# Summary

As
highlighted on several occasions, this section on science represents
a personal learning process, which is why most pages are presented as
a discussion and not authoritative statement of facts. However, as a
learning process, electromagnetism can be fraught with many difficulties
that extend, not only out of the different forms of
*mathematical notation*,
*units* and plethora of
*concepts*, but also from the fundamental ambiguity
that is associated with idea of wave-particle duality.

So is the EM spectrum a science of wave or particle interactions?

Clearly, a *number of pages* in this section have tried to present
a visualisation of an EM wave in the form of a transverse sine wave,
which might be viewed as a composite of an electric [E] wave and a magnetic
[B] wave travelling in-phase, but perpendicular to each other. Of course,
this description of an EM wave seems to be predicated on an assumption
that a source charge was originally undergoing acceleration, although there is
still the question of a degree of ambiguity surrounding the propagation
of electric and magnetic field, when the charge is on subject to constant
velocity, e.g.

How do electric and magnetic fields associated with a moving charge propagate into the surround space?

Despite our apparent ability to* ‘see’* EM waves in the form
of light, we cannot actually see these waves in transit or for that
matter necessarily describe them with any certainty. Of course, if it
is not a wave, then issue of the *structure of a photon* seems to also
be missing from most mainstream references. However, we might realise
that trying to *‘visualise*’ these waves in 3-dimensions, even
if they exist, cannot be resolved in the form of a
*mechanical wave*,
as previous discussed. First, there is the obvious problem of the propagation
medium, when discussing EM waves in a vacuum, plus a more fundamental
problem of geometry. Previously, we have discussed the basic nature
of a *3-D mechanical wave*
in the form of a sound wave, but with the understanding that the underlying
propagation mode was longitudinal, not transverse.

Is it more logical to describe light as a wave or as a photon?

Today, we are aware of the *wave-particle duality* issues that subsequently
arose following the wider acceptance of the quantum model. However,
at the time of Maxwell and Larmor, the consensus had switched from Newton's
corpuscular theory to a general acceptance of a wave theory underpinning
all electromagnetic phenomena, inclusive of the propagation of light.
Only after the work of *Planck* and
*Einstein* at the beginning of the 20^{th}
century would the particle analogy begin to return in the form of a
photon. However, from a more pragmatic perspective, we know that light
from our own sun, and the stars beyond, must have an ability to transport
energy through the near vacuum of space and over its vast distances.
As such, a macroscopic model in which we know the sun radiates EM energy
in the form of light, in all directions, might serve as a good starting
point for this summary. Therefore, in order to help visualise the discussion,
the following animation is attempting to show the path of 6 EM waves-photons
radiating along all [±xyz] directions from a central radiating body,
e.g. a star. Initially, we might visualise each path as an EM wave being
continuously sourced by an oscillating charge or, alternatively,
as a finite pulse of EM energy, i.e. a photon.

As such, the 6 representative dots expanding in each direction are
only illustrative of a line of radiating energy that travels along every
conceivable outward path from a centralised star at the speed of light
[c], not just the 6 shown. Equally, the boundary of the sphere only
represents the distribution of the energy waves/pulses over an expanding surface
area, which increases in size with time. At this point, it might be
useful to present some actual figures associated with the ability of
an EM wave-pulse to transport energy between the stars. For this example,
let us assume the source of EM light is a sun-like star that is 100
light-years from us, which has the same properties as our own sun:

Parameter |
Value |
Units |

Radius | 6.96*10^{08} |
metres |

Mass | 1.99*10^{30} |
kg |

Age | 1.58*10^{17} |
seconds |

Temperature | 5777 | Kelvin |

Luminosity | 3.84*10^{26} |
W or J/s |

In the context of astronomy, the luminosity of an object, like the sun, corresponds to the amount of electromagnetic energy a body radiates per unit of time. However, the temperature [T] and the luminosity [L] of the sun are linked together by the following equation:

[1]

The value of luminosity in the table above has already been calculated
from the known temperature of the sun and its radius [R], where the
symbol [σ] corresponds to the Stefan-Boltzmann constant. Now,
on the basis of this description, the luminosity defined in [1] corresponds
to the electromagnetic energy radiated per second. As such, we might
initially try to link this equation to the Larmor formula as discussed
in *Electromagnetic Radiation*:

[2]

