What are electromagnetic fields made of? I am trying to understand electromagnetic fields so I have two question related to them.


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*What is a electromagnetic field made of? Is it made of photons / virtual photons?

*How about a static electric or magnetic field?
 A: When thinking about fundamental entities, it's quite easy to ask a question that, upon reflection, is contradictory.  The questions of this kind take the form:  What is [some fundamental thing] made of?
The contradiction here is that there can only be an answer if the fundamental thing isn't fundamental!  
The electromagnetic field is one such fundamental entity.  It's not made of anything else, it just is what it is.  In the context of QFT, photons (real and virtual) are, loosely speaking, "excitations" of this entity.  Real photons are associated with the long range propagation of energy and momentum, i.e., electromagnetic waves.  Virtual photons are associated with the electromagnetic force, i.e., the Lorentz force, as well as evanescent waves, and near field antenna radiation. 
A: Fields are more fundamental compared with particles (as fundamental as string).
Particles, such as electron, are the excitation of Dirac fields (sorry for that). Possible related discussion from M Strassler which may be helpful. http://profmattstrassler.com/articles-and-posts/the-higgs-particle/360-2/       Some points are the following :" A field is something that
1.is present everywhere in space and time,
2.can be, on average,  zero or not zero, and
3.can have waves in it.
4.And if it is a quantum field, its waves are made from particles.

So for example: the electric field is a part of nature that is found everywhere.  At any given point in space, and at any particular time, you can measure it.  If it’s non-zero on average in some region, it can have physical effects, such as making your hair stand on end or causing a spark.  It can also have waves — visible light is such a wave, as are X-rays and radio waves.
Ok, so, what is a particle?

A quantum field’s waves cannot be of arbitrary intensity.  The least-intense possible wave that a field can have is called a particle, and it often behave in rough accordance with your intuitive notion of “particle”, moving in a straight line and bouncing indivisibly off of things, etc., which is why we give it that name.
In the case of the electric field, its particles are called “photons”; they represent the dimmest possible flash.  Your eye absorbs light one photon at a time (though it typically waits for several photons to arrive before sending a signal to your brain.)  A laser produces very intense waves, but if you shield a laser with a screen so that only a tiny fraction of the light gets through, you will find, if you shield it enough, that the light passes through the screen in little blips — single photons — all of them equally dim."
A: Electromagnetic fields, which include static electric and magnetic fields, are indeed made of photons. From a particle physics perspective the Quantum Electrodynamics as a model of particles carrying electric charge interacting via photons has a spectacular agreement with experiment. The thing is, those experiments are very special in that we are sending in 'free' particles with a ton of energy and treating the interactions with the electromagnetic field as a very small perturbation on the free particles. So the picture we draw in our heads of particles interacting via exchange of a single photon is a simplified case that works very well in this situation:

Now, to make the answer more precise for something like a static electric field, to my knowledge is pretty much impossible. To see this we can look at something much simpler, coherent states (see http://en.wikipedia.org/wiki/Coherent_states) . These states don't even have a well-defined photon number, so while they are clearly ' made' of photons as the state is a linear combination of states of well-defined photon number:
$ |\alpha \rangle = e^{ \frac{- |\alpha|^2}{2}}  \displaystyle\sum\limits_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}|n \rangle  $
the the probability of detecting n photons is:
$ P(n) = e^{-|\alpha|^2}  \frac{|\alpha|^{2n}}{n!}$
which clearly isn't a delta function for n, which it would be if n was always the same number. 
And as far as I can tell, a state which produces a Coulomb-type field ($\frac{k q}{r^2}$) is going to be even more complicated than the coherent states, so it seems hopeless to try and phrase it in these terms. Note that this is in stark contrast to say, the electron number, which is always well-defined. Thus thinking about an electromagnetic field as made up of photons as the same way a block of metal is made up of electrons and other particles is probably a bad analogy to stretch very far. 
A: A magnetic field is a essentially a cloud of virtual photon "place-holders" in a state of flux; it's what the electrons that produce the field "owe" to other nearby electrons (which have gained real photons), for having their spin-charge moments aligned in the same direction.
A magnetic field is even more fundamental an entity than particles such as electrons, protons and neutrons.
A: The most fundamental thing in physics, is the way that we conduct phyiscs. And physics will be as good as we are conducting it.
We conduct physics by using 1) logic and 2) the scientific method.
In that context - a scientific theory is just us Humans, trying to give an accurate description of our unerstanding of our reality and our experiments - by creating mathematical models that best fit that understanding.
If at any point our 1) description, 2) understanding, 3) experiments are flawed, so will our physics.
So, the best model that we have today, is that of a Universe where fields are fundumental and they are not "made of" anything simpler.
That doesn't mean that the Universe is that way, we are just describing, that way.
If the real Universe turns out to agree with our theoretical one then that's good and the more it does the more accurate our predictions about this world would be... And ofcourse if the theory turns out to not describe our reality at all, then our theory is not so usefull.
So fields in this model are not made of anything. But maybe in the future, a model that has fields made of something rather than nothing will prove itself even more accurate at describing our understanding of our reality...
So we don't know what fields are made of, so far in our models they are not made of anything.
A: If I understand QFT correctly, you can think of the electron field as an all pervading bubbling cauldron (called vacuum fluctuations) of premature, not-ready-for-prime-time (potential?) electron energy balls, having energy below the minimum (but not zero) quantum energy required for an adult electron. If they get whacked with enough energy they can start vibrating at the minimum adult electron quantum energy level which causes a ripple in the field (like a stone dropped into a still pond). This ripple wave is what we call a real electron. 
Now I realize that is an unconventional idea, but if each elementary particle has its own vacuum field (photon field, electron field etc..) then there must be something unique about each field that enables it to create its associated particle. It's not enough to say "the field is what it is". So there has to be something specific about each field, and that is what I am saying above in order to try to give some type of theory about what the fields are "made of". Since nobody knows as of now, why not make an attempt.  
As Brian Skinner has said http://www.ribbonfarm.com/2015/06/23/where-do-electric-forces-come-from/ "for a physicist, the construction of such “origin stories” is perhaps the very most important part of the profession.  It is absolutely integral to physics that its developers never be satisfied with any level of description of reality.  To every law or equation or theorem, we must always ask “yes, but why is it that way?”  This impertinent questioning, where it succeeds, ultimately always turns one question into another question.  But along the way it can rewrite very fundamentally the way we perceive nature.  And, when those revisions succeed, they pave the way for significant new insights and discoveries while recapitulating all the results that came before."
A: Electromagnetic fields are made out of photons. The magnitude and direction of the electromagnetic field intensity at any point is directly proportional to the magnitude and direction of the force of the photon at that point. The field lines are the paths the photons take (and since according to quantum physics, a photon take all possible paths, then field lines of that photon would also have more than one direction).
Since photons are only emitted when the charge accelerates, a stationary electromagnetic field is not possible. A stationary charge will not have a field.
However, there seem to be two exceptions to this. Why do charges moving with constant velocity still attract or repel each other when they aren't emitting photons?
And on the surface of earth, if one magnet is on the ground and the other is suspended above it, both are at rest relative to each other, but when the suspended magnet is let go the magnets still repel.
Also, this view explains repulsion between two accelerating charges, but doesn't explain attraction.
