How to define the electric and magnetic fields of an EM Wave in a vacuum?

You've come to a long time phyilosophical question: does a field actually "exist"? We cannot measure fields.

There is no instrument able to measure a "field". The instrument you use to measure electric field is actually based on forces. The instrument detects an electric force and then it deduces the value of the $E$ field.

• In an empty space, there's nothing.
• With one charge, there's electric field everywhere $(E=Kfrac{Qq}{r^2})$
• As soon as there are more than one charge, two forces appear between every pair of charges. Those forces are such that $F=qcdot E$

And there are tons of charges everywhere. The measurign instrument itself is made up of charges. What the instrument measures is forces between charges. Then $E=F/q$.

What I mean is that we cannot observe fields, we can only detect their effect on matter.

So, regardless of whether they actually exist or not... the reality works LIKE if they were real.

And a field is nothing more than a mathematical idea. It's a function of space,

$$vec{E}: mathbb{R}^3 rightarrow mathbb{R}^3$$ $$vec{r}in mathbb{R}^3 rightarrow vec{E}(vec{r})$$

For every point $(x,y,z)$ in the space, there is a vector $vec{E}$ in that point (unless there is a charge in that point).

So fields are just vectors in space, in the same way as you study geometry in mathematics: there are ideal objects (points, lines, planes...). In the same way, there are electric force vectors, defined at some point in space.

And we cannot observe that vectors, they are mathematical constructs. We can observe their effects on matter. Do those effects come from that vector? We don't know, but it's LIKE IF they did.

I understand that EM waves are self-perpetuated due to the interactions between changing electric and magnetic fields as described by Maxwell's third and fourth equations, but I'm stuck conceptually on what they are if there are no electrons or conductors around.

people were stuck like you begininning of the 20th century thinking the EM waves required a medium. But the medium couldnt be detected, so Einstein said there was none, then said it was spacetime.

The magnetic force is a relativistic effect of the electric force, so we are left with the electric force.

but I'm stuck conceptually on what they are if there are no electrons or conductors around.

The electric force is a concept like eveything in physics (what is potential energy you could ask similarly), and the concept is defined by its effects (the electric force accelerates charged particles). I'd be more interested in knowing why it sticks to electrons rather than what it is without it. Finally I doubt you'll get the meaningful answer you are looking for, simply because I don't think we understand what's below electromagnetism yet, for now it's just a primitive unbreakable lego brick of physics

Electromagnetic radiation is the emission of photons from excited subatomic particles. The light from a heated wire is an example of the stochastic emission of photons from the excited electrons in the wire.

The concept of EM radiation as a stream of quanta was a by-product of Planck's equation for blackbody radiation. He did not initially believe that the energy packets introduced were real particles. Later, Einstein interpreted the photoelectric effect with the existence of such quanta, later called photons.

If you move far enough away from such a heat source or dim the radiation with filters, you end up measuring single incoming photons. They could be made visible by photoemulsion plates or electronic devices (CCD chip).

It has long been known that light does not need a medium. Otherwise, light from celestial bodies would not reach us. Light travels through the vacuum, every single photon does so.

In short, photons are indivisible particles from emission to absorption and spread out in empty space between emission and absorption.

The fact that photons consist of an electric and a magnetic field component can be observed in radio waves. Synchronously accelerated electrons in the antenna emit photons with an electric field component all aligned parallel to the antenna and with a magnetic field component aligned perpendicular to the antenna. Hertz was the first to measure these components.

How to define the electric and magnetic fields of an EM Wave in a vacuum?

In a vacuum, the electric and magnetic field components of each photon in EM radiation are perpendicular to the direction of motion and perpendicular to each other. The Cartesian coordinate system is useful for illustration. If X is the direction of propagation of the photon, then Y and Z are the directions of the field components.

For photons from thermal sources, the direction of the Y-Z pair is random with respect to X. For radio waves, free-electron lasers or cyclotron radiation, for example, the electric field component is parallel to the acceleration direction of the electrons involved.