However, without further information about [q] and [a] at the surface
of the sun, it would seem that [2] cannot really be progressed any further,
at this stage. Of course, with a unit charge per particle of 1.6*10^{-19}
coulombs, there is the obvious suggestion that a lot of charged particles
must be involved, as well as some possibly phenomenal acceleration.
Therefore, you might wish to check whether anybody has ever verified
the Larmor formula at the macroscopic scale or whether the acceleration
of the charge is actually a direct cause of the radiation predicted.
This said, let us return to the main thread of the example, as we now
need to estimate the luminosity, i.e. the energy per second, arriving
at a distance of 100 light-years from the star in our example. In this
case, the energy associated with the luminosity, as given in the table
above, has to be averaged over the surface area of an expanding sphere,
which is 100 light-years in radius in this specific example. We shall
call this flow of EM radiation its flux [F]:

[3]

As we might expect, the flux or power passing through 1 square metre,
at a distance of 100 light-years and subject to the inverse square law
is very small. However, it might be worth highlighting that equation
[3] is also illustrative of why the EM energy radiated aligns to the
same inverse square law reduction as a 3-dimensional mechanical wave
and why this has nothing to do with the mechanism of energy transport,
simply a correlation to the same geometry, i.e. the surface area of
a sphere. However, this geometry might also being suggesting that any
single path of what we might call a *‘ray
of light’ *cannot lose any energy in transit. At one level, we
might reflect on the fact there is no obvious mechanical medium in the case
of a vacuum in which to lose energy, which can then be tied in with
the basic assumption of energy conservation, i.e. energy cannot be created
or destroyed. However, it is not clear whether this model can be aligned
with the classical image of the EM waves propagating as in-phase sine
waves with the practical reality of the near chaotic random motion of
billions of charged particles near the surface of a star. Again, if
we make reference to the Larmor formula in [2], we can see that this
sort of visualisation seems to be a fairly impossible task at the macroscopic
level.

*So what about the pulsed energy/photon model?*

If we step outside the timeline of classical electromagnetism, we
know that the work of Planck and Bohr would lead to the idea of
*quantised energy*, such that the
energy of an EM wave is now described in terms of a discrete quantity
defined by the equation [E=hf]. As such, we might realise that the model
of a *dipole antenna*,
subject to an alternating constant sine wave voltage, may not be applicable
when considering the near random chaos of charge movements near the
surface of a star. In this context, the charged particles, which we
are assuming are still the sourced of any EM wave-photons, would accelerate,
collide and change direction constantly and therefore be more likely
to produce discrete pulses of energy radiating in all different directions.
However, within this generalised model, we might still make some 1^{st}
order approximation of the number energy pulses/photons required to
deliver the same energy per second per unit area as required by [3].
To do this, we need to make the very broad assumption that the photons
arriving from a distant star are all in the visible yellow light range
of the electromagnetic spectrum. If so, we can estimate the energy per
photon via Planck’s equation [E=hf] and if we then divide this figure
into the value provided in [3], we will have an estimate of the number
of photons arriving per second:

[4]

As an aside, it might be interesting to try to determine the total
effect on the star’s mass given that it has been losing this energy-mass
in all directions over the age of the star, e.g. 5 billion years. We
can do this by first converting the photon energy to an equivalent mass
by equating Einstein’s and Planck’s energy equations [mc^{2}=hf]

[5]

However, the figure given in [5] only accounts for the photon-mass lost per second per square metre at a distance of 100 light-years. Therefore, we have to reverse the calculations to get the total mass lost at the surface of the star.

[6]

Finally, we have to multiply the result in [6] by the age of the star, which we have assumed to be 5 billion years and then compare the result as a percentage of the total mass of the star, as given in the table above:

[7]

Of course, we do not really need the photon model to do this calculation,
as we could have achieved the same result by rearranging Einstein’s
equation [m=E/c^{2}], where the energy per second corresponds
to the luminosity figure in the table and the result again multiplied
by the age of the star:

[8]

The result in [8] aligns with the result in [6]. So while the mass lost per second is large on a human scale, the total mass lost by the star due to electromagnetic radiation in the visible spectrum only accounts for 0.03% of the star’s mass, even after 5 billion years. However, while we now have some better idea of the scale of energy being transported within this example, we do not necessarily have any clearer idea of how this energy is actually being propagated within the wave-pulse-photon model.

*So what is in a wave?*

In the case of a *mechanical wave*,
a given model can describe the interaction between the potential and
kinetic energy associated with the displacement of the physical media.
However, Maxwell’s equations support the conclusion that electromagnetic
wave are self-propagating in a vacuum, although we now also know that
subsequent developments led back to the duality of a wave-particle description.
In the previous discussion of *electromagnetic
wave propagation*, we showed that [9] is a solution of Maxwell’s
equation for a plane wave:

[9]

It was also shown that a sine wave is a possible solution to [9]
as it will provide the constant acceleration required by a charge to
radiate. Equally, we might recognise that this solution aligns nicely
to the model of an oscillating
*dipole antenna*
being driven by an alternating sine wave voltage. As such, the following
animation supports the classical image of an EM wave propagating outwards
from a source charge oscillating under acceleration in space.

