# How to understand that the electromagnetic wave propagates?

Don't the electric field and magnetic field have infinite range? When a charged particle moves, the electric field vectors at two different locations A and B should start to change at exactly the same time. Yet A and B can be infinitely far from each other. This is different from the ripple in a pond that makes A vibrate first then some time later makes B vibrate. For the ripple, it is easy to understand that the wave propagates, since there is a latency when A and B start to vibrate. But how should we understand the propagation of electromagnetic wave?

• – ACuriousMind Aug 2 '15 at 14:48
• @ACuriousMind, thanks. It is indeed a similar question! – ayuanx Aug 3 '15 at 12:33

I believe that you are asking this question because you are thinking that the electric field of a point charge is given by Coulomb's law:

$$\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi \epsilon_0} \frac{q}{|\mathbf{r} - \mathbf{r}_0|^2} \hat{(\mathbf{r} - \mathbf{r}_0)}$$

where the charge is located at $$\mathbf{r}_0$$, and then you are noticing that if we make that charge position $$\mathbf{r}_0$$ dynamic, then $$\mathbf{E}$$ changes at each point in space in perfect synchrony with the change in said position.

The problem in your understanding is that Coulomb's law only works for static charges: charges that are not moving. Coulomb's law is not, by far, the fullest description of electromagnetism and it does not generally give the correct electric field for moving charges, at least not for charges executing arbitrary motions. In order to properly determine the fields around a charge executing an arbitrary motion, one has to use the full power of Maxwell's equations, and if one does so, one will find that the disturbances produced do, in fact, propagate as electromagnetic waves.

So the answer is simple: you are relying on a mathematical model that is based on an assumption that is wrong in the situation you are trying to think about. That assumption is that the charge generating the field is not moving, and your question is precisely about the negation of this assumption. Hence, the model will start generating nonsense.

When a charged particle moves, the electric field vectors at two different locations A and B should start to change at exactly the same time.

Why do you say that?

That's instantaneous action at a distance and contradicts the speed of light limit.

When the pictures of Pluto came back to Earth, we had to wait hours after they were sent to receive them.

But how should we understand the propagation of electromagnetic wave?

In his answer below, John puts the answer to your question very well, the only addition I want to add to my answer is we should understand it as a wave travelling at the speed of light, rather than any analogy with ripples in water.

Here is the wikipedia animation of how a polarized ( for ease of concept)electromagnetic wave propagates:

Electric and magnetic fields both become zero at the node, so it is not that one generates the other.

Don't the electric field and magnetic field have infinite range?

In classical electrodynamics, the charge sources generate the electric and magnetic fields, and those yes, in the theory have infinite, even though of smalll amplitude, range.

The electromagnetic wave electric and magnetic fields obey also Maxwell's equations but are limited in range by the amplitude given by the wave equations.

In addition the electromagnetic wave obeys Lorentz transformations which give inherently the velocity of light in vacuum as c.

There is no ripple in a pond, as the Michelson Morley experiment showed there was no medium on which electromagnetic waves propagate. Only the sinusoidal structure of waves is common between water and acoustic waves which propagate energy on a medium.

In any case when amplitudes become very small due to the spreading of the wave, one has to go to quantum electrodynamics which describes classical light waves as emergent from the superposition of zillions of photons with energy h*ν where ν is the frequency of the wave.

So to understand the propagation of electromagnetic waves you have to enter into the mathematics of Maxwell's equations.

I'm not quite clear on your question, ayanx. The field is the electromagnetic field, and as per your title, it's an electromagnetic wave. It propagates from A to B at the speed of light. Things at locations A and B don't start to change at exactly the same time.

Perhaps there's some issue here with the way many depictions of an electromagnetic wave give the impression that there's two orthogonal waves present, one electric, one magnetic? You can even see this in the animations on the Wikipedia electromagnetic radiation article. This is unfortunate, because it really isn't like that at all. The E wave doesn't generate the B wave which generates the E wave, and so on. Instead it's an electromagnetic wave, and it propagates much like any other wave. If you look further down in the Wikipedia article you can see this:

"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..."

It isn't totally unlike the ripples on the pond. If it was a water wave and you were in a canoe, the tilt of your canoe would be E and the rate of change of tilt would be B. There aren't actually two waves present, just one. An electromagnetic wave.