In some respects, this page is an extension of the previous discussion of electromagnetic energy, but now the focus is on the specific issue of 'radiated' electromagnetic energy. As a starting point, energy might be described as radiated once its propagation is independent of the source - see EM propagation. However, in this discussion, we shall try to derive an expression of the energy per second or the power radiated by an accelerated charge, which corresponds to Larmor’s formula, when relativistic factors are ignored. This equation was first derived by J. J. Larmor, in 1897, and along with Maxwell’s equation and the Lorentz force equation, might be thought to underpin the classical theory of light. However, before tackling the derivation of this formula, we need to provide some background to this particular discussion. It has been shown that the wave equation, as presented in  below, is a solution of Maxwell’s equations based on resolving either the 1st or 2nd derivative with respect to time and space. In the generic form shown below, (A) can be substituted to reflect either the strength of the electric [E] or magnetic [B] fields in space [r] or time [t]:
As such,  might suggest that the amplitude of the electric [E] and magnetic [B] fields must correspond to some form of acceleration in time and space. If so, the charged particle, being the source of both the electric and magnetic fields, might not only have to be moving with a given velocity, but also subject to an acceleration in order for an EM wave to be initially propagated. While we will proceed based on this assumption, it is one that we may eventually have to challenge.
It has been suggested that EM wave propagation might be predicated on a charge particle being subject to acceleration. While the actual mechanism of how a charged particle might be accelerated is not really the subject of this discussion, it may be informative to provide some sort of visualisation of how a charged particle may be caused to oscillate up and down with simple harmonic motion (SHM) under the influence of an alternating sinusoidal voltage [V]. In addition, this approach may also allow us to draw some parallels with the previous discussion of SHM in connection with mechanical waves and the associated wave equation, which has a form very similar in nature to .
The diagram right tries to illustrate the electric and magnetic fields about a short vertical antenna driven by a sinusoidal current. This concept seems to parallel the description of an electromagnetic wave originating from an accelerated electric charge, at least, when applying an alternating voltage that causes the charged electrons to accelerate up and down within the dipole antenna. The acceleration of the electrons associated with the up-down oscillation within the antenna could then be considered to be acting as the source of an electromagnetic wave, which radiates sideways out of the antenna at the same frequency as the alternating voltage applied to it. In the context of free space, the diagram only shows the upper half of the electric (E) and magnetic (B) fields that extend symmetrically along the whole length of the antenna. The magnetic field is circular about the antenna, which has to also be perpendicular, at every point, to the electric field, and proportional in intensity to the magnitude of the electric field, as in a plane wave. The diagram also reflects that the projection of the electric [E] and magnetic [B] fields that only extend outwards in axial symmetry, i.e. the fields fall to zero in the direction of the oscillating motion.
While there are still some open issues regarding how the energy associated with the EM wave can be radiated into space by an accelerated charge. Observation and experimentation has shown that a static charge particle only has a spherical electric field [E], i.e. there is no magnetic field associated with a charge particle at rest, as illustrated on the left below. On the other hand, a charge particle moving with velocity [v] will have both an electric [E] and magnetic [B] field, as shown on the right below, although it is said not to radiate EM wave energy.
Therefore, we need to consider a mechanism that might help us visualize how an accelerated charged particle might also generate an EM wave that can radiate energy in the form of a self-propagating wave. So let us start by considering a stationary charge at rest at time [t=0], emanating electric field lines, as shown above on the left, but which is then subject to an acceleration for a period [t], but viewed at some later time [T], as reflected in the diagram below.
