Therapeutic light boxes purport to deliver 10,000 lux to a user, typically sitting ~24" from a box. I was wondering how many lumens are necessary to achieve this illuminance, e.g. what lamps would be necessary to build such a box.
Suppose the light box has a flat, square diffuser, where the square has a total area $A$. The box emits a total of $l$ lumens. Given the distance $d$ to the light source, can we have a formula for the illuminance received by a viewer centered in front of the light box, facing it directly?
We need to first determine the luminance $L$ of the box diffuser's surface (lm/m^2/sr). It has a luminous exitance of $l / A$ (lm/m^2). Apparently for a Lambertian (diffuse) surface, we should divide by $\pi$ to get luminance: $L = l / A / \pi$ (lm/m^2/sr).
Multiplying this luminance by the area $A$ of the box, it has a luminous intensity of $L \cdot A = l / \pi$ (lm/sr). The box is viewed from distance $d$ away, so its apparent size will be approximately(1) $A / (4 \pi d^2)$ steradians. Multiplying luminous intensity by solid angle, the viewer receives $l/\pi \cdot A / (4 \pi d^2)$ (lm).
But wait -- the units seem wrong now. I should have gotten lm/m^2 (lux), not lumens. A single point only receives an infinitesimal amount of lumens. Where did this go wrong?
(1) In reality, the corners of the boxes are farther, so it appears a smaller area. I wonder how much effect this has.