What is the wavelength of electrical current?

Question

I am trying (as an exercise) to determine the force between two wires, in a similar nature to this question here. This suggests a solution based on the wavelength of the wave. However, the wavelength is different in the antenna to that between the antenna - the current is moving at a slower speed in the conductor and to that of light in a vacuum or air.

Wikipedia points out that that the wavelength depends on the Velocity factor of the material (which reduces the velocity to about 70% of the speed of light), which creates a long wavelength. Alternatively, other books/articles, like this here, suggest the velocity (and therefore wavelength) is orders of magnitude lower.

So which wavelength should I use to determine the magnetic field changes along the wire?

Also, if the answer simply turns out that the first is the propagation rate of voltage and the second is the propagation of current, then does that mean that V=IR doesn't work with AC?

EDIT I'm editing the question because I can see a potential misunderstanding of what the question is, when thinking through the lens of the configuration of the referenced question. That was my fault for not being explicit or simple enough with the extraction of my question from out of the details that inspired the question. It may be the case that my question has nothing to do with the wavelength formula given in the referenced question. So here goes my simplification.

If one were to take a snapshot of a wire with an AC running through it, one could assign a current and voltage to every position along the wire, and these follow a sinusoidal function. Therefore it has a wavelength that is associated to the velocity of the wave through the medium. But I have read of two different ways to determine the velocity of the wave through the medium, with very different wavelengths resulting (orders of magnitude different). What is the correct way to calculate the velocity of the wave through the medium, if one wants to measure the (instantaneous elemental) magnetic field along the wire?

Answer 1

If your wire is in air, the "velocity factor" will simply be equal to 1 - so the wavelength of the current in an antenna in air will be roughly equal to that of the wave in air / vacuum.

This is why we can think about things like "quarter wave dipole antennas" which are resonant because the time it takes for the current to flow to the end, reflect with phase inversion, and return, is exactly the period of the oscillation when the length of the antenna is 1/4 of the wavelength in air.

When the wire is surrounded by a medium with a relative dielectric constant greater than one (as is the case in the Wikipedia article you link), then the wave propagation speed will be slower as the square root of the relative permittivity, $\sqrt{\epsilon_r}$.

Note - the other article you link talks (indirectly) about the drift velocity of electrons in conductors. This velocity (of the individual electrons) is indeed MUCH slower than the speed of light - but as one electron moves, the field with which it pushes electrons ahead of itself moves at the usual speed of light, so the effect of the (small) motion of the electron is "felt" far away, nearly instantaneously.

This means you should ignore the "electrons are slow" argument when you consider the question you ask: current in one part of the wire will result in current in other parts of the wire almost instantaneously (although different electrons are carrying the current in the different parts of the wire). The only thing that matters is the dielectric constant of the material surrounding the conductor(s). See also Telegrapher's equations

Answer 2

Actually, both the geometry and the material properties depends on which specifications the antenna will have.

If you are making an exercise, it is enough by now to consider a perfect conductor, and do not deal with other effects, such as cable thickness, thin geometries, materials, etc.