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Our eyes don't see light; they detect vibrations in the 400-800 THz range that we call color. Since our eyes can detect the color we call black, what is its frequency?

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Black is not a color, it is a shade.

In physics, we call something "black" when it does not reflect any of the incident light. However, all black bodies radiate. The frequency of that radiation, the black body spectrum, is a function of the temperature of the object and follows Planck's Law:

$$B(\lambda, T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda k_B T}}-1}$$

Plotting this for a well known "black" body (the sun) you get the following (normalized):

enter image description here

We recognize that as "white", but really it is black.

Even white objects only appear white when we illuminate them with white light - if you shine red light on a white object, the object doesn't change color, but you perceive it as "red".

So there are three things going on here:

  1. When you look at an object, what you see is the interplay between the color of the incident light, and the surface reflectivity
  2. "black" means "does not reflect light"
  3. black bodies have a color corresponding to their temperature: colder = redder
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