# Is my illuminance calculation correct?

I have been reading about illuminance, illuminous flux, lumens, lux, irradiance, etc on wikipedia. I made this toy problem to test my knowledge. I am posting it here to test if I'm correct. (note, this really isn't a homework problem, I generated the problem myself).

Problem. Suppose we have a room of 5 by 5 by 5 metres, and we have a point light (i.e. a lamp) in the middle of the room. How many lumens must the lamp radiate, and how many Watts lightbulb (of a typical LED lamp) do we need, to produce the equivalent ambiant illuminance of daylight on a typical day?

Derivation. The SI unit of illumance is lux, and the ambient illuminance on a typical day is $$10\;000$$ lux (wikipedia). Assume the lightbulb radiates in all directions equally, and therefore produces $$4\pi\cdot x/r^2\approx 12.5 x/r^2$$ lux on the surface of a sphere of radius $$r$$ around it, where $$x$$ is the lumens of the lamp. If we approximate the room by a sphere of radius 2.5, then we have approximately a lux of $$2x$$ per square meter. So we need a luminous flux of approximately $$5000$$ lumens. Given that a typical LED lamp produces about 100 lumens per watt, this means we need about 50 watt LED lightbulb, or about 8 typical 6 watt lightbulbs. Answer: 5000 lumens, 50 watt.

Is this derivation correct? Does the answer pass the sanity check?

Assume the lightbulb radiates in all directions equally, and therefore produces $$4\pi\cdot x/r^2\approx 12.5 x/r^2$$ lux on the surface of a sphere of radius $$r$$ around it, where $$x$$ is the lumens of the lamp

First, the units here used inconsistently here. $$r$$ appears to be dimensionful (radius), while $$x$$ is dimensionless (luminous flux divided by lumen). Let's instead use only dimensionful variables, and employ the steradian. Let's denote luminous flux by

$$L=x\,\mathrm{lm}.$$

The $$4\pi$$ should be augmented with $$\mathrm{sr}$$. Your formula would then state (denoting unchecked correctness with $$\stackrel{?}{=}\!\!\text{''}$$):

$$I\stackrel{?}{=}\frac L{r^2}4\pi\,\mathrm{sr}.\tag{∗}$$

Does the answer pass the sanity check?

Dimensional analysis is the first tool to try to answer such questions. It would now give us the following representation of $$(∗)$$:

$$\mathrm{lx\stackrel{?}{=}\frac{{lm\cdot sr}}{{m}^2}},$$

or, transforming to base units,

$$\mathrm{\frac{cd\cdot sr}{m^2}\stackrel{?}{=}\frac{cd\cdot sr\color{red}{\cdot sr}}{m^2}}.$$

Here's your first mistake: you needlessly multiply by $$4\pi\,\mathrm{sr}$$, because luminous flux is luminous intensity already integrated over solid angle.

So, the corrected formula would be

$$I=\frac L{r^2},$$

pretty much the definition of illuminance.

For your spherical room to get $$10^4\,\mathrm{lx}$$ on its walls, the light source should then emit

$$L=Ir^2=10^4\,\mathrm{lx}(2.5\,\mathrm m)^2=62500\,\mathrm{lm}.$$

Thus, you'll need $$625\,\mathrm W$$ from light bulbs of efficiency of $$100\,\mathrm{lm/W}$$ to get such illuminance, or about 104 bulbs each consuming $$8\,\mathrm W$$.

Sounds too many? For comparison, in my $$4.5\,\mathrm{m}\times 3\,\mathrm m$$ room I have 5 bulbs each consuming $$11\,\mathrm W$$ (nominally $$880\,\mathrm{lm}$$, didn't actually measure), and typical illuminance measured by a luxmeter in the center of the room is about $$300\,\mathrm{lx}$$. If we take into account that the bulbs don't emit light isotropically, the order of magnitude of this figure then appears to be in agreement with the above calculation.