Outside of the sphere of radius [R=cT], the electric field lines still point towards the original position of the charge, i.e. the small grey shell, because the information about the acceleration has not yet moved farther out than [R]. In contrast, inside of this sphere the field lines point towards the location the charge had after the acceleration, i.e. the small red sphere, although any subsequent drift due the implied velocity after acceleration is being ignored within this general presentation of the primary concept. Since the electric field lines inside and outside the sphere of propagation have to be connected, there must also be a small region of width [ct] in which the electric field [E] has a non-radial component [ET] as shown in the inset above. This process will assume a constant acceleration [a], which can be defined by the simple relationship:
From the diagram above, it can be seen that it represents the situation at the time [T], after the period of acceleration [t], which defines the radius of the sphere [R=cT], where [c] again represents the propagation speed of the electric field emanating from the charge at the speed of light. However, it can also be seen that this expanding sphere will have an implied thickness, which corresponds to the period of acceleration [ct]. For the purposes of this derivation, the focus is on just 1 electric field line and, at this stage; no mention of the magnetic field need be introduced, although it will be shown to be an equal component of the energy propagated by the accelerated charge. So, returning the focus to the electric field line shown within the expanded inset, we see that the electric field [E] within the spherical shell region must have two components associated with the radial [ER] and transverse [ET] vector components of [E]. From the information shown in the diagram, we can express the ratio of these two components as follows:
However, if we substitute  into , we get an expression that includes the acceleration [a], which is an important focus of this discussion:
We might realise that the radial component [ER] conforms to an electric field as described by Coulomb’s law, which we have already shown to be:
As such, we may now substitute  back into , while highlighting the radius to be [r=R] and [T=R/c]:
Equations  and  now appear to lead to some key insights about the relative strengths of the electric field [E] component vectors, i.e. the radial component [ER] is proportional to [1/R2], while the transverse component [ET] is proportional to [1/R]. We might also note that the strength of the transverse electric field [ET] is directly proportional to the acceleration [a] of the charged particle. However, the possibly bigger implication of the proportionality shown in  and  is that as time passes, and the radius [R] of the spherical shell increases, the transverse electric field [ET] becomes much stronger than the radial electric field [ER]. Ultimately, as [R] continues to increase, the effects of the radial component [ER] becomes increasingly small, such that it can be dropped all together. At which point, we are left with only the transverse component [ET], which is dependent on its angle of propagation away from the accelerating charge, as illustrated right. When the angle [θ] is at right angles to the accelerating charge, the transverse component [ET] will be at a maximum, but when the angle [θ] is parallel to the accelerating charge, the transverse component [ET] falls to zero. Therefore, it suggests that there is no EM propagation along the axis of charge motion, which reflects the axial symmetry of the EM field shown in the earlier diagram connected with a dipole antenna. At this point, we shall make a cross-reference to an earlier discussion concerning the energy density [η] of an EM wave in terms of its energy per unit volume:
Based on the equivalence between the electric field [E] and the magnetic field [B] in [7a], we can relate the entire energy density [η] in terms of just the electric field [E], which based on earlier arguments reduces to [ET] as the radius [R] increases. As such, we may now substitute  into :
As such, this transverse electric field seems to correspond to a pulse of radiation, which travels at the speed of light [c] and carries energy away from the accelerating charge. At this point, we might recognise from the diagrams that the total energy density must be associated with the sum total of the transverse field energy residing within the volume of a spherical shell of radius [R=cT] and thickness [ct]. However, before we can use the result in , we need to obtain an average energy density aggregated over all values of [θ].
While the actual distribution of energy is reflective of the axial symmetry shown in the diagram above, we can calculate an equivalent average of this energy, which allows the sine function to be eliminated. We can proceed to do this by using the implicit symmetry of a sphere in Cartesian coordinates [x,y,z]:
In order to take advanctage of this symmetry, we simply have to position our Cartesian coordinate system such that the origin of the sphere aligns to the charge particle moving along the x-axis. Now we can define the angle [θ] in terms of basic trigonometry and introduce a standard trigonometric function:
However  also allows us to substitute for [x] in  such that we get:
We are now in a position to express the total energy density for all values of [θ] aggregated over the volume of the spherical shell, which is the product of the surface area of the shell [4πR2] multiply by its thickness [ct]
We can normalise  to unit time, which is essentially an expression of the energy per second or the power radiated by an accelerated charge, which is the common form of Larmor’s formula:
For clarity, we should remember that we have already accounted for the magnetic field component of the accelerated charge in , which highlights that both the electric and magnetic fields contribute to Larmor’s formula based on the caveat that the velocity of the accelerated charged particle is always much less than the speed of light [c].
So have we explained all the open questions within the constraints of classical electrodynamics?
