What are electromagnetic waves really

[aaaa202]

My book takes a look at the maxwell equations in free space, shows that E and B satisfies the wave equation and then guesses on a plane wave as a solution such that you get a tranverse wave with E and B perpendicular to the direction of propagation.
I don't know if I am too slow in realizing things, but to me the above is pretty weird. What does this electromagnetic wave derived represent?
And why was it allowed to take for granted that it must be a plane wave. Who says electromagnetic waves are not spherical or has some other weird shape? After all there is considerable freedom in the structure of the wave, if you just know that the electromagnetic fields satisfy the wave equation.
And furthermore, what is it that actually produce these electromagnetic waves? I can see that the essence of them is somehow that the E and B fields sustain each other, but overall I am clueless as to what produces them. If you have an electrostatic situation, is there electromagnetic waves there? I think not, but why not? Surely the E and B fields are always a solution to the wave equation.
Phew, that was a lot of questions.. :)

 

[clamtrox]

Have you studied the theory of partial differential equations? The wave equation $\ddot{u}+ c^2 \nabla^2 u = 0$ is solved by a large class of functions u, like you suspect in your post. For example, a spherical wave is entirely possible. However, the simplest possible case, a plane wave, can often times approximate these solutions fairly well, if the wave's wavelength is small enough. For example, a spherical wave can be locally approximated with a plane wave, if the radius of the sphere is much larger than the wavelength.

As for what causes the waves, that's just how the electromagnetic interaction works. If you have a changing magnetic field, that induces a changing electric field and that again induces a changing magnetic field. I don't think there's any more "physical" picture for it in classical electrodynamics.

 

[aaaa202]

okay but what is the true wave emitted by an accelerated charge? The wave equation satisfies a number of functions as you say but surely there must only be one kind of electromagnetic wave?
My discomfort with the plane wave approximation is that these waves are used to derive the laws of optical geometry, snells law etc. Does that mean these are only approximations?

 

[clamtrox]

surely there must only be one kind of electromagnetic wave?

Indeed. Like with all partial differential equations, it's the initial and boundary conditions which tell you what kind of wave you got.

My discomfort with the plane wave approximation is that these waves are used to derive the laws of optical geometry, snells law etc. Does that mean these are only approximations?

Of course :)

 

[aaaa202]

okay, thanks for the answers - they helped!
Now, we're at it I might as well ask about another thing that is bothering me. The electromagnetic waves sent out will carry energy with them. My question is where this energy comes from in a situation like this:
Two charges of same sign which are accelerating away from each other. They will send out electromagnetic waves I assume, since the field E and B field is changing. On the other hand, all problems I previously did with situations like this you assumed that their potential energy was converted completely to kinetic energy - which makes sense, especially if you take the situation of mass in a graviational field as an analogy.
But that forces the question: Where did the electromagnetic wave get its energy from?

 

[rude man]

It's good you ask to get a physical feel for e-m waves rather than just go with the math.

E-M waves are time-varying electric and magnetic fields. Same fields as you get in electrostatics and from a permanent magnet.

You need accelerating charge to produce an e-m wave. For example, a constant current in a wave does not generate an e-m wave, but an ac current does since there you have accelerating charge (i = dq/dt = i0sinwt so d²q/dt² = wi0coswt).

No, plane waves arent the only type. In fact, a pure plane wave in space does not exist since the source would have to come from infinity. But in the limit you do approximate a plane wave and the math is easier that way. In reality, all e-m waves have a spherical wavefront as Huygens described back a few hundred years ago.

Where you do kind of have to have faith in the math is the way an e-m wave sustains itself by exchanging energy between the e and b fields. The maxwell equations themselves are really just distillations from previously observed phenomena, except for the displacement current term he added which makes e-m waves possible (he did that to account for the magnetic field around a capacitor since no current flowed across his capacitor plates).

As to your 2nd question, here I'm on less solid ground, but even a static charge has an E field and there is energy associated in that e field = 1/2 εE² joules per cubic meter where ε = permittivity. So as the charges move away from each other, so do their E fields. The total energy including the initial E fields and the generated e-m wave does not change. There may be some quantum-mechanical effects involved here with which I'm not intimatelyfamiliar though.