Why is the candela dimension J, not W?

Question

According to the table at the bottom of the Wikipedia page for the candela, the dimension for candelas is J (joules). Why is this not W (watts)?

The luminous intensity for light of a particular wavelength λ is given by

$I_v(\lambda) = 683.002·\bar{y}(\lambda)·I_e(\lambda)$

where $I_v(λ)$ is the luminous intensity in candelas, $I_e(λ)$ is the radiant intensity in W/sr and is the standard luminosity function.

Since $\bar{y}(\lambda)$ seems to be unitless and between 0 and 1, and $I_e(\lambda)$ is expressed in W/sr, why is $I_v$ not also W/sr (or J/S since sr is dimensionless)?

Answer 1

Variables, units and dimensions are fundamentally different concepts. When writing symbols it is conventional to use different fonts to avoid confusion.

Quantity symbols (variables) are written in italic font, e.g. $A$.

Units are written is upright serif roman font, e.g. $\textrm A$.

And Dimensions are written is sans serif roman capital font, e.g. $\textsf A$.

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According to the The International System of Units (SI) [Section 1.3; page 11]:

The seven base quantities (dimensions), and their corresponding S.I. units are

1. length $\textsf L$, the metre $\textrm m$,
2. mass $\textsf M$, the kilogram $\textrm {kg}$,
3. time $\textsf T$, the second $\textrm s$,
4. electric current $\textsf I$, the ampere $\text A$
5. thermodynamic temperature $\textsf{}\Theta$, the kelvin $\textrm K$,
6. amount of a substance $\textsf N$, the mole $\textrm{mol}$
7. luminous intensity $\textsf J$, the candela $\text {cd}$.

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Hence in the example above, the Luminous Intensity of a specific light source $I_v$, has the dimension of luminous intensity $\textsf J$ , and the S.I. unit of measure is the candela $\text {cd}$ (not joule).

A joule is the derived unit of energy $\textrm J = \textrm m^2\, \textrm{kg}\, \textrm s^{−2}$ and has the dimensions of $\textsf L^2\, \textsf M\, \textsf T^{-2}$. Clearly $\text J$ is not the same as (or even comparable to) $\textsf J$, as one is a unit and the other is a basis dimension.

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The Radient Intensity $I_e$ has units of $\textrm W\,\textrm {sr}^{-1} = \textrm J\,\textrm{sr}^{-1}\,\textrm s^{-1}$ and thus the dimension is $\textsf L^2\, \textsf M\, \textsf T^{-3}$.

A lumen is a unit equivalent to a candela steradian $\textrm{lm}= \textrm{cd}\,\textrm{sr}$, such that the luminous coefficient, approximately $683\,\textrm{lm}/\textrm{W}=683\,\textrm{cd}\,\textrm{sr}\,\textrm{W}^{-1}$

Putting the units into the equation for the luminous intensity; $$I_v(\lambda) = 683.002\,[\textrm{cd}\,\textrm{sr}\,\textrm{W}^{-1}]\cdot\bar{y}(\lambda)·I_e(\lambda)\,[\textrm W\,\textrm {sr}^{-1}]$$ (and cancelling units) we see that luminous intensity has units of candela $\textrm{cd}$ (and dimension of $\textsf J$) as expected.

Answer 2

I believe (as per John Rennie's answer) that $\mathrm{J}$ in this case does not stand for Joules.

The Wikipedia page for the International System of Units (SI) makes a distinction between "unit sybols" (the familiar $\mathrm{m}$ for metres, $\mathrm{J}$ for joules, etc.) and "dimension symbols". The difference is that there are dimension symbols only for the SI base units (metre, kilogram, second, Ampere, Kelvin, mole and candela) and not for any of the derived units like Newtons and Joules. Confusingly, the dimension symbol for candela is $\mathrm{J}$, even though this is also the unit symbol for Joules.

The symbols you mention on the Candela page are dimension symbols rather than unit symbols, so in this case the $\mathrm{J}$ stands for candelas rather than Joules.