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I am currently working on a project that involves the use of LEDs. According to the datasheet, the LEDs I am using have a maximum brightness of 10,000 mcd. In my project, though, I will not be using the LEDs at full power. I was wondering how reducing the power would change this brightness measurement of the LEDs. I know that this will decrease the brightness, but by how much? For example, if I were to double the resistance of the LEDs while maintaining a constant voltage (cutting the power in half), what would the brightness be? Is there a formula/law that could be applied in this situation?

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Under the assumptions that (i) the efficiency of the LED doesn't change with load, and that (ii) neither does the spectrum, its luminous intensity (what you refer to as 'brightness') is proportional to the power it draws. Therefore, if you cut the power in half, the luminous intensity will be half as bright.

(However, you should keep in mind that human vision is not necessarily linear, and humans will not necessarily perceive a decrease by half in the luminous intensity of a source (i.e. its rating in candelas) as a decrease by half in brightness. Here, though, you go into more subjective regimes of how the human eye and brain respond to stimuli.)

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