watts. . . . . . (A5)
cylindrical conductor carrying current in both cases, direct In other words,
and alternating. P (tot) = IV watts. . . . . . (A5)

In order to simplify the calculations and not to worry This is exactly the total Joule heating through the
about the details of the electric circuit, we limit our analysis wire in Joules/s i.e. (Watts).
in the two different cases A and B to a long cylindrical wire b) Which means that the heat losses in the wire are
traversed by a current. provided by the electromagnetic field in the outer
A. Case IIA space of the wire.
a) DC current carrying conductor Further, the E/H ratio at the wire surface r=a is given
Consider the simplest case of pure resistive DC by,
circuits, a uniform cylindrical wire carrying DC E/H = (V/I) * (2π a /L) ohms
hence the length of the wire L and the radius a [4]. Since,V/I is the total resistance of the wire R in ohms
Here we arbitrarily define the electro dynamic which varies with the physical properties of the
quantities in SI units as, conductor, so we have,
Voltage across wire = V volts E/H= R * 2πa /L . . . . . . . . . . . . (A6)
Current in the wire = I amps and not to be confused with the impedance to
And the electromagnetic field strengths due to the propagate electromagnetic waves which is 120 π or
current in the wire, 376 ohms in a vacuum.
H = I / (2 π r) ampere m^-1 . . . . (A1)
The wave impedance of an electromagnetic wave
For r equal to or greater than a. is the ratio of the transverse components of the
π =3.1416 …… dimensionless. electric and magnetic fields E/H denoted by Z0
Z0 = Sqrt (μ/∈) where μ = 4π× 10−7 H/m (Henries
And, per meter) is the magnetic permeability and ∈ =
E = V / L volt m^-1 . . . . . (A2) (1/36π) × 10−7 F/m is the electrical permittivity.

For r equal to or less than a. Z0 Approximation at 120π or 376 ohms.

Furthermore, we have shown in a previous article [2]
while E=0 outside the wire, that is to say for r when Maxwell's equations are combined with the
greater than a. relativistic mass-energy equivalence relation
E=mc^2, that the total energy of the magnetic field in
The two vectors E and H are perpendicular to entire space corresponds to the kinetic energy of a
each other as shown by equation 4. single charged particle producing this field, which is
also true in the case of a current-carrying conductor.
And, the Poynting vector P is the vector
multiplication of two vectors, E & H, Assuming that u is the drift velocity of electrons
P= IV / (2 π a L) watts/m^2 . . . . . (A3) in the wire which is directly proportional to the
electric field E,
At any point on the surface of the wire r = a with u=KE,
its direction of power flow normal to the E&H plane, Where K is the material-electron mobility,
i.e. parallel to the axis of the wire. And assuming that e, Me and Ne are respectively the
charge of the electron, its rest mass and the total

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Volume 7, Issue 3, March – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
number of electrons in the wire, simple calculations And the total Poynting vector power traversing the
can show that the total magnetic energy in the whole entire wire P(tot) is given by,
space Ut(H) is , P(tot) = ImVm . sin(wt) sin(wt-Phi) watts. . .(B3)
Ut(H)=Ne Me u^2/2 . . . . . . A(7)
This means that the total energy density of the What exactly is the instantaneous heating per joule in
magnetic field Ut(H) corresponds to the drift kinetic the wire.
energy of the electrons Ne.
In other words,
B. Case IIB b) Heat losses in the wire are ensured instantaneously
a) AC current carrying conductor by the electromagnetic field in the outer space of the
In case IIB we consider a uniform cylindrical wire wire.
carrying an alternating current, where the length of the wire The ratio E(t)/H(t) = Sqrt (R^2+X^2 )=Z ohms varies
L and the radius a [4,5]. with circuit impedance Z but their pointing vector
multiplication ExH remains matched to the
Case B is similar to case A except that the voltage V electromagnetic power flow through the wire.
and the current I and therefore E and V vary slowly over
time, which does not affect the calculations. If we need to calculate the time average of the
Poynting vector Pav, then,
Consider an inductive-resistive R-L alternating current Pav = 1/2 Vmax.Imax. cos (Phi) /2.π.a L watts/m^2.
electrical circuit where the reactance X is the opposition . . . (B4)
presented to the alternating current by inductance. Here we
assume that both I&V are time dependent sine waves. In equation B3 we have used the property of sin
functions, the time average of sin wt .sin (wt-Phi)
Following similar procedure to that of section A we from 0 to 2 Pi/w , for the Phielement of [0, π/2] is
have, equal to ½ cos Phi.
Voltage across the wire V(t) = Vmax sin wt volts
Current in the wire I(t)= Imax sin (wt-Phi) amperes Similar to case IIA, the time average of the total
Where w is the angular frequency and Phi is the offset power flow through the entire wire area P(tot)av is
or phase angle called the phaseshift measured in radians. obtained by multiplying P(t) by the wire area, i.e. 2 π
a L, hence,
Phi=jwL / R in the complex phasor diagram. P (tot)av = P x total passage area A
Where the imaginary unit j, defined by its property j^2 P (tot)av = ½ Imax Vmax watts. . . . . . (B5)
= −1, is introduced to simplify the calculations.
Imaginary time is a mathematical representation of III. INTERPRETATION OF THE RESULTS
time that appears in many approaches such as special
relativity and quantum mechanics. Mathematically, Poynting's vector theorem P = ExH is one of the
imaginary time is real time that has been rotated counter- universal laws of physics that applies to electromagnetic
clockwise so that its coordinates are multiplied by the fields in AC and DC circuits. The analysis of two arbitrary
imaginary unit j. cases II-A & II-B of electromagnetic fields of DC and AC
It is obvious that w=2 Pi f where f is the temporal circuits proves that Poynting's law P = ExH applies equally
frequency of I and v. well to AC circuits as to DC circuits generating
I(t)=V(t) /Sqrt (R^2+w^2.L^2) . . . . . . . . (B1) electromagnetic fields that they are time-dependent or
Obviously, stationary.
H(t) = I(t) / (2 πa) ampere / m
And, The assumption that a bundle of electromagnetic
E(t) = V(t) / L volts / m energy called a photon in quantum mechanics has an infinite
On the surface of the wire. lifespan relative to n' any reference, greater than 10^18
Again, the two vectors E(t) and H(t) are perpendicular yearsis beyond our imagination, whereas an alternative
to each other by Eq5. understandable can be explained in the following[6,7],
 Although the above hypothesis or mathematical model has
When we apply P(t) = E(t) x H(t), (vector had reasonable success in solving modern quantum
multiplication of two vectors, E and H, we have, mechanical problems, it is inadequate in electromagnetic
P(t) = Im .Vm / (2.πa L) sin(wt) sin (wt-Phi). . field problems.
watts/m^2 . . .(B2) In electromagnetic field problems, it is preferable to apply
Maxwell's equations and Poynting's universal theorem.
At any point on the surface of the wire r=a.  The photon cannot live forever and an alternative to the
assumption that the photon has an infinite lifespan is that
Applying the Gaussian divergence theorem again with the photon, defined as a bundle of electrical and magnetic
the orthogonality property of the E & H fields, we get, energy in EM theory, is annihilated or created by its
P(t) = ImVm . sin(wt) sin(wt-Phi)/2 π a L. . watt/m^2 . interaction with electromagnetic field of bound or free
. .(B2) charges such as electrons in case IIA&IIB.

