Electronic – How exactly are radio waves produced from a current in a circuit itself

Is amplitude of the waves modulated by the amount of current?

The amplitude of the current signal is half of the difference between the maximum and minimum of the current in each cycle of the oscillation.

For example, if the current over time is described by the equation

$$i(t) = A \sin(\omega{}t+\phi)$$

then A is the amplitude of signal because the current oscillates between +A and -A (and A has units of amps).

In the circuit shown, there is nothing modulating the current. Modulation happens when some other signal (like an audio waveform) changes the amplitude (for example) of the current waveform. This would require a more complex circuit than what is shown in your example.

Frequency is modulated by the frequency of the capacitor release of energy, correct?

Again, your circuit shows no mechanism for modulating the frequency. If you used a variable capacitor and the capacitance was controlled by another signal, that would cause frequency modulation. But it would probably also cause some undesirable parasitic amplitude modulation. To see some typical frequency modulator circuits, you can do a google image search for "frequency modulator" and look at the schematics that are found.

That would mean that using a capacitor with a lower farad rating should produce a higher frequency?

Generally the resonance of an LC tank circuit like in your example is given by

$$\omega_0=\frac{1}{\sqrt{LC}}$$

So the oscillation frequency can be increased by reducing either the capacitance or inductance value.

How would one go about keeping a sustained current to the inductor since it's constantly losing current without messing up the cycle?

You can use an oscillator circuit. Generally this means adding some kind of amplifier to the circuit to add energy to the waveform as quickly as it is lost to parasitic resistance and to feeding the load.

You may find this of interest. This is a spectrum allocation map, specifically the one for the United States, as determined by the FCC. It spans the entire radio spectrum.

Frequency Allocation Chart

Crowded isn't it? We are, as of October 2011, effectively allocated the entire RF spectrum. You probably noticed that it spans 9 KHz to 300 GHz.

It of course extends all the way down, but frequencies below 9 KHz are not practical to use, so we ignore them. But why does it end at 300GHz on the high end?

Answering that question is going to take a little context. We will get to the answer though, I promise!

1. Bamboo Lightbulbs

Beyond 300 GHz, we stop calling electromagnetic waves 'radio' and start calling it infrared light. There is no fundamental difference, the entire electromagnetic spectrum, from radio waves to high energy gamma rays, are just different frequencies of the same thing. Any time an electromagnetic charge carrier is accelerated, such a wave is produced.

Matter, which has charge carriers bouncing and vibrating around and otherwise doing things that definitely qualify as accelerating, emits electromagnetic waves. The frequencies emitted are determined by the temperature of the matter emitting them. The colder the matter, the slower the vibrations, and the lower the frequency.

The very first commercial lightbulb used bamboo as the filament. It could withstand the heat well enough, at least until the superior tungsten filaments were invented. How could bamboo and tungsten both do the same job of creating light?

Actually, anything can do that job. Everything made of matter, regardless of it's phase, will begin glowing visibly at the same temperature. They all start at a dull red, become more and more like white light, then become white with more blue. This is color temperature. It's the color that matter at that temperature glows. At 2700K, its the light we know as 'warm white'.

Below visible light, this still happens, people show up on infrared because their body temperature makes their people meat release infrared. Things that are very cold 'glow' in radio.

Radio, light, x-rays, they're not different. A tsunami and a ripple in a pond are both still waves and both still water. So too, is it for the electromagnetic spectrum.

Except when it isn't.

2. Oh noes, reality isn't analog?!

Nope. It's not. Reality is not continuous, it is quantized. It comes in discreet, minimum values. That is the basis of what we call as a whole, quantum physics.

Electromagnetic waves are quantized into packets of energy called photons. Because of this, all energy of that wave is packed into one wavelength. Imagine electromagnetic waves as wiggles that grow at one end, but shrink the exact same amount at the other, and wiggle up and down, tracing the path of a sine wave. This is not at all an accurate description, but its a good analogy.

A shorter wavelength means a higher frequency, which means a much sharper acceleration produced it, and not only does increasing the frequency also increase the energy of the photon, it decreases the size of what that energy can be delivered to.

3. Domains - now with rationale!

This energy aspect is what gives electromagnetic waves such different behaviors and properties, despite being nothing more than different frequencies of the same thing. Armed with this context, we divide the spectrum up in what is actually a very non-arbitrary way:

1. Radio. Radio waves are so large that their energy can only be delivered to large groups of charge carriers, which in a conductor can be thought of as behaving like a plasma. This is what we see as current induced in an antenna - its the wave pushing on a large number of electrons in a conductor, and the larger the wave, the larger an antenna you need to get a reasonable push. You can use a smaller antenna, but the amount of power/signal you can receive at that frequency will be diminished.