However, we are now trying to decide whether this model is realistic in the context of what must be the near chaotic random motion of billions of charged particles at the surface of a star.

*So how might we perceive the nature of the energy
propagation within such a large-scale electrodynamic system? *

This might be another point where we might gain some additional perspective,
if we step back from the specific issue of electromagnetic waves in
order to consider some slightly tangential issues. Normally,
*SI/MKS units* are generally used
to define such quantities as the electric [E] field and the magnetic
[B] field. However, there are other systems, such as the Gaussian/GCS
units, which resolve the units of composites quantities, such as the
[E] and [B] field, based on different assumptions. In fact, these unit
systems can lead to 2 different presentations of Maxwell’s 3^{rd}
and 4^{th} equations:

[10]

As such, we might question the meaning of [c] in these equations and consider the suggestion that [c] is only being used as a conversion factor for changing the unit of time to the unit of space in-line with the Lorentz transformation. Of course, this then raises the wider issue of how we have come to think about composite quantities, such as energy, force, fields etc, and whether we always realise how these composite quantities have been specified in the first place. So let us ask the previous question in an extended form:

H*ow might we perceive the nature of the energy
propagation within such a large-scale electrodynamic system in terms
of the most fundamental units?*

In this context, the *fundamental
units* are length, time, mass and charge. Of course, some might
argue that length and time are not separate fundamental units, when
viewed as spacetime. However, almost all basic physics still accepts/requires
the separation of these units. At one level, we might describe these
units as the ‘*stage*’ on which the *‘actors’* perform; where
the ‘*actors’* in this case are the other 2 fundamental units,
i.e. mass and charge. However, again, it is possible to question the
fundamental nature of these units, for example, if we equate E=mc^{2}=hf,
it might be argued that [h] and [c] are both constants and therefore
the implication is that mass is somehow equivalent to frequency. On
a slightly more philosophical note, we might also question the very
nature of a particle at the sub-atomic level, where we are unable to
name the substance the ‘*particle*’ it is made of. As such, frequency
may yet be proved to be a more fundamental concept than mass. Still,
at some level, we have the tangible perception of ‘*particles*’
and therefore the concept of rest mass becomes a useful unit of containment,
but not so when considering energy in the form of an EM wave, which has no
rest mass. Of course, questioning the true nature of sub-atomic particles
could also cause a problem when considering the *fundamental
idea of charge*, which classical physics considers being an
attribute of a mass particle, i.e. electrons and protons. In addition,
we may also wish to reflect further of the idea that charge cannot really
be said to exist in isolation, i.e. it is only a concept that exists
between 2 charged ‘*particles’*. Of course, even if we accept the
4 fundamental units at face value, we still need to
*introduce the concept* of force,
with units of m.kg/s^{2}, and energy, with units of m^{2}.kg/s^{2},
in order to describe how ‘*action*’ takes place. However, reverting
to the fundamental units does not really seem to help describe these '*
composite
quantities*’ and it may be more physically meaningful to
consider force as something that results from energy and energy as
something that ultimately comes in 1 of 2 basic forms, i.e.
potential and kinetic. In this context, potential energy might
simply represent the state of a system that is not in energy
equilibrium and kinetic energy is the manifestation of a system
trying to restore equilibrium, either via particle motion or wave
motion. Of course, if the particle’s mass is related to frequency,
then there may be only one real mechanism at work. See '*Scope
of Speculative Ideas*' foe a more expansive discussion of
such issues.

[11]

We can see from [11] that the force equations associated with gravity and the electric field share a very similar form based on the inverse square law. Equally, if you integrate the force [F] with respect to distance [R], the implication is that gravity and the electric charge field both correspond to a form of potential energy. Now, within this classical model, we might consider a system that consists of a large central and positively charged mass and a much smaller negatively charged mass falling radially towards the centre from a great distance. At the start, it would seem that the negative charge can tap both gravitational and electric field potential energy, but in doing so, it would acquire kinetic energy, i.e. velocity, and as such, a moving charge would have a magnetic field that would surround the moving negative charge with an axial symmetry. However, the negative charge would also be constantly accelerating under the influences of both gravitational and electrical forces. Therefore, according standard texts, the accelerating negative charge must also be emitting EM radiation, which Maxwell’s classical equations describe as a wave, but modern physics would prefer to describe as a stream of photons. While the mechanisms may be debated, at a general level, we can stated that this system only started out with access to gravitational and electric field potential energy, ignoring the rest mass energy, but ends up with kinetic energy of motion, a magnetic field that can exert a force and therefore has an ability to do work, i.e. transfer energy, while also radiating EM wave-photon energy due to its acceleration. However, despite the apparent complexity of this description, we might try to generalise this complexity to one issue:

