How can one meaningfully say that one field generates the other in an EM-wave?

Question

This is a follow up question to: Do the electric and magnetic components of an electromagnetic wave really generate each other?

Clearly there are nuances of how one states the "mutual induction" explanation for EM-waves. My question is, of how strong can the statement in this direction be if one insists on that there should be a mathematical proof for it starting from Maxwell's equations. Nevertheless the statement should be simple enough to serve as (correct but) "popularized" view on the EM-field and it should be close to what is often said about EM-waves and mutual generation of the field in undergraduate and high school physics courses.

If one says that one field generates the other, one should distinguish between a causal relation and a logical one (assuming Maxwell's theory).

Let's start with the second (I guess weaker) interpretation:

From Maxwell's equations in vacuum:

$$ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\,.$$

one can see that if one field is changing with time, logically necessary (assuming the validity of Maxwell's equations) there must also be the other field (not necessary both in the same point, but somewhere in space, i.e. it can not be identically zero since this would imply that the curl is zero and thus also the time derivative of the other field).

This seems to be a very weak interpretation of the mutual generation thing. One guess to make the statement stronger may be not to talk about existence but say something like: If one field $\mathbf{E}$ is changing with time, there must be a $\mathbf{B}$-field which is also changing with time (and vice versa).

This seems to be wrong, take for example

$$ \mathbf{E} = 2\hat{\mathbf{z}}t $$

and

$$ \mathbf{B} = y \hat{\mathbf{x}} - x \hat{\mathbf{y}} $$

This satisfies the Maxwell equations in vacuum. $\mathbf{E}$ is changing with time, but $\mathbf{B}$ is not. So the stronger statement seems to be wrong.

Another idea to make is stronger is to say that if the time derivative of one field is high, the value of the other field is high. This is clearly wrong if one talks about the same point in space, as an in phase EM-wave shows.

Clearly one can rephrase the Maxwell equation, but speaking directly about the curl is not what I am looking for since it should be something which you can express in simple words...

Up to now this is all logically not causally. How can one interpret the statement in a causal way such that it is correct? Since electric and magnetic fields are "just" different components (in a fixed reference system) of the electromagnetic field tensor I guess that there will be no correct causal statement at all, but I am not sure.

Answer 1

How can one meaningfully say that one field generates the other in an EM-wave?

You can't because they don't. The electromagnetic wave is an electromagnetic field variation. See Wikipedia where you can read this: "Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole — the electromagnetic field".

Or see section 11.10 of Jackson's Classical Electrodynamics where he says this: "one should properly speak of the electromagnetic field F_μν rather than E or B separately".

Or see Jefimenko here: "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents".

It's a popscience myth that an E wave generates a B wave which generates an E wave. One of those "lies to children" that ends up being widely believed, even by professional physicists.

Clearly there are nuances of how one states the "mutual induction" explanation for EM-waves.

I'm afraid there's no nuance at all Julia. The "mutual induction" explanation is wrong. E and B are in phase because they're space and time derivatives. See this answer where I used a canoe analogy.

My question is, of how strong can the statement in this direction be if one insists on that there should be a mathematical proof for it starting from Maxwell's equations. Nevertheless the statement should be simple enough to serve as (correct but) "popularized" view on the EM-field and it should be close to what is often said about EM-waves and mutual generation of the field in undergraduate and high school physics courses.

There is no mathematical proof. The equals sign in $\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} $ and in $\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\ $ does not indicate cause. Instead you should read it as "is another aspect of". Or simply "is".

If one says that one field generates the other, one should distinguish between a causal relation and a logical one (assuming Maxwell's theory).

Maxwell unified electricity and magnetism to give us the electromagnetic field. But here we are 150 years later and people still will talk about E and B as if they're two totally different things. Tut.

Let's start with the second (I guess weaker) interpretation: From Maxwell's equations in vacuum: $\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} $ [and] $\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\ $ one can see that if one field is changing with time, logically necessary (assuming the validity of Maxwell's equations) there must also be the other field (not necessary both in the same point, but somewhere in space, i.e. it can not be identically zero since this would imply that the curl is zero and thus also the time derivative of the other field).

There aren't really two different fields, there's one field, and two derivatives. Spatial and time.

This seems to be a very weak interpretation of the mutual generation thing.

Agreed. When you look into it it just doesn't stand up.

One guess to make the statement stronger may be not to talk about existence but say something like: If one field $\mathbf{E}$ is changing with time, there must be a $\mathbf{B}$-field which is also changing with time (and vice versa).

That's not quite right because there's only one field there. Electromagnetic field interactions typically result in linear and/or rotational force. When we only see linear force we typically talk of an E field, when we only see rotational motion we typically talk of a B field, but these result from F_μν field interactions.

This seems to be wrong...

It is.

Up to now this is all logically not causally. How can one interpret the statement in a causal way such that it is correct?

You can't, because it isn't.

Since electric and magnetic fields are "just" different components (in a fixed reference system) of the electromagnetic field tensor I guess that there will be no correct causal statement at all, but I am not sure.

Shrug. There is no correct causal statement at all.

Answer 2

A changing electric field does generate a magnetic field, ( hence electromagnets) and a changing magnetic field generates an electric field. The solution for radiated energy is given in terms of electric and magnetic fields in a given reference frame.

This animation is instructive for radiation:

Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. Note that the electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together

Bold mine.

Here electric and magnetic fields increase and decrease together, how can one be creating the other? The wave solution of the combined maxwell equations differs from the solutions for stand alone ones for electric and magnetic fields.

Aside:

At a specific $(x,y,z)$ at time $t$ both electric and magnetic fields are zero, nevertheless the light beam carries energy :

\begin{align}S&= \frac{1}{c\mu_0} E_m^2 \;\overline{\sin^2(kx-\omega t)}\\&= \frac{1}{c\mu_0}\frac{E_m^2}{2}\;.\end{align}

The rate of energy transportt S is perpendicular to both E and B and in the direction of propagation of the wave.(A condition of the wave solution for a plane wave is Bm = Em/c )

It always bothered me that zero $E$ and $B\;,$ where is the energy at that point? One can sympathize with the need for a luminiferous ether, which was disposed of experimentally. The averaging seemed to gloss over that zero.

But the particulate nature of light, photons, in some sense compensates for the non existent ether. The classical beam emerges from a confluence of photons which individually carry the energy $h\nu.$ The electromagnetic wave's $E$ and $B$ are built up by zillions of photons which are excitations on the photon field and add up in the necessary way to produce the classical beam.