# Minimization of Illumination

1. Oct 27, 2015

### Paul I.

1. The problem statement, all variables and given/known data
Two light sources of identical strength are placed 8 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 d meters 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. (Assume the constant of proportionality is 1.)

(b) If d = 4 m, use graphs of I(x) and I'(x) to find the value of x that minimizes the intensity.

(c) If d = 8 m, find a value of x that minimizes the intensity.

(d) Somewhere between d = 4 m and d = 8 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Find this exact value of d.

2. Relevant equations

3. The attempt at a solution
I'm not sure how to approach the problem at first glance. I started by writting dow this for (a):
$$I(x)= \frac{1}{d^2}$$
I'm pretty sure I'm wrong with this. I just need enough to get me started.

2. Oct 27, 2015

### Samy_A

$I(x)$ should depend on (the square of) the distance of the object to each of the two lamps.

3. Oct 27, 2015

### Staff: Mentor

That should be the reciprocal of the distance squared. As you have written it above, the intensity would be greater for longer distances.

4. Oct 27, 2015

### Ray Vickson

I(x) is NOT 1/d^2, because 1/d^2 does not depend at all on x---so no matter where you locate P, the illumination would be unchanged. Does that sound right to you?

Hint: go back and re-read the question in detail; make sure you pay attention to every single word!