Although the main discussion surrounding the last question will be deferred to the summary, the animation above attempts to provide some initial context to the remaining issues. We might imagine that the animation represents a charged particle being switched up and down in a very strong electric field, such that the shape being traced out in time aligns to an approximate square wave. The ovals reference lines drawn to the left and right of the charge correspond to a cross-section through the doughnut toroid, as illustrated in the previous diagram. Based on the criteria of the Larmor formula, when a charge is subject to acceleration [a], i.e. during the transition of positions, it radiates power, as per , but reference to  also suggests that the energy density [η] is also subject to the angle [θ] with respect to the axis of charge motion. As such, the energy density is reflected by the depth of the yellow shading, symmetrical about the axis of motion. However, the intention of left-right sides of the animation is to be somewhat illustrative of wave-particle duality in that the left reflects the electric field lines, while the right reflects the 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. Finally, the oscillating red lines on left reflect the total electric field [E=ER+ET] as a function of distance based on  and . So what you see is the effects of [ER] reducing by [1/R2], while [ET] only reduces by [1/R] and so quickly becomes the dominant field as the radius from the charge increases. In this respect, the left-hand side of the animation is broadly supportive of the basic assumptions that led to the Larmor formula; although you might question the existence of [ET] in the photon stream model.
What about the effects of maximum and minimum acceleration?
Actually, an animation based on a square-wave is not best suited to discuss this question, which is why the previous animation is replaced in the summary discussion with a sine wave motion. However, according , no power is radiated when [a=0], which corresponds to a maximum velocity [v], and the magnetic field being a maximum. In this context, the photon stream model might be thought to better reflect this situation in the sense that there are no photons emitted when [a=0]. While this overview of this subject is not in a position to talk with any authority on the complexity of electromagnetism, it is in its remit to highlight a number of issues that appear questionable or confusing, which the reader may wish to research further for themselves. Therefore, let us raise the question, which many may assumed is already answered by classical electrodynamics:
You might rightly think that the derivation of Larmor’s formula has already answered this question positively. In fact, we might cite equation  as proof, which we arrived at by substituting the expression for the transverse electric field into the general energy density expression, as shown in  & . As such, equation  clearly suggests that the energy per unit volume is proportional to the square of the acceleration [a] of the charge and the inverse square of the radius [R]. In the final form of the Larmor formula, as shown in , the inverse square dependency on the radius [R] disappears because it aggregates the total power radiated to an expanding spherical shell, which is itself proportional to the surface area of a sphere, i.e. 4πR2. However, further consideration of this result may be needed. For example,
- A charged particle at rest on the Earth’s surface is also subject
to a gravitational acceleration of 9.81 m/s2. However,
direct observation from a collocated position on the Earth’s surface
would suggest that this apparently stationary charged particle does
not radiate energy.
- A charge electron is required to lose energy when entering a
magnetic field, which is orientated normal to its trajectory and
would follow a circular path. However, this circular path is also
subject to a centripetal acceleration [a], which will require the
electron to radiate energy and, as a consequence, the electron would
actually follow a decaying spiral trajectory and slow as it loses
kinetic energy. This process is called ‘radiation damping’
and was cited in the context of the
Bohr atomic model, which led to the conclusion that
electrons in atomic orbits do not radiate energy.
- Historically, X-rays were generated by firing electrons, at
high speed, into a metal target and it was initially believed that
the radiation was generated by the sudden deceleration of electrons
as they struck the metal target. However, at this time, the atomic
model was more of an amorphous sphere in which both the protons
and electrons were evenly distributed, which was only later revised
by Ernest Rutherford, in 1911, showing that most of the volume of
an atom is essentially empty. Today, the description of X-ray generation
is considered more in terms of an interaction between the charge
particle and the intense fields inside an atom.
- There is an outstanding issue as to whether an electric [E] field has an implicit propagation velocity [c] in respect to 'action-at-a-distance'. If so, changes in the E-field strength must still propagate into the surrounding space, even when subject to only constant velocity.
Many of the details for and against the issues raised go beyond the
scope of this discussion. Therefore, no conclusion is being drawn at
this stage as to whether, or not, it is the acceleration of the charge
that causes energy to be radiated. However, for those interested in
learning more about the details of this debate, then the following link
may be of interest:
Uniformly Accelerating Charge Radiate?