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The LC oscillator is most commonly used to generate radio waves for practical use and the frequency $\omega$ of the LC oscillator equals that of the electromagnetic wave so produced. So, can they in principle be used to emit visible light?

The frequency of visible light is on the order of a few hundred terahertz, and the frequency of an LC oscillator is

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

I admit, the product $LC$ does become very small (on the order of $10^{-30}$) when the numbers are plugged in, but making an inductor and a capacitor with small values isn't difficult, is it?

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Actually, creating extremely small inductances (and capacitors) is quite difficult: even a simple wire will have about $\mu_0\approx 10^{-9} H/mm$, so how will you ever reach something of the order of $10^{-15} H$? – Ansgar Esztermann Jun 30 '14 at 13:09
up vote 11 down vote accepted

Stop and actually think about it. Small-value inductors and capacitors are possible, but you also need extremely small size else the lumped system approximation your equation relies on becomes invalid. Let's say the wavelength of the light to produce is 700 nm. A distance of half that (350 nm) will cause a inversion (180° phase change). Generally you want to keep the maximum dimension to 1/10 the wavelength to consider it a lumped system without having to worry about that further.

Your inductor and capacitor therefore have to all fit roughly within a 70 nm dimension. With everything that close to everything else, there will be a lot of parasitic capacitance. Getting a low enough capacitance and still have the deliberate capacitance accross the ends of the inductor dominate over all the distributed stray capacitance won't be easy. Even if you could do that, you still have to drive it with a 430 THz signal somehow. Where are you going to get that from?

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Are you sure there aren't microscopic structures with optical frequency plasmon modes? If so, it would be easy to describe an effective inductance and capacitance. Your answer is written as if the notion of optical resonance is ridiculous, but I wonder if it really is. The notion of a lumped electrical LC optical mode is completely unfeasible, as you said, of course. – DanielSank Jul 1 '14 at 7:50
@daniel: I didn't say optical resonance is unfeasable. I only addressed the question, which was about a lumped L-C resonant circuit. We may yet some day be able to create resonant cavities at the few 100 nm scale, but this is not what the OP asked about. – Olin Lathrop Jul 1 '14 at 11:21

If, by LC oscillator, you mean a circuit composed of an 'ordinary' inductor and capacitor etc. then the answer is no.

The equations for the LC oscillator are derived within the context of ideal circuit theory which is the limit of a number of assumptions.

To apply the results from ideal circuit theory to physical systems, the physical systems must approximate the assumptions of ideal circuit theory.

One of these is that physical circuit elements can be represented with lumped element models.

And, for that to hold, the wavelengths of the signals of interest must be much larger than the dimensions of the physical circuit elements.

When this assumption does not hold, we must use the distributed element model such as in, for example, transmission line theory.

Another related consideration is that physical circuit elements have parasitic properties that cannot be avoided. Thus any physical system of conductors possess parasitic inductance and capacitance that become significant at high enough frequencies.

For visible light, the wavelengths are so small (hundreds of nanometers) that the assumptions of ideal circuit theory are not remotely valid so, unless one generalizes the notions of "inductor" and "capacitor" (and the other circuit elements) far outside the ordinary, the answer is no, an LC oscillator cannot produce visible light.

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What do you mean by 'far out of the ordinary'? – Gerard Jun 30 '14 at 14:01

During WWII, British radar researchers struggled to develop very short wavelength radar waves (on the order of 10cm). These would be useful because they would (among other things) permit small airborne antennas for night fighters, bomber defense and navigation, etc. They couldn't do it with ordinary electronic circuitry, and had to develop the cavity magnetron to generate such high frequencies and high power levels necessary, which uses much different principles. Now take that difficulty, and try to generate Infrared waves, much less even shorter Visible light. That's not even considering the problems of building an antenna that tiny!

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An electric oscillator for light exists indeed. They are called LASER diodes. Due to the works of a number of Nobel prizewinners (e.g. Einstein predicted the working in 1911), you can buy then in a store and simple powers then with a DC power source. It's a sort of oscillator, not with electric current, but with light.

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Lasers are inherently quantum mechanical devices, though, and their principle of operation cannot be explained in terms of an LC circuit. – Ted Pudlik Jun 30 '14 at 19:51

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