Minimizing Illumination with Two Light Sources on a Parallel Line

[Paul I.]

Homework Statement

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.

Homework Equations

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.

 

[Samy_A]

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.

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

 

[Mark44]

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

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

 

[Ray Vickson]

Homework Statement

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.

View attachment 90884

(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.

Homework Equations

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.

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!