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Volume 7, Issue 3, March – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
 The photon (EM field bundle)-electron interaction is REFERENCES
possible in several ways, such as DC and AC circuits
generating stationary or time-varying EM fields where the [1.] Relativity, Special and General Theory, Through,
energy of the magnetic field matches the kinetic energy of Albert Einstein, Ph.D. Professor of Physics at The
the electron of mass Me. University of Berlin, Translated By Robert W.
 There is an obvious analogy between the EM energy Lawson, M.Sc., University of Sheffield, New York,
bundle in EM wave propagation called photon and the EM Henry Holt and Company, 1920.
energy bundle of AC&DC circuit fields. The latter has two [2.] I, Abbas, A rigorous reformulation of Einstein
components of two time-varying electric and magnetic derivation of the special theory of relativity, IJISRT
fields, both of frequency f and normal to each other and review,Mar 2022.
having a phase difference Phi. The E/H ratio differs and is [3.] Maxwell’s Equations and the Principles of
equal to the impedance Z of the electrical circuit while Electromagnetism (Physics (Infinity Science Press))
their product ExH still represents the EM power flow. 1st Edition by Richard Fitzpatrick
[4.] I.Abbas,Why the Poynting theorem P = ExH is quite
Photons are expected to decay or completely transform valid for direct current circuits, Researchgate , October
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happens in the physical phenomena of Compton scattering [5.] I, Abbas, Why the Poynting theorem P=ExH is quite
and the creation of electron-positron pairs. The lifetime of valid for DC circuits,PartII,Researchgate, Mar 2022.
the photon, which seems like an eternity, is in fact limited [6.] Cosmology Research Update, What is the lifetime of a
by the chances of interaction with free or bound charges. photon? ,July 24, 2013 Tushna Commissariat.

In the context of special relativity, taking into account [7.] I, Abbas, how can photons of light have mass and
the relativistic dilation of time, that is to say that the photon momentum ? , Research gate, April 2020
crosses a relativistic micro instant with respect to its own
reference frame.

However, 4D EMW bundles bounded by obstacles of

electrical charges in empty space can be assumed to have a
mean free path or mean lifetime regardless of duration, but
not infinite.

IV. CONCLUSION

A rigorous analysis of EM fields from AC and DC

circuits can show that Poynting's theorem P=ExH is a
universal law of physics that applies to both stationaryand
time-varying electromagnetic fields.

The extensive study of EM power flow in the simple

case of DC and AC circuits, a uniform cylindrical wire
carrying current validates this proposition.

In addition, the present study suggests that photons

from electromagnetic wave radiation and electromagnetic
energy bundles cannot live forever. An alternative to the
assumption that the photon has an infinite lifetime is that the
photon, by analogy to a bundle of electrical and magnetic
energy in EM theory, is annihilated or created by its
interaction with the electromagnetic field of charges free or
bound.

The EM field-electron interaction is possible in several

ways, examples are DC and AC circuits generating
stationary or time-varying EM fields where the energy of the
magnetic field matches the kinetic energy of the free
electron.

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