Getting a given wavelength radio signal given an antenna with real-world constraints

Question

Supposing you are given a transmitting antenna of whatever type of metal is most commonly used these days, and supposing that you are applying an AC current with the intent of transmitting a 1 m wavelength radio signal, about how far is one electron likely to make it in one direction along the antenna conductor in one half cycle from a negative peak to a positive peak, factoring in the likelihood of the presence of antenna material atoms affecting the electron's path? Can individual charge carriers actually make it very far (on the order of meters) through metal?

But if the answer is that individual charge carriers actually only make a microscopic or very small displacement in space, how could a 1 m electromagnetic wave be produced? I don't see how a large number of charge carriers each only making a very small displacement in space can add up to a 1 m wave.

This site, especially Fig. 2 there, is where I'm getting my basic understanding of how alternating current gets an EM wave transmitted. From that is where I got the feeling that charge carriers would actually need to make a displacement in space on the same order as the intended EM wave.

Answer 1

The average displacement of an electron in the presence of an electric field is given by it's 'drift velocity' (link here). You can see right at the end of the page that the average displacement about a mean position for an electron is about $ \approx 10^{-6} m$. This is clearly a tiny amount (in macroscopic terms) to be displaced. So how exactly does a 1 m wave get produced?

Well, firstly there's the equation that relates the velocity, frequency and wavelength of the wave given by $$v = \nu \lambda$$ where $\nu$ is the frequency, $\lambda$ is the wavelength, and $v$ is the speed of propogation of the wave. For electromagnetic radiation, this $v$ turns out to be $c$, the speed of light. So from here it's straightforward to see that an AC voltage of some frequency $\nu$ will create a wave of some wavelength $\lambda$.

To answer your question about the displacement of the electrons - The wavelength of the emitted radiation not given by the length of the wire. As the website that you referred to explained, accelerating charges emit electromagnetic radiation. Therefore, when each electron is accelerated, it will emit radiation. When you apply an AC voltage across a wire, what is happening is that the electrons are oscillating around a central position (since the direction of the voltage changes every half a wavelength of the applied AC voltage), and will radiate according to the above equation. So whether you have a tiny wire or a large wire, the wavelength of the emitted radiation will be determined by the frequency of the AC voltage applied, and not the wire length (as long as the wire isn't a few micrometres long!).