# Do the electric and magnetic components of an electromagnetic wave really generate each other?

Frequently when EM waves are taught, it is said that the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth.

But from my understanding of basic electromagnetics, it is not necessarily a changing electric field that creates a change in magnetic field, but instead an accelerating charge. and it is this charge that creates both the electric and magnetic field. Again, from my understanding, a changing magnetic field is not generated by a changing electric field, but instead just happens to always be present perpendicular to a changing electric field due to the laws of electromagnetism.

Am I wrong? Or is the "mutual generation" concept between the electric and magnetic components of an EM wave an actual thing?

• This is a common misconception. In reality, the electric and magnetic parts of the wave are in phase. I've also been confused by this in the past, since some depictions incorrectly show it as out of phase (with peak in electric being a trough in magnetic and vice versa). – Luaan May 6 '15 at 8:14
• @Luaan: I actually thought the correct version was one in which one wave grows as the other one dies, but I guess I was wrong... – user541686 May 6 '15 at 11:34
• A charge doesn't have to accelerate to create a magnetic field. – user121330 May 6 '15 at 14:58

As you say, a changing magnetic field is always associated with a changing electric field, and in fact in relativity they are finally revealed to be the same field. So at this level it cannot be said that the one field generates the other, as they are merely two aspects of the same object.

But maybe you still want to look at it from the perspective of "naive EM", and see what sense one could make of the statement that one field generates the other.

Now, if you look at the fields of a plane wave at a fixed point in space, you'll see that they oscillate in sync, and both reach maximum and $0$ intensity at the same time. In fact what you could say is that increasing $E$ field is trying to reduce the $B$ field and the increasing $B$ field is trying to reduce the $E$ field. You could follow the appropriate equations and you'll see this is analogous to the equations of a vibrating membrane.

But the point is, that in fact the fields try to mutually reduce each other rather than generate. In fact they do that so well, that there is an overshoot, and the cycle repeats. This dynamics turns out to make the energy flow through that point in space, to the fields in the neighbouring space. In fact, at some moments there is no EM field at that point, which means there is no energy at that field.

I guess the best description would be that a moving charge generates a disturbance in the electromagnetic field, in the sense that there is an energy transfer from its kinetic energy to the EM energy. This disturbance than propagates through space and time.

• Here's a link illustrating that indeed the fields are (causally!) generated by charges, but not by fields themselves. – Ruslan May 6 '15 at 6:16
• The relativistic aspect is often omitted, certainly in undergradute texts, but I found it to be the key to understanding EM phenomena. Similarly the role of charge in initiating the whole thing is often understated, to the student's detriment. – xenoclast May 6 '15 at 11:25
• @xenoclast, it was surely one of the finest moments of my undergraduate when I was asked to Lorentz transform from the lab frame where a wire carries a constant current to the frame where the charges are at rest! – Andrea May 6 '15 at 12:19
• +1 because of "This dynamics turns out to make the energy flow through that point in space, to the fields in the neighbouring space." The zero E and B at the nodes has been puzzling me, except if one thinks of the synergy of photons in building the wave and the HUP everything is OK :) . Maybe we would have to invent the photons in order to conserve energy , instead of sticking to average fields. – anna v May 6 '15 at 13:18
• @AndreaDiBiagio Can you link your point about the mutual reduction and overshoot more close to the mathematical theory? I don't see this from the equations. Only that if there is a changing magnetic field the curl of the electric field is not zero, which just implies that the electric field is not zero everywhere and the other way around. So I see only that the change of one field with time implies logically (which is not necessary causally) that the other field must exist (but doesn't say directly something about "generation" or "reduction"). – Julia Jan 11 '16 at 14:20

Do the electric and magnetic components of an electromagnetic wave really generate each other?

No they don't. Like Andrea said, they're two "aspects" of the same thing. And like you said, it's an electromagnetic wave. See the wiki article for electromagnetic radiation where you can read that "the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time". One's the spatial derivative, the other's the time derivative. If it was a water wave and you were in a canoe, the tilt of your canoe is E and the rate of change of tilt is B.

