(a) Find an expression for the intensity I ( x ) at the point P . (Assume the co

Question

(a) Find an expression for the intensity I(x) at the point P. (Assume the constant of proportionality is 1.)

(b) If d = 6 m, use graphs of I(x) and I'(x) to find the value of x that minimizes the intensity.

(c) If d = 12 m, find a value of x that minimizes the intensity.

    (d) Somewhere between d = 6 m and d = 12 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Find this exact value of d.

Two light sources of identical strength are placed 12 m apart. An object is to be placed at a point P on a line scripted 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 scripted 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. Find an expression for the intensity I(x) at the point P. (Assume the constant of proportionality is 1.) If d = 6 m, use graphs of I(x) and I'(x) to find the value of x that minimizes the intensity. If d = 12 m, find a value of x that minimizes the intensity. Somewhere between d = 6 m and d = 12 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

Label the coords of the sources S(-9,0) , T(+9,0)
and P(x,d)
Then the squares of the distances are:
|PS|

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