Do electromagnetic fields are already present all over the space? Consider a region $R$ in space without any source of electromagnetic field. Now put a source $S$ of electromagnetic wave in the vicinity of $R$ so that at time $t=0$, $S$ starts radiating electromagnetic wave. The wave moves with a finite velocity $c$ and hence cannot reach at a distance $x$ at time $t=t1$, if $x>ct1$. Now consider a general solution for electromagnetic wave equation is as follows:
$$
 E=E_0 \sin (kx-\omega t)  ...............(1)
$$ $$
    B=B_0 \sin (kx-\omega t)...............(2)
$$
Then at time $t=0$   at any distance $x$ the values of $E$ and $B$ are not zero except at the $nodal$ $points$. Hence, 


*

*either the solution is not satisfactory for any values of $x$ and $t$ (in that case, what will be a satisfactory solution?) 


or 


*electromagnetic fields are already present all over the space even if there be no source of the fields (in that case, how is it possible to have fields without sources?).


or


*electromagnetic fields are already present all over the space because a space free of electromagnetic fields cannot be created. (in that case, what is the reason behind this?).


If you have any other opinion different from the above three, you are welcome to express it and explain as well.
 A: The solution you have written down is the solution for an infinite plane wave. The wave described by this solution has existed for an infinite time and will continue to exist for an infinite time, and it has an infinite extent in space.
This is an idealised solution that doesn't exist in the real world because in the real world EM waves are created, propagate for some time and are then destroyed. However in many cases the infinite plane wave can be a useful approximation for a real wave, and that's why you are taught it in physics classes.
A: I am asking whether the field preexists?
Imagine a particle which is at rest atfirst.The particle has electric field around it to infinity.(As we know electric field has infinit range and the particle has no magnetic field).
NOW lets suppose somehow the particle begin to oscillate.
And also imagine a region far away from the charge.
An observer in that region will not see a magnetic field(magnetic field due to the oscillation of the charge), instantaneously(otherwise that would violate special relativity).
 That apparently saying observer will see the magnetic field when magnetic field propagate and reach that region.
Thus this says field doesnt always pre-exist it also propagate.
A: What you wrote is a solution normally used far away from its source and in a limited volume. These solutions make sense as an external force in the equation of motion of another system with charges. This wave may well be absorbed and turned into heat. In other words, we always deal with emitters and absorbers so the wave equations are not "free" and are accompanied with equations of matter. Only all of them determine the whole picture.
