Intensity of Light The intensity of illumination at a point...

Intensity of Light The intensity of illumination at a point varies inversely as the square of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are $6 \mathrm{~m}$ apart. At what point between the two lights is the total illumination least?

Key Concepts

Inverse Square Law

This law states that the intensity of light (or any point source radiation) decreases in proportion to the square of the distance from the source. It is a fundamental principle in physics that explains how energy spreads out in space, making it crucial for understanding how varying distances affect observed intensities.

Inverse Variation

Inverse variation is a mathematical relationship where one quantity increases as another decreases such that their product remains constant. In problems involving light intensity, it characterizes how intensity diminishes with increasing distance, following the formula I = k/d².

Calculus-based Optimization

Calculus-based optimization involves using derivatives to find the points where a function reaches its minimum or maximum values. This technique is essential in problems where multiple effects—such as overlapping intensities of different light sources—need to be balanced to find a point of minimum or maximum total effect.

Transcript

00:01 We have two light sources six meters apart.

00:03 So let's say this is our light source one.

00:08 And let's say it has an intensity of l1.

00:11 The other light source is here.

00:15 We'll call that l2.

00:17 And we're told that this second light has an intensity that's eight times that of the other.

00:23 Okay, so we're going to say that this one is equal to 8l1.

00:28 And we know that the distance in between is six meters.

00:34 So the question is, where in between these two light sources will the total illumination be the least? okay, so let's say this point here.

00:44 So we'll call this the distance between this blue point to l1, x, which means the other one must be 6 minus x.

00:53 And we're told that the intensity at a point, it varies inversely, as the square of the distance between the point and the light source.

01:03 So that means, so let's call the intensity at this point i.

01:10 Well, the total intensity there is going to be the intensity from light source one, which we will call i1, plus the intensity from the second light, which we will call i2.

01:24 So inversely proportional just means that this is equal to the illumination from light one, so that's l1 over the square of the distance and the distance between l1 and our point is x right so this would be x squared and then we multiply by some constant k so this is i1 and we do the same thing for i2 so now this is going to be the intensity from the second light which is l2 over the square of the distance which is six minus x squared and then we multiply by the same constant okay so now we know what l is in terms of l1.

02:15 So let's substitute that in.

02:16 So we get this as equal to.

02:18 And then we can pull the k to the front.

02:21 So that's k times l1 over x squared plus 8l1 over 6 minus x squared.

02:33 Okay.

02:34 And that is equal to the total intensity.

02:37 And this is now only in terms of x.

02:42 Okay.

02:43 And remember that l1 is just a constant.

02:44 Okay...