# Three Clear Questions on the Nature of Electromagnetic Radiation

## Main Question or Discussion Point

Hello, this is a first year undergraduate student speaking, thanks for taking the time:

1) Is it unphysical (in a perfect realm, etc) to say "one EM wave"?
With which I mean: an infinite mathematical line with for each point an E and B-vector defined varying sinusoidally. In other words, would there be any physical laws I'm breaking by saying there's only "one wave"? (Is there a need of an infinite multiplicity to make sense?)

2) How do I relate an EM wave with a photon?
Obviously an EM wave is a mathematical idealization, correct? A photon is not periodical. Is "one EM wave" a continuum of an infinite amount of photons on one line? Am I far off? Is this is a crazy analogy? Is it acceptable? If the two are not connectable: is there a mathematical description of a photon; is it similar to that of a linear EM wave?

3) The (average) intensity (= average Poynting vector) of (one) linear wave is constant, that of a spherical wavefront is not.
This implies a spherical wavefront is not just a sum an infinite amount of linear waves originating from one point outward. Intuitively that is how I imagine it. Maybe that is because I'm secretly thinking about photons flying from a point source. If you want to describe the waves from a spherical point source, you have to start from scratch? Just like a plane wave is an infinite sum of linear waves, is a spherical wavefront a sum of something? What is one such element, if so?

I hope I was clear. I welcome all replies,
mr. vodka

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1) It is unphysical but it is consistent with Maxwell's equations. The infinite plane wave is a bread and butter tool of solving 3 dimensional EM problems.

2). A photon obeys Maxwell's equations. You can indeed launch a photon that is a single quantum of a planewave field. It would have a definit energy and momentum and no position.

3). Just as you can decompose a time varying wave into s sum of sinusoids in time, you can write an arbitrary wave in three D as a sum of planewaves going in various directions including imaginary directions. This is sometimes called k-space planewave decomposition or in physics this would constant-momentum planewave superposition.

But the simple reason for spherical waves diminishing is the inverse square law.

Dale
Mentor
1) Is it unphysical (in a perfect realm, etc) to say "one EM wave"?
With which I mean: an infinite mathematical line with for each point an E and B-vector defined varying sinusoidally. In other words, would there be any physical laws I'm breaking by saying there's only "one wave"? (Is there a need of an infinite multiplicity to make sense?)
It sounds to me like what you are describing is not a solution to Maxwell's equations. Antiphon thinks you are describing a sinusoidal plane wave:
$$A\;sin(k_xx-\omega t)$$
But to me it sounds like you mean something more like:
$$A\;sin(k_xx-\omega t)\;\delta(y-y_0)\;\delta(z-z_0)$$

Could you clarify? The former is a solution to Maxwells equations while the latter is not.

1) (to Antiphon and DaleSpam) Hm, indeed (in relation to DaleSpam's post), I meant only one line (say the x-axis) not having the E and B zero at all times (with line of propagation the positive x-axis). This is wrong in the mathematical-physical sense, then? You can only speak of a whole plane wave? That is, "one EM wave" is actually a plane wave, defined all over 3D space? But this is weird, when we fire one photon, we don't have E and B-vectors defined (related to that photon) all over 3D space, do we?

2) "a photon that is a single quantum of a planewave field" So if you have a plane wave propagating in the positive x-direction, then a photon can be seen as a "slice" of the whole plane-wave from x = a to x = b? (with a and b moving at the speed of light) But in relation to the previous question: a plane wave extends into infinite sideways (in the yz-direction), but a photon is not (we can only sense the photon if it strikes us, right? Not when it passes us). Is this not a problem?

3) Okay thank you. I suppose for this question (for now in my studies) I'll have to accept that if you start with the maxwell equations and demand you have a point-source and then solve it, you get a r²-dependence. That's all right, much like a spherical sound-wave. I just wanted to make sure I couldn't use my plane wave solution to somehow derive the intensity for a point source.

Thank you both.