# ELECTROMAGNETIC FIELD CALCULATIONS

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16 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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4th revision 13 March 2006

By Charles A. Hobson BA, BSc (Hons)

of radio waves in space. It all says pretty much the same thing: The
electric and magnetic f
ields are transverse, at right angles to each other in the
x and y planes and travel in the z direction at the speed of light. Maxwell’s
equations with the derived wave equations and the Poynting Vector (
E X H)

provide eloquent mathematical descriptions of

these phenomena, which
Heinrich Hertz actually demonstrated in 1888.

Because these waves were invisible with few free electrons in space to
support such descriptions, Maxwell came up with the notion of “Displaced
Current” to mathematically handle the si
tuation. However, the
E X H

Poynting
Vector expressing power intensity (Wm
-
2
) in the z direction puzzles me.

The Poynting Vector expects the
E

and
H

fields to be in phase to get
Watts/m
2

average real power flow. But how can this be when it is believed
th
at the two fields support each other as they travel through space? The
energy travelling in space would have sinusoidal variations in amplitude. If this
is the case, then where is the energy coming from and going to as it travels
through free space? Does t
his violate the conservation of energy law? I
maintain that there is a phase difference of 90
0
between the
E

and
H
fields.
Further to that, I believe that EM energy moves between the
E

and
H

fields
supporting each other as they travel. And lastly, that the

sum of the
E

and
H

field energy densities would be constant over complete cycles and decrease
by 1/r
2

from the energy source. See the equations and waveforms below:
(Note that in figures 1A and 1B,
E
0

and
H
0

represent the magnitudes of the
Electric and M
agnetic fields respectively, whereas the red U
0

and blue U
0

represent the magnitudes of the Electric and Magnetic field energy densities) .

Figure 1A

Figure 1B

Mind the units: ½ x Cm
-
1

x V
2
m
-
2

= ½ x V
2
Cm
-
3

=

U
E
m
-
3

(energy density)

M

If you are having difficulty coming to terms with the 90
0

phase difference and
constant energy notions, consider this. Charge a perfect capacitor to some
voltage. It is well known that the stored energy in the capacitor wou
ld be U
C

=
½ V
2
C. Now connect the capacitor across a perfect inductor. The capacitor
discharges its energy into the inductor. When this is complete, the capacitor
voltage will be zero and the discharge current in the inductor will be at its
maximum indicat
ing the inductor has received all of the capacitor’s energy.
When this occurs, the energy will be in the form of a magnetic field expressed
as (U
l

= ½
I
2
L) The inductor will try keeping the current flowing and in the
process start recharging the capacitor
in the reverse direction. When the
inductor finally returns its energy back to the capacitor, ”
I”

will be zero and “V”
will be at its original value with opposite polarity. In other words, L and C
oscillate the energy at a sinusoidal rate. “V” and “
I”

will

be 90
0

out of phase
with each other, but more importantly, the sum of energies in the capacitor
and inductor will always be constant. (perfect inductor and capacitor
assumed) Hence, energy is conserved. I believe that something equivalent to
this occurs w
ith EM radiating in free space. The only difference is that the
energy is trapped in the LC circuit but not so in free space where it is fanning
out. Could such a notion be compatible with the quantum theory?

Now, back to the EM concept with the magnetic
and electric field energies:

U
E

and
U
H

The total energy is:
U
T

=
U
E

+
U
H

which is of course

in units

of
Joules per cubic metre.

Since energy travels at the speed of light, (300 x 10
6

/ sec.), 300
x 10
6

cubes of energy
U
E

would pass through one face
of the 1
metre cube volume in one second.

And since Watts = energy rate per second (J/s), then the power
flow through one face of the cube (rate of energy per sec.)
would be: W
E
/m2 = 300 x 10
6
x
U
E
/s for the electric field energy.
It follows that: the ma
gnetic field energy would be W
H
/m
2

= 300
x 10
6

x

U
H

/m
2

The total EM power passing through a square metre in space
(Wm
2
) = W
E
/m
2

+ W
H
/m
2

However, this is reactive power as it is
not seeing a resistive load in space. Even a receiving antenna is
reactive
. That is, it consists of mainly L and C elements and the
unavoidable tiny resistive bits. However, the usual antenna
output matching resistive load will see the real power.

Finally, what’s happened to the Poynting Vector
EXH
? This
cross product does
not seem to work with the
EH

vectors when
they are 90
0

out of phase as I suggest they are. That is, one
does not multiply such vectors together easily. One must add
them. Since kinetic energy is always on the move, I believe that
the transverse waves must
always travel directly away from their
source (antenna) at the speed of light.

EPILOG

My treatment of electromagnetic radiation in terms of energy was just a notion
of mine conjured up while sitting in my garden watching birds and dealing with
a nice
cup of tea. It would be unacceptable for me to say that my view on EM
phase relationships is correct and that other views regarding this matter are
wrong.

No part of the electromagnetic spectrum has ever been seen. We know of its
existence from the eff
ects it causes. These effects: light, x
-
rays, microwave,
high frequency, etc. are well known things and their explanations abound. But
perplexing mysteries persist. For example, electric and magnetic fields are
considered to be the consequence of moving

charges but there is no
evidence of free charges in space. Maxwell, through his equations, explained
that there must be displacement current (charges) to support EM radiation in
space. Yet no one has ever measured displacement current to prove its
existen
ce.

Quantum theory says that radiation is made up of photons with their energy
being a function of frequency. The plot thickens because the theory also
states that photons have momentum but have zero rest mass. Both wave and
quantum theories agrees on on
e thing though: electromagnetic energy, waves
or photons, travel at the speed of light. Then Einstein comes along and says
that E = Mc
2
.

This theoretical stuff boggles the mind. Perhaps a “Unified Field Theory”
would put the various pieces of this myster
y together into something
consistent and acceptable. At the moment there seems to be no acceptable
“Unified Field Theory”, which appears necessary to fully understand