*How does energy move in space and time?*

While this question might generalise the problem, it does not necessarily make the answer any easier, but let us try to make some attempt at an answer. It would seem that energy moves around either in the form of matter particles or waves, where each form is subject to certain fundamental restrictions:

- Matter cannot move at the velocity of light due to its rest mass.
- While due to its light speed, EM waves cannot have rest mass.

While not wishing to run too far ahead of the timeline under discussion,
*relativity* places some unusual conditions on our earlier pulse/photon
model. In later models, the idea of the [E] and [B] fields oscillating
as 2 in-phase sine waves runs along side the general concept of pulse-photon
of energy that can also propagate through the vacuum of space. As such,
the energy pulse/photon would represent an energy density moving between
2 points in space and time. Now, irrespective, of whether we describe
this energy as a wave, pulse or photon, there seems to be common agreement
that this *‘thing’* must move at the speed of light [c]. As such,
it cannot have any rest mass, although it is allowed to have momentum
via the relativistic energy equation:

[12]

Now with reference to our example above related to the energy
radiated by a star, the first animation above along with the
subsequent description may have given a mental image of a photon
being a very small, but finite particle-like ball of energy.
Equation [12] may also appear to allow a visualisation of a photon as a particle-like object and conclude
that it might continue to propagate in a straight line, through vacuum,
solely by virtue of its momentum. However, appending Planck’s law [E=hf]
to [12] reminds us that the energy of a photon is still defined by its
frequency and we are forced back towards some sort consideration of
the wave model. However, there does not seem to be any clear picture
of how the electric and magnetic fields of an EM wave ‘*fit within’*
the constraints of a particle-like photon model. For example, we might
perceive the photon corresponding to some integral number of wavelengths,
e.g. 1, which in the case of yellow light is of the order of 600*10^{-9}
metres. As such, we might perceive a photon as being very, very small,
but this image falls apart within the long-wave radio spectrum, which
can have wavelengths measured in kilometres. At this point, we might
ask the obvious question:

Well, certainly the idea of a radio photon being over a kilometre
wide is problematic to say the least. However, special relativity may
address this problem by suggesting that a photon has no physical length
in propagation, even though it would appear to represent some form of
energy density [joules/m^{3}]. The argument being that due to
its implicit propagation velocity [c], the concept of *
length contraction* would result in
the observer of a photon not being able to measure any physical length,
although it would presumably continue to have width. To be honest, any
discussion of EM waves and photons seems to breakdown at this point
and classical electrodynamics may have to simply defer the problems
to *quantum physics*. However,
before we leave the world according to classical physics, let us consider
once more the microscopic model of a single charge oscillating under
the influence of an alternating sine wave voltage:

The left side of the animation above shows the effect on the electric
field lines, while the right shows an equivalent view based on streams
of photons being emitted by the charge. The field lines or photon streams
are shown at different angles, e.g. 0, 30 and 60 degrees, from the maximum,
which is always perpendicular to the axis. The oscillating red lines
on the left reflects the total electric field [E=E_{R}+E_{T}]
as defined by the *Larmor
formula*, such that you see the effects of [E_{R}] reducing
by [1/R^{2}], while [E_{T}] only reduces by [1/R] and
so the latter quickly becomes the dominate field as the radius from
the charge increases. However, the photon view on the right might better
illustrate the situation as the charge passes through the centre of
oscillation, i.e. maximum velocity, minimum acceleration. Under this
condition, i.e. a=0, the Larmor formula would suggest the charge should
not emit any photons due to zero acceleration. Of course, over an extended
period of time, the statistical distribution of photons will still conform
to the radiated power predicted by the Larmor formula as suggested by
the intensity of the yellow shading. However, it is difficult to explain
how the photon carries the electric and magnetic field components as
it propagates, especially in view of the fact that a photon’s trajectory
is said to unaffected by the electric and magnetic fields of another
charge. In contrast, the wave-like view on the left of the animation
suggests that the electric field is changing at every point in space
and time around the oscillating charge.

