Calculate Optimum lamp height for maximum illuminance

A lamp is on point B (eg dining table) be suspended so that the illuminance at point A (eg table edge) is at a maximum. Thus, the optimum lamp position with maximum brightness would be found at the table edge.

At what height h the lamp should be suspended over the table?


Bild "Bilder:sketch_lamp.jpg"

The illuminance is proportional to Bild "Bilder:BelStk_Formel_02.gif" and the projected area Bild "Bilder:BelStk_Formel_08.gif" .
Thus: illuminance Bild "Bilder:MatheFormel023.gif" . k is a proportionality factor.

Furthermore, the following applies: Bild "Bilder:BelStk_Formel_07.gif" .
Illuminance: Bild "Bilder:MatheFormel024.gif" .

Forming the 1st derivative to find the maximum:
Bild "Bilder:MatheFormel002.gif" .

Thus, the first derivative is zero, the counter must be zero.

It follows for the extremes:

Bild "Bilder:BelStk_Formel_03.gif" .
(Note: That Extrema is a maximum because of the extremes on the h-axis pass through a sign change of + ("uphill") to - ("downhill") takes place alternatively use 2nd derivative..)

Hence the optimum lamp height:

Bild "Bilder:BelStk_Formel_04_big.gif" .

Example "lamp on a table" has h = 70 cm above the table when the table has a diameter of 2 m (a = 100 cm).

Have fun hanging lamp.