The Poynting vector $S$ gives the energy flux density of an electromagnetic field

$\boxed{S=\frac{1}{{\mu }_0}E\times B}$ (Poynting vector)

along with the linear momentum density $g$

$g=\frac{S}{c^2}={\epsilon }_{0}E\times B$(2)

where $E$ and $B$ are the electric and magnetic fields, respectively, while ${\epsilon }_0$ and ${\mu }_0$ are the electric permittivity and magnetic permeability and $c$ is the speed of light (we use SI units throughout).

Because it is only the divergence of the Poynting vector that appears in the conservation law from which it is derived [4] [6] , one can limit oneself to looking at

$\nabla \cdot S$(3)

Now, using the vector identity

$\nabla \cdot \left(E\times B\right)\equiv \left(\nabla \times E\right)\cdot B-E\cdot \left(\nabla \times B\right)$(4)

and then applying Faraday’s and Ampère’s law in free space

$\nabla \times E=-\frac{\partial B}{\partial t}$ (Faraday’s Law)

$\nabla \times B=\frac{1}{c^2}\frac{\partial E}{\partial t}$ (Ampère’s Law)

we arrive at

$\nabla \cdot \left(E\times B\right)=-B\cdot \frac{\partial B}{\partial t}-\frac{1}{c^2}E\cdot \frac{\partial E}{\partial t}$(5)

So, we find from Equation (5) that there is no contribution from the Poynting vector in the conservation law from non-time-varying fields, and therefore, such components can be dropped. That they should be dropped can be seen from the wave equations:

${\nabla }^{2}E=\frac{1}{c^2}\frac{{\partial }^{2}E}{\partial t^2}$(6a)

${\nabla }^{2}B=\frac{1}{c^2}\frac{{\partial }^{2}B}{\partial t^2}$(6b)

The second time derivative of $E$ or $B$ cannot be zero for a wave, so constant fields will not give rise to radiation and therefore should not be included in the Poynting vector. This is the classical version of the simpler quantum perspective that zero-frequency photons don’t exist. Thus, we see that non-parallel electric and magnetic fields alone are not sufficient to produce an energy flux: they must also change with time. We are not the only ones to be concerned about the contribution of static fields to the Poynting vector; some are aware of paradoxes arising from these contributions [7] and others have tried to cancel them out with ‘hidden momentum’ [8] .

This exclusion of non-time-varying fields allows us to get the Feynman form from the Larmor form as we showed previously for Equation (1).

The Larmor version of power gives rise to non-physical equations of motion that are resolved by the Feynman form

Using $P=F\cdot v$ we can include the radiation reaction force to get

$ma=F-\frac{P}{v}$(7)

and so, for the Larmor power (using a constant pushing force $F_c$), we get

$\left(F_c-m\stackrel{\dot{}}{v}\right)v=\alpha {\stackrel{\dot{}}{v}}^2$(8)

where $\alpha \equiv q^2/\left(6\pi {\epsilon }_0{c}^3\right)$. This has pathological behavior, namely, if the particle starts at rest, the acceleration is then also zero, and thus, nothing moves despite the application of a finite pushing force.

On the other hand, for the Feynman power, we get the Abraham-Lorentz equation

$F_c-ma=-\alpha \stackrel{\dot{}}{a}$ (Abraham-Lorentz)

whose solution is

$a=\frac{F_c}{m}+ℂ\exp \left(\frac{m}{\alpha }t\right)$(9)

where $\alpha \equiv q^2/\left(6\pi {\epsilon }_0{c}^3\right)$, but this is non-pathological if the integration constant $ℂ$ is zero.

While the Larmor form for power can never be negative, this is not always the case for the Feynman form (this possibility was not mentioned in the original work by Feynman [5] ), since if $\stackrel{\dot{}}{a}$ and $v$ are in the same direction, the predicted Feynman power is negative, and since this is not possible, the Feynman form might be more correctly expressed as

$P_F=\{\begin{array}{ll}-\frac{q^2}{6\pi {\epsilon }_0{c}^3}\stackrel{\dot{}}{a}\cdot v\hfill & \text{if}\stackrel{\dot{}}{a}\cdot v<0\hfill \\ 0\hfill & \text{if}\stackrel{\dot{}}{a}\cdot v\ge 0\hfill \end{array}$ (Feynman power)

This ties into the run-away solution for the Abraham-Lorentz solution (Equation (9)), where the integration constant $ℂ$ must be chosen to be zero on physical grounds, and hence one has zero Feynman force when $\stackrel{\dot{}}{a}$ is in the same direction as $v$, namely no negative power is a compelled and consistent choice. One might be concerned that this exclusion will make a difference for sinusoidal motion, but this is not the case since when $x~\sin \omega t$ this leads to a Feynman power that goes like ${\cos }^2\omega t$ and thus has no negative component; however, for exponential motion ( $x~{\text{e}}^t$) the suppression mechanism comes heavily into effect, but the easiest way to avoid radiation while speeding up a charge is simply to maintain a constant acceleration.

Dirac [2] contemplated a charge/monopole that, despite being stationary, is perceived as having angular momentum $L$ due to the electromagnetic momentum density $g$ (see Equation (2)); this is illustrated in Figure 1 .

However, indiscriminate use of the Poynting vector (those not excluding the constant part) can lead to various problems, namely a stationary electric charge together with a magnetic charge (monopole) will have an angular momentum that does not depend on their separation as originally tackled by Dirac [2] but more easily seen in the texts by Jackson [4] or Griffiths [6] . A stationary system having angular momentum is in itself strange, but even worse, if the charge and monopole are approaching each other, the angular momentum flips sign as they pass, and thus violating the conservation of angular momentum.

These problems can be resolved once it is recognized that a time-invariant $E$ and $B$ field represent photons with zero frequency, which, since they cannot exist, need to be removed from the original Poynting vector, so once it is recognized there is no momentum contribution from non-changing fields, the angular momentum that would have arisen is no longer present. In this way, both paradoxes cease to exist once the original Larmor formula for radiated power.

$P_L=\alpha a^2$

is replaced by the Feynman form

$P_F=-\alpha \stackrel{\dot{}}{a}\cdot v$

where $\alpha \equiv q^2/\left(6\pi {\epsilon }_0{c}^3\right)$; $q$ being the charge, $a$ the acceleration and $v$ the velocity.

Start with a solenoid being turned on and feeding into a conducting ring followed by a test coil, as illustrated in Figure 2 .

One can now see that, due to the negative of Lenz’s law (the induced magnetic field opposes the change of the original magnetic flux) that the original field $B$ is weakened by the induced field $B_{ind}$. Using the example of an exponentially increasing current makes conceptualizing the mechanism easier since the derivative of an exponential is again an exponential, albeit of differing strength, so all fields vary in sync. The test coil on the right experiences a reduced induced voltage due to the intervening ring since Lenz’s law has a negative; if it was positive, energy would not be conserved in this counter-example.