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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 meter 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.

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

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

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

P d 10 m