Chapter 3.7, Problem 80E

Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line $\mathcal{𝓁}$, 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 $\mathcal{𝓁}$ 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.

(b) If d = 5 m, use graphs of I(x) and I'(x) to show that the intensity is minimized when x = 5 m, that is, when P is at the midpoint of $\mathcal{𝓁}$.

(c) If d = 10 m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint.

(d) Somewhere between d = 5 m and d = 10m there is a transitional value of d at which the point of minimal illumination abruptly changes. Estimate this value of d by graphical methods. Then find the exact value of d.