A vertical shaft of light from a projector forms a light spot `S = 100 cm^(2)` in area on the celling of a round room of radius `R = 2.0 m`. The illuminance of the spot is equal to `E = 1000 1x`. The reflection coeffiecient of the celling is equal to `rho = 0.80`. find the maximum illuminance of the well produced by the light reflected from the celling. the reflection is assumed to obey Lambert's law.

Explanation

The illuminated area acts as a Lambert source of luminosity `M = piL` where
`MS = rho ES =` total reflected light
Thus, the luminance
`L = (rho E)/(pi)`
The equivalent lumonous intensity in the direction making an aangle `theta` from the vertical is
`LS cos theta = (rho E S)/(pi) cos theta`
and the illuminance at the point `P` is
`(rho ES)/(pi) cos theta sin theta//R^(2) cosec^(2) = (rho ES)/(piR^(2)) cos theta^(3) theta`
This is maximum when
`(d)/(d theta) (cos theta sin^(3) theta) = -sin^(4) theta + 3 sin^(2) theta cos^(2) theta = 0`
or `tan^(2) theta = 3 rArr tan theta = sqrt(3)`
Then the maximum illuminance is
`(3sqrt(3))/(16 pi) (rho ES)/(R^(2))`
This illuminance is obtained at a distance `R cot theta = R//sqrt(3)` from the ceiling. substitution gives the value
`0.21` lux