# Why do we say electromagnetic waves are self-propagating if they follow an inverse square law?

Electromagnetic waves are frequently described as "self-propagating", implying a mode of propagation distinct from that of electrostatic fields; but as I understand things, both have strength proportional to the inverse square of the distance from their source. Let me lay out what one ignorant of wave propagation and ignoring the magnetic field expects to see from a moving charge:

1. Suppose I am some distance $$r$$ away from a charged particle moving away from me with constant velocity $$v$$. Then at time $$t$$ I will perceive an electric field of strength proportional to $$\frac{1}{(r+t\cdot v)^2}$$.
2. Suppose instead that the charge is oscillating along the vector pointing from it to me, with period $$P$$ and amplitude $$A$$. Then I expect to see an electric field of strength proportional to $$\frac{1}{(r+A\cdot \sin(t\cdot \frac{2\pi}{P}))^2}$$.
3. Suppose rather that it oscillates perpendicularly to the vector connecting us. Then I expect to see an electric field whose direction wobbles between right-ish and left-ish with period $$P$$ and whose magnitude is proportional to $$\frac{1}{r^2+A^2\cdot \sin^2(t\cdot \frac{2\pi}{P})}$$.

Edit Rephrased the below because I forgot that I was dealing with inverses.

In both situations (2) and (3) the electric field where I stand is the sum of a constant and a periodic function (in case (3) two periodic functions along perpendicular axes), purely as a result of the oscillation of the source charge--no magnetic or special "propagation" effects needed. Obviously I have neglected the finitude of the speed of light in these calculations, which would introduce a tiny bit of distortion.

The periodic component is something like the multiplicative inverse of a squared sine wave, shifted so as to stay finite; some fancy trig likely makes it sinusoidal, since it's pretty dang close. Here are graphs of, respectively, the transverse and longitudinal components of (3), using r=1, P=1, and A=0.1:

Is it the case that the electromagnetic wave produced by Maxwell's equations in (2) and (3) will lose amplitude at precisely the same rate as this "inverse wave" that derives trivially from the inverse square law and the charge's motion? How, then, do we consider the wave "self-propagating" if it has no special powers to resist decay and acts just like the rest of the electric field?

Related desired elaboration: Apparently the Maxwellian wave will have the same frequency as the inverse wave, so how/why do their phases/amplitudes differ? And where do we get the energy for this extra wave?

• I would not say that em waves are "self" propagating. Who does? Electrostatic waves are not propagating at all. And what do you mean by strength in "strength proportional to the inverse square"? Are you familiar with Maxwell's equations? – my2cts Aug 19 '20 at 6:19
• Everyone seems to say so, from Wikipedia to academic websites. By "strength proportional to the inverse square" I mean the force experienced by a charge will decrease with distance squared from the source (F=k*c1*c2/r^2). I am only very slightly familiar with Maxwell's equations, since I haven't taken physics courses in college to work with specific solutions and develop an intuitive grasp of the full phenomenon. My familiarity is showing up understanding curl/divergence/etc. and going "oh, it's like charges are helixes and the magnetic field is an incompressible fluid" and similar thoughts. – Julian Kintobor Aug 19 '20 at 6:26
• You are lumping radiation intensity and field strength of a point charge together. Each falls off with r$^{-2}$ for a different reason. – my2cts Aug 19 '20 at 7:36
• "the electromagnetic wave produced by Maxwell's equations in (2) and (3)" These expressions do not describe waves. – my2cts Aug 19 '20 at 7:39
• The wave density falls of as $r^{-2}$ – Superfast Jellyfish Aug 19 '20 at 7:44

The description of EM waves as self-propagating is misleading. There's no causal connection between changing/curved electric and curved/changing magnetic fields: Maxwell's equations simply state that whenever you detect a changing electric field in empty space, there's also a curved magnetic field at the same spacetime point, and vice-versa; they have the common sources: charges and currents.

This fact is nicely summarized in the Jefimenko's equations, which reformulate EM fields (and potentials) as functions of charges and currents at retarded times, with all the fields and potentials being completely independent from each other.

Wave intensity falls off as r$$^{-2}$$ because of energy conservation. The field of a point charge falls off as r$$^{-2}$$ because it is the gradient of the potential which falls off as r$$^{-1}$$ as described by Coulomb's law, not because of a conservation law.

The inverse $$r^2$$ intensity you are talking about is just geometry. Whether it is light intensity, gravitational field intensity, or electric field intensity, the amount of the field intercepted by a detector falls off as inverse $$r^2$$. The sum of intensity over the entire sphere of radius $$r$$ will be the same as the source, unless there is something between the source and the detector to attenuate it. The inverse $$r^2$$ intensity has nothing to do with the properties of light, gravitational force, or electrical force.

In the case of light, it is easy to see because the measured light intensity is directly proportional to the detector's area. Integrating over the entire $$4 \pi r^2$$ spherical area, you will get the same constant for all $$r$$. The inverse $$r^2$$ intensity fall-off is strictly due to the geometric spreading out of the beam and has nothing to do with the wave nature of light.

In the case of gravitational and electric fields, the geometric nature is easily seen with Gauss Law. In the case of the electric field:

$$E\ A=q/\epsilon_0$$

where for a spherically symmetric charge distribution, $$A$$ is the same $$4 \pi r^2$$ area that light spreads its energy into.

Gauss Law for gravitation has the same form with $$F/m$$ replacing $$E$$ and $$4\pi GM$$ replacing $$q/\epsilon_0$$.

In all three cases the field intensity falls-off by inverse $$r^2$$, because the field is spreading to an area that increases as $$r^2$$.

If you were able to focus a beam of light so it never spread out, and a laser comes pretty close, the intensity would stay the same with distance.

• What is "amount of field"? – my2cts Aug 19 '20 at 8:56
• @my2cts I imagine the "amount of field" he means would be the surface integral of the force experienced by a unit charge due to the field over the intersection of a radius-r sphere centered on the source and an infinite, solid half-cylinder surrounding a ray from the source. – Julian Kintobor Aug 19 '20 at 9:05
• Yes, that's what's so confusing to me; everyone talks as if light were a special propagating phenomenon. So the answer is "there's nothing whatsoever self-propagating about an electromagnetic wave, it's exactly like any other fluctuation in the field due to the movement of a charge, we're just emphasizing the interplay of the electric/magnetic fields when we talk about it"? Is the transverse wave in (C) exactly the E component of the Maxwellian wave emitted by the charge then? Or is there a second wave that emerges independently from the dynamics of the field itself when the charge moves? – Julian Kintobor Aug 19 '20 at 9:08
• @JulianKintobor there's no "interplay between the electric and magnetic field" in the EM wave. See the Discussion section of the Wikipedia article Jefimenko's equations. – Ruslan Aug 19 '20 at 10:10
• @Ruslan THAT is a beautiful thing! In fact that is EXACTLY the sort of thing I was hoping to find out about when I created this question. Those equations are exactly the generalization of what I did above, with no interfield causation... wow. – Julian Kintobor Aug 19 '20 at 19:36

Electromagnetic waves are frequently described as "self-propagating", implying a mode of propagation distinct from that of electrostatic fields; but as I understand things, both have strength proportional to the inverse square of the distance from their source.

You seem to have a misunderstanding. EM radiation fields fall off as $$r^{-1}$$ not $$r^{-2}$$. The energy density is proportional to the square of the fields, so for the radiation the energy falls off as $$r^{-2}$$, not the fields. In contrast, the energy density of a Coulombic field falls off as $$r^{-4}$$. More importantly, for radiated fields the flux falls off as $$r^{-2}$$ while for electrostatic fields it is 0.