How High Should a Street Lamp Be for Optimal Road Illumination?

[skrat]

Homework Statement

Above the center of the street a lamp is hanged. A lamp radiates equally in all directions. How high does the lamp have to be so the edge of the road is as bright as possible?

Homework Equations

The Attempt at a Solution

I'm kinda lost here...

$j=\frac{P}{A}$ where $A=4\pi r^2$ therefore

$j=\frac{P}{4\pi r^2}$.

Now the idea is to get $j(h)$ but nothing seems to work... For example: Let's say that the street is $2a$ wide, therefore $r^2=h^2+a^2$.

$j=\frac{P}{4\pi (h^2+a^2)}$

$\frac{\partial j}{\partial h}=-\frac{P}{4\pi}\frac{2h}{(h^2+a^2)^2}=0$...

Hmm, what am I missing here?

 

[TSny]

Maybe you need to take into account the angle at which the light strikes the road surface.

See http://en.wikipedia.org/wiki/Illuminance