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Microwaves can induce current in a coil of wire why cant visible light do the same?

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Like radio waves, the electric field of the optical light acts on the electrons in a material and can cause them to move. But visible light has a much shorter wavelength than radio waves (about $10^6$ shorter) and so it doesn't drive macroscopic currents. Moreover, the conductivity decreases, which we can see in the Drude Model:

$\sigma (\omega )={\frac {\sigma _{0}}{1+i\omega \tau }}={\frac {\sigma _{0}}{1+\omega ^{2}\tau ^{2}}}-i\omega \tau {\frac {\sigma _{0}}{1+\omega ^{2}\tau ^{2}}}$

where $\omega$ is the light frequency, $\tau$ is the relaxation time (due to scattering and related to the carrier density, temperature, etc.), and $\sigma _{0}=\frac {nq^{2}\tau }{m}$ is the effective DC conductivity ($n$ is the number density and $m$ is an effective mass which can be different for electrons and holes depending on the crystal).

We can see that when the frequency $\omega$ is large (like for optical waves) the conductivity goes to zero. This happens because at really high frequencies the electrons do not have time to respond to the field and so they no longer (appreciably) move back and forth. Above this frequency the material becomes transparent (identical to the behavior at the critical frequency in plasmas).

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There is the developing field of nanoantennas. It is now possible to manufacture arrays of Yagi-Uda antennas on micrometer length scales. And nanoclusters like red gold in glass have been made since Roman times.

The problem is a bit what to do with such currents. There is no electronics to act as a radio tuner at such frequencies. But there are ideas to make rectifying nanowires of GaAs and use them to harvest solar energy.

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