2. Far infrared (300GHz - 214THz). At 300 GHz, the waves become small enough that all their energy can be delivered to a certain molecules that are large enough - namely, H2O. But the amount of energy is too low to do anything more interesting than rotate or generate heat in those molecules. The water content of our atmosphere absorbs 300GHz to some THz electromagnetic waves so strongly that our atmosphere becomes effectively opaque. That is why radio ends at 300GHz - not even the atmosphere is transparent anymore, and it stays dark until you get into much higher frequencies.

3. Infrared (214THz - 400THz) This is where things get interesting. At this size, molecular vibration emits electromagnetic radiation of this frequency, and conversely, this frequency of electromagnetic radiation has enough energy packed into a small enough photon that it can vibrate molecules. And it has also grown small enough that it can squeeze between gaps between water molecules, so while it is still absorbed, much of it is able to make it through, and the atmosphere becomes transparent once again. Remember color temperature? Because molecular vibration is one of the primary components of heat, this is where the radiation from matter becomes much more meaningful, as a warm body emits A LOT of infrared.

I'm ending the list, because now, we get into something special.

4. Light up the night (with your jellyfish)

Visible light (400THz - 790THz). Despite what I said earlier, light is actually special. Up until now, even in infrared, we can see how the waves are just getting tinier and interacting with matter primarily due to the change in size, and can see how its just a continuation of the radio spectrum.

With light, we get something totally new. The waves are smaller still of course, but there is so much energy packed into one photon now, it can do something amazing - it can excite the surface electrons of molecules. This excitation is quantized - there are specific energy levels an electron is able to be excited to. This sharp discontinuity means that only certain electromagnetic wavelengths can excite it, and once excited, it can only release that energy in quantized amounts as well. This means that matter suddenly starts interacting with electromagnetic waves based on these energy bands, and interacts with waves selectively, and most importantly, can absorb a wave in the form of an excited electron, and emit a new electromagnetic wave (or several) of similar magnitude but lower frequency. This lets you determine the nature of the energy bands of any bit of matter, and we perceive this as color. And electromagnetic waves are no longer interacting or being absorbed or reflected, they are being re-emitted and modulated in ways dependent on the material itself.

It's much more information rich. To do this with radiowaves, you'd need to build passive devices that are powered by radio waves alone, and retransmit new waves at different frequencies. This can be done, but requires high power radar, and its large. Light does this with matter itself. The visible range is not accidental or arbitrary, we see the range we do because it is the range that electromagnetic waves excite molecular surface electrons.

If alien life evolved in environments where there is a full spectrum available, they would still 'see', if vision was useful, in a range very similar to us. There is life on earth that has involved in environments with no light at all, and much of it evolved the ability to emit light on its own in the absence of it. It can't be environmental, as there is no light. It's the physics that make light different, and also universal.

5. It's over 9000

Above visible light, the power levels become dangerously high. Our vision ends at the transition to damaging light. Ultraviolet is made of waves so small and energetic that they no longer excite electrons - they can rip them away entirely. Or excite deeper valence electrons on the outside of atoms. Valence electrons are what causes chemistry and molecular bonds, and excitation of these can cause interactions in the chemical domain. Now, matter can be changed chemically by electromagnetic waves.

Xrays are further distinguished by there ability to excite or eject electrons deeper than even the valence electrons, core atomic electrons. This causes matter to ionize, and we call this ionizing radiation.

Finally, we reach gamma rays. Gamma rays are further distinguished still by their ability to also excite and even eject atomic nuclei. Gamma rays are so powerful, than can destabilize atoms and transmute them into new isotopes or even other elements. At this point, matter can be ripped apart into its smallest units, and cannot withstand electromagnetic waves of this magnitude. The death star's laser was almost certainly largely gamma rays, as Admiral Ackbar states the truth - that they cannot repel fire power of that magnitude. Nothing can repel gamma rays. Or even withstand them.

But, the spectrum goes higher still. We come to the end, the last distinction. High energy gamma rays. These are photons so mind-bogglingly energetic that they can create particle-antiparticle pairs. When the wave is so powerful it can create matter, more energy just creates more matter. This is the end of effects worth distinguishing.

So that is what happens when you increase the frequency of radio waves. I hope that answers your question - or even better, has given you new questions to find answers to!