Frequently when EM waves are taught, it is said that the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth.

Yes, and the people who say this tend to say this is why electromagnetic waves don't need a medium. But they do need a medium. Space is the medium. It isn't a medium like water or the ground, but it isn't nothing. See LIGO and note that they're trying to detect a length-change in the arms of the interferometer. That's essentially space waving. Also see this where Robert B Laughlin talks about quantum vacuum and likens space to window glass rather than Newtonian emptiness.

and it is this charge that creates both the electric and magnetic field

IMHO you should avoid charge for now and stick to electromagnetic waves.

Is the "mutual generation" concept between the electric and magnetic components of an EM wave an actual thing?

No it isn't. Check out this Wikipedia article about Jefimenko's equations:

There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave (electromagnetism). However, Jefimenko's equations show an alternative point of view. Jefimenko says, "...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".

• This is correct, but one shouldn't think that Maxwell's equations will give zero solutions in the absence of charge. Although in physics radiation almost always comes from accelerated charges, the equations do admit nontrivial solutions even when $\vec j(\vec r,t)\equiv0$. – Ruslan Oct 20 '17 at 13:50

This plane polarized wave from wikipedia may help

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.

It is plotting the mathematical functions, solutions of Maxwell's equations, that describe a propagating wave.

$$\text{electric field}: E = E_m\sin(kx - \omega t) \\\\\\\ \text{magnetic field}: B = B_m \sin(kx - \omega t)$$

To be consistent with Maxwell's equations, these solutions must be related by

$$\frac{E_m}{B_m} = c$$

$E$ and $B$ are perpendicular to each other and in synchrony growing and diminishing as the wave propagates. The statement

the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth

refers to this graphic, where electric and magnetic fields increase and decrease in synchrony.

The acceleration of charged particles generates electromagnetic waves and then the waves become independent of the source of the electric field that generated it and propagate according to the equations.

This is the classical framework.

In the quantum mechanical framework the wave is built up by an enormous number of photons generated by the accelerated electrons in the antenna , but that is another story.

• I've read this paper by Jefimenko where he said that since electric & magnetic fields are simultaneous, they can't create each other. Okay, but isn't simultaneity a relative concept; that is, if for me, the event is simultaneous, it may not be simultaneous for another frame, isn't it? I don't know relativity much but so, could you clarify Jefimenko's statement about the simultaneity, please? – user36790 Dec 18 '15 at 15:31
• He was probably talking of the electromagnetic field where E and B are in synchrony for a plane wave as seen above and in the link. en.wikipedia.org/wiki/Electromagnetic_radiationThe EM only changes wavelength depending on the frame as it moves with velocity c. – anna v Dec 18 '15 at 16:14
• So, apart from EM waves, is his assertion that electric field & magnetic field don't create each other, wrong? after all, simultaneity is relative. – user36790 Dec 18 '15 at 16:34
• @user36790 A changing electric field creates a magnetic field, that is experimentally true. Also a changing magnetic field creates an electric field and they are solutions of maxwell's equation. These in different relativistic frames can differ. The electromagnetic wave is a different solution of the maxwell's equation. If he is not talking of EM waves he is wrong, imo – anna v Dec 18 '15 at 17:38
• @Anna v there is another solution for $$\frac{E_m}{B_m} = c.$$ Take $$\text{magnetic field}: B = B_m \cos(kx - \omega t)$$ and one get a perfect solution with no vanishing energy. – HolgerFiedler Jan 11 '16 at 21:08

Maxwell's equations in vacuum are: $$\nabla\cdot\mathbf{E} = 0$$ $$\nabla\cdot\mathbf{B} = 0$$ $$\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}$$ $$\nabla\times\mathbf{B} = \frac{1}{c^2}\frac{\partial\mathbf{E}}{\partial t}$$ It's the last two of these that give rise to the interpretation that a changing magnetic field generates an electric field and vice versa. But you could always take the curl of both these equations, use the other two, and get simple, straightforward wave equations separately for $\mathbf{E}$ and $\mathbf{B}$: $$\nabla^2\mathbf{E} - \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$$ $$\nabla^2\mathbf{B} - \frac{1}{c^2}\frac{\partial^2 \mathbf{B}}{\partial t^2} = 0$$ While these wave equations mean that $\mathbf{E}$ and $\mathbf{B}$ propagate as ordinary waves, they are still constrained by the original relations; we haven't magically made that dependence disappear.

