What is the Definition of a Candela?

[SamRoss]

TL;DR

Trying to wrap my head around the definition of a candela.

I'm trying to wrap my head around the definition of a candela:

"The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×10^12 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian."

I'm aware that this definition is from 1979 and there is a new definition which went into effect in 2019 but I would still prefer to work with the 1979 definition. Here's what I think I get so far...

- Watts are joules (a measure of energy) per second.
- Radiant intensity is watts per steradian - so joules per second per steradian.
- Luminous intensity is also measured in watts per steradian, but it does not take into account all energy (per second per steradian), only energy from light of frequencies which are visibile to the human eye and it is weighted based on how well the human eye responds to those frequencies. For this reason, the luminous intensity is found by taking the radiant intesity of a certain frequency and multiplying it by some weight (which could be zero if the frequency is out of the visible range). For light of more than one frequency, the weight would be replaced by a luminosity function which is integrated over all frequencies.

Procedure: So to define a candela, we start with a green light (specifically, light with a frequency of 540×10^12 hertz), then measure how much energy is coming out of that light per second per steradian.

Now, I know I must be wrong about something because with my "understanding" as I've outlined it above, the definition of a candela seems redundant. If radiant intensity and luminous intensity are both measured in watts per steradian, then why can't the definition just be "The candela is the radiant intensity of a source that emits monochromatic radiation of frequency 540×10^12 hertz" which happens to be 1/683 watt per steradian. I am also implicitly assuming here that since the frequency of light used in the definition is a frequency that is perceived very well by the human eye, the radiant intensity and luminous intensity should be the same (which is why I used "radiant intensity" in my definition and did not mention multiplying this by the luminosity function in my procedure). Is that also incorrect?

 

[Dale]

Summary:: Trying to wrap my head around the definition of a candela.

Luminous intensity is also measured in watts per steradian

No. Radiant intensity is measured in W/sr. Luminous intensity is measured in cd. They are non commensurate units.

The difference between the two is human perception. Two sources have equal luminous intensity if they visually appear the same brightness (all other things equal). Two sources have equal radiant intensity if they provide the same power density (all other things being equal).

 

[SamRoss]

No. Radiant intensity is measured in W/sr. Luminous intensity is measured in cd. They are non commensurate units.

The difference between the two is human perception. Two sources have equal luminous intensity if they visually appear the same brightness (all other things equal). Two sources have equal radiant intensity if they provide the same power density (all other things being equal).

Is it correct to say luminous intensity = radiant intensity x luminosity function?

 

[Dale]

Yes, but note that the luminosity function is not dimensionless in the SI unit system.

 

[SamRoss]

Yes, but note that the luminosity function is not dimensionless in the SI unit system.

Alright, so that's one piece of the puzzle. I was assuming the luminosity function was dimensionless. So I suppose its dimensions are cd sr W^-1 ?

 

[Dale]

So I suppose its dimensions are cd sr W^-1 ?

Yes, exactly.

 

[SamRoss]

Yes, exactly.

Okay, so I get that the units are not the same. There's still something that's bothering me, though. If the light is of only one frequency and the radiant intensity is within the limits of what the eye can handle, wouldn't the "weight" that we multiply the radiant intensity by to get the luminous intensity just be 1? In other words, isn't all of that light getting through to the observer?

 

[Dale]

If the light is of only one frequency and the radiant intensity is within the limits of what the eye can handle, wouldn't the "weight" that we multiply the radiant intensity by to get the luminous intensity just be 1?

No, the weight is $683\text{ cd sr/W}$ at $540\text{ THz}$, and it varies from there.

 

[hutchphd]

The radiant intensity is the power detected by a perfectly color unbiased calibrated detector. The photometric and radiometric detector will agree (in Watts/whatever) if presented with only the magic green light (is it 556nm? I don't care} . Otherwise the eyeball is considered attenuated by insufficient sensitivity.

 

[SamRoss]

No, the weight is $683\text{ cd sr/W}$ at $540\text{ THz}$, and it varies from there.

Right. Okay, I think I'm seeing where my thinking went off the rails. If we have light of many different frequencies then the luminosity function is sort of a filter which picks out the frequencies we can see and attaches different weights to them. I was confused about the need for the definition of candela to use the words "luminous intensity" at the beginning, thus implying the luminosity function, if the use of the function is so trivial for light of one color (I mistakenly said the weight would be 1 but a weight of 683 is just as trivial; the point is that the whole apparatus of a filter seems to be overkill when there's only one color to begin with). I suppose the wording is necessary simply because we want the measurement to be in units of candelas. Now I see that your earlier statement about the incommensurability of the units for luminous and radiant intensities is a key point. Once again, Dale, I owe you a debt of gratitude. Thank you.

 

[Dale]

You are more than welcome! It is always enjoyable to see someone pick up a concept quickly like this