*So what are Maxwell’s time-dependent equations
really telling us? *

The following equations partially replicate the form of Maxwell’s time-dependent equations, as shown in [10], but use Gaussian units and have been re-arranged with respect to the time derivate:

[13]

In isolation, the first equation in [13] suggests that any change in the E-field with respect to time causes a corresponding rate of change in the magnetic field with respect to space with the suggestion that this equates to a propagation in space with a velocity [c], The second equation in [13] then implies a complementary relationship between the magnetic and electric fields. Now the source of the equations in [13] are linked to Maxwell's time-dependent 3rd and 4th equations, which unlike the time-independent 1st and 2nd equations, make no direct reference to the field strength being subject to an inverse square law with distance. As such, we might assume that [13] describes a process of self-propagation that can continue indefinitely. In contrast, the space surrounding a charge in motion with velocity [v], but no acceleration [a], would still presumably create a change in the magnetic field [B] and we might still assume that the strength of the E-field would be linked on the electrostatic equation:

[14]

However, even in this case, there seems to be a suggestion that any change to the E-fields
would still propagate outwards at the speed of light [c]. This statement
does not seem to be explicitly supported by Maxwell’s equations, but
arises as a consequence of relativity, which requires all *action-at-a-distance*
forces to propagate within the restriction of [c]. Now based on the
following relationships between energy density [η] and the strength
of the [E] and [B] fields, any propagation of a change to the E-field
would also lead to the suggestion of a propagation of potential energy:

[15]

At first, this suggestion would appear to contradict the Larmor formula,
which requires the charge to be accelerating for energy to be radiated.
However, there may be some advantage in considering this situation in
terms of the Gaussian system of units in which [E] and [B] fields are
both defined in terms of the same fundamental units. Although the fundamental
units of [E] and [B] in the Gaussian system do not immediately convey
any obvious meaning; [E^{2}] and [B^{2}] both align
to the *units of energy density*,
which possibly raises an interesting question about the photon model:

*What is the energy density of a photon, which
is said to carry energy [E=hf], but cannot specify its volume? *

At this stage, we will only table this question for *
another discussion*, as we need
to return to the implication that a moving charge, subject only to constant
velocity, propagates energy. Possibly the best way of resolving the
apparent contradiction is to simply look at the E-field strength and
the associated energy density [η=ε_{0}E^{2}]
with respect to [E_{R}] and [E_{T}] as defined by
Larmor's formulation:

[16]

We know that (a) is derived from the time-independent form of Coulomb’s
laws, which by itself does not account for the measured power, in any
practical system, due to its additional sensitivity to the radius [R].
In contrast, (b) can account for the measured power, but is said to
require the charge to be accelerating in order to generate [E_{T}].
Let us try to put this situation into some perspective in respect to
our earlier example related to energy received from a sun-like star
some 100 light-years from Earth. In this example, the actual energy,
calculated in [3] based on (b) above, would fall from 3.42*10^{-11}
joules/second to 3.83*10^{-47} joules/second, if based solely
on (a). In practice, we might also realise that the chaotic movement
of charged particles near the surface of a star would more closely approximate
a model in which billions of charged particle are subject to extreme
acceleration and then collide and change direction. As such, the energy
propagated by any charge moving with constant velocity would be simply
swamped by the energy propagated by all the other charged particles subject to
acceleration. However, this statement would probably hold true for almost
any real world system. Therefore, we might wish to look, yet again,
at the Gaussian form of Maxwell’s time-dependent equations for a more
general description of how the [E] and [B] fields interact in order
to self-propagate.

Within a plane-wave model, we might assume that the E-field oscillates
in the y-plane, while the B-field oscillates perpendicularly in the
z-plane, which leads to propagation in the x-plane. Therefore, as [E_{Y}]
changes with time, it causes a change in [B_{Z}] along the x-axis.
By the same token, any change to [B_{Z}] along the x-axis must
also be seen as a change of [B_{Z}] with respect to time and
therefore causes a knock-on change to [E_{Y}] along the x-axis.
Therefore, these rates of change are self-propagating and must propagate
perpendicular to [E] and [B] with velocity [c]. In the case of [E_{T}]
dominating, the direction of energy propagation would be radial, as
shown in the animation. However, we might realise that any energy propagation
due to [E_{R}] would be subject to an additional inverse square
factor and in a direction perpendicular to [E_{T}]. As such,
we might again be describing a system that is seeking to
*restore its energy equilibrium* by radiating away its excess
potential energy. Whether the process of energy radiation takes the
form of an EM wave or photon remains a subject of further debate