Is the reach of the E- and B-field in an EM-wavefront infinite?

Question

In my textbook, the wave-equation for EM waves was derived by using Maxwells' equations in integral form on a EM propgagation in $x$-direction (in vacuum), with E-field in $y$-direction and B-field in $z$-direction.

First, to see if this wave was consistent with Maxwells' equations, one imagined a gaussian closed surface in the shape of a cube, facing in the $x$-direction. $\oint E\cdot ds = Q_{encl}=0$, but $\oint E\cdot ds = A_{y}\times E - A_{-y}\times E$. Therefore must E be equal on both sides, and furthermore homogenous in each $yz$-plane.

But one can in principle make this gaussian surface as streched in the y-direction as one want, and not necessarily centered on the axis of propagation. If you create a surface with $A_{-y}$ just below the $x$-axis and, let's say, $A_y$ three lightyears over the $x$-axis. Then, by the same logic, the E-field has to be of equal magnitude over there...

So does, as the wave-propagates, the electric and magnetic field instantly appear in an infinitely large plane perpendicular to the direction of propagation, or am I misinterpreting something here? I probably am...

Answer 1

The solution being considered here, a plane wave, is indeed constant at any time on a plane orthogonal to the direction of propagation. It's safe to say there are no "real" plane waves in the world. However, they are still important because

1. They are easy to analyze.
2. Within a restricted region of space-time they are a good approximation to real waves.
3. Through the magic of Fourier analysis, real solutions can be "decomposed" into plane waves.

Answer 2

Why is this on mathematics? Seems like more a physics question, but I'll answer anyway.

First, something must make an EM wave. If the wave is a plane wave, then the analysis is correct, it is a plane moving through 3D space. However, how do you make a plane wave? Not from a point charge, that's for sure... meaning that in order for the plane wave to get there, you need to have a source (think of what type of source is required to make a plane wave). Now, plane waves are good to study because at far distances from say a point charge (which gives off a spherical wave), the wave can locally be approximated as a plane wave, just like Earth is approximately flat for everyday stuff.

Typically, a wave is produced by an accelerating charge, spinning dipole, etc. These waves are not plane waves, but spherical (or approximately spherical) waves.

Try not to over think it - treat EM waves just like you'd treat waves traveling on a string. You need a source first and foremost, and without any dissipation, the wave travels at a constant speed off to infinity. If anything need clarification, let me know. Hope this helps.