# Electromagnetic Wave Propagation

In order to describe the propagation of an electromagnetic wave,
we need to tie the derivation of any *‘wave equation’* to Maxwell’s
equations. However, when discussing the propagation of EM waves in vacuum,
Maxwell’s equations can be simplified by the assumption that the charge
density [ρ] and the current density [J] can be set to zero. The idea of a self-propagating EM wave will also require
a departure from the description of an electric field [E] being directly
linked to a charge particle [q], either stationary or in motion, which
was then complemented by a magnetic field [B] that only exists when the charge moves
with velocity [v].

*Note: Newton’s 2nd law of motion implies that a charge particle
moving with constant velocity cannot be subject to a force [F]. Therefore,
it is difficult to explain how either a static charge, or one moving
with constant velocity, might continuously radiate energy without any
obvious energy being put into the system. Of course, a charge particle
undergoing acceleration [a] must be subject to a force [F=ma] and the
displacement of the charge equates to work done or energy being input
into the system, such that it may subsequently radiate this energy in
the form of EM waves. *

From a classical perspective, we might initially make the assumption that the propagation of a continuous EM wave, which radiates energy has to be linked to a charge subject to acceleration. As such, we might wish to tabulate some of the assumptions implied in previous discussions:

- A static charge only has an electric field.
- A charge moving with velocity [v] has both an electric and magnetic field.
- Only an accelerated charge radiates energy as an EM wave.

Of course, even if we simply accept 3) as the source of an EM
wave, we still have to reconcile the description of an EM wave
conceptually propagating indefinitely,
and without loss, through the *‘vacuum’* of space. For example,
if we assume that a charge particle is subject to a short burst of input
energy, such that it accelerates and produces a corresponding burst of EM radiation, we
might initially assume that the EM waves propagate outwards in all directions.
However, while this statement will need to be qualified in a
following discussion addressing *EM
radiation*, we might initially assume that any
energy radiated into the vacuum cannot be destroyed, although
its density will become dissipated over an ever expanding surface
area, as in the case of any *3D outward wave*. So while energy is not
lost, and its measurement will quickly fall with radial distance,
conceptually this radiation is said to self-propagate outwards infinitely, even thought the charge particle
is assumed to stop radiating energy as soon as its acceleration stops. However,
unlike a 3D mechanical wave, there is no obvious media associated with
the vacuum of space on which the EM wave can propagate independent of
the originally accelerating source charge.

*So how do Maxwell equations explain self-propagation?*

Well, we might start by summarising the four Maxwell equations using the simplifying assumptions for a vacuum outlined at the start of the discussion:

[1]

We can also introduce another simplification by proceeding on the basis of an EM plane wave solution, where the E-field and B-field are perpendicular to each other, such that we can write the field strengths in terms of its Cartesian [xyz] components:

[2]

It is possibly worth clarifying an impression that the various diagrams and animations of propagating EM waves might create, i.e. that the E-fields and B-fields extend into [yz] space. What [2] is implying is that the vector quantities, i.e. the [E] and [B] fields, when measured at some specific point in [xyz] space, at time [t], act in the [y] and [z] directions respectively, while propagating in the [x] direction at velocity [v=c]. Of course, having made this clarification, we still need to understand how Maxwell’s equations explain this propagation. So using [1] and [2] as our frame of reference, we need to resolve the dot and cross products of the [E] and [B] fields, where the dot product can be expanded into the following generic form with [A] being replaced by either [E] or [B]:

[3]

As such, we can now simplify the form of Maxwell’s first two equations
in [1], based on [3], plus the assumptions in [2]. The *
first equation *defines* *gauss’ law for the electric field [E]
and reduces to:

[4]

The *second equation** *
defines Gauss’ law for the magnetic field and reduces to:

[5]

As such, [4] and [5] define the time-independent description of the electric and magnetic fields, which then leaves the time-dependent description predicated on the cross products, which can be expand using he following generic formulation, where [A] again represents either the [E] or [B] field in [xyz] space.

[6]

We can now solve [6] for the [E] and [B] field based on the assumption
in [2] plus substitute for the reduced forms in [4] and [5]. *The
third equation* defines Faraday’s law of induction
and becomes:

Finally, the* fourth equation *
based on Ampere’s law as modified by Maxwell becomes

Based on an initial examination of equations [7] and [8], we can see that any value of [E] or [B] will depend only [x] and [t], such that we need only focus on Maxwell's time-dependent equations. However, before commenting on the solutions in [7] and [8], we might continue by taking the 2nd derivate of [7] respect to [x] and [8] with [t]:

[9]

By noting the equivalence of the magnetic field terms in brackets leads to the wave equation for the electric field [E] in isolation:

[10]

By reversing the order of the 2^{nd} derivatives applied
to [7] and [8] leads to the wave equation for the magnetic field [B]:

[11]

