The perception of colors in a rainbow and the colors observed when using color filters over white light are different phenomena with distinct explanations.
When we observe a natural rainbow, such as after rain showers or when sunlight passes through water droplets in the atmosphere, we typically perceive a continuous spectrum of colors. This spectrum includes various shades of colors blending seamlessly into one another. The seven colors often associated with a rainbow are red, orange, yellow, green, blue, indigo, and violet, which were historically described by Sir Isaac Newton. However, this division into seven colors is somewhat arbitrary and serves as a mnemonic device rather than a strict scientific classification. In reality, the colors in a rainbow form a continuous spectrum with an infinite number of colors blending together.
On the other hand, when we use color filters over white light, such as in the case of a prism or a glass of water with colorants, the filters selectively absorb certain wavelengths of light while allowing others to pass through. The filters act as a kind of "gatekeeper" for specific colors, blocking some and transmitting others. Depending on the characteristics of the filter, different wavelengths of light are absorbed or transmitted, resulting in a limited set of colors being observed. This process can yield a more discrete and defined set of colors, distinct from the continuous spectrum of a natural rainbow.
In summary, the difference between the colors observed in a natural rainbow and those seen through color filters arises from the distinct optical processes involved. Natural rainbows display a continuous spectrum, while color filters selectively transmit specific wavelengths, leading to a narrower range of colors.