Lambert's Law differential equation

[dgoncha]

22. According to Lambert's law of absorption, the percentage of incident light absorbed by a thin layer of translucent material is proportional to the thickness of the layer. If sunlight falling vertically on ocean water is reduced to one-half its initial intensity at a depth of 10 feet, at what depth is it reduced to one-sixteenth its initial intensity? Solve this problem by merely thinking about it, and also by setting it up and solving a suitable differential equation.

From: Differential Equations with Applications and Historical Notes: Third Edition by George F. Simmons

I figured out the mental part and initially found that the equation for the intensity was I=1/2^(T/10) - (10, 1/2) , (40, 1/16). I had trouble finding a differential equation that match what was described and has the same solution. I went backward and got I=I₀e^(ln(1/2)*T/10). I used this and got I=e^(ln(1/2)*T/10 + I₀) -> ln(I)=ln(1/2)*T/10 + I₀ -> ∫1/I dI = ∫ln(1/2)*1/10 dT -> ln(1/2) = k, 1/10 = n -> 1/I dI = kn dT -> dI = knI dT

I checked my answer, using (0, I₀) as an initial value and it got me the same solution as I got mentally.

My question is shouldn't the question say, "the percentage of incident light absorbed by a thin layer of translucent material is jointly proportional to the thickness of the layer (T) and the intensity of the light (I)." Intuitively it seems to make sense that the intensity of a light would affect how much of it is absorbed as well as the thickness of the material.

 

[BvU]

22. According to Lambert's law of absorption, the percentage of incident light absorbed by a thin layer of translucent material is proportional to the thickness of the layer. If sunlight falling vertically on ocean water is reduced to one-half its initial intensity at a depth of 10 feet, at what depth is it reduced to one-sixteenth its initial intensity? Solve this problem by merely thinking about it, and also by setting it up and solving a suitable differential equation.

From: Differential Equations with Applications and Historical Notes: Third Edition by George F. Simmons

I figured out the mental part and initially found that the equation for the intensity was I=1/2^(T/10) - (10, 1/2) , (40, 1/16). I had trouble finding a differential equation that match what was described and has the same solution. I went backward and got I=I₀e^(ln(1/2)*T/10). I used this and got I=e^(ln(1/2)*T/10 + I₀) -> ln(I)=ln(1/2)*T/10 + I₀ -> ∫1/I dI = ∫ln(1/2)*1/10 dT -> ln(1/2) = k, 1/10 = n -> 1/I dI = kn dT -> dI = knI dT

I checked my answer, using (0, I₀) as an initial value and it got me the same solution as I got mentally.

My question is shouldn't the question say, "the percentage of incident light absorbed by a thin layer of translucent material is jointly proportional to the thickness of the layer (T) and the intensity of the light (I)." Intuitively it seems to make sense that the intensity of a light would affect how much of it is absorbed as well as the thickness of the material.

Yes, but they don't talk about the intensity itself. They talk about the percentage of the intensity. So if the intensity absorption is linearly proportional to the intensity, the percentage is independent of the intensity. Check out the derivation