So, to summarise, [7] and [8] reflect a 1^{st} order rate
of change of the [E] and [B] fields with respect to space [x] and
time [t],
while [10] and [11] reflect an independent 2nd order rate of change of
the [E] and then [B] fields with respect to space [x] and time [t]. Now,
at this point, it might be useful to, at least, cross-reference earlier
discussions of the ‘*propagation of mechanical waves’ *in order
to consider both the similarities and differences with respect to EM
wave propagation. Now, in the case of mechanical waves, it was generally
shown that wave propagation involved the transport of potential
energy through an exchange of potential and kinetic
energy within a physical medium, which is difficult to reconcile
with the description of in-phase [E] and [B] fields propagating in
a vacuum. We also have the issue as to whether ‘*only an accelerated
charge radiates energy as an EM wave’ * and how this energy might
continue to self-propagate. Now, in many respects, the ‘*idea of
a charged particle’ *might be eventually challenged in later discussions
by a '*speculative*' suggestion that all particles are only a manifestation
of a localised energy density that moves through space-time as a wave. While we
will not consider such speculative ideas in any detail, at this stage,
it may not be unreasonable to entertain the idea that [10] and [11]
describe an accelerated rate of change of the [E] and [B] fields in
the form of the 2^{nd} derivative with respect to space [x] and
time [t]. If so, the forward propagation of the [E] and [B] fields may
linked to this ‘*accelerated*’ rate of change at any point in space
[x] and time [t]. We can also see that [7] implies that a 1^{st}
order rate of change of the [∂E_{y}] field, at a given
region of space [∂x], will cause a 1^{st} order rate of
change of the [∂B_{z}] field in time [∂t]. Likewise,
[8] implies that a 1^{st} order rate of change of the [∂B_{z}]
field, at a given region of space [∂x], will cause a 1^{st}
order rate of change of the [∂E_{y}] field in time [∂t].
Based on the assumption in [2], the energy associated with the [E_{y}]
and [B_{z}] fields cannot radiate out in all directions, but
only propagate in the [x] direction. However, this is only an arbitrary
mathematical constraint as, in practice, the original source could equally
radiate in any direction, e.g.

[12]

In the case of [12], the propagation of the [E_{x}] and [B_{y}]
fields would be constrained to the [z] direction, while another permutations
could propagate in the [y] direction. However, while there is an implication
that [7], [8], [10] and [11] all conform to some form of wave description,
we have not really quantify what type of wave is involved. Therefore,
in order to progress towards some sort of wave based description of
propagation, we need to assign some form of waveform to the [E] and [B]
fields,
which is also a solution in terms of both the 1^{st} and 2^{nd}
derivatives, e.g. a sine/cosine function. However, it is highlighted
that any wave associated with the [E] and [B] fields does not have any
physical amplitude that extends in space and therefore the following
animation might be a better visualisation:

Based on the animation above, the idea of a traverse sine wave propagating through space might be misleading. For, in this case, the sine wave simply represents the change in the [E] or [B] field strength as a function of space or time as represented in the bottom section of the animation. With this clarification highlighted, let us proceed to consider the implication of a sine wave function:

[13]

Now [7] and [8] must both provide an equivalent solution, such that
we will need to determine the 1^{st} derivatives of [13] in-line
with Maxwell’s 3^{rd} equation, as per [7]:

[14]

On the basis of [14], we can now re-arrange Maxwell’s 3^{rd}
equation, as per [7], as follows:

[15]

However, it can be seen that the cosine function on either side of [15] can be cancelled out, such that we can now physically interpreted the implication of [k] and [ω]:

[16]

From a generic perspective, any wave with a frequency [f] and wavelength
[λ] implies a propagation velocity [v], although we have not yet
quantified the value of [v]. Of course, the wave functions in [13] can
also be solved as required by Maxwell’s 4^{th} equation in [8]:

[17]

On the basis of [17], we can now re-arrange Maxwell’s 4^{th}
equation, as per [8], as follows:

[18]

Again, the cosine functions in [18] can be cancelled out and the result re-arranged in terms of its fundamental physical attributes:

[19]

At this point, we might make reference to an earlier discussion related
to the ‘*system of units’ *in
order to interpret the implication associated with permittivity [ε_{0}]
and permeability [μ_{0}]
of free space.

[20]

We may now substitute the result in [20] back into [19]:

[21]

Of course, if we are to reconcile the first result in [16] with [21], we are led to the following equivalence of velocity [v] and the speed of light [c]:

[22]

So, at one level, Maxwell’s equations appear to provide some sort of mathematical
model that supports the propagation of energy as an EM wave. However,
while outside of the timeline of science being discussed, the
*idea
of a photon* would also develop and renew the *
wave-particle debate*
at the start of the 20^{th} century and, as such, we may still
have to question whether the EM wave model or the photon model can both
be true descriptions of the propagation of EM energy between 2 points
in space and time. We may also want to make further reference
to the *system of units* used to defined Maxwell’s 3^{rd} and
4^{th} equations in Gaussian units rather than SI/MKS, i.e.

[23]

In many respects, the formulation in Gaussian units seems to reflect a more logical symmetry, although the ratio of [E/B] shown in [22] then ceases to align to a velocity [c] and becomes just a numeric number. However, at this stage, we shall defer further discussion until a few other aspects of the EM model have been reviewed.