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?
sketch
![Bild "Bilder:sketch_lamp.jpg"](/en/kategorien/Bilder/dateien/sketch_lamp.jpg)
The illuminance is proportional to
![Bild "Bilder:BelStk_Formel_02.gif"](/en/kategorien/Bilder/dateien/BelStk_Formel_02.gif)
![Bild "Bilder:BelStk_Formel_08.gif"](/en/kategorien/Bilder/dateien/BelStk_Formel_08.gif)
Thus: illuminance
![Bild "Bilder:MatheFormel023.gif"](/en/kategorien/Bilder/dateien/MatheFormel023.gif)
Furthermore, the following applies:
![Bild "Bilder:BelStk_Formel_07.gif"](/en/kategorien/Bilder/dateien/BelStk_Formel_07.gif)
Illuminance:
![Bild "Bilder:MatheFormel024.gif"](/en/kategorien/Bilder/dateien/MatheFormel024.gif)
Forming the 1st derivative to find the maximum:
![Bild "Bilder:MatheFormel002.gif"](/en/kategorien/Bilder/dateien/MatheFormel002.gif)
Thus, the first derivative is zero, the counter must be zero.
It follows for the extremes:
![Bild "Bilder:BelStk_Formel_03.gif"](/en/kategorien/Bilder/dateien/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"](/en/kategorien/Bilder/dateien/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.