Two light sources of identical strength are placed 16 m apart. An object is to b

Question

Two light sources of identical strength are placed 16 m apart. An object is to be placed at a point P on a line 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 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. (Assume the constant of proportionality is 1.)

(b) If d = 8 m, use graphs of I(x) and I'(x) to find the value of x that minimizes the intensity.

(c) If d = 16 m, find a value of x that minimizes the intensity.

(d) Somewhere between d = 8 m and d = 16 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Find this exact value of d.

Explanation / Answer

If the origin is taken as the mid-point O of the line AB joining the two sources and P
(on the parallel line) is displaced by distance x from O. This makes
(AP)^2= d^2+(8-x)^2 and (BP)^2=d^2+(8+x)^2. This gives
I(x)= 1/[d^2+(8-x)^2] + 1/[d^2+(8+x)^2]
But I(x) is clearly an even function of x and I(x)-->0 as x-->+ or - infinity so
the maximum intensity (perhaps) is when x=0.

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