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:
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:
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  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  and  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]:
As such, we can now simplify the form of Maxwell’s first two equations in , based on , plus the assumptions in . The first equation defines gauss’ law for the electric field [E] and reduces to:
The second equation defines Gauss’ law for the magnetic field and reduces to:
As such,  and  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.
We can now solve  for the [E] and [B] field based on the assumption in  plus substitute for the reduced forms in  and . 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  and , 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  and , we might continue by taking the 2nd derivate of  respect to [x] and  with [t]:
By noting the equivalence of the magnetic field terms in brackets leads to the wave equation for the electric field [E] in isolation:
By reversing the order of the 2nd derivatives applied to  and  leads to the wave equation for the magnetic field [B]:
So, to summarise,  and  reflect a 1st order rate of change of the [E] and [B] fields with respect to space [x] and time [t], while  and  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  and  describe an accelerated rate of change of the [E] and [B] fields in the form of the 2nd 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  implies that a 1st order rate of change of the [∂Ey] field, at a given region of space [∂x], will cause a 1st order rate of change of the [∂Bz] field in time [∂t]. Likewise,  implies that a 1st order rate of change of the [∂Bz] field, at a given region of space [∂x], will cause a 1st order rate of change of the [∂Ey] field in time [∂t]. Based on the assumption in , the energy associated with the [Ey] and [Bz] 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.
In the case of , the propagation of the [Ex] and [By] fields would be constrained to the [z] direction, while another permutations could propagate in the [y] direction. However, while there is an implication that , ,  and  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 1st and 2nd 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:
Now  and  must both provide an equivalent solution, such that we will need to determine the 1st derivatives of  in-line with Maxwell’s 3rd equation, as per :
On the basis of , we can now re-arrange Maxwell’s 3rd equation, as per , as follows:
However, it can be seen that the cosine function on either side of  can be cancelled out, such that we can now physically interpreted the implication of [k] and [ω]:
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  can also be solved as required by Maxwell’s 4th equation in :
On the basis of , we can now re-arrange Maxwell’s 4th equation, as per , as follows:
Again, the cosine functions in  can be cancelled out and the result re-arranged in terms of its fundamental physical attributes:
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 [ε0] and permeability [μ0] of free space.
We may now substitute the result in  back into :
Of course, if we are to reconcile the first result in  with , we are led to the following equivalence of velocity [v] and the speed of light [c]:
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 20th 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 3rd and 4th equations in Gaussian units rather than SI/MKS, i.e.
In many respects, the formulation in Gaussian units seems to reflect a more logical symmetry, although the ratio of [E/B] shown in  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.