# Minimizing Illumination with Two Light Sources on a Parallel Line

• Paul I.
In summary: The Attempt at a SolutionI'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!
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.

## 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.

Paul I. said:

## 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.

Samy_A said:
##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.

Paul I. said:

## 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.

## 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!

## 1. What is minimization of illumination?

Minimization of illumination is the process of reducing the amount of light or illumination in a given space or environment. This can be achieved through various techniques such as using dimmer switches, changing light bulbs, or adjusting the placement of light sources.

## 2. Why is minimization of illumination important?

Minimizing illumination is important for several reasons. First, it can help reduce energy consumption and save on electricity costs. Additionally, it can help create a more comfortable and relaxing environment, especially for those who are sensitive to bright lights. Lastly, minimizing illumination can also help protect artwork, furniture, and other items from fading or damage due to excessive light exposure.

## 3. What are the benefits of minimizing illumination in the workplace?

Minimizing illumination in the workplace can have several benefits. It can help reduce eye strain and fatigue, leading to a more productive and comfortable work environment. It can also help save on energy costs, which can positively impact a company's bottom line. Additionally, minimizing illumination can also improve the overall aesthetic and ambiance of the workplace.

## 4. How can I effectively minimize illumination in my home?

There are several ways to effectively minimize illumination in your home. One way is to use dimmer switches to adjust the intensity of light in each room. Another option is to switch to LED or CFL light bulbs, which are more energy-efficient and produce less heat. You can also strategically place lamps and other light sources to create a more balanced and comfortable lighting scheme.

## 5. Are there any potential drawbacks to minimizing illumination?

While minimizing illumination can have many benefits, there are also potential drawbacks to consider. Some people may find dimmer lighting to be too dull or gloomy, which can negatively impact their mood. Additionally, minimizing illumination may not be suitable for certain tasks that require bright, focused lighting, such as reading or detailed work. It's essential to find a balance that works for your specific needs and preferences.

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