# If electromagnetic fields don't actually propagate in plane waves, how do they propagate?

When electromagnetic waves were introduced to me, they were introduced through the use of Maxwell's equations in a vacuum, which are \begin{align} \nabla \cdot E &= 0 \\ \nabla \cdot B &= 0 \\ \nabla \times E &= -\frac{\partial B}{\partial t}\\ \nabla \times B &= \mu_0\epsilon_0\frac{\partial E}{\partial t} \, , \end{align} and then looking for solutions to $B$ and $E$ of some form. The form introduced to us was the plane wave form $E(t,y) = \hat{z}E_0\sin(ky-\omega t)$ and $B(t,y) = \hat{x}B_0\sin(ky-\omega t)$(with these ones being arbitrarily chosen as ubiquitous on the xz plane).

Ok, so when this is introduced and when we solve problems using solutions of this form, the problems have sometimes had a plane wave that is present through all of space at once on the plane, and moves in the direction of the Poynting vector at the speed of light, $c$.

The assumption that the field ubiquitous on a plane seems very unphysical. How does an electromagnetic wave actually propagate when it is produced (by any known method to produce EM waves)? Is this phet.colorada.edu simulation accurate (but in 3d of course: https://phet.colorado.edu/sims/radiating-charge/radiating-charge_en.html)?

Also, whenever I've seen EM waves described, I've seen a picture where the electric field and magnetic fields propagate and change sign in a sinusoidal fashion. The only method I know to produce EM waves is to 'accelerate charged particles', with some spherical perturbation, the electromagnetic field, emanating out from the particle as it gets accelerated (as illustrated in the simulation above). Why would the sign of the electric field ever change as this happens? The particle carries the same charge as it oscillates, so why would the sign of the electric field at some point ever change as the particle oscillates about?

• perhaps you are interested to read about What are photons, what is electromagnetic radiation and what are EM waves Jul 2 '17 at 19:12
• You're aware that, since (classical) electromagnetism is linear, any reasonable electromagnetic wave can be expressed as some linear combination of planes waves, right? Jul 3 '17 at 0:16
• Retarded potentials in the Lorentz gauge Jul 3 '17 at 3:15
• It's worth noting that another simple analytic solution to Maxwell's equations in homogenous media is the Gaussian Beam, which you've unknowingly encountered if you've ever played with laser pointers. Jul 3 '17 at 3:42

You are right that perfect EM plane waves cannot exist in nature because they are infinitely extended in space, so they can't be produced by a localized source. But they are a useful approximation for two reasons.

First, if you "zoom in" enough, then arbitrary EM radiation locally looks like a plane wave (for example, far away enough from a point source that you can ignore the curvature of the wave front, or deep in the bulk of a transmission line or waveguide).

Second, and maybe more importantly, EM plane waves form a basis for the set of solutions to the vacuum Maxwell's equations. So we can express any EM field in the absence of charges as a linear combination of plane waves, so they're the "elementary building blocks" of all solutions.

If you've taken quantum mechanics, the situation is exactly analogous to the case of momentum eigenstates.

Do you know the following joke: A physicist is ask to calculate the volume of a cow. The physicist says: "Assuming that the cow is a sphere ...".

This joke captures a key point: Most of physics is not about accuracy, but to develop a simple model, which yields a good estimate. There will always be a regime, where the model is a good approximation. This is exactly what the idea of a plane wave is about.