Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line l parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on l so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.

(a) Find an expression for the intensity I(x) at the point P. (Your answer may depend on x,d, and k where k is the product of the proportionality constant and the strength of the sources).

I(x) =

(b) If d = 5 m find the smallest value of x which minimizes the intensity.

x = m

(c) If d = 10 m find the smallest value of x which minimizes the intensity.

x = m

(d) Somewhere between d = 5 m and d = 10 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Find this value for d.

d = m