So, while it is correct to say that changing magnetic and electric fields generate each other (simultaneously), it is also correct to say that each propagates as a wave of its own right, but is everywhere related to the other by the original Maxwell equations.

Indeed, when a charge is accelerating, it does generate both a changing electric and magnetic field at its position. You can then say that they both propagate out as ordinary waves, but (repeat) are always constrained by Maxwell's equations to be interdependent.

• would you say that the concepts of being generated and being accompanied coincide for EM fields at the level of classical electromagnetism, but then become different in special relativity? – Andrea May 5 '15 at 19:43
• @AndreaDiBiagio Of course: in special relativity, it is all concisely expressed in terms of the electromagnetic tensor $F_{\mu\nu}$. And the wave equation is better expressed in terms of the 4-potential - there is no 'accompanying' there. Update: what do you mean by 'coincide'? – AV23 May 5 '15 at 19:47
• What I meant was: would there be a way to meaningfully say that the E field generates the B field and vice versa (for example by looking at energy transfer between the two), or are they just always appearing together and that's all we can say? – Andrea May 6 '15 at 9:20
• There is no energy transfer between the two. The effect of the "cross-equations" is simply the relations: $k\mathbf{E} = -\mathbf{k}\times c\mathbf{B}$ and $kc\mathbf{B} = \mathbf{k}\times\mathbf{E}$, where $\mathbf{k}$ is the wave vector, which constrains the relative magnitude and direction of the fields. – AV23 May 6 '15 at 9:42

"a changing magnetic field is not generated by a changing electric field, but instead just happens to always be present perpendicular to a changing electric field due to the laws of electromagnetism."

So ... it is due to but not caused by. What is the difference?

Short answer: it is not only "a thing" it is a correct thing.

This is much more clearly stated in the differential form of Maxwell's equations:

$$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$ $$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{j} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right) \,.$$

And while magnetic fields can be cause by either current or changing electric fields, light will happily travel in charge-free regions of space, so there is no question at all about which term is at work in electromagnetic waves.

Jefimenko believed that time varying electric and magnetic fields did not cause one another because Maxwell's equations only give the relationship between the fields at the same instant in time. He said the equation for the E field did not predict a future value for the B field - it just tells you what the B field is if you know the E field. This is unlike the calculation of E and B fields from earlier charge positions and movements using the retarded field equations which are patently a causal relationship.

However, taking Maxwell's equation:-

$$\nabla\times\mathsf{E_{present}} = -\frac{\partial\mathsf{B_{present}}}{\partial t}$$

Re-write the rate of change of B as:-

$$\frac{B_{future}-B_{past}}{dt}$$

Substite this expression and re-arrange the terms in Maxwell's equation to give:- $$B_{future}= B_{past}-dt{(\nabla\times\mathsf{E_{present}}})$$

Similarly using Maxwell's equation for curl B gives:-

$$E_{future}= E_{past}+c^2dt{(\nabla\times\mathsf{B_{present}}})$$

Written in this way the two Maxwell's equations are now seen to be predictive and causal. This is not just a mathematical trick as there is a numerical computational technique, known as the finite-difference time-domain (FDTD) method, which by alternately using each equation in a time stepping sequence can calculate exactly how the fields will propagate. It does not require any information about the charges or currents which produced the initial fields, just the starting distribution of the electric and magnetic fields. An FDTD computer program provides a good demonstration of how the electric field generates a magnetic field and the magnetic field generates an electric field.

• Everything right. However one should also stress that both electric and magnetic field at time $t$ do generate magnetic and electric field at time $t + \Delta t$. – GiorgioP Feb 17 '19 at 22:48