Two light sources of identical strength are placed 14 m apart. An object is to b

Question

Two light sources of identical strength are placed 14 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 k = 1.)
I(x) =

(b) If d = 7, use graphs of I(x) and I'(x) to find the value of x that minimizes the intensity.


(c) If d = 14 m, find the value of x that minimizes the intensity.


(d) Somewhere between 7 m and 14 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Find this value of d.

Explanation / Answer

The minimum will occur at the half way point between the two light sources so x = 9 meters. (c) The minimum occurs at the end points so x = 0 or x = 18. (d) The transition occurs when the center value equals the end point values. 2/(81 + d^2) ..................... intensity at center 1/d^2 + 1/(18^2 + d^2) ...... intensity at end points 2/(81 + d^2) = 1/d^2 + 1/(18^2 + d^2) 2d^2(324 + d^2) = (81 + d^2)(324 + d^2) + d^2(81 + d^2) 648d^2 + 2d^4 = 26244 + 405d^2 + d^4 + 81d^2 + d^4 648d^2 = 26244 + 486d^2 162d^2 = 26244 d^2 = 